On the Dynamic Behavior of Thin Elastic Plates Citation O'Sullivan, Michael John (1968) On the Dynamic Behavior of Thin Elastic Plates. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2095-RQ32. https://resolver.caltech.edu/CaltechETD:etd-10022002-113310 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Two wave propagation problems are considered: the propagation of acoustic waves in a fluid slab and the propagation of elastic waves in an elastic slab. When formulated in terms of nondimensional variables these problems depend explicitly on two small parameters [epsilon] and [delta]. The parameter [epsilon] provides a measure of the thinness of the slabs considered and the parameter [delta] measures the impulsiveness of the applied excitation or loading. Approximation solutions of the problems considered are obtained consisting of several parts, each part having the form of a power series expansion in the parameters [epsilon] and [delta]. The most important result obtained is the development of the approximate theories - the plate wave equation and the Euler-Bernoulli plate equation - directly from the full equations of dynamic elasticity using a rational perturbation expansion technique. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Mechanics and Mathematics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Mechanics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Knowles, James K. Thesis Committee: Unknown, Unknown Defense Date: 8 March 1968 Record Number: CaltechETD:etd-10022002-113310 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10022002-113310 DOI: 10.7907/2095-RQ32 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3863 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 03 Oct 2002 Last Modified: 22 Mar 2024 19:52 Thesis Files Preview PDF (O'sullivan_mj_1968.pdf) - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

ON THE DYNAMIC BEHAVIOR OF THIN ELASTIC PLATES. Thesis by Michael John O'Sullivan In Partial Fulfillment of the Requirements For the Degree of Doetor of Philosophy California Institute of Technology Pasadena, Califoruia 1968 (Submitted March 8, 1968) it Acknowledgement The author wishes to thank his research advisor, Dr. James K. Knowles, for the invaluable advice and assistance offered during the investigation of the problems considered in this thesis. Lii ABSTRACT. Two wave propagation problems are considered: the propagation of acoustic waves in a fluid slab and the propagation of elastic waves in an elastic slab. When formulated in terms of nondimensional variables these problems depend explicitly on two small parameters « and 6. The parameter € provides a measure of the thinness of the slabs considered and the parameter 6 measures the impulsiveness of the applied excitation or loading. Approximation solutions of the problems considered are obtained consisting of several parts, each part having the form ofa power series expansion in the parameters e€ and $6. The most important result obtained is the development of the approximate theories - the plate wave equation and the Euler-Bernoulli plate equation — directly from the full equations of dynamic elasticity using a rational perturbation expansion technique. iv TABLE OF CONTENTS. INTRODUCTION. rT. AN APPROXIMATE SOLUTION OF THE WAVE EQUATION IN A THIN DOMAIN, i. introduction a. Formulation of the Problem 3. Formal Inner Approximation 4. Formal Boundary Layer Approximation 5. Wave Front Approximation 6. Summary of Results If. AN APPROXIMATE SOLUTION OF THE DYNAMIC EQUATIONS OF ELASTICITY 10. ll. IN A THIN REGION. Introduction Formulation of the Problem Formal Inner Approximation Formal Diffusive Approximation Farmal Boundary Layer Approximation Fucthez Consideration of the Inner Approximation and the Diffusive Approximations Formal Wave Front Approximations Formal Dilatation Wave Front Approximation and Associated Boundary Corrections Formal Shear Wave Front Approximation and Associated Boundary Corrections Further Approximations Sumnary REFERENCES. ll. 17. 23. 31. 35. 38. 44, 56. 68, 77. 81. 85. 98. 108. lll. 118. INTRODUCTION. This thesis is concerned with the discussion of two wave propagation problems: the propagation of acoustic waves in a fluid slab and the propagation of elastic waves in an elastic slab. These two problems can be regarded as particular cases of a general class of problems which arise in the investigation of the physical behavior of a material contained in a geometrically tnin region. Such problems are often described mathematically by a boundary value problem or a boundary-initial value problem for a partial differential equation. In many cases it is difficult or impossible to obtain an exact, explicit solution for such problems and therefore it is desirable to be able to find approximations to the exact solution. Usually the most significant behavior of solutions to boundary value problems for such thin regions can be described in terms of a thickness average of some relevant physical quantity. For example, we are usually more interested in the average flow of fivid through a pipe,.than the details of the velocity distribution across the pipe. Consideration of such average quantities allows the number of geometric- al dimensions of a problem to be reduced. Thus the static displacement of a thin three-dimensional elastic plate can be approximately describ- ed by the average displacement of the plate considered in the corres~ ponding two-dimensional region. Similarly the three—dimensional flow of fluid through a thin pipe can b2 considered approximately as a one- dimensional problem for the average flow. Various methods have been devised for obtaining "approximate" theories which describe the average behavior of solutions for a thin region, Many of these methods involve assumptions whose validity is difficult to assess. Perhaps the most systematic formal method consists of formulating the problem in terms of a small parameter e which measures the thinness of the region. This is followed by a formal perturbation expansion procedure in powers of € which yields many simpler houndary value or boundary-initial value problems. The lowest order term derived by this procedure is found to be some significant thickness average satisfying an equation suitable for use as an "approximate" theory. A difficulty arises here in that the solution derived by using this pertur- bation scheme is often not uniformly valid; in the case of the static deflection of an elastic plate the corresponding approximation cannot satisfy all of the originally prescribed boundary conditions specified at the edges of the plate. This difficulty is to be expected as the problem formulated in terms of e« can be considered as one of a singular perturbation type. To improve the solution predicted by the approximate theory further corrections are necessary. The derivation of these corrections also allows the rational development of boundary conditions and initial conditions suitable for use with the approximate theory. The perturbation expansion approach discussed above has been applied to the problem of the bending and stretching of thin elastic plates by Friedrichs [5], Friedrichs and Dressler [6], Reiss [35] and Laws [21]. The deformation of a cylindrical shell was investigated using this method by Johnson and Reissner [14] and Reiss [34]. Green [7,8], Green and Naghdi [9] and Green and Laws [10] have extended the perturbation procedure to thin shells of arbitrary shape. Fox [4] and Knowles [17] used the same procedure to inves— tigate the potential problem for a flat plate with Dirichlet boundary conditions and Westbrook 38 ] extended this work to the potential problem for a cylindrical shell. For the problems considered in these three papers the authors demonstrated rigorously that the solutions obtained by using the perturbation procedure are uniformly valid. An investigation proving the accuracy of the results obtained from an application of this perturbation procedure to a nonlinear shell probicm was carried out by John [12,13]. Some estimation of the worth of the technique is inherent in the work on Saint-Venant's principle by Knowles [18,19], Horvay [11], Johnson and Little [16], Novozhilov [30] and others. The present work is concerned with the application of this perturbation expansion procedure to dynamic problems for a "flvid plate”, dealt with in chapter I, and an elastic plate, considered in chapter II. In both instances a boundary-initial value problem is considered for a thin two-dimensional rectangular region. Initial quiescence is assumed and impulsive excitations or loads are prescribed at one end of the rectangular region. In addition slowly varying excitations or ioads are prescribed on the other boundaries. When these problems are formulated in non-dimensional form two small parameters c«, 6 emerge. The parameter « measures the thinness 4 of the region considered and 6 measures the ' suddenness’ of the applied impulsive disturbance. Approximate solutions are obtained consisting of several parts, each in turn a perturbation expansion in « and 6. A more precise discussion of the problems considered is given in the introduction to the separate chapters. I. AN APPROXIMATE SOLUTION OF THE WAVE EQUATION IN A THIN DOMAIN. 1. Introduction. In this chapter an approximate solution of a boundary value- initial value problem for the two-dimensional wave equation in a thin rectangular domain is constructed and discussed. The wave equation mathematically describes many physical situations. Two examples, similar in certain respects to the more complex elasticity problem described in chapter II, are the small motion acoustic behavior of a fiuid, and the behavior of an elastic mediim sheared in such a way that the particle motion occurs only in one direction (the so-called SH wave). The former of these two cases is used as a model to allow some physical insight into the mathematical procedure that follows. In physical terms the problem we wish to consider may be described as a waveguide problem. We consider an initiaily quiescent slab-shaped region of fluid suddenly excited on its boundaries, and we try to determine the subsequent motion of the whole region. For simplicity it is assumed that the prescribed excitations are such that they induce a motion which does not vary along the slab. This means that the motion is two-dimensional and we accordingly speak of a " rectangle of fluid " to connote a typical cross section of the corresponding three-dimensional region. We shall consider a "thin" rectangle of fluid. That is, a section of a plate whose thickness is small compared with its length and width. In this dynamic case another length scale besides the physical dimensions of the plate is introduced by the time dependence of the applied boundary excitations. It is assumed that large changes in these applied excitations only occur soon after the start of the motion. The duration of these sudden effects multiplied by a characteristic velocity of the problem (the velocity of sound in the fluid) provides a quantity with the dimensions of length which is a measure of the suddenness of loading. We call this quantity the "excitation length". The relative magnitudes of the thickness of the rectangle, the length of the rectangle and this excitation length greatly influence the solution. We do not attempt to solve the problem for completely general boundary conditions. but rather specify in advance that the excitation applied to the long sides of the rectangle is slowly varying with a corresponding length comparable to the excitation length of the rectangle: and we specify that part of the excitation of the ends is slowly varying. In addition there may be a brief initial effect at the ends with an excitation length small in comparison with the length of the rectangle. This rather vague description of the problem is made more precise with the detailed formula- tion in the next section. We carmot expect an approximate solution for this problen entirely analogous to those obtained for static problems (Knowles [17], Friedrichs and Dressler [6], Reiss [34], Reissner and Johnson [14] for example) since the dynamic problem for a fluid plate is concerned in an essential way with the phenomenon of propagation. The effect of boundary data specified at the ends of the rectangle will not remain localised in boundary layers near these ends. However for the particular type of boundary excitation considered here it might be expected that propagation effects are mainly concentrated in a pulse travelling along the rectangle and that this pulse could be approximated without having to obtain an exact solution of the problem. This idea is found to have considerable merit and an approximate solution is constructed consisting of three parts: an "inner approximation satisfying boundary data on the. sides of the rectangle, "quasi static" boundary layer approximations valid near each end of the rectangle and a wave fronc approximation describing the pulse propagation. For this problem some assessment of the quality of the approximate solution obtained is possible by rigorous comparison with the exact solution. 2. Formulation of the Problem. The two-dimensicnal equation of motion for the acoustic behavior of a fluid may be written in terms of a velocity potential ¢ in the following form: + |: - Yay T yy Te The velocities U,V in the X,Y directions respectively, are given by the formulae: The constant c is the speed of sound in the fluid. We consider the open rectangle R in the X,Y plane consisting of those points for which 0 < X <« 2, 0 <-‘Y < h. It is required to find a function 0(X,Y,T) satisfying the differential equation (2.1) for (X,Y) ¢€ R, T > O and fulfilling the following boundary and initial conditions: O<X <2 , O<T< 2%; V(&,h,T) = 0, J U(O,Y,T) = F(¥,T) + A(Y,T), O0<Y¥<h > 0< T< U(2,Y.T) = Q, | @(%,Y,5) = 0, (X,Y) € R. In the construction of an approximate solution for this problem it is found that the prescribed excitation at X = 0 produces two effects: a quasi-static boundary layer and a propagating pulse (not including the propagation of the average excitation). We have anticipated this result by writing the prescribed excitation in two parts: F(Y,T) is a slowly varying function. of T responsible for the quasi-static boundary layer near X = Q, and A(Y,T) represents a sudden excitation. It is assumed that A(Y,T) is of limited duration vanishing outside an interval (0,7 ] say. The details of the assumed smoothness of the functions P(X,T), FC(Y,T) and A(Y,T) will te discussed when they become important in later calculations. For the present we assume that given data ara 9 such that a solution for 4 does exist which gives rise to continuous welocicies on the closure of R for each T > 0. Tne rectangle Ris thin if the parameter ¢ = 2 is small in comparison with unity. In order to exhibit the role of « explicitly we introduce new independent variables x,y,t defined by the change of scale: -x = =. £L x= , ye » tet (2.3) Nondimensional dependent variables ¢,u v are also introduced by the following definitions: . 1. g p(,y,t3e,6) = he 2 @x,hy, c t) ? hi L, z u(x,y,t3e »6) h U(2x,hy, © t) 3 (2.4) 1 . 2 vix,y,tje,6) = pV exny, c t) . The parameter §, small compared to unity, is defined by the relation cT Oo : , : . . . 6 =~ , where To is the length of time for which the given excitation 2 at x = 0 has a large variation (the time derivatives of the prescribed excitation are not 0(1) in comparison with e for 0<7<T_). We are interested in the particular case where the applied excitations depend on ¢« in such a way that the limit problem as ce > 0 will produce a displacement in the x direction, u, which is 0(1). Accordingly we introduce new functions pt, z 0) and a through the definitions: 1 1 g € >! ) Get) = HP ax, 2 0 ’ (0) e-ton, & a) vy, 5) = = A(hy, % t) , 10 More general applied excitations, say Li oP (ax, ct p(x,tse) = ep (x,t) + e2p ) (xt) Fisee sy require more cumbersome calculations with no fundamental change of method and will not be considered here. Using this new notation the problem for ¢ may be written in the following final form: Pry + e* ($555 - dor) =0,90<x, y<1,0<té< ™+(2.6) 1 . bs, (x,0,t3.,6) = ep‘ ) (x,t) ’ y O<x<1,0<t<o, (x,1,t3e,6) = 0 Oey : > (2.7a) 1.(0 1 0 t 2, (0,y,tse,5) = 2 ¢y,t) + * a‘ diy, ra) P) oo sly. tse,5) = 6 ’ O<y<l, O<t<o, (2.7b) o{x,y,0 ) = 0, ; QO<x, yel . $3, (x,y,0) =O, (2.7¢) Once the potential ¢ is known, the velocity components follow from the relations: u= Edo, ’ V= 9, . (2.8) This problem may be solved exactly by elementary means. However, since we are only interested in the case where ¢ and ¢ are small, we shall attempt to obtain an approximate solution for $, as an asymptotic expansion in « and 6, directly from the differential equation, boundary conditions and initial conditions without first finding the exact solution. While this scheme is of interest for its own sake, it il also provides a pilot problem to suggest procedures in more complicated cases where an exact solution may be difficult or impossible to obtain. Such is the case, for example, in the elasticity problem described in chapter II. 3. Formal Inner Approximation. The most obvious first approach to the problem (2.6), (2.7) is to investigate approximations having the form of power series ine. We wish to find an approximation for the velocity u which is 0(1) in the limiting case ¢ > 0. Consequently we assume a formal expansion © Gy.t38) + cul? (x u, (x,y,t3e,5) =u syot36) + «6. . (3.1) The subscript "i" has been added as an abbreviation for "inner" as a reminder that we expect this expansion to only partially fulfill the requirements of the problem. -We see from equation (2.8) that the expansions for 5 and vy associated with (3.1) are: 9, (&y,t3€,6) = 2g) (9,056) + 0 Gyse36) +... (3.2) v, (x.y.tse.d) = 2 yD Oy t38) + vi Gey.t38) +... (3.3) We now substitute the expansion (3.2) into the problem (2.6), (2.7) and consider the equations associated with each particular power of ©€ as individual prublems. We first obtain the following equation for of: (-1) $ isyy 9 O<x , ysl, O< tK< *, (3,4) Solutions of this equation certainly cannot, in general, fulfill given conditions at x = 0,1. However, the boundary conditions arising from 12 (2.7a) merely require: i ro) 7 day (1) O, 0,t36) O< x<l O< t < »; S*5 ’ , (3.5) bg y Go1,t36) ll oa These may be fulfilled by a solution of (3.4) which is a function of x,t only. We use the notation: of Desystss) - rie (#5058) . (3.6) The displacements derived from this potential are: v, = 0 . (3.7) The equations (3.4) and (3.5)alone do not constitute a well set problem (-1) . . ; - ,{-1) for o, °° Having determined the y-dependence of 0; » it is not yet ~(-1) obvious what further conditions to impose on Oy - Equations, similar to (3.4) and (3.5), determine the y-dependence of 9, gD) : ee. , and also supply more information about go ~1) (0) toes (-1) $4 satisfies equations identical to those for o, in (3.4) and (3.5). Therefore we conclude that: 6 tx,y 536) = oe (x,t 38) ; (3.8) where at present see is an arbitrary function. The associated displacements are: uy? = rae , vy = 0 . (3.9) In the first order system arising from (2.7) and (2.8a) (1) (-1) (1) there is some interaction between by and cn 3 in fact os satisfies the following differential equation and boundary conditions: 13 @Q) +90 ~ gD =0, O< xysl,0<t<e, (3.16) VY i,xx i,tt g61) (x,0,t35) = pl) (x,t) i,y . Qs x<1,0<t<», (3.11) = 0 go) (x,1,£56) _aisy Equation (3.10) and (3.11) can be used to determine the y-dependence of of) and to obtain a condition on eed as follows: QD Geyser) = Chey -YpD Gye) +9) Gye38) (3.12) The term - dp) (x,t) is not immediately suggested by equations (3.10) and (3.11), but is added to the arbitrary function $1) (x,t) to arrange 1 . 1 that, | oL) (x,y, t38) dy = (1) (x, 35) . In writing equation (3.12) we 0 have used the condition: 1,xx i,tt which arises from the fulfillment of the boundary conditions for p@) at i y = 0. Condition (3.13) still does not fully determine 9(1), Boundary conditions at x = 0,1 and initial conditions at t = 0 are required. From the prescribed initial quiescence condition it is natural to require: _(-1) —{-1) oi (x,y,0;6) = 0, bar (ayy, 036) = 0 . (3.14) These conditions may be derived more rationally from investigations of an initial approximation where they are requirements to ensure bounced- ness of the approximation. Since we are not concerned with non zere initial values and the resulting vibratory type of initial approximation we are obliged to assume conditions (3.14). The question of boundary (1) () conditions at x = 0,1 will be considered later. Displacements up avy 14 are associated with the expression for p@) obtained in (3.12): i 1 1 y2 (1 —(1 wy (xy,t38) = C-5 ty - Fp Gt) + Fp aw ess), (3.15) 1 (yp Pct) ve yy, t58) Proceeding through successive orders we can evaluate o}) 66), 6D), in terms of the prescribed boundary excitation ep) (x,t) and a sequence of functions of x,t only, rie os oe ... , Satisfying different- ial equations similar to (3.13). Rather than carry out this program in detail, we shall employ a different procedure to obtain the higher order teriis more quickly. The alternate procedure is a symbolic one similar to that used by Knowles [17] for a second order elliptic boundary value problem. 2 2 Introducing the operator L* = 9°. - 3°. we formally consider the 8 P xe vr) y equation (2.6) as an ordinary differential equation: ¢ + 2129 = 0 > (3.16) 9vY : witn the boundary conditions: »¥ (3.17) = 0 ’ y=l1ss > 2¥ This- has the ‘symbolic’ solution: cos(l-y)eL (1) ¢* LsingcL P - . (3.18) The formal expansion for 4 in powers of ¢« is obtained by using the Taylor series expansion (cf. Abramowitz and Stegun [1], p.804): i-v oe 2n . z sin z 2’ nl ny n=0 2n where Bf) is the Bernoulli polynomial of degree k. Substituting 15 2 2 Zo = - we obtain the expansion for ¢: e= y BL 2 (-1)n_2n- Z~ ee yn-1,(Dx,e) (3.19) no 2 2 “ We interpret the cerm ( - 2, 0eue in the fullowing way; w y2 2 3 t) - —(a- Sy - Sere Moe =] Pe@Vegey , n=o —(-1 — —(- where 3 ) satisfies the differential equation p- pe > ; > > and a) satisfies the differential equation ra - a =O, j>_0. ? 2 Then we can rewrite the expansion (3.19): foe) y Be (x ra 2" (-1)Re 2n- te, _ sel (1) + ) els 1g(a-l) . (3.20) - n ‘Ox- n=O The validity of expansion (3.20) of course depends on the differentiability properties of ph (1) . For example, if p (x,t) is a polynomial in x,t, expansion (3.20) becomes a finite sum. it is easily verified that the expansion (3.20) formally satisfies the differential equation (2.6) and the prescribed boundary conditions at y = 0,1 (2.7a). Also the first few cerms of expansion (3.20) agree with the terms of the inner approximation obtained in equations (3.6), (3.8) and (3.12) (with the subscript "i" added in (3.20)). With further calculations we could show this agreement to be valid in general. Often all that is required for the inner approximation is the first few terms of (3.20). We assume that the derivatives Ga, - snp) for lgn<M, all exist for O ¢x<¢1, 0¢ t <m and are o compared to €, We write: °, ~ YL (4) Vaaa me 1)Re2n- G5 a7 syn lp C1) (x, t)+ ay cartg tnd), n=o0 "an % al ‘ax? n=0 * (3, 21) 16 The rigorous interpretation of the relationship connoted by (3.21) will be investigated later. Let us reexamine equation (3.17), which may be written in the form: (L sine L)? = (cos (1-y)eL) p VY (x,t) (3.22) This may be considered as an inhomogeneous, infinite order, partial differential equation for ¢. Then expansion (3.20) is a particular integral of this equation to which we must add a complementary function satisfying the equation (L sin cL) ¢ = O and fulfilling the additional constraint that $= 9 at y = 0,1. Together, the particular integral and complementary function must satisfy initial conditions and boundary conditions at x - £,1. Since z sin z has the infinite product expansion: z sin z we may formally assert that the complementary function satisfies the differential equation: eo 2 ‘ 27 3 2 2 fe 8 22 = ft fe C2 vee, n“Il4+]¢ = 0 . n=o The significance of this equation in relation to the boundary layer and wave front approximations will become clear in subsequent sections. The inner approximation bas written in (3.21), is an approximation to a particular integral of (3.22) and it is found that the boundary layer and wave front approximaticns together form an approximation to the complementary function of (3.22). To assess $. as an approximate solution we substitute it into the equations (2.6), (2.7). We find that b; satisfies the specified 17 conditions at y = 0,1 exactly and satisfies the differential equation 2M) +1 with an error which is O0(¢€ 1 }. However, in general >, is a poor approximation to the boundary conditions at x = 0,1. The smoothness conditions assumed on px, t) are that the derivatives (37-5) p: forl < n < My all exist and are 0(1) for O<x<1,0<t <o, This means that all the derivatives of order <Mj-l are zero at t = 0. Thus the only contribution to the initial values ¢ sy205 , 8) and O 5 ery05 & 8) arises from the functions 96) We impose the initial conditions (similar to conditions (3.14)): 7 (n) sc) o ~(n) .f) a 5 (x,0;6) 0 Py Of 205) 0 ° (3.23) —(n) We cannot fully determine ¢; yet as we do not know what conditions to prescribe at x = 0,1. This difficulty will be resolved in the next section. 4. Formal Boundary Layer Approximation. In the previous section we observed that in general the inner approximation o5 does net satisfy the prescribed boundary condit- ions at x =0,1. To explore a possible boundary layer correction near x = 0 we introduce a bowmdary layer variable ¢t defined by the scaling: and we write: otx,y,t3c,5) = o (ayy, t3e,6) + o)(osy,t3e,5) , (4.2) where % represents a tentative boundary layer correction near x = Q. Substituting this expression fcr $ into the original differential equa- tion, and using the property of ¢4 written in equation (3.23), we obtain 18 the following equation for oo: + ¢ ~ = 2M_+1 where from equation (3.23) we can show that y is 0(e ) . Since we are only interested here in the boundary layer near x = 0 and since we shall require these corrections to tend to zero as ¢ increases, it is natural to replace the domain 0 < ¢ < ‘, O<y<i1,0<t <-@ of the differential equation (4.3) hy the region 0 <7 <e, O<y<1,0<t <®. We also require that 5 + by fulfills the boundary conditions except at x = 1. This places the following boundary conditions on o,: Og py 06 20stse56) = 0 a 0<¢<*7, O0<t<e@, 4.4 Oy y(Golstse,6) = 0 (4a) (0 0 o, -C.ystse,5) = £° Yoy,t) + a (y,t/5)-e6, _(O,y,t5e,8). 9G 1,X (4.4b} In addition we impose a decay conditicn: oy» c (Gsystse,8) +0 as 07% (4.4c) > We assume an expansion for bo: 0 1 , o,(c.y.tse.é) = os )(yy.t38) + co! cy, t38) +... (4.5) so that, following standard proredure, pO satisfies the preblem belew: ‘ . O24 64 =0,0<¢<%,0<y<1,0<t<-», (4.63) 9) (0,055) =0, (a) rf) <6< %, 0 sts yy, (4.6b) $ (¢,1,t;5) = 0 > Rsy 19 60° @ny,t38) = £O yt) + a GW, t75) - TEP Ot38) (4.6c) (0) Oeic 7 0 as 57% (4.6d) Here it should be remembered that oo) ex t38) is still not 1 determined. Without actually solving the system (4.6) we can derive a necessary condition for the existence of a solution with the right pro- perties. This-condition provides a boundary condition for aie ) at x =0. We integrate the differential equation (4.6a) over its domain: leo (0) (0) 0 = [Jos Lsyy + o9 pp acdy 2 fete) 0 s (#,y,t3d)dy nm Ne ae ) (Oy, 03 6)ay + of 0 ov i) —_ + bn (c,1,t3s)de - os (¢,0,t3d)dc ° ou 8 ott Ot——38 OH Substituting from the boundary conditions (4.6b), (4.6c) and the decay condition (4.6d) we obtain the following condition: 1 fe cyt) + a Gy,t/6) - EP Oto ay -0 fe) That is: 1 a? c,t38) = feo + a vy ,t/8) ay » (4.7) 9X o This equation provides us with a boundary condition for $. OY Gest t3é6) at x = 0. A similar condition at x = 1 is derived by consideration of the boundary layer there. 20 We can obtain a solution for yo having the form: (0) _ © 20) , (0), @ BMS Py ah Gs ray ny “oo TY " oe8) where Ll Q) oz (_ (0 a\ 1 = ja! dy, t/s) cos nny dy ° ° and 1 ey = 6 yy cos nry dy . d oO Note that the condition (4.7) ensures that there is no term in the series solution (4.8) which is independent of y and hence non decaying as (0) ¢ >, The variable t only occurs in the solution o; (0) & as a parameter and hence tne behavior of @ (0) (0) with respect to t will be similar to the behavior of £ and a with respect to t. 1) ( sce . . At the next order we find that %e satisfies equations which acre the same us (4.64), (4.6b) and (4.6d). However at ¢ = 0 we now have (1) . —(1) 9g Oryst56) = Op 3 (00858) . Proceeding as above we obtain the condition: Fy 20,858) - 0 (4.9) and the solution for oe: pY) = 0 , (4.10) At the second order some interaction with g 0 appears. oO is a solution of the following problem: (4. 11a) 21 (2) il Q (¢,0,t36) Q9< o< wm O< t < w > (4. L1lb) Hit Oo (2), * s2st56) > 2 1 ye 1 1 . a O,y,t38) = -C 3 ey yp‘ (0,t)-0072? (0,58) ay 9X LyX . (4.11c) 4 62) > ~» 09 wo” a * (4.114) Writing the differential equation (4.1la) assumes that 6 . is 0(1). (0) Therefore we cannot include the function a (0) (a) approximation; a is O(1) whereas a tt 3 in the boundary layer is Gr x). This is why the prescribed boundary condition at x = 0 is written as the sum of 60 vy) and a) vy t/8) . ie is found that a boundary layer approximation near 0) ,@ My) x = Q which includes terms og, oy eee yg Oy can only accomodate > m. (0) a function 2 dvy,t) for which the derivatives g — ,O<m< 2M, exist ot and are 0(1) in comparison with c, for 0 <y<landO<t < o, (0), We rewrite the solution for op : “aTG 6 (,y,t) = - ) 6 (ey & cos uty (4.12) n=1L From the equations (4.11) we obtain the condition: 1 4 FD (9.428) = i Yo), = 4.13 950000050) = -|C > + y - pp Ly’ O,t)dy = 0 (4.13) fe) and the solution for ye? ), 2 0 eis ) dc, y,t) = 5 ce ) Gtnrs) - ap Co, t)) Saoe cos ally . n=1 (4.14) More complicated higher order boundary layer corrections may be found by procecding systematically in this manner. The foilowing condition is obtained: 22 of 0,058) = 0 , n> (4.15) providing a boundary condition at x = 0 for successive orders gm) . 2M 6 2 LMA oye. (4.16) n=0 A similar correction say dors where 2M 3 - Oot = z eng) dcx »y3t) : (4.17) n= can be developed to approximate the solution near the end x = 1. This process will also determine boundary conditions for e(-)D, HO aes, opm) at x = 1, These together with boundary conditions (4.14), (4.13), (4.9) and (4.7), plus the initial conditions (3.25) determine solutions of the differential equations for oC), 0) == 2 (2M) . In fact 900s ods wee gM) = 0. In the next section the initial wlues of boundary layer approximations will be signi ficant. With this knowledge we make a further assumption that the prescribed data at x = 0 may be divided into £(0) (y,t) and al) (y,t/8) so that alG) vy t/8) includes all the initial effects. Thus ae oo) = 0, 0<m< 2M and this ensures that the boundary layer approximations contribute nothing to the initial walues at the orders considered. We now consider the convergence of the series represent- ation of the solutions for 6), g@) in equations (4.12), (4.14). From these two orders it is obvious that progressively higher and higher powers of —L are introduced into the Fourier coefficients of the higher ni order boundary layer corrections, say of), This means that we need not 23 prescribe very restrictive smoothness conditions on £9) (y,t) for O<y< 1. We assume that £09) (yt) has a continuous y-derivative for O<y<landQ<t< oe, We assume that the series co 2M, __ (0) . 8 78)” Ce) ) ary cos nity n=] dt. 2M, 12M, converges uniformly to the sum 2“! (yt). | 9 “f(y, t) dy . gp cit2 4 ,2M2 lo dc It should be emphasized that in this section we have obtained boundary conditions for gin) and demonstrated the inability of the boundary layer to handle prescribed data like a (0) (y,t/8) without finding the boundary layer approximation explicitly. 5. Wave Front Approximation. The composite approximation do, + ®» + bor satisfies the differential equation (2.6), the boundary conditions (2.7a), the initial conditions (2.7c) and part of the boundary conditions (2.7b), all with 2M an error which is 0(e 1 ). Tne remaining part of the prescribed bound- ary conditions at x = 0 is the rapidly varying portion a) (y,t/8); this is expected to produce a pulse propagating along the rectangle. It is reasonable to invescigate scalings of ¢ and x which make the wave 2 2 operator (2 - 3 4 2 ax* at? This procedure produces approximations to the wave front behavior which ) the most important part of the equation (2.6). are not uniformly valid. To obtain an approximation having wider applic- ation we introduce the following variables suggested by the boundary ut condition at x O and the differential equation: t= tox n = —2 (tex) . (5.1) 6 e2 We write an approximation for 4: 24 - eV. pk j l-x . p(xsy,tze,0) = 0 Hoystse Sty) Csystscdthy Co sys tie dbo sy sG5e 55) 5 (5.2) where $, represents the pulse approximation. Substituting this form for » into the original system (2.7) and (2.8) we find that %¢ satisfies the following problem: 2M _ 1 bevy 7 eye, = Oe , (3. 3a) tr [Co Nop fh for E<yn<~r,O<y<1,0< & < @ Q, (,0,83e,86) = 0 ’ fy (5. 3b) ¢ y Mahr83e56) = 0 > > for oye or “resn< 7, OS F<; M 2 5 (0 2 be pv org 27 BO GE) + 0G6e ) » (5.3e) B) ? > n= Se, O<y<i1,0<&<~, We know that no disturbance can travel faster than the acoustic velocity; thus as the fluid is intially at rest d= is zero for € < QO. Also we require that d¢ be continuous at € = 0. That is: o¢Msys05e,6) = 0 > Os ns »Ox ysl, (5.4) We do not attempt to enforce a boundary condition at x = 1. The effect of the solution we shall obtain for be at the end x = 1 will be used as a boundary condition for another wave front approximation, The most important aspect of the development that follows is our assumption that the ratio ee small, Then at the lowest order we can approximate the domain described in equation (5.3a) by the region O<y<1, 0< § <@, 0 <n < ™. And we apply the boundary condition (5.3c) at c= 0, 25 This places a further restriction on the prescribed end conditions. The problem for $e as written in (5.3) and (5.4) does not depend on 6 and € separately but only their ratio. We introduce a new parameter Vv = © and we shall consider only the case where , is small. Using this new parameter y, we can write the problem for ¢ as follows: 2M, - = 0 ; be yy 898.0 ,¢ f ) : for y2—E<c< 0, 0O< y< 1O< E<@ 3 9 yG 2085) = 0 ? b¢ yield sv) = 0 ’ for y2& <n <, O< & < »} M - 2 =~) ai?) 2 es, g v o£ in ~w a (y,é) + Oe ) > for n =v2E, 0«< y< 1, O< E < »3 oe .y,05v) = 0 : for v*E<n<e” ,O<y<1, We now assume an expansion for be of the form: 1 2 eM sy.8 sv) = vos Gaye) + v25 mr aysE) + vee Following the usual procedure we can establish that 6 Gayd satifies the equations below: (1) @) _ . Seryy Hen OS NS O<y<1,0<&<-e, u fo) ¢ GD) | £ yl 2058) - 0O<n<«, 0 < E<oos | fo} rf oro 21.6) = / ~ (5.5a) (5.5b) (5.5c) (5.5d) (5.6) s@Poye) =O.) , Os ysl ose<e: ia 66D & y,0) = 0 »O sns*, 0s ysl. Note that we have replaced the boundary condition at 4 = ve by one at n = 0. This is a procedure similar to the replacement of the domain QO<y<1,0<€&< * by the semi infinite strip 0< y< 1, 0< &< oe in the analysis of the boundary laycr in section 4. However in this case errors produced by this approximation of the domain are not negligible at higher orders. We can now solve the above system for o) obtaining the solution in the form: o § | oo? =~) [a0 (ays, conn 2) az . cos niy (5.7) n=l, where 1 0 0 . a‘ @) = [a‘ (y,£) cos nny dy > Oo and J is the Bessel function of order zero. Displacements corresponding to this potential are: . & 2) = a (yt) + } face) SUR 5, (an/n G2) }dz.cosnry. n=l 2VE-z (5.8) 4 ve?) = Jag) 5, (n/n E=2) )dz.nn sin niy. n=1 fa) n Before considering the convergence of the above expansions and behavior 27 far from the wave front, of the functions they represent we examine the (2 ) yD procedure required to obtain higher order terms t¢ Pe pene ie (2) (3) _opaeee £ £ satisfies the As the system (5.5) is even inv, $ does not occur. > following equations (3) (3) . . $¢ SY ~ Abe ne = 0 »>O<y<1,0< FE <%,0<yn<@; bg (M5056) = 0 , O<n< 7, O0K< & < #3 (3) = - 5.9 be ytd) 0, ( } 667 Osy,8) = lin da (8)-0p OE, Ys Eten we s708) } veo” O0< &E <a, O0<y<i; 6°) .y,0) = 0, Q<n<e, 9s yell. (3) The condition on $ at n = 0 requires further explanation £6 If the procedure we are following is to be of use, the approximations (1) (3) Vo > v36, 3++. Must be successively of lower and lower order uniformly inn ,y, and &. Also at x=0 (n= v2e) 1 ake BOE) Gy £)- Gre Oe tv OY) O, (3), 2k (Ze-1) +? ($5 gg ote. ety be, > must be uniformly of 0(y2**?) we examine hee det a fy E)- o£) 28,958) 298) O2,y 83) in detail. We find the result: . 3 mee) = ) | 20.0? (2)—BE— — 3, (nmvv= (E-z} )dz.cos niy. n=]. . 2vE tE=a) + . G. 11) Using the following inequality: 28 [ 5, ny WCE=2Y | < meen ; (), we can estimate the magnitude of E Ee - ney? z|a° (2)| a2 , fe) We have prescribed that a wie) = 0 forg > 1. Therefore 1 je <& yn? [ zl2 ©) (2)| az . (5.12) n=1 ° AC present we assume that this sum is convergent and then we conclude je is uniformly 0(2). Writing the Bessel function J, in equation (5.11) in power L series form we find that zZ pl) -¥ l, n2 72 | z a) (a)az . cos my + O(v4) n=l fe) These results then provide a boundary condition for » S) at y =v%e which > we approximate by the following condition at n = 0. z 3 , i & 0 os *? Osy.8) =F » oat z af ) (z)dz.cos nny 5 (3) We can then obtain a solution for be 3 gs? =. d orf a0” hf Z 3 1 (atvn E~-2z) )dz.cos ary. £ Ly 2 ! (5.13) Proceeding to higher orders we can obtain a solution o & k ped o - 7 6) a ( a) (2) 2%)? 5, (on/AETz)ae- coe nny. ug (5.14) 29 We also obtain an estimate for the error p (241) | ~ 1, + k+1f 2” 0 | g2kH1) < vk 2 1 & | 2 | as ) (2)| dz » (5.15) Oo which is unifornly 0W2*) provided the series above converges. We now take up the question of the convergence of the series in the crror estimates (5.12) and (5.15) and in the solutions (5.7), (5.13), (5.14). We assume that a ¢y42)-fa Gy edey has continuous k+l th y derivative fe) and piecewise continuous k+2 th y derivative for 0< y< 1 and O<é <1, These conditions ensure the convergence of 1 (nt yt) >| Also 5 (nar) converges to. vy GHD) a0) ¢ n=l sin n KL ay uniformly for 0 < y< 1, for all O< € < 1. We further assume that the convergence of ) (nr yi Dei true, for example, if a0 ey) d= at yal” (y), The above conditions (1) ; RO geet) are a ee} ktl ae ) cos n1y a) is uniform for 0< — < 1. This is are sufficient to ensure that the errors E uniformly 0(v2), Ov)... 0 (eet?) (1) | 6 og CRD are all uniformly convergent. respectively. Also the series solutions for ¢ (0) From equation (5.15) we can see that the condition on a 0 a (y,£) = 0 € >, is unnecessarily strong. The condition, a vy 6) = oO 5) for large €, is sufficient. 6 (2k-+1) We are interested in the behavior of b¢ far behind the wavefront. That is for large © (but for @ < dy) For large 6€ rol 1 it 2 a) (2) E% J, (anYn (E-z) dz .cos riTy, 2k, (2k+-1 2k ¢ /ni vi) 2 PK GE 30 Using the assumed properties of a vy 2) and the fact that aE co < + k + we can see that although the approximate solution ) 9 3 1) produces j=0 errors at x = 0 uniformly at o(y?*) the separate orders are not uniformly of 03) for all E,n,y. We can demonstrate this fact by substituting g = £® and n= S (t+) in part of p PRD), k k ky zee _ ES @ny' 7 a) (2) Ee82)? J, (anv E=2) dz n=1 (5.16) X cos any. 2k Ok) i uniformly of ot) and decays We can readily verify that v like g7ll4 far from the wave front. The assumption that a (y,5) = 2 G.0-] a) (ye )ay possesses k+l continuuus derivatives on the open interval 0 < y < lis not a very restrictive practical condition. However the assumption that this condition also holds at the end points y = 0, y = 1, is rather severe. As a result we can only prescribe functions a vy 2) fer which the odd y derivatives of order less than or equal to k+l are all zero at y=OQOandy=1. If 2 (y,8) does not satisfy these conditions we have to consider "corner approximations". We have been unable to develop an. asymptotic procedure for examining these "corner approximations", ‘he solution we have developed for $. was obtained without using any boundary condition at the end x = 1. The solution te first effects the end x = 1 at time t = 1. We use the values of $¢ at x = 1 and t > 1 as new data for boundary layer approximations and wave front approximations similar to these of section 4 and this section. This proccess must be repeated after time t > 2 at the end x = 0, time t > € at the end x = 1 and so on, 31 In practice we will probably. only be interested in the approximate solution of the problem up until the first reflection. 6. Summary of Results. In this section we summarize the results of earlier sections. We have developed an approximate solution of the problem (2.6), (2.7) of the form: g(x,y, t3e,6) = 6, (x,y,t3e,6) +o) Eystse) + 1-x 6 t-x. 6 ba. ray atselto, Crlttx) v.53 2) (6.1) The usefulness of this expression depends on the prescribed properties 0 as . es of the excitations pP, ¢ 0) and a‘ , Tae inner solution o5 satisfies the boundary conditions at y = 0 and y = 1 (2.7a) exactly. Also os describes the propagation of the y-average of the excitation prescribed at the end x = 0 (2.7b) exactly. However, the accuracy with which 04 satisfies the differential equation (2.6) and the initial conditions depends respectively on the smoothness and initial growth of the excitation oY ct) (the smoother oi) , the more accurate Os). The functions » and Por describe the quasi-static boundary layer effects at the ends x = 0 and x = 1, respectively, arising from the slowly varying excitations prescribed at the ends (except for the y-average of the excitations whose propagation is described by the inner solution 5). The function %¢ describes the propagation effects stimulated by the rapidly varying part of the prescribed excitation at the end x = 0. The development of the terms Op and ¢ and the order of accuracy with which they satisfy the differential equation (2.6) and the end condition (2.7b) depends on the prescribed excitation at x = 0. This excitation is 32 represented as the sum of two functions 6 vy 46 Cy t/5). Here £O vy by represents the slowly varying part of the prescribed excitation (0) and the development and accuracy of bo depends on the smoothness of f (0) (the smocther f* °, the more accurate be [cf. section 4]). The function a (y 78) represents the rapidly varying part of the prescribed (0) excitation. The development of bs depends on a’ “ vanishing for t > 6, where © is small (that is the sudden part of the excitation must be over in a time shorter than that taken for an acoustic wave to travel across the rectangle). The accuracy of $, also depends on the smooth- (G) (0) ness of a with respect to y (the smoother a is with respect to y the more accurate is b- {c£. section 5]). The division of the prescribed data at x = 0 into functions 6 (+) and ay £75) is a rather arbitrary process, There is no very precise mathematical way to define the admissible class of excitations at x = 0. From practical considerations we know that the impulsive excitations considered here are described by functions of time which vary rapidly for a short time and then ocillate more slowly back to some final static value. Given the graph of such an excitation (say obtained from some measuring device) we must decide on a time scale for the rapid excitation, say TS" We smoothly join the section of the prescribed excitation for t > Ty to a zero value at the origin t = 0. This determines our functions a vy t/8) and gO oy ey, The method developed is only applicable if the time scale T5 leads to a parameter 6 (rapidity of excitation) which is small compared with ¢ (thinness). : a § . «gs tae Ever if the condition 2% 1 is satisfied, the other smoothness conditicns 33 required of the prescribed excitations, especially initial smoothness of the function po Gx, t) and smoothness of aD) Ot /6) near y = 0,1 limit the generality of the procedure. However in practice the leading term in cach approximation may be all that is required, and these can be obtained for prescribed data of a reasonably general nature. We have established that the expression for $¢ written in (6.1) satisfies the conditions of the problem (2.6), (2.7) accurately. If we write p = exact -¢ , then we have shown that y satisfies a system similar to (2.6), (2.7). This is: M - Vovy + e* (Vs - Voi) Se A(x,y,t3¢,6) > (6.2) for O< x< 1, O< y< 1, O< t< Il Woy O6s0,05656) 0, Voy @otstse58) = 0 > (6.3a) for O< x< 1, O< t < »; M e B(x,y,tse,6) , Ws, (.y.tse.8) M (6.3b) ¥s,,Clsy,tse,5) = ¢ C(x,y,t;e,6) ’ for O< y< 1, 0O<t<; yx, y,03¢,6) = 0 > Vo, Gry,05e,5) = 0 > (6.3c) 2M, 2M, 2M; V for O0<x<1, 0<y<i. Here Pa is the largest of ¢« » = ; and A, B, CG are functions of x,y,t determined from the expressions for 1 1 ] bys dy» pr and ee It is found that { A dy = | B dy = | C dy = 0. In °o oO oO order that ¢ should be considered a good approximation to the exact solution exact the solution of the system (6.2), (6.3) for jj should be oe"), 34 This result can be obtained using elementary methods to solve (6.2) , (6.3) exactly; or better, by using the results obtained by Shield [36]. 35 It. AcT APPROXIMATE SOLUTION pF THE DYNAMIC EQUATIONS OF ELASTICITY IN A THIN REGION. 1. Introduction, In this chapter an approximate solution is constructed for a boundary-initial value problem in the linear two-dimensional plane strain . theory of elasticity. We consider an initially quiescent elastic slab suddenly loaded on its boundaries, and then try to determine the subse- quent motion of the whole slab. For simplicity, it is assumed that the prescribed loads lead to a state of dynamic plane strain in the slab. This means the motion is two-dimensional, and we accordingly speak of an "elastic rectangle" to connote a typical cross section of the actual three-~dimensional region. Furthermore in this thesis we are interested in thin plates and therefore we consider an elastic rectangle whose thickness is small compared with its length. | In a dynamic problem certain characteristic time scales (for example: a pulse duration, or a period of oscillation) are introduced by the prescribed loads. Suppose Te is an example of such a characteristic time. This time, Tos multiplied by a characteristic velocity for the problem (say the shear speed), gives a quantity having the dimension of length. We call this length the "loadiag length". The relative magni- tudes of the length of the rectangle, the width of the rectangle and various loading lengths associated with the prescribed loads, greatly influence the nature of the solution. We do not attempt =o solve the problem for completely general prescribed loads. We specify in advance, that the stresses applied to the long sides of the rectangle are slowly 36 varying with a loading length of the same order of magnitude as the length of the rectangle. We further specify that part of the loading at the ends is similarly slowly varying. In addition there is a brief initial impulse at one end which has a small loading length compared with the length of the rectangle. This rather vague description of the problem is made more precise with the detailed formulation in the next section. It is well known that two types of waves, the so-called shear and dilatation waves, arise from a disturbance in an elastic medium. These waves interact near a surface to produce a surface (Rayleigh) wave. The differential equations and boundary conditions required to describe these phenomena are naturally more complicated than the simple wave equation and boundary conditions of the acoustic problem of chapter. I. However some of the features of the approximate solution obtained in this earlier work occur in the more complex and difficult elasticity problem of this chapter. It 1s possible to construct an inner approximation which satisifes the specified loading conditions on the long sides of the rect- angle and describes the propagation effects arising from a slowly varying average normal stress at the ends in a manner exactly analogeus to that used for the inner approximation of chapter I(or the inner approximation for the deformation of an elastic plate c.f. Friedrichs and Dressler [6é]). Again it is found that part of the slowly varying end loads, with certain averages having zero values (zero average normal stress, zero average shear stress, zero bending moment), produce effects which are important 37 near the ends and can be described by quasi-static boundary layer approximations (c.f. Novozhilov [30]). To describe the effects generated by the average shear stress and the bending moment applied at the ends, a "diffusion approximation" is introduced. This has no analogy in the acoustic problem or the static elasticity problem. We use the term “diffusive” to describe the spreading of the slowly varying part of the average shear stress and bending moment into the interior of the elastic rectangle. Furthermore, propagation effects produced by the impulsively applied loads at the end of the rectangle require the development of three wave front approximations associated with dilatation waves, shear waves and Rayleigh waves. As will become apparent in this chapter, the struciure of these wave front approximations is more compli-~ cated than that developed for the acoustic problem of chapter I. Except for particular preblems corresponding to very special end loads, "corner approximations" are required. These have not been obtained in what follows because they are connected with the unsolved problem of a dynamically loaded elastic quarter space. Since these corner approximations and the wave front approximation moving with the Ray leigh wave speed are inter-related, the latter is also not completely developed. The main difference between the methods developed in this chapter for the elastic rectangle and those of chapter I developed for the fiuid rectangle is that in the elasticity case the slowly varying effects and the rapidly varying effects interact through the boundary couditious, The capidly varying effects in the elastic rectangle cause other effects at the boundaries which may not all be rapidly varying. In 38 the acoustic case considered in chapter I it was possible to completely separate the slowly varying and rapidly varying portions of the solution. In spite of the deficiencies mentioned above, this work may help to elucidate the various phenomena occuring in an elastic plate after impulsive loading. Also it may help to show more clearly the significance of the approximate theories - the plate wave theory and the Euler-Bernoulli plate theory - in relation to an exact solution of a dynamic elastic plate problem. In this respect the present work can be regarded as a supplement to the work of Miklowitz [23,24,25,26], Mindlin {[28,29] and others (summarized in a review article by Miklowitz [27]). The results obtained in subsequent sections are summarized in section 11. The reader may understand the development of this work more readily by first reading section 11. 2. Formulation cf the Problem. The two-dimensional equations of motion for an elastic medium may be written in terms of the three components of stress o and the two displacements U,V, in the X and Y¥ directions, respectively, in the following forn: o xx xt oxy > PU sap ’ (2,1) o = oV we +o , XY,X -YY,Y *TL where p is the density of the medium. The stresses are given in terms of the displacements and the Lame constants 4,u by the following consti - tutive equations: o = (At2)U +i Vy. 39 Syy = wet uy > (2.2) of = U, YY + Q42u)V,, > X The problem may also be considered in terms of the two potentials % ¥ for which: U = ®,, +¥ »y 5 (2.3) V=o,)-¥5, , (c.f. Love [22], p. 47). In terms of these potentials the equations of motion are: $ + 4 _-4 ¥ xx °YY c,? rT = 0 ? (2.4) Us - - = Yo yg FY vy eye OTT 0 where cy is the speed of propagation of dilatation waves, and Co speed of propagation of shear waves. These velocities can be expressed in terms of A, Lu, Pp: c2 = AtZE 62 ob i p ; 2 2» ° We consider the open rectangle R in the X,Y plane consisting of points for which O< X< 2, -h < ¥Y< h. It is required to find functions U(X,Y,T) and V(X,Y,T) satisfying the differential equations: ary 7 -_ = OF + OHI, + WU = Uso = 0 , or. Nh . mn a HVS, + Gnu, + Ot2Vs ~ OVismq = 0 : for (X,Y) ¢ R, T > OG, and fulfilling the following boundary and initial conditions. 4&0 ST yy& ,Y »T} = P, &,T) > 9 (kK hsT) = Q, &.T) > O<xK <%,0<T<~», (2.6b) Gy bk h eT) = Qy CST) 9 F(Y,T) + A(Y,T Hi a 0,Y,T xx! ) -h< ¥<h, O< T < ~,(2.6c) o (&,Y,T) = 0 XX a (0,¥,T) = G(Y,T) + B(Y,T) 4+ XY ~h< Y¥<h,QO<T<-, o (%,Y,T) = 0 (2.64) XY - u«&,Y,0) = 0 a V(X,Y,0) = 0, . \ (X,Y) ¢ RB. (2.68) Vp &,¥,0) = o, , Here PoP 29 20,2 F GAB are prescribed functions. The division of the boundary conditions at the end X = 0 inte two parts is a notational convenience which anticipates results developed in later sections. It is found that the slowly varying parts of the applied stresses at X = Q, which we denote by F and G, and the rapidly varying parts of the stresses denoted by A and B, produce quite different effects. The properties required of F, G. A and EB depend on the order of accuracy needed for the approximate solution and will be discussed subsequently. At this state we assume that the sudden part of the loading has a limited duration, 41 RMNamely A and B are zero for T 2.7): The smoothness properties of Pi>Po2Q and Q) are also discussed in later sections when they become important in the construction of the approximate solution. For the present we assume that the given data are such that solutions for 64 and ¥Y exist which give rise to continuous displacements on the closure of R for each T > QO. The rectangle R is thin if « = 7 is small compared with unity. In order to exhibit the role of « more explicitly, we introduce new independent variables x,y and t, defined by the change of T x ¥ “2 scale x = uey*Reo tery . (2.7) Another length scale Cot, is formed by multiplying the pulse duration, T., of the end loading by the speed of propagation of shear waves, c ra) 2° We define a parameter 6 measuring the suddenness of loading by the cot, equation 6 = a Non-dimensional dependent variables U,Vst > Thos T yy? o and xy y are introduced by the following definitions: u(x,y,tse,5) = *u(2x,hy, et) > rf fo . i a. vix,y,tse,5) = FVQ&x,hy, —— t) , 2 J) . i chy eet Tax Oy st3258) = yy (hx, hy, C5 t) 1 . g Ty Hoy t5e,5) = ib Cyy Pe hy cy t) ; (2.8) T (x,y, tse 6) = 46. (2x,hy, t) + yy HK. YY Co 1. & ¢ {,y,t3e,6) = qe 2 @x,hy, Cc. t) > 2 h* 2) 42 We are interested in the particular case where the given loads produce displacements u and v which are O(1) in the limiting case e + Q. This occurs when the loads, written in terms of scaled variables, have the orders of magnitude displayed in equation (2.9) below. The prescribed loads are given in terms of the new functions Pp?) pi?) ql?) ; 1 af) £0) a1) , g(3/2) and b(3/2) when written in scaled variables. Thus: 22) et) ely tim bt). eee @iety ak a € Py (x,t) . a rae € Po (x,t) : Pe (exe t) : 2, (2) -l g 2,(2) =i x oe ed (x,t) : 0, ee) e % (x,t) 7 Q, Ox rere) ’ (0.9) ef (y,t) = L rthy St) , ca (y,e/8) = £ acny£t) , a fo u Co e/2g/2 (yt) = 2 cthy,£t) , c8/*pO/4) (y, 478) = 4 Bany, St) . u cy u Cy Here the superscripts on the above functions refer to the associated power of «. As in the acoustic problem of chapter I, we have assumed the given data to have a very special dependence on ¢, the thinness parameter, when expressed in scaled variables. The problem associated ‘with the prescribed loads of (2.9) can be considered as one of a4 whole hierarchy of problems associated with more general loads. The essential features of the dynamics of an elastic plate are included in the problem with prescribed loads given by equation (2.9). ‘Using this new notation, and defining a parameter a by the n 2 equation a* = me » we are lead to the following finai formulation: c 2 Us + e(atl)vs + e*(a?u, - Us) = 0 (2,10) x22 1 1 + -SOTt + 2 ad = 0 5) Ve yy ee “oxy ° oa Voce 7 32 ee? , 43 for 0 <x <1, -L <y<1, 0<t <o The boundary conditions and initial conditions are: 2 Tyy & st tye, 6) = 2g Ge t) ; (2). (2.11a) Tey He-Lst3e, 6) =e7q5 (x,t) . for O<x<1,0<t < 2 Tyy ol. t3e,8) = eps (x,t) $ (2.11b) 2 Tyy Gnd, tse, 5) = pe (x,t) ’ for O< x<1,0<t < »} tpg Oryatsesd) = cf YG, e) + ca GY, c/6) . (2.11c) Ty (bey st5e55) = 6 9 for O< y<1,0<t <4; 3/2 (3/2 . 3/2, (3/2 Ty Ory, t3e55) = ¢ / 3! / yt) te / pf / Uy ,t/s > (2.1143 Tyy Loy. tie,6) = 0 ’ for O<y<1,0<t< >»; u(x,y,03¢,4) = 0 3 vix,y,0;c,5) =0, (2.11e) us, (x,y,0se,6) = 0, v5, ,y,03e,8) = 0 3 for O0< x< 1, -l<y« il. The stresses are given in terms of the displacements according to the equations below: 44 The = (a?-2)v,, + ca* UW» t yy = Yrs + EVs. ’ | (2.12) = 2 - Tyy avs, + e@ 2)u, . The potentials » and p associated with the displacements u,v are cal~ culated from the equations: u= Voy + EVs, > (2.13) |The problem (2.10), (2.11), (2.12) can not be solved explicitly and exactly by simple methods. | Thus the construction of an approximate solution directly from the differential equations,boundary conditions ‘and initial conditions inthe form of an asymptotic series in ¢ and 65 will be of interest. We attempt to carry out this scheme in later sections. 3. Formal Inner Approximation. In this section we utilise a procedure: previously developed for the static elasticity problem [c.f. Friedrichs and Dressier [6j) utilized in Chapter I for the wave cquation. First, approxim&tions having the form cf power series in ¢€ are examined. Furthermore, as stated in the last section, we are interested in the problem where the displacements u,v are O0(1) in the limiting case « + OQ. We call the approximations investigated in this section us and Vis and we* assume expansions for them as follows: 45 i] u, G,y,tse,5) 20 (yy ,ts8) + cul) (x,y,t38) +... > vi.) (yy st36) + eve? (x,y,t38) +... (3.1) v, Gy, tse,5) Here the subscript "i" is used as an abbreviation for "inner". We also assume expansions for the associated stresses 1_.., T,., and t_. of 4 XXL xyl yyi the form: = OL, xxi xxi xxL (2 OL, xyt xyi xyt (3.2) T = (0) + er “) + o.ee yyt yyt yyt We now substitute these expansions into the system (2.10), (2.11) and (2,12) and soive the resulting equations, by considering the coefficients of each power of € separately. This leads to the following differential equations for a and vy: (0) = 0 “Layy ° , (3.3) v6) = 0 ’ Lyy for O0< x <1, -L< y <1, 0< t< %*. Solutions of these equations cannot in general satisfy the prescribed boundary conditions at x = 0,1. This statement is also true for the solutions of the equations for wD yO) yO) VQ atten are provided by this procedure. Consequently in this section we are content to fulfill the prescribed boundary conditions at y = 1. The approximations so chtained are not fully determined, and have the capability of accomodating only special types of boundary conditions at x = 0,1. These conditions are developed 46 in the later investigations of the “boundary layers". (0) (0) The boundary conditions for u, and v5 at y = 1 which follow from (2.1lla), (2.11b) and (2.12) are: (0), 4 ~ 60) LS) a Tyg So? 1,t36) = By yy Ko t58) 0 : (3.4) 0 0 ty e,st1, ct; 36) = vs? Gxy#1,056) = 0 > for O< x< 1, Q0<t<oe, We conclude from (3.3) and (3.4) that wf, vi) have the form: ui) (x,y,t36) = ay”) ac, £38) : (3.5) v (y,t38) = 7 x, t38) ; i i a) and vy ) are as yet arbitrary functions of x and t. Corresponding stresses are all zero, that is, 0) = 0, Tt (9) =O, 7 0) = 0. To obtain xxi "yi "yyi more information about u(0) and v9) we must investigate higher order terms. At the first order there is some interaction with the zeroth order. The differential equaticns and boundary conditions for ult), vi) resulting from (2.10), (2.11) and (2.12) are: a). yh). i,yy , i,yy ° for 0< x<i1, -l< y<1,0<t < »; (1) vey a ph ce ty ergy a yee tq psay e Tyg Fo*1,059) = Uy yy eeth t58) + Vie jr35) = 0 > (1) t 2 1) ¢, + & (1), £ : = Tyyi & 1,t3é) = (a ~2)us 3 & 1,t; } +02 Vay 1t;é) 0, 47 for O< x<1,0<t<», The most general solution of this system is: 1 —(0 —(1 uf ) Gey ,t38) = ye?) (x,t38) + us ) (x, t38) > (3.6) vo) (x,y,t36) = -y 2 OG, t36) + ve (x, t36) ; where us a vy ) are arbitrary functions of x and t. The corresponding stresses are: ) . 4@%-1) 5@) |, . g (1) Lg | Taxi * a “iy x” "xyi > Tyyi ° (3.7) This procedure will determine the structure of the y~dependence of successive terms uf? vy, a, vy ».», but only in terms of the ~(0) =(@) -Q) =) arbitrary functions Up os Ve oy Us sy Va once e However, at the next ~(0) =(0) order, further restrictions on the behavior of u, “s Vj" are obtained. (2) The differential equations and boundary conditicns for us (2) and V5 are: uO) 4 aw + 02a - uO = 0 Lyyy L,xy “i, XX “i, tt vy g-1 1) ,l.0) _1 ,@) _ Lyy tu YG sxY To i,xx ” a? Va,et” 0 , for 0< x< 1, -l< y< 1, 0<t<»®; a?) = £2) 2 y@ 4 yO Gtsy-1 . 2 2 g2ayuD 4 gty@ 2 PY? GE sy =a, “yyi i,x isy ee, t),yrFoe-l, for 0< x <1, O<t< , After substituting the earlier results from 48 from (3.5) and (3.6) we can solve the above system to obtain the solutions: (2) - y2 pat-2 FCO) 2 (g(2) ~ 9 (2)y) ~ yp) ~ beg (204 9 (205 4G) a airy mer ae ia Oe he (2) oy. p(n (2)-3(2))_ 92-25(0) yy pa2-25 ()- 10 (2)4,(2)y 14 FC2) i 2 Parc Po? ri yer; 202 Baz Py po “i (3.8) where u@) and y(2) are arbitrary functions of x and t. The integration of the differential equations governing ul?) and vi?) leads to the result that they are both quadratic functions of the variable y. Using this information in the boundary conditions we find the solutions (3.8) and also the following restrictions on the behavior of WO) and YO): i i (0) _ 4(a2-1) —(0) _ (2) _ g (2) Si itt a i xx + Gal qo yD; (3.9) ~(0) _ 1 7,(@2) _ (2) “i,tt 2 (Py Pods (3.10) When recast into physical variables, equation (3.9) is readily identifiable as the "plate" wave equation (the plate wave speed is 1 i c 2_ + 2 ry Laon) = [b Ste shy* ; Eis Young's modulus and co is Poisson's ratio (c.f. Kolsky [20], p.81). Equation (3.10) can be interpreted as a statement of Newton's law of motion for the mean behavior of each section of the plate. It is interesting to note that, as with the static elasticity problem, the inner approximation can be identified with one one of the approximate theories; in thie case, the plate wave equation. We assume that the inner approximation satisfies the initial quiescence condition (2.1lle). Thus «7 (0) satisfies the conditions: i 8) Q vs Gesy,058) = 0, vpPLGcys058) = 0, 49 and we can integrate equation (3.10) to obtain: t t vo = x { a | [pe (x,t, =p (x,t,) Jat, . (3.11) ro) fe) However, the initial conditions: —(0 —(0 us ) (x,y,038) = 0, 0°) Gx,y038) = 3 are not sufficient to determine a ) as a solution of equation (3.9); boundary conditions at x = 0,1 are also required. If the prescribed normal stress at x = 0 was just a function of t (that is, constant across the thickness of the plate) then, using equation (3.7), we could write a ZO) | suitable boundary condition for u,; In general this is not the case and (0) it is not immediately obvious what condition to specify for us —(0) and x = 1. Suitable boundary conditions: for uy at x = 0 are determined in the - course of the boundary layer investigations « of section 5. (2) (2) 3 The stresses associated with uu, and Vv, ! ,@) . y (ooh “=2. (, @)_ p?))4 AD y 7O 42 Xp @), (2) 54 Lo -1)=(1) Text V5, KX “4x 7 « X (g?) - a) +$ (q » tay”) ; (3.12) xy 1) i (p?) - (2), @) yyi +P) po?) + + 50} (6) Proceeding through successive orders it is possible te evaluate uy yO). 4G) > (2) _ (2) qi? (2) gor Bye Vy sees in terms of the boundary data Py > Po L 249 4. : oo oe . =O «5 “ay 7) and a sequence of Functions of «x and t only, US aa Vis Ups Vp “eres satisfying differential equations similar to (3.9) and (3.10). It is 50 found that the successive orders of the displacements depend on success- ively higher derivatives of the boundary data (as was the case for the analogous terms of the inner approximation derived for the acoustic problem of chapter I). Thus the smoothness of the prescribed loads de- termines the order to which the process may be continued. To examine the structure of successive orders in more de- tail we now proceed tc the third and fourth order equations. By exam- ining these higher orders we can discover the relation between the —(@) —(2) functions uj “and u; °, and show the inability of the inner approxi- mation to deal with the rapidly varying part of the applied end loads. The differential equations and boundary conditions govern- (3) (3) ing the behavio~ of uj andvy = are: uf) + (a 2-1)vi2) +a 24Q) -~ yl) = 9 1,yy 1, xy i,xx i,tt 2 (3) asl (2) (I) _y@) - Vv +> u + Vv Vv =0O , L,yy a i,xy az 1,xx i,tt for O<x<1, -l<y<1,0<t<.; 1@) = y+ y% 20, OD = 22)0 + ty «0 xyi isy | “i,x “yyi 2X i,y for 0<x<1, 0 <t< «, After substituting the earlier results from » y* 71, (3.5), (3.6) and (3.8), and performing some tedious manipulations we can solve the above systen to obtain the solutions: i 6 a* i,ttx a i,xxx + 27 | al) _ 302-2 gil) - at 2o1 Pl + p?))) + 2 i,tt a* i,xx 2a7- + y[- 724 see] G10) - 2¢a°1)710) 1+ 0 (3.13) i,x ae i ,ttx a* i,xxx 51 (3) _ ae -(0) , 3a2-4 700) y7a2-2 —(1) , 1 =(1) V4 7 “4, xttt io) “i, xxxl 2 [ ae Vise! a ‘a, tt wha 214 el + y[- a2 a?) 42 otal 7 (0) 202 su ZO) 7 4 y@ vat “a, ttx “is xx! i @, (2) (2) (2) In writing these solutions we have assumed that p > Po > 4 > do are at least differentiable, and their first derivatives with respect to x are O(1) in comparison with ¢«. The development of the solutions (3.13) also produces conditions restricting the behavior of ue a na Vo" > These are: s(1) _ AG@o®-1) =) _ 22 | (2) (2)) “ai,tt a “i,xx 2a (PT x * Po) , (3.14) FO = de® + af) “i,te 2°41,x The right-hand side cf (3.14) represents the squeezing action of the normal stress applied on the long sides. The right-hand side of (3.15) shows that the shear gradient produces a lateral deflection of the plate. It is worth noting that both of these effects are important in the development of some of the "approximate"plate and rod theories. The Rayleigh correction to the plate wave equation: based on squeezing motion (or lateral inertia) is described by (3.14) and the Timoshenko shear correction to the approximate beam equation accounts for the effact described by equation (3.15) (o.f. Miklowitz [27]). The system of equations for ul) and vi) were investigated and solutions obtained. These are lengthy and contain no significant new information and hence we shall not record them here. As part of the (4) (4) investigation of the solurions for uy and v, ° we can derive the . 2) —(2 following equaticns governing the behavior of Fc and ve , 52 762) 4 (a2 -1) 7) 4 (2) (2) 7at=14024+8 2) oS? Yate. {a (4 fy tr ~ 42 tt 607 ( ) co “i, xx" 0 1, xx 42 »XX -4 (a? =1) (a 2-2)? —(0) a® 4a, XXXX 0 , (3.16) > (2) L_ @@ _ ,@ 3aP=2 (2) _ 52) ) +4 Cs =1) —(0) Vaytt + 124° ( 1,tt ~ Po, ee) 12a ( 1,xx 2 xx) Vi, XAXK =0. (3.17) 2) (2) (3) 1,tt’ 41, xx? Pritt? —(0) Here we have assumed that the second derivatives, (2) loxx? ttt? exist and are 0(1). Also it is clear that u, b) must be at least four times differentiable with respect to x to allow for TQ) a continuous solution for uy. A more precise statement of this fact is made later when suitable boundary conditions for (3.9), (3.14) and (3.16) are discussed. This will be done in section 5 which is devoted to the investigation of "boundary layers" near each end of the plate. In principle it is possible to proceed to higher orders for GS) _G) Solutions for Ups Ve “sees 3 at the same time we can develop further i a, yi) equations restricting the behavior of u i? Yq »e. » However, the calculations involved would be prohibitively lengthy. In this case a symbolic procedure similar to that used in section 3 of chapter I in dealing with the inner approximation. for the wave equation problem is of little use. A "symbolic" solution analogous to equation (1.3.18) can be developed but no convenient expansion similar to (1.3.19) is found. ‘ (OF i However for a practical problem the first three terms u @ may be all that are required. We shall adopt a truncation of the original expansion (3.1) as our inner approximation: u, = uO) 4 eu) #07ul) i _ vi) ; (1) (3.16) i* + ev, + ety, Y? Vv 53 where 10) yO) wu) v6) uy and v0?) i i i 4 4° VE are determined from equations (3.5), (3.6) and (3.8). In full these are: Bu, = 3) + ef{-y vi + uD He2{ ypucsd 2 1. bq? 4} + Gy + a) aaa ray , (3.19) a0 ¢ ve? He Lie (pt (2) (2))_ at e2 = 7) 1 Vv, =v yo ) + ef{- ~y HG “Py Va xx ® (2) a2 —2 TQ; 5 (2) * yiggr (p * Po )- “ge © i,x! VG } Here wi), vO, a, we, ay”? and vw? are all functions of x and t only, satisfying equations (3.9), (3.10), (3.14), (3.15), (3.16), (3.17), respectively. We require that the inner approximations given by equation (3.19) fulfill the initial quiescence condition. That is: u, (x,y,03e56) = 0 > u, _(sy,05e,6) = 0 v, &,y,03e,5) = 0 , Var V2 03e58) = 0 (3.20) This condition provides the initial conditions for a ) ao p 00s 2 70) . FO) 27M. TM) 2FO.T@ 260, tee, i “i,t i “yt i “ast 50) 3) 2 FO) 2) 272) HD, 20 -eo. OP i i,t “a i,t “a. i,t ; ? for O0< x<1, -l< y<i1. The equation (3.20) also requires that the 2 prescribed functions pe? ) po, ae? ) and al 2) all vanish at t = 0 and have vanishing first time derivatives at t = 0. 54 The stresses corresponding to the inner approximation for the displacements written in equation(3.20) are: = ¢ { 44021) 700) 5 4 62 a?-2 (2). 9 (2)) 4 402-1) FO) 74 ext et at "4, x et yee (Py Po @ Vioxx! o2-2 ( (2), .(2)) , 4€02-1) —(1) 3 + [ 02 (P; + Py ) + 02 uy dt 9¢e9) = e2@{z (2)_ (2) (2) (2) ‘yi 7 { py > Py) + 50} + py )} + (3.22) 3 f(y2- a2-1 (((2)_ (2) 2(a? a1) (0° 2) =;(0) Fe GSD “gaz (Sy, x 903) 7 at “iy xxx! , = 2d (2)_ qi” i (24. qi?) Txyt { 544; ) + AG 9 bt . q2~ 20 _ . Note that the terms jin e? in t o9t are written explicitly to show . xyi yyi that the boundary conditions at y = +1 are satisfied by the inner Sin t | will not be used and approximation (3.21). The term in ¢« . : XXL therefore is not written explicitly. We can now use the results of (3.21) and (3.22) to test Uys ve as solutions of the original problem (2.10) and (2.11). We finda: 4- 2. + 2 2 ~ = Of 3 My yy e(a WY 4 ny e“Ca Ms xx Bee fe?) > ay. + e(a%-l)u, + c2( v, - Vv, ) = 0(e3) > L;59Y L,xXy 1,Xx% 1,tt x ( 2 Toys Hebe C5E.8) = c2qi?? ’ Toye Hebets es 8) = Pq ) > t ((x,1,t3e,6) = e2p (2) ’ tT ((x,~-1,t3e,6) = e2p (2) yyi 1 yyi 2 55 Thus the prescribed boundary conditions at y = #1 are satisfied exactly by u,, v5 and the differential equations are satisfied with an error of i O(e3). However we can see from equation (3.22) that in general u, and Vv, cannot fulfill the prescribed stress conditions at x = 0,1. As discussed following equation (3.20), the inner approximation satisfies prescribed initial conditions. Before proceeding to the later sections of this chapter, where we obtain corrections to us and vy to improve the approximations near x = 0,1, we briefly examine some aspects of a formal "symbolic" approach used for finding higher order terms of the inner approximation. |? As in chapter I we introduce the notation L = and D = Q ct * |” and formally consider the equations (2.10) as differential equations in the variable y: uy, + (a7-LeD v,_ + o%e*D u - c2L*u = 0 > yy y Voyy + Gz eD u,y + or e* Dev - or e“L*v = 0 ’ 1 for -1 < y < 1, with the boundary conditions, 2, (2) = wy + eDv {" qy > y=l > c2q?) 2 y= -1 s 2, (2) =1 (a2=-2)eDu + aev, -( on ; y , (3.24) e*p, > yrol . Now "symbolic solutions" for u,v analogous to (1.3.18) can be obtained. These "symbolic solutions" divide naturally into odd and even functions of y. The calculations are lengthy and will not be included here. x It is found that the part of v which is even in y, (Voven? ; end the part 56 of u which is odd in y, (u aa? both satisfy equations of the form: ) eVD2~ L2/a% (3.25) 2 - AD bP L2 sinevD*- L* cosevD2 L2/a* Woven Zee. eC, When we put € = 0 in (3.25) we obtain.equation (3.10). Assuming an = v6 + evl) expansion v even even even +... , and equating the coefficient of € equal to zero in (3.25) leads us to an equation similar to (3.15). It is interesting to examine a truncation of each side of (3.25) at O(z?). The left hand side becomes: eeey 2 2 2 24 £7.34 1 2 _ (4€at-1) 2n2 4. 4¢a%-1) py [L* + 3 {G+ 5-7 VL (Sy + 2)LEDS + = D } ] V owen This equation has a strong resemblance to the Timoshenko equation (cf. Kolsky [20], p. 55) for the dynamic behavior of plates, which in our present notation may be written: eae ry 2 fy 2 248 (Lys _ AOSD 5 Lyre 4 Ses Dps - [L° + 3! K L ( a + ae be + a pit] Veven where K is a "correction factor" usually given a value between 2/3 and 1. The above discussion is net intended to be a rational development of the Timoshenke equation. In fact the arbitary truncation process has no rational basis within the scope of the perturbation techniques used in this thesis. It suggests however that the Timoshenko equation is a combination of the first two orders of the inner approximation. 4. Formal Diffusive Approximation, In general the inner approximation (3.19) cannot fulfill the prescribed etress condition at x = 0,1. To explore the transfer 57 of the prescribed loads into the interior of the plate we must study the solution near the ends in more detail. We expect the rapidly varying prescribed loads at x = 0 to produce propagating pulses which can be described by wave front approximations (see section 6). Using the results of chapter I as a guide we might exyect that the slowly varying prescribed loads at x = 0 produce effects which can be described by a'boundary layer" approximation, important only near the end, plus the inner approximation. A preliminary "boundary layer" investigation was conducted near the end x = 0. It was found that the slowly varying part of the prescribed loads produces an effect local to the end only if certain conditions are satisfied. These are: (i) The prescribed average normal stress is zero, (ii) the prescribed average shear stress is zero, (iii) the bending moment resulting from the prescribed normal stress is zero. In general the prescribed loads do not fulfill these three conditions and this means that effects other than a quasi-static boundary layer are produced by the slowly varying prescribed loads at xX = 0. The a average normal stress at x = 0 can be used to provide a suitable boundary condition for the arbitrary functions a, ay? : uy? in the inner solution. Thus the average normal stress will propagate with a velocity equal to the "plate" speed. So far we have no means “of dealing with the average shear stress and the bending moments. Thus before examining in more detail the “boundary layer" avproximation near x = 0 and its interaction with the inner approximation,we study the effects produced by the average shear stress and the bending moment prescribed at x = 0. o We consider the idea that the part of the total solution given 58 by the slowly varying prescribed average shear and bending moment at. x = 0, is important near the end but decays away from the end. However, this decay is more gradual than that of the boundary layer approximation and we allow the possibility that the decay depends on time; in other words perhaps the effect of the average shear stress and bending moment "spreads" or"diffuses" away from the end. In order to examine a neighborhood of the end x = 0 on a length scale longer than that associated with the boundary layer of the next section, but shorter than that of the inner approximation we introduce a new variable z, by the equation: x . , 2-7 . (4.1) ¥E The reascns for the choise of this scaling are ost obvious, Trial and . : a error methods using arbitrary scale factors, say « , showed that a = holt was a significant value allowing the development of an approximation having the required properties. We assume approximate solutions for the displacements u,v of the form " u(x,y,tic,6) = u,(x,y,tse) + u,(z.y.tse) . (4.2) v(x,y,t3e,6) = v, syste) + v,@,y,tse) , where Uae Vy represent the diffusive approximation describing the effect discussed above. We do not inciude the parameter $ in both the inner approximation and the diffusive approximation since we shall find later that these approximations can cnly haadle the slowly varying part of the prescribed loads. 39 We require that u = u; tui, ve vy +v, satisfy the differential d’ d equation (2.10) and the boundary conditions at y = #1, (2.llab). Also we require that the stresses derived from these displacements, which we write as T = T + T > Tt =T ,+tT > T =T ,tT » are capable XX xxi xxd’ “xy xyi xyd*® yy yyi yyd > * of fulfilling the macroscopic features of the prescribed loads at x = 0 “(average normal stress, average shear stress, bending moment). However as we have not yet examined the boundary layer near x = 0 in detail, it is not yet obvious what boundary conditions to impose on the diffusive approximation and inner approximation at orders higher than that at which the loads are prescribed. In this section we determine scme properties of the diffusive approximation but do notdetermine it completely. The boundary layer investi- gation of the next section will demonstrate how to completely determine both the inner approximation and the diffusive approximation. Using the properties of u; and Va derived in the last section (see (3.19), (3.22)) and introducing the new variable z we can derive the following system of equations governing the behavior of u, and v.: d d 2. 2 _ 2 - 3 Ya yy + ve(a Da ye + €a Ud az E Yate Oe") 5) v + fe S54 u + ety v ~ ett, y = o(e2) d,yy a d,yz af d,zz ac d,tt , fordO<2< ~1 < <1,0<t.<.e@;: —-? Y ’ 3 ve thyg@o*t tse) = 0 ’ Te peg Cee*l tie) = 0 » (4.4) 1 . . for 0 < z <= > 0< t < ~; here the stresses are given in terms of the displacements by the following equations: 60 2 = 2— fT oe . a * fa 2)vy y t¥e Ua Tx : t xyd 7 “day te vax, Tyg = ave y + Ye (a?-2)uq . (4.5) Since we expect u to be small for large z, that is, far away d and Va from the end, it is natural to replace the domain of equation (4.3) by the domain 0 < z2< oa, -l<y<1, 0 < t < Then, instead of a boundary condition at the end z == » we impose the following decay € condition: lim uj, Vg = 0 > -leye1,0<t<o . (4.6) Zo We shall discuss the errors introduced by this approximation of domain at a later stage. The appropriate formal expansions for u, and v, are: 1, d d (0) D u,(z.y,tie) =u, (z,y.t) + vr ug (Z.y.t) +... (4y (4.7) vi (yt) + fe va (z,y.t) +... , V4 (Zsyetse) and proceeding in the standard way the following differential equations and boundary conditions for The and vi are obtained: (0) (0) u = 0 ; Vv = 0 dsyy d,yy , for 0< z2<, -l< ye 1, 0< teow} (0) T yya 1,t) = 0 ’ Oe, 4,t) = 0 > for 0 <z <a 0 <t <“;3 a), Ly lim d ?° “ad , Zo 61 for 1s yS1,0S5t<*%. We do not expect this system to determine ui), vo; however it does predict that the solutions have the following form: 0 , (0 0 Ww (e,y,e) = 1 (zt) » Ve Gaye) = WO ze), (4.8) where uy”? ard vy ) are arbitrary apart from the condition: Lim 70” a (4.9) Zc To —(0). —(0) obtain more information concerning the functions us and Va owe must investigate higher order terms. Unfortunately to develop ali the conditions governing the behavior of veo ) possible 5 this section, we ) @) have to investigate the form of the solutions fur ua > v5 dp eee 5 2 2 22), y@) Before proceeding to the next order we note that the stresses corresponding to uO), wv? all vanish: ~ _ 6) (9) (4.10) Txxd ~ 0 ; "xyd ~ = 0 ? vyyd ~ ° The differential equations and boundary conditions for uD on 1 C ; Va" are: (1/2) 2 2 (0) y /2) af=1 0) CL d.,yy + (a -Dva szy” 0 > Ve oy + “3 UG .Zy 0 for 0< z2<¢, -l<y<1,0<t< 9%; 1/2 1/2 a ) (2, #1,t) = 0 > ae ) (2, #1,t) = > for 0<z<~, O< t < %; (1/2) (1/2) _ Ug 2 Yq “ > for -l<y<1, 0<t<-°,. Using the expressious for stresses in terms 62 (0) of displacements written in (4.5) and the results for ug (1/2) gy (1/2) d of (4.8) we can obtain the form of the solutions for UG (0) and Va Suggested by the above system. These are: wl) (yt) = -y Wo) Gt) + wel) (ge) ’ (4.11) vOl2) 2 y,t) = -y £2 Oe, t) + vel) »t) , Here at) ¢2 | t) and v2) ty are arbitrary except for the condition: lin ul?) > as = 0 5 -O< t<o . gre (4.12) The stresses corresponding to u (1/2) and vill?) are: . 7/2) 4(o ay 7(1/2)_ ~ (1/2) © Toexd . “5 i > "xyd 0, “yyd = 0 > (4.13) We now move on to the next order and examine the following (1) (1) differential equations and boundary conditions for WG and Va ug) + (2D $e + 2a) = 0 ——_ yh) 4 2-1 0/2) , 1 (0) djyy 0% “a ZY a” ‘d,zz > for 0< z< ~, -1 <« ye l, O< t < 3 (3 t1t)=6 1) @,#1,t) = 0 ; for O0<2<*, 0<t < »; lim u za for -l<y<1,0<t< », After substituting the expressions obtained (0) _ (0) pl, y t/2) Yd earlier for Uys Vy > (4.8), and u >» (4.11), we can find the form of the solutions of the above system to be: 63 uit) (z,y,t) = -y vee) (2, t) + we) (, t) : (4.14) 2 yg 209 — ~ vg? eayst) =F GE VO ceyey - y M2 TOD eet) 4 FD In deriving these expressions we obtain restrictions on the previously us, arbitrary function u The satisfaction of the condition rl) (z,+L,t)=0 requires that: xy (0) _ Bae (4.15) This differential equation, together with the decay condition (4.9), provides the following solution for a P 7 = uy 5 . (4.16) The stresses corresponding to ue a nd ve? are ~@. 2-1) yoo) + 42 a1) 7G1/2) 7M. (1) _ "xd “4 a Vajzz" ae dyz > Tavd 0; yyd 0. (4.17) is partly an odd function of y, it may be able to describe (1) Since ¢ xxd the prescribed bending moment at x = 0 (for example, by taking 1 Veo, Ost) = = CSD) ly ey ejay). In (4.14) the functions at ) -1 and wit ) are arbitrary except for the condition: lin of”, vo) = 0 ’ ~lsys1,0<t<%. © (4.18) gre — —_—f 7 To find out more about vo ) ul?) voll?) geee, We proceed with the above scheme. The system of equations fcr u (3/2) > 3/2) is (3/2) 2 (1) 2, 04/2) _ Ud yy + (a “lv, ae ao Ug a 0 , (3/2) | of-1 yp) y(t/2) _ Vasyy + “aed 229 * ary d,zz 0 ; for O0<2<", -ley<1,0<t<»; 64 1/2) 2 21) = 0 (3/2) _ Txyd Td (z,#1,t) = 0 for O<z<~,0<t <4; Lim u (3/2) v (3/2) _ Zoe for -l<y<1,0<t<-, ‘Using the results obtained from the lower (3/2) (3/2) order systems we find that a, » VG have the following forn: (3/2) ~ x 302-2 -(0) 1), 2(a?-1) 00) = (3/2) 4 (2,y,t) = 6 at Vd,z2z ( djz? at Va jz22? a (3/2) = o=2 -(1/2)_ a? 2 C1) (3/2) vq (z,y,t) -” 9 Ya, 22 ¥ ae Vd,z + vq ° (4.19) (3/2) og y6a/2) Here ua d are arbitrary functions of z and t satisfying the condition: lim a?) ; vol?) = 0 . (4.20) go —(1/2) In deriving the results of (4.20) we find a restriction on Uy . The condition (3/2) (2 1,t) = 0 requires that: xyd . ape = 0 ; (4.21) ZZ This equation with the decay condition (4.12) has the solution: ~(1/2) _ us = 3 (4,22) a result exactly similar to (4.16). The stresses ccrresponding to ede vol?) are at ~ 22/3) _ y A@2-2) 5/2) , 42-1) 7Q) Taxd a* "d, 2Zz an d,z ~@/3) 2 ge 2 {07 1) 500) Txyd = (y*-1) a: ‘d,zzz , (4.23) 73/2) "yya 65 Tae form of these stresses corresponds to an associated non-zero bending moment and non-zero average shear stress. The system of equations governing che behavior of a7) and v2) are: 02) 2/2) 4 gaa _ d,yy + Co va ay “d, ZZ “d, tt. 0 ; v6) 4 Mek ,G/2) , Ld) 1) Vasyy™ a= “dyzy tory d,zz ae “dytt ~ 0 ? for 0<2<o, -lL<y< 1, 0<t<»;: 13 @ #15) = 0 > AACE t1,t) = : (2) v2) a 9 Lim ua 5 d = 3 Zo for -l<y<i1, 0<t< ™, Using the results of earlier orders we can (2) (2), obtain the following forms for uy and vio? (2) ; _ oie 342-2 yale ) vol?) 4 20eF-1)=(1/2) —(2) (a,y,t) = 6 ey Va ,22Z ~ yg ay Va, aun + va > vi ae »y5t) = ya “=z vs) ot ree aa “ + $02 1) (3 5)¥y. den) 2.9 — — y #2 73/2), 52) a G,z d In deriving these results some interesting restrictions are obtained for —(1 (1) (1) ua satisfies the differential equation: Wo) Vd,zz and ve , The boundary condition 12 @,41,¢) = 0 , requires that With the decay condition (4.18) this means that we) = 66 Satisfying the boundary condition tyes 1,t) = 0 requires the following —(0), restriction on v a: —(0) , 4 o2-1 —(0) = . 2 d,tt 3 at Va zzzz 0 (4.27) We can write this in the original physical variables as follows: —(0) , he - -@) _ Ya,tr + 3 pG-o2) “dy xxxx =? ; (4.28) where E is Young's modulus, o is Poisson's ratio, p is the density for the material of the plate, and 2h is the plate thickness. Equation (4.28) is easily recognizable as the Euler-Bernoulli equation for the dvnamic bending of a plate (cf: Love [22], p. 496) . To fully determine solutions for 70) from (4.27) we require two boundary conditions at z = 0 as well as the decay condition (4.9}. We expect the conditions at z = 0 to be connected with the average shear stress and bending moment of the applied loads. These will be discussed in detail in the next section. Initial conditions are also required to determine v0 and we assume that Ug and Va fulfill the initial quiescence condition. Then we find that: ¥0) (2,0) = 0 > V0) (2,0) =O. (4.29) We cannot say much more about the diffusive approximation wntil the investigation of the boundary layer approximation is completed in the next section. S8efore temporarily leaving the study of the diffusive approximation (we reconsider the inner and diffusive approximations in section 6) we summarize the results obtained so far. We know that Wa Vg up ta O(¢2) have solution which produce stresses having the following forms: a 67 4(a2-1) —(0 a2-1) —(1/2 3g ) gear ve) Ite “I -y es v3 ae Te? pet vy, O a Va, ZZZZ + yc 4G WED + HPD Gatare +79] , 2 cytayA@ VFO) teri 22S GR Vl), 30) "yyd nc? ES 307" yt a Vy eee! ° In a similar manner we can develop a diffusive approximation important near the end x = 1 by introducing the scaled variable z' defined by the equation: 7X 2! ~ ix ve ° (4.31) If we write this diffusive approximation as Us and Vgi it can be shown that the lowest order term, vi » is independent of y, say ve dz" ',t) and satisfies the differential equation: yO) 4(a2-1) 7) _ 39 Vai watt 30 Val ztatzta2! = 0 ? (4.32) for 0 < t <<», 0 < z2'< » $ and the initial conditions vo (2" 0 30) = 0 > ww 62' 20) = 0 ’ for 0 2z' . (4.33) vl), Similar results can be found for higher order terms v yi) 68 5. Formal Boundary Layer Approximations. The approximate solution consisting of the sum of the inner approximation and the diffusive approximation cannot in general satisfy the prescribed boundary conditions at x = 0,1. To explore the transfer of the prescribed loads at x = 0 into the interior, and to investigate the solution near x = 0, we introduce a ‘boundary layer' variable defined by the scaling: x , = 2 . (5.1) The following investigation will provide boundary conditions to fully determine the inner approximation and the diffusive approximation. We assume approximations of the form: HT] u(x,y,t3e,6) u, Gy, tse) + u, Frystse) + u, (sy, tse) , (3.2) x v(x,y,tie,é) v, Gy, tse) + Va Geryetse) + v_Gsy,tse) > where uy, Vp represent tentative boundary layer corrections near x = 0. We have not included the parameter 6 in any of the above approximations. This anticipates the conclusion that the inner approximation, diffusive approximation and boundary layer approximation can only describe slowly varying effects. We require u = u,tuztu,, v = v,tvgtv, to satisfy the differential equations (2.10), and the corresponding stresses to. satisfy the boundary conditions (Z.llabcd}. Then using the results of sections 3 and 4, we obtain the following differential equations and bowndary conditions for u, and v,: R & u + (a%-1)v + a2u ~ o3u = O(e3) , Lsyy L, Cy 2,57 R,tt (5.3) atv + (a2-1)a +v - e%y = O(c3) , Lyyy R,ty 2,5 2,tt for 0 <¢<ife , -l<y<l1, 0O<t < ™; 69 Tyy 95> #1, ts ©) = 0 > Tyy g 6s #1. tse) = 0 > (5.4a) 1 . for 0 < Zz <= 0 < t < @ 1, CO.y.tse) = cf Y (y,t)-1, (vy. tie)-7,,(Ovystte) Bf (3/2) oy bye Tey, O.¥.tre) = xyi Orystied-t. gry, tie) ~ (5.4c) for -l < y <1, 0<t< +. Since we expect such a boundary layer approxi- mation to be important cnly near the end x = 0, it is natural to replace . the domain of the differential equation (5.3) by O< ¢@<-, -l<y<il, 0<+t< ©, Thus instead of boundary conditions at 5 = + we impose the following decay condition: lin p70 "xxl? Taye Tyg 9 2 “tsysts Os Bx (5,5) (For the analogous static problem, the error introduced in this way turns wx/ey) | out to be O(e We now assume expansions for us , of the form: U(o, ¥etse) = eu, seyte 17 u'9/ yy) + see > : (5.6) v, Gsy,tie) = ev syst yre /2yGI2) | syst) tere. Following the standard procedure we obtain a system of differen- tial equations and boundary conditions, which, written in terms of the (1) @) GQ) stresses T ’ xt? *xy2?Tyye » is: OD) OD 25, Q) , @) xxL 0 ayh yy Taya c mS f XyYL55 YYLoy ° 9-7) 70 for 0< 37 <%&« , ~i < y< 1l,O< t<om 3; xy yy for 0O< 7 <o, 0 <t<a;} Deoy,t) = ED oye) - AED. GO 0,e)- 2 1, (5.98) MVo,y,t) =o , (5.9b) for -l< y<1,0<t< 2; (1) (1) qd) oe “ wack 5 Tye? yye 0 5 (5.10) for -l<y<1, 0< t<o«. These equations are merely the equations of static elasticity with t appearing as a parameter. No convenient explicit solution for the stresses can be found (cf. Johnson and Little {[16]). However by integrating equations (5.7) over their domain, we obtain a condition (restricting the prescribed stresses at c= 0 ) necessary for the existence of a solution for the above system. From (5.7) it can be observed that if a solution exists, then: 1 = (1) 7) ° L ic "exe, r TRy2,y ]dedy = 7) 7) 0 = dcd ie ic “xylt “yy! y | oad Assuming that we can ceyerse the order of integration where necessary, using the boundary condition (5.8) and the decay condition (5.10) we find: 1. 1 [ <Doyea-0 , | tDooy,eyay-o0 . (5.11) If a solution existe the first equation of (5.7) shows that: After integrating by parts we can simplify this equation using the second result of (5.11) and the boundary condition (5.7). We find that: ts @ | yt (0,y,t)dy = 0 . (5.12) -1 xXxX2 Conditions (5.11) and (5.12) are satisfied by the prescribed stresses (5.9a), (5.9b) if: _ 2 1 uy”) (0,t) = Ta ~1) £ (y,e)ay + (5.13) VO © t) = - 302? EY ta 7 14) d,zz-°’ 8(a2-1) > Yo ¥ > . for 0 < t < », These conditions provide suitable boundary conditions for wi) and v0) at the end x = 0. The equations (5.12) and (5.14) are the main results of this section. We do not attempt to develop explicit solu- (1) (1) 2 2 tions for u and v, °, but merely note before proceeding that Johnson and Little [16] have obtained useful results in relation to eigenfunction expansion solutions for the system of equations (5.7), (5.8) and (5,10). To demonstrate the complex interaction bewteen the diffusive approximation, inner approximation and boundary layer approximations at higher orders we now investigate the next two orders. (3/2) (3/2) 24 , 3/2) exh Tye yyt satisfy the following The stresses T equations: 73/2) 5, G/2) _ xxk ,T xy yy , (5.15) (3/2) (3/2) _ * xyf 4% * "yy2ey =o 72 for 0 <<, -l<y<1,0<t < % wn Ce 41,8) =0 , oe eal, t) = 0 , (5.16) for 0 < 7 < *7, O<t ¢ % 1/2) ,y,6) = - - y Met vl? 0, t) (5.17) 21) 9,0) = 8) Ge) G-yABGY FO) 0.0), (5.18) for -1 <y <1, 0<t < ce Text Tye Tye TO 6-19) for -l <y<1, 0< t<™, Using wethods similar to those used in inves- tigating the system (5.7)-(5.10) we can find a necessary condition for existence of a solution of the above system. It is: i 1G? ) (o,y, tay = 0 > (5.20) -1 i : Toe Oy st)ay = 0 , (5.21) 1 (3/2) | | Yiyyg Osythdy =O , (5.22) ~-l for 0< t< , Substituting the prescribed conditions (5.9ab) into the equations we obtain the results below: 709) _ 307 (3/2) (5.23) Vg zzz Ot) = BeG2aay [ g (y, t)dy : yOl/2) - ; Va, zz (0,t) = 0 2 (5.24) 73 for O< t < » , These results provide suitable boundary conditions fur 30) , =(1/2) d d We consider one additional order before examining i. we XxX 2), in more detail. The system of equations governing the behavior xy of uf), vi?) interact with uD) , vil), We can write them in terms of stresses as follows: 7 (4) + 762) = yh) (5.25) (2) (2) (1) T + T =u ’ xy%,o yy2,y R,tt XKR,C xyh,y 2,tt 3 for O<t<e,-lLey<1dAa<t<o; Wen =0 . Mea eo', (5.26) for0<t<.%7,O0<t<@; (2) = -yp 607-2) (2) (9-4) - 92) (9.4 4 4Ca*=1) 309) 4g _ Seg CO0V 9) = VETER O58) = By One ME ne O28) - o=3 (p2) (0,t)+p°2) (0,t))+ Aa") BO (0, £)]- LAC SDV) (One) ~ p- 4007-1) FO) (9-4) 4 202-1) (304-1 1024-7 oO) ! a4 Ya ze‘ >t) 3° 1) (3 ° OVE es (0,091 , (5.27) (2) = ~ ¥¢,(2) ¢ +)_,(2) _ 1/,(2) (2) . Tye (Oey) ¥(as*? (0,t)-a5*" (0, £)) 56 (0,t)+q) (0,t)) 5 2 _ ~ 421 2(a*-1) (1/2) O.t 5.28 (y ) aren ere ) ’ ( ) for -l Sy <1,0 <t<o ; lim 1(2), 7(2), 12) = 0 , (5.29) tom Oxxh xy yys for -l <y <1, 0 <t<o. Even at this order interaction between the 74 boundary layer approximation, the diffusive approximation, and the inner approximation has become quite complex, Proceeding as before we obtain a necessary condition for the existence of a solution of the above system as follows: 1 1 f | 1?) (o,y,t)ay = Tw ce,yse)asdy > (5.30) -1 sm]? 1 1 f | oi) ay st)dy = J Po cexy stacey (5.31) -1 ow 1 1 F yal uy? (Soy st)dady ~ | f [we (1) (z,y,t)dzdtdy, 1 (2) = J Qh ey.eday = “alo -l’o (5.32) for 0 <t<%* We assume at present that the infinite integrals on the right hand sides of equations (5.30), and (5.32) all converge (this assumption is discussed subsequently). We can therefore substitute the stresses prescribed at & = 0 to obtain the following conditions: 1 _ 2 Woe 2S" 5p eo, rd+p.? (0, £)]- a] [ a) Coy.todtay, (5.33) 1 —(1/2) gaz (2) (2), 3 a2 (| Vaze2 = TEQeaqyld, Otitag WH I+gQz zy] O, [ ¥s pete Ost) d Gey, (5.34) —_ 1 Lys veh) ,t) = | fo unt 1) pbosyst)didy - -| | [ vo 1) p(asyst) dad ty + ° ~Lia -llore a2 tyieeplP 0, t)- «, t)l+F5 (6435.02 “5515 a®)y > en(Oot)* + v0 Carey , | (5.35) i,xx 75 These equations show that at this order, interaction is oc- curring between all three of the approximations considered (inner, diffusive and boundary layer). Until now we have assumed that the boundary layer approxi- mation is incapable of dealing with rapidly varying part of the pre- scribed loads. We are now in a position to show why this assumption is (2) (1) L walid. The solutions for u and v only depend on t as a parameter. & ' (1) (1) Thus if the prescribed loads depend on t/6 then Uy. and vy also de- (1) and yh) g, tt g,tt would not be 0(1) in comparison with e. pend on t/$ and u This would make (5.25) invalid, since in writing these differential (1) (1) att and Yott all 0 <t <®. From these facts we conclude that the function gl) (1}.- (1) & “2 equations we tacitly assumed that u exist and are 0(1) for occurring in the boundary conditions for u must have a second time derivative which is 0(1) for 0 <t <0. A similar argument appli- (3/2) (3/2) (3/2) 2 7% second time derivative which is 0(1) for 0 < t < wo. If greater accur- (2) (2) 5/2) 5/2) acy was required and the corrections u > Vv 5 & % & Q then we would require 6 vy ¢) and g(3/2)(y,t) to possess fourth time ed to u shows that the function g (y,t) must have a obtained derivatives, 0(1) 0 < t < ~, and so on. We now briefly consider the error introduced by the approxi- mation of domain and the related question of the convergence of the integrals in equations (5.33), (5.34) and (5.35). The results of 1 1 _ Knowles [19] show that the stresses + ) and ) are Oe Le xxk xy and therefore need not be considered in any analysis at the end x = 1. Jatr=ls A similar result can be deduced from the form of the eigenfumction expansion representation for solutions of (5.7)~-(5.10) obtained by 76 Johnson and Little [16]. These authors found that each term of au (1) xd eigenfunction expansion, for say Ten? involved. a decaying exponential (1) g in the variable ~. The overall exponential decay of T and the con- vergence of the integrals in (5.33), (5.34) and (5.35) can probably be proved as a consequence of the exponential form of the eigenfunction expansions; however this result has not yet been established. It would be even more desirable to use the more direct energy estimate methods used by Knowles[19] to prove the convergence of these integrals. Results similar to those discussed above can be developed for the end x = 1 by introducing the variable ;' defined by the equation: =~ 1x . (5.36) € and consideriny the boundary laye: corrections ups (cr ayt) and Var" syst). We will not consider these corrections in detail as the analysis is substantially the as that for u and vee However, it is of interest to note that we would obtain the following boundary conditions to use with the inner approximation and the diffuse approximation at xe: —(0) _ us Get) = 0 ’ (5.37) 70) Vat atgt (Ort) = 0 2 (5.38) —(0 Vet gt gtgt (Ost) = 0 ° (5.39) for 0S t<, These last two conditions, together with the initial condition (4.33), and the differential equation (4.32) show that (1) var = O. In fact the lowest order boundary layer terms at x = l, ae and Vor are also identically zero. 77 6. Further Consideration of the Inner Approximation and the Diffusive Approximation, With the additional information obtained in the last section we are now able to completely determine the functions BO) ’ ail) >» which L are part of the inner approximation, and the functions KO WG), vO) WO), WON), vl which occur in the two diffusive approximations. First we consider (0) and erie) » In section 3 it was shown 1 1 that uO) satisfies the following differential equation (3.9): =(0) _ 4(a2-1) =(0) 21 ¢9(2)_ 9(2) “i.tt ae ey ey MYT 4) for 0< x< 1, 0< t < ~3 in section 5 we found the following boundary conditions (5.13), (5.37) : > a a (0,2) = _ fF eDyy,tyay, Wat) =0, i,x 8(a*-1) Je i,x for 0 < t <@: and we have assumed that u (0) satisfies the initial conditions: WO (xo) =0 ., DOC =0 , oa i,t for 0< x< 1. The above system of equations completely determines w(0) which can be found explicitly using standard techniques. We shall i not exhibit the solution for u(9) here. It can easily be verified that i the derivatives U(0) and 709) are 0(1) in comparison with ¢, and the i,tt 1,xx development of C0) valid, only if the rapidly varying prescribed. normal i stress is excluded from the boundary condition (5.13). We can summarize 78 the results for uas a solution of the following system of equations: i (1) 4@2-1) (1) — 92-2 (2) (2), “i,t ~ a? Yi xx = 2a" (P1\x + p2 ee, Fy for 0< x< 1, O< t<om ; a1) 2-0 (2) (2) - (1) us"? (0,t) = TER, 0! {p\“? (0, t)+p, (0,t)] ep) Ms t a y,t)dzedy, Jo lio > (1) 2-07) (2) (2) wl) (1,t) = RES; bp (1,t)tp'(1,t)] for O0< t<-™ ; WD) = 0 , wHe,t)=0 , i i > for O<x<1. From these equations it may be observed that the first term of the.boundary layer approximation, u(t) , must be obtained before we con solve for ul) i The lowest order term in the diffusive approximation near x = 0, vO, is a solution of the following system (cf. (4.27), (5.14)): — (0) a4 g?-1 —(0) Yd,tt * 3 2 Vg. zzzz 7 9» for O<z<0,0cect<ow; —(0) 304 4 (1) Vdizz =~ 8(a2—1) | £ (y,t)dy ’ yO) 2 (Ort) = 30 (" 9/2) (y r)a “d,zzzz.” 8(a2=1) i Fok tey for O<t<@ ; —(o — . lin uv ) (z,t) = 0 , #£4Ilim yO (2) =O , d d zo Ze for O<t <@ ; 760 —(0 5 gO, Vyee=0 79 for 0 < 2<o, This system completely determines vi? and a solution can readily oe found using standard techniques. In an earlier section —(0) we assumed that the effect of the diitfusive approximation Va (Was small at a > (x1). To verify this assumption we must examine the solution for “Oo ) in detail. We can obtain a solution for vs © aving the following form: =(0) _ x2[t!sC/) (er) 2 ol, 2 Vane = -—T372 M*~? (t-1)s sin (4 zat ; (6.1) Sato) ve ket 5 het where the notation below has been used: 2 1 yO) (t) = CEES) he ¢ 4) (y,t)dy , 2 g (3/2 30 3/2 a7=1) _° The behavior at the end z — » is found by substituting this value for € zion (6.1). We find that: _ 5ft/o (3/2) ny) Dery) v9) Bey ek 2 SE | ED cing at. (6.2) 222 fe Jo it 2k? ve | © For finite values of t, using the Riemann-Lebesgue lemma (Apostol [2]) (3/2) 1, and the properties of aD y ty, g t) discussed earlier (existence of two t-derivatives for 0 < t < ©) we find that the right side of 3/2, | (6.2) is at least as small as O(€ This result is obtained by using integration by parts in the right-hand side of (6.2). This estimate of Vaswe , t) is not uniformly valid. In fact for t as large as 906 e : 1, oo e : . : . O(=) it is not valid. But we are only concerned with times before the first wave (the dilatation wave, see section 8) travels the length of the 80 rectangle. Thus the original assumption that the diffusive approximat- ion VO) does not produce significant effects at the end x = 1 is valid. We can also show that v9 (- ,t) is smaller than O(e). Thus . 9222 YE v eo »t3€) is smaller than 0(69/2) and t e xyd Ye : xxd Ye 0(e5/2), These stresses contribute to higher order effects at the end »t3€) is smaller than x = 1 but only at orders beyond those of interest: for the present work. Further discussion of the contribution of ¥00) at the end x = 1 for long times can be found in section 10 of this chapter. The next term in the expansion for v, is v/2) which satisfies the following system of equations: —(1/2) 4 a%-1 _ (1/2) Va,tt + 3 of YVdizzzz = ’ for 09<zZ<67,0<t<o;3 —(1/2) ; Vd,zz (0,t) = 0 >: —(1/2) 302 (2) (2) 302. fl {? (1). Vd,2z2(0,t) = - T6(q2-1)laz (0,t)tq2 (0,t) I~ §(q2=7) | Ve erdody -lo for O< t <o@ ; lim vO/2) (zt) = 0 , goo dl for 0O< t <©@ ; VA/2 (2,0) =0 , Vv/2 (2,0) =0 , d d,t > for 0 < z2 < ™ 3 note that the boundary layer approximation term yl) ~ R must be determined before a solution for y (1/2) can be found. d . — 2 A similar system of equations can be written for vl), the lowest order term in the diffusive approximation at the end x = l. This completes the consideration of the inner approximation, the diffusive approximations and boundary layer approximations. These 81 approximations together describe only the effects produced in the elastic rectangle by the slowly varying part of the precsribed loads. We now consider effects resulting from the rapidly varying part of the prescribed loads. 7. Formal Wave Front Approximations. In constructing the approximations of the previous sections, we have taken into account all the conditions of the original problem except that part of the applied load at the end x = 0 which varies rapidly. In an elastic plate we expect such loading to give rise to significant stresses propagating in pulses associated with three different speeds: the dilatation speed, the shear speed and possibly the Rayleigh speed (we use the symbol 8 to represent the ratio of the Rayleigh speed to the shear speed). The rapidly varying parts of the prescribed loads are functions of t/é and we expect this scale to be important in describing propagative effects. Consequently we introduce a wave front coordinate € defined by the equaticn: & = Six a (7.1) ) At present the constant c is undetermined. We will deduce the signific~ ant possible values for c in the course of the construction cf the wave front approximations and at present assume only that it has a value in the range 0 < c <a. We introduce another new variable emphasizing the wave character of the problem. This variable,» , is defined by: R= g_ Cet+x) . (7.2) e2 The choice of scale in the equation above for n ensures that wave front approximations obtained later are valid everywhere in the plate and 82 not just near the wave fronts. It will turn out that the scheme developed in this section is of use only when the parameter $ is small in comparison with ce; that is, when the rapidly varying part of the load has a shorter duration than the time required for a shear wavw to travel across the thickness of the plate. ' Additional corrections u,. v are added to the approximate c ¢c displacements u, v as follows: u(x,y,t3e,6) = u, Gy, tiedt uy Bry .tiedt Got F%,y,c5e)+ + us (F sy, tse)t+ ua (Fy, tse)+ a (nsys65e55)5, v(x,y,t3e,5) =v, (x,y,tse) + v, E,y,tie)t vy AES, y,t3e)+ 1- + Va Jeryrtielt Vat Cay stse)t v, (nsysb5es8) Fy where Yer Ve represent certain wave front approximations. We require u tu tu tutu tu and v tv tv tv tv_,tv_ to satisfy the differential i 2 e' d d'ic i zg e' dd'ic equations (2.10), the boundary conditions at y = *1 (2.lla), (2.11b), the boundary conditions at x = 0 (2,11e), (2.11d), and initial condition (2,lle). At first solutions are considered only until the first wave has reached the end x = 1. Here it is found to be convenient to consider the problem fér u ,v in terms of the corresponding potentials ce c¢ ¢ , defined by: c c u =p —~£6 + 5% , c c,sy 6 c,& € Cyn (7.4) v=o -£y + Sy . c csy 6 cyé — c,n Using the properties of u_,v_..., dewloped earlier we fird ii that ¢ ,) are solutions of the Hllowing problem ee 83 22 2 . <2 52 ? oF Ge a) #0, eEt Pcsyy™ (14 G2) cent 22> 92) %e nn = 0(e*) , (7.5) 2 ; 2 fr (i-c?)p ty -2(1te?)p + Sh (d-e?)y = OC 9) o= c,E— cyyy c,En e* e,nn , 2 for Se <n<o ,-1 < y < 1,0<&<e; € + ¢ = t bd = 2 for —&<n<7,0< F<; 2 ~ 2 1 Tee Sb 832,8) = © aM (y,E/e) (7.6b) 2. 3/2 2 Tye Cen ¥ 965855) = 69/74 3/2) ye 70) > for -1 <y <1, 0 < — <@. Here the stressest +7 37 are xxc xyc yyc given in terms of the potentials » and p by the following equations: c c te = SOW. ee 2% eyt 2076. ea 24. yt = 2. nyt om , laye 7 SPW, en ED, et UCD, eat 26 ynt SCe2)4, , "yye 7 S(e*-2)0, cet Ee seyt 2CC2¥6 com Bo yal BO 2¥ (7.7) Throughout the problem posed above, except in the definition of che domain of interest, the parameter 46 occurs only in the combination é/e, For the approximations developed in this section it is assumed that the parameter v defined by: is small. 84 Two obvious special choices for c arising from a study of equation (7.5) are c = a and c=1. We shall find that the dilatation front approximation (c = a) describes the propagation of the prescribed normal stress from the end into the interior of the rectangle, and the shear front approximation (c = 1) describes the propagation of the given shear stress from the end into the interior of the rectangle. In the study of these wave front approximations we find that they generate high order effects important near the sides y = :1. In fact we find that the dilatation front approximation generates higher order shear stresses at y= #1 and that the shear wave front approximation generates higher order normal stress at y = *1. To find a method suitable for dealing with the ‘secondary’ stresses we must examine the sides vy = +1 more closely. Near y = 1 we introduce a ‘boundary variable!: y= ay (7.9) Vv Together with each wave front approximation'$ .,v, we introduce wave front boundary corrections ¢ gn sr.Bse,¥), Y og (UE .E5E,v)~ These correc- tions must satisfy the differential equations (7.5) and certain stress boundary conditions at r = 0. That is, 9 00 and V oe satisfy the following differential equations: 2 2 2 4 a ' xe 24. & = sz (- WD oh EE ot peel 24 watch en (1 [pty Oo, (7.10) 2. 2 tu" Ci-c WY oe san = 9 . SELAH oy eee pp 2CO Wy 6 We consider these boundary corrections to the wave front approximations in specific detail in the next two sections. 85 Before proceeding it is necessary to define the properties of « and 6. We do not attempt to deal with the problem where ©. and 6 are independent small parameters. In general it is assumed that v = e*, a>0O, and expansions for oo in powers of eP are assumed where b is the highest common factor of a and 1/2. We shall only consider the special case a = 1/2 in detail. It is easily seen that the methods dev- eloped for this special case also apply for other values of a. We now consider the two cases c = a,l in detail. Since the nermal stress at the end x = 0 is given at a lower order than the shear stress, the case c = a is considered first. 8. The Formal DilatalLion Wave Frout Approximation and Associated Boundary Cerrections. First consider the wave front approximation associated with the choice of ¢c = a in equations (7.5)-(7.7). We denote the correspond- ing potentials as bey? and use the independent variables bang defined by the equations: atx _ at+tx ba ° 3/2 ; Vy = ve ° (8-1) (ef. equations (7.1), (7.2)). As was stated at the end of section 7, only the special case v = Ye is considered. It is required that te should satisfy as many of the following conditions as possible: a = 0(e3 Payy “areung OP (8.2) 2 G-a?)y : +0 - 2(1+02)4 + e(l-a2)p = O(e3) , By Eq bo ayy By TigSg Bs aya for €&y < tig < @, “kh < y <1, 0 < Gy < *3 86 tT (n ,t#1,€ 3c) = 0 5 t (n_,41,& se) = 0 xya* ‘a a . yya- oa °o , (8. 3a) for e& £1, < 720 < & < 9} 1 Teg (CE yV2E se) = © al y,-%) : ; ) E (8.3b) (3/2), (3/2 Tey (Cbg 27 2EQ3) = € b (y,=8) 3 for -l<y<l1, 0 £8, < ? 3 u(nysy,05e) = 0 - v(nyy0se) =O, (8.3c) for ef Sn ses -1 < y < 1; where the stresses Taxa? Txya? Tyya and the displacements Use vo are given in terms of the potentials bo and Wy by the folluwizg equations: 1 » 1 2 er T = 174 - 2 +2024 .. -26 — +¥E2y + ca2p) ; KO € 4558 Ve a,b 7 aE Ny osVY. On y MeN - a 4(02-2)y ror 144 +2(a2+2)y. +524 te (a2-2)y oy xy E no ’ OEY aE ny a2 Y aA t= A(a2-2)6. + 4 +2 (024+2)¢° -2VEy tela2-2)— 4 —yya € O56 6. ve OEY Ose Asyny Os "ou ae Yasy ° *osn, 1 . . v= = + -~Ve . Ct ve Oye Pay . Yasn, Expansions for 9 Py of the following form are assumed: 2 5/2 (5/2 - " oy (nyo b 5) 296 (nay eS) + of o / (nay &) teen, (8.5) 2) ee / 5/2 ¥Ongsys 652) 26 (nays) + Pas 2G6/ 6m oF 98) +... 87 Then adopting the standard procedure we find that g) and y@) satisfy the system of equations written below: g(2) 4 ¢@) 26 asyy % Ny Sy p(2) =o , a,E8) < -~ ° for 0 < ny < @: 1l1<y<il1, O< Ey < 0 Qn 41,2.) = 0 T ? Ha & xya Na 1 forO<n <m~,O0<& < om; . a Qa : i ry o fe Ne o-oo te e| om R ee <2) (0,y,8,) - 1 oO ww (1) Taya O2¥2 Ea) - for -l<y<1,0<€ <@3 a u 63/2) (n, ,y,0) =O , v63/2) (n, ,y,0) =O , for 0 <n <*% “Le y <1 « Note that the boundary condition at Ny = & & has been replaced by a similar one at Ny = 0 and the domain of the problem modified accordingly. This is a procedure similar to the approximation of domain used in deriving the diffusive approximation and boundary layer approximations of earlier sections. Now the errors produced by the approximation of domain are relatively large. However, provided that the prescribed rapidly varying normal stress al) ¢y,4/8) satisfies certain conditions (discussed later) we can show that these errors. are uniformly smailer than a(eo/24 in the ealculation of the . .(2 : ? ton4 : stresses. Wa can now solve for gh?) and pO) obtaining solutions: . 4. 88 6?) = ) =a Ss, (any¥y, E72) dz. sin nay a nel a fo) oo be + ae i Ss a? 3,284 mn 2) dz. cossat my (8.6) oO yp Ya, om = 0 (8.7) where 1 gO) (2) = (1). 2. aon @) = [a (y {sin niy dy , a) & =| Poe Jeos*Bo# ny dy , and Jy is the Bessel function of order zero. The subscripts "o" and "e" ase) refer to the fact that the part of a #on ‘OL en. (1) is an even function of y contributes only to aon Cy ,£/a) which (1) : and the part of a which is an odd function of y contributes only to a, Here we solved for the derivatives ton and yo) rather than the potentials 0, ), y 2 9 themselves since only these derivatives are required for the calculation of stresses and displacements at this order. The corresponding stresses and displacements are: E a , « an VE TH * a) (5 /a)- ) at) 2) 5, (only CE 3) )dz-0in nry n=l} 8 % Ove =z ey % (2n-1)aVE~ - ) alt) (2) \ _O 5. (= 1 Ape az) ee. cogabzh ny n=} 8 8 ave =z 712 2 (a) 7) 2 9 xyG (1) 2 ia?-2 (1) yya af XxXO 89. gE a . wl3/2 -) sea) 5, (awh CE -add2- sin niwy n=] 6 wo fey - te az 4 a) a, a = vn, (6,72) dz. cos 3 why, (8.8) n= . 43/2) - 9 a gE This solution is useful only if the prescribed load 2) (y=) has the property that: ug aft) (1,9) = 0 . (8.9) Equation (8.9) requires that the rapidly varying part of the stress Ta prescribed at the end x = 0, vanishes at the +cvrers y = +1. In general this condition may not be fulfilled. We shall aiscuss how to deal with loads having non-zero values at y = +] in a later section dealing with "corner approximations". At present it is assumed that the prescribed normal stress at the end x = 0 fulfills equation (8.9), or that we are considering the part of the normal stress which fulfills condition (8.9). | We now consider the error introduced by the apprceximation of domain. At ny = eb 3 ao: (1) _ a) (2) (2) (2) 3.9, (2) Toy ba. ty, “ay ea “by -e* (207 Seen Te ea yg J-Era¢ xx XO a 56 ey A sE Ny Ob SVY aan If our procedure is to be of use then Tt Kee _yy,é sejrer ¢ dee )- Pp " xno 5? 7 Pq? yee. (2) is uniformly of higher order thane. Substituting the solution for de obtained above in (8.6) we find the following result: 90 wey . 2. 1) = xxo, eo g 29 Ey 3e) 7 ET co (eb ¥ 2 by) ~ 3/25 | De 8. fa 1/2 — € nt (2) | ——2— + 0 St") 5, (nvvevE CE ~z)) 67 3/2 . - ¢ E ) J, (nnvevé (€ -2z))]dz.sin nay + Oo a “a 2a a a) 2 2 8, Po 7 1/2 +} Ta’ Ole + DG), CEL le Ee )) n=1 - do VE (6,72) o — LEI, 202 EATER) Ite.co0 2) vy! E 300. «TE aS? ome Bessel functions satisfy the following inequalities (cf. Watson [37]): 15, Ge) | oe » jxkreal, k>l , |5, | <= » x real. 2 Using this information we can obtain the estimate below: |z [eS ETD et (ee y.8)/ < Ve +2/72(1 1+ 5) Jaz 2/2 nT (L) ,2 au, 1) 1? Ste n=1 Oo ni 2 ®en Sa (8.10) The property that ay, Ey Ja) = 0 for a > a has been used in the above formula (8.10). This property ensures that the integrals in (8.10) eer tas 1} converge. Of course a less restrictive condition on a’ ’ could have been : (i) ; 1 ; prescribed; for example, a (y,€/a) ~“ “ik for large eae However insis- (1), 8 Sa aa . 1 a ting that a is zero for — >a is a more cenvenient practical criterion 91 for the division of the given data into slowly and rapidly varying parts. To ensure that the estimate (8.10) is uniformly valid we must prove that the series on the right hand side of (8.10) converge uniformly for ~1 <y< 1, A suffielent condition for this 1s that the prescribed . (1) - . ss . was function a (y,€/a) has a continuous second derivative with respect to » (1) _ (1) 4 4 y and a y (1,6 /o) =a (-1,6 /a). This means that we can only admit > ba, quite smooth functions of y (where our definition of smoothness is the existence of continuous setond derivatives), It becomes apparent that in (5/2) order to develop higher order terms oo »+ee, we would have to know if 1), - . . . . a! (y,€ 14) has continuous fourth derivatives with respect to y. We do 4 not investigate this matter in any further detail. We now briefly consider the next order, that is the functions 9 g65/?) yl), These functions satisfy the following system of equations: (5/2) (5/2) (5/2) - 4 -~. = 0 p - = 0 Posy 7 Payne. ase 6, (8.11) tor 0 < My <> -l<y<il1, O< by <7 3 (3/2) + _ (3/2) + = Txya (n> 1,6) = 0 > "yya (nye 1,8.) 0 > for O<n <e%,0< & <@ 3) (8.12a) — oa — °o (3/2) - \ _ dim 1 ,, (1) ~_ 1) To 6099284) = esq i fa’ (ys 6 /agnt (eb syb)t € for0<é > <@ , -l <y <1; (8.12b) (2) (2) _ u (n ’ »0) = 0 2 Vv (n »¥39) = 0 a a oo ° (8.12c) for 0 < Ny <o ,-l1<y< 1. Herve, as in the previous order, the domain e& <n < ~, -i <y <1, 0 < € < ~, has been approximated by the domain 92 ~O<n< mw -lL< y<i1, 0< & < © To verify that this approximation of domain is valid at this order the properties of tim 4 fa (yy fo) - CY (ce yyy] 240 Ve a xxa > “O a must be determined in detail. In particular we must know the behavior of this expression for large Bae It may be necessary to divide it into a slowly varying function and a rapidly varying function before proceed- ing. However our primary interest in considering the equations for ¢@/?) and yor?) is not to find explicit solutions for these functions but to demonstrate that both boundary conditions at y = +1 cannot be satisfied. We can write the shear stress (3/2) in terms of potentials xya g69/2) ,yG/2) : 7 (3/2) = -292) + (q2-2) yS/2) a a xya a, Egy Oy Ey Eo, From equation (8.11) it is clear that yl?) = 0, and we have already a 7 70° determined g(2) completely in equation (8.6). It is obvious from (8.6) a that in general 13/2) does not vanish at y = +1. We now develop xya ‘boundary approximations’ to deal with the stress (3/2) generated at xy y = *1 by the fimction (2) , o We first consider the side y = 1 and introduce the coundary variable r defined in equation (7.9). The dilatation wave front boundary corrections ¢ (1 ,r,& 3€) and W (Mm ,r,& 3€) are introduced. ak a 0 ak a a From equation (7.10) it can be observed that these function satisfy: Ly . - 46 en 0 € ak,ri "a ote 7Sala (8.13) + fy - (a%-1)y } ~ 2(1+a?)y - e(a2-1)p = 0, € ak rr ak, aQ?a a > alg ORs Ney Ney ee <n <*, O<r< Loo<€ <~, We also require that ¢ and W a, a ve (07 ak al deal with the stress 1{3/2) generated by the wave frent approximation “Kya 93 at r= 0 (y = 1). That is: T (n ,0,€ ) = 3/2 297) (ye) , xyak fe a,euy a O for c& <n <2, O< & <o@, There must be no interference with a7 a7 — “a the normal ‘stress condition at y = 1 and consequently: u fo) « T (n ,0,€ se) yyak a a for ce <n.<e ,0<€& <o. It is assumed that the displacements S 5 | Qa a7 u_, Vv corresponding to the potentials ¢ and wy must be continuous. ak ak OR ad Other conditions are required to determine 4 a? Vy) a) however they are a fe} not introduced until after the differential equations (8.13) are considered in more detail. We assume expansiers of the form: wc) = go/2,(5/2) 343) : > 9 Oy FE 3e) € one (n2t9E) +e ose (note) +... y (8.14) 5/2 (5/2) 3,, (3) a € vee (n toe) te vee (note) toa. y yp mo Pe ers se) aL a a and thence, substituting in equation (8.13), we find that the lowest (5/2) and yo!2) order terms oe ‘satisfy the following equations: 40/2) - 0 > yore - (a 2 eee =O , (8.15) ak,rr OR,E 8 ; for 0O<&e~- ,O<re2%,O047n «< . We might expect that the effect produced by the secondary stress 7@/2) at y = 1 would remain . xya. local to the side. However the second equation of (8.15) shows that effects propagate even though they occur on the same scale as r. Thus there is no decay conditicen for large +. We assume that the solutions gO72) yf yO") remain bounded for large r. a, ; 94 The boundary conditions at r = 0 require that: fl 13/2) (ny 50, om ) 20 (yn ,l,6), xy od Egy 2 a (3/2) (ny 056 ) yyak il fan) e forO<n <#,O<€ <o. Here the stresses (03/2) and 7103/2) are a a xyad yy ar given in terms of the potentials as follows: 4(3/2) = 24 (5/2) + (a 2-2) (5/2) xyar rt » Sgt a bby by 1(3/2)= (q2-2)9(5/2)_ ~ 2965/2) yy ak ak, EySyg vee 9 Sgr The only solution for g6/2) which is bounded for large r and which will a allow higher ozje,r terms 63) 97/2) >++. to be bounded for large r is o. g(9/2) = 0. Since no propagation in front of the wave front — = 0 is fe) possible, the solution for pC5/2) must Le a function of fa r, and ad va7-1 thus it is possible to specify only one condition at r = 0. We require y a to deal with the shear stress at r = 0 as we have already seen that a the wave front approximation (5/2) , (5/2) cannot. Then the equation: a oO 0) ; Sak r< QO , Fe y(5/2) = rarct (8,18) Ms*a @) So oy Eq $e (n > i, — - ft 2 _ - Tz 0 > yee > (a -1) o Vo2-1 Vq2-1 provides a function which satisfies the differential equation (8.15) and the shear stress boundary condition at y = 1. Similarly, near y = -1, if we define a boundary variable: r' = ty , (8.17) ve 95 and a corresponding boundary cerrection aie at? we find that: , A ; g 0 9 —4-r'<0, ¥o2=1 y/2) = (8.18) mel sb e 6 5 9) G sok, - ®-r'), @_-r'>o, a*-2 “ary Vor=1 Vo2=1 ¢65/2) = 9 . ak" Of course y5/2) and yS/2) do not satisfy the normal stress condition ak ad at y = 1 and y = -1 respectively. Before considering this fact im more detail we consider the shear stress produced at y = -1 by yl?) and the _ Oo shear stress produced at y = 1 by pO/2), At y=1: ad 0 —& < 2. Yae -l , a ve 13/2) (, 2,2) = xyon! “we a E 26) (,1,—%- 2), ¢ > ert, aye. a Vaz—1 Ve a and at y = -l: 0 e < 2yar-1, fo} € “xyal aye 4 z _ 29) (my ,-l, —*% - 4), Ee >2 VorL. oy a Voz-1 Ve a We can see from the above equations that the boundary corrections per) a y (3/2) are only cf use for a limited distance behind the front = = 9. ax" , a) In physical terms what this means is that the dilatation wave effects are generating secondary waves propagating diagonally across the rectan- gle. These secondary waves reach the other side of the plate a distance behind the wave front given by — = 2, The wave fronts of these second- a ve ary waves are given by the equations: Y & y q In physical variables these equations are equivalent to the following wave fronts: c;t - X - Yet/ce5-1 (h-x) = 1 tT - ¥¢e 1/5 2-1 (h+¥) = 0. 0, Cy cy fnese fronts appear to be related to the Stoneley waves occuring foll- owing a moving disturbaace iin an elastic half space. (5/2) (5/2) 9 For bo 2 — va*-l the boundary corrections Yo > 5 gt ad € not suitable for dealing with the shear stress at y = 1. We can improve (5/2) (5/2) are veg Yaar by taking the following expressions instead of (8.16), (8.18): [ 0 ; Ga - vo2-1 x < 0 > y?/2) p22) [I> [I 2 Yor rE =\ 42-2 Foo Marte bam af—1 x) > 0 a bo ~ Yat=-l ir < iz 5) 2 Voc. 2 2 Id, of ( en, 21,6 ~Yar—1 x)- "> <5 of ro eb veerd rp) £0< Ey ~ vaz—-1 x < 4 ; (8.20) Ve ve~ (° ; fy 7 voz-IT v' < 0 , pO/2) : 5 0? (n ,-1,& ied ) , O< €-VeTr' <4 , abt, E “1 Q2no G ~ fr _ mye 0 (noch om -Va2=1 r > = Oy 1,6. -vo2-1 x'- 2), , 2 [I 4 = <€ -vof-lr'< = , ve ve There are obvious extensions of these expressions for GO in the ranges —t< &- Va*=1 < 8 etc. Thus the dilatation wave front approximation ve Ve ; is accompanied by wavelets bouncing back and forth across the rectangle behind the primary wave front. 97 512) ang 65/2) The expressions for Yo written in equation (8.20) do not produce zero normal stresses at y= 1. Also if extended yO?) yor ) far enough these expressions for and show that these boun- dary corrections produce both normal and shear stresses at the end x =0. (2) a,€6 y in fact decays for large Sa? if the given rormal stress a vy t7/8) has However these stresses are at most 0(e3/2) since ¢ is bounded (and a continuous second y-derivative). Using the information we have obtained about yl?) and yl?) we can now rewrite the equations gover- ning the behavior of gol) which were previously written as equations (8.11) and (8.12abc). The potential 60!) satisfies the equations: (9/2) (5/2) _ ew 7 4oaE no 0, (8.21) for O<n<e% ,-l<y<1,0< € <™3 a a (3/2) 7 _ 3/2) va (3/2) Tyya (nyt, Sy? = “Tyyok (ny O,€ > t yo. (n> 1,6) = Tryant Ng 0,é we? for 0 <n < > O< E <@ 3 (8.22a) — "a = Sy ey Fen OsYs6q) = o>! vel® D6 yyy) > 0 xa(€6yr9>6q)]- (8.22b) ~ 13/2) (eg sVs bu )- 13/2) eg ss ey 3, Txxak for O<n <7, -lLley <1 3 2 uf (ns¥20) =O , (8,22c) for O<n <%, -1 <y<l. We shall not attempt to find an explicit solution for gOl2) it should be roted that the end cundition (8.22b) may be incompatible with the boundary conditions at y = ti (8.22a) making 'corner' approximations unavoidable. We now examine a method suitable for dealing with the shear stress at the end x = Q. 98 9. The Formal Shear Wave Front Approximation and Associated Boundary Corrections. The wave front approximation associated with a value of c =l in equations (7.5)-(7.7) is now considered. Corresponding potentials are written % and v° We use the independent variables Bony defined — by the equations: 51 7 "3/2 , € (cf. equations (7.1), (7.2) and (8.1) ). Again the special case v =ve is considered. Then O99, satisfy the following equations: = Pee, ¥ oo) yy 7 2OF em + eG- Pan, =o; (9.2) Yhayy | ang, a for eg) < ur <e,-l<y<1,0< 54S oO Tyr My? bE 5) =Q , Tyr (My #15 E58) =o , ~ (9.3) for c£) <n, <*™, 058 < 23 (9.4) .-) = 3/2 (3/2) _ ~ Typ OPE V9 5) E b (y.&,) ley (ea, .y.08,) - - E yrt.ak Teyag (OE To 8) Tyan Cb ot 2064) , for -l<y<1,0<&< oe, -Here the stresses T oT aT are xxl = xxyl yyl given in terms of potentials % and a by the following equations: 99 1 1 T ==$ - —2y +26 ~2¢6 +Ve2y yted ; - xxl € 1,658, Je 1, 1,8)n, l,yy 1, nyy 1, Ty Ny tr, =-4y. - he +61). +V¥e2 -ey » (9.5) xyl e*l 615) ve 1,659 1,74 Lyn; Lynyny Ted ® - 45, + Vy ¢ yt6o, ~Ve2y, -eb5 . We assume that the disturbance arising from the shear stress at the end = 0 (cf. equation 9.4) cannot travel along the rectangle faster than the shear speed. Thus for continuity of displacements at the shear wave front we require that: u, (nj, »¥,05e) =O , v, ¢ y2¥s0se) = G , (9.6) for eb) <7 » “L<y <1. The displacements uy and Vv, are given in Ll < terms of the potentials oy and Wy by the following equations: 1 =-— +), +e M1 Oe Me ay Pay ny , (2.7) v,=sy +>, -vew 1 Ve 1,8, lyy ° 1,n, . We assume expansicns for ty and vy of the form: 5/2, G 2 G | (9.8) 5/2, (5/2 3 ¥, (ny oys6,3e) = € / us 22 ny1¥s5,)te ys (ny syeB) tee and following the standard procedure find that 26/2) and yl?) satisfy the system of equations written below: 160 (a iO =o, yOl2)_ 4y S12) - 0 *Loyy 95404 for 0 <n, < % ~l<y <i, 0<é) < 7/2) = 13/2) ¢, . u fan] uw for O< ny < % O< & < 93 Pe, 4) = p 3/2), LE Pines “378 ay (3/2 1?) (cok, s¥ 08) + we > leat sr, a&,) + Te) (cag st" a8) for -L<y< i, 0 S$ 8, < 23 2) 59,0) = 0, vy?) (nj»y,0) = 0, for 0 £ ny < %, ~l<y< 1. Note that we have replaced the condition at ny = eb, by a similar one at ny = 0.and have modified the domain ‘above accordingly. This is a procedure similar to. that used in the previous section and the error introduced by this procedure is discussed later, . We can now solve for sO an a yO! obtaining the solutions: (5/2) $ = 0 1.8, by ie) ~ L q ’ n=] 1‘ @ [*1 2a 2n-1 -j ben z } Ly 472) )dz. cos Ty fe] 101 where “ “ A 1. bon te) -[ 3 ” (y,z)sin nny dy 5 5) (2) | bY (yz) cos* toy dy and J, is the sessel function of order zero. The subscripts "o'" and "e" on De, ), b ae ,) refer to the fact that the part of pi diy,e p> which De? *(1) is an even function of y contributes to db, only and the part of b “(1 1) which is an odd function of y contributes to bo only. We solved for (5/2) (5/2) rather than the potentials por, yl?) 1, Sy 1,8, 1 themselves since we require only these derivatives for the calculation of and p derivatives $ stresses and displacements at this order. The corresponding stresses and displacements are: 7 (3/2) _ "xxl =O 9 > Si JET a nt aT 7 (3/2) = py 36 Dr 5 Bare 15 poaryn 14 - -z)dz.sin ny "x 1 Y Mnel) ave -Z E o | 7], Ome ~ ) bo) ca) 1 5, a EH?) )42.cos tet TY, =] 4VEs-2 . “ 73/2) = 0 *yyl ° (9.10) ~ aT vy?) =- ) |’ raya (avn 1 Ey7 -z))dz.sin nry n=1 0 aT 1. - } }° bd (2 J, (Fats a z))dz. cos TV. n=1 fe) This solution is only useful if the prescribed load 53/2) (ye Dv has the property that: 7» (3/2 pf EL,e) = 0 . (9.11) 102 ~(3/2) In general b (y, 5 ) does not satisfy this condition. We shall (3/2) discuss how to deal with prescribed stresses b naving non zero . . : . : ' : . ' vaiues at y = 1 ina later section dealing with corner approximations, “(3/2) At present we assume that b fulfills equation (9.11) or that we ~(3/2) are considering the part of the shear stress b waich fulfills(9.11). Using methods similar to those of the previous section deal- ! ing with the dilatation wave front approximation we can estimate the error introduced by the approximation of domain. At the end n, = €€ 1 1 we find that: _3/2) (3/2) ¢y _ 3/2. Gi) = . a ; -(5/2) ; ; . uaiformly in ¢,aad y if b (y,5 ) has coatinuous secona y-derivat- (3/2) , ives, ana if b (755) decays capidly euvugh for large 4° Tc is - (3/2) , difficult to decide if bt (y.8)) satisfies these conditions. Oaly . (3/2) . those prescribed loads b (y56,) which satisfy these smoothness and decay conditions are admissible; however b ducludes the shear stresses resulting from lower order dilatation wave “effecte at x = 0, (3/2) _, as well as the prescribed shear stress b (ys }. These generated shear stresses may not be sufficiently smooth or may not decay fast enough to allow the validity of our procedure involving the approximat- ion of the domain. Once we have worked our the details of the function- 3/2 3/2 3/4 al form of the stresses - /2) a! ) ) this issue is decided. 3 ? 1 xy a xyak “xy ar Unfortunately this seems to be practically impossible. At the moment 2) we just assume that b (y,&)) has a continuous second y derivative : 1 nos and decays faster than —= for large er" Any shear stress waich oy 103 does not satisfy these conditions must be dealt with in other ways. Shear stresses at the end which are not sufficiently smooth with respect to y give rise to 'end waves! propagating back and forth across the ends. We shall discuss them briefly in a later section dealing with corner approximations. Shear stresses at the end which do not decay sufficiently fast must be broken up into slowly varying portions and rapidly varying portions and then used in the appropriate sections (sections 3,4,5,6 for the slowly varying end stresses, and sections 7,8,9 for the rapidly varying end stresses) as end conditions. Apart from the difficulty of determining all the properties n (3/2) fb ,» further trouble arising when we proceed to the next order. (3) (3) and attempt to develop solutions for oy and at . This difficuity ce) is exactly similar to that encountered when the systems of equations ~ (5/2) (5/2) ° and ¥) (3/2) there that the condition "xy (Ny, 71,84) = 0 could not te satisfied. In the present case we find that: (8.11) and (8.12) for was examined. It was found (2) _ ,,,(5/2) “yy * Ley In general, the stress.defined by the above equation cannot satisfy (5/2) the condition x x n> *1,6)) = 0 CY, Ey is fully determined by Py > 1 2 y equation (9.9) and is net zero at y = +1). Therefore we must develop another method for dealing with this norme] stress. In the previous section we examined the side y = 1 by intro- ducing the variable + defined in equation (7.9). In a similar manner 104 we introduce here shear wave front boundary correctians ¢ X63) _¢ wey and yy (n ,r,6 3€). Equatiun (7.10) shows that these functions must 1 21 satisfy the following differential equations: 1 1 1 - gl - ore 6, ae yer 2 a? Pie, n, a Fanny =O; Ley - eM ler ye ein = 0 , (9,12) nN 2 alii .; a _ <o, Note that wa allow < ny ss O<ts< Jy ote "1 by eg 1 to be negative in this case. Although it is assumed that ro shear effects can propagate directly down the plate taster than shear wave velocity, we now allow effects produced by the interaction of the shear wave and the sides to appear in front of this wave. It is also (2) in yy¥ generated by the wave front approximation at r = 0 (y = 1). That is: reguired that °) and Y oe are capable of deaiing with the stress ' 2 In this case we shall find that the effects produced by this secondary 1 stress are important near the side y = 1. At present wa require that and 4%. are bounded for r large. Expensions for fe and Yy o£ "ie 12 the following form ere assumed: . we) = 63463) 7/2, (7/2) , . . bp (Mypstebyse) = Eds)’ (yer sEy) FE by, (ayeksSy) tere > 4 a 3,(3 _. 7/2 (7/2 7 V3 ip Eseyy + e// wv (ay or sty) t+ saws il Wi, (nm, s,6,35e) = € 12° 1 1 105 After substituting these expressions into equation (9.12), we find 3 that the lowest order terms ) and oe satisfy the following equations: (3) (3) _ (1- in, E451 + P42, rr ° ; (3) (9.14) Viewer = 0 , for 0 < Ny < % O< r< eo, -o < by < o, These equations can be solved in conjunction with the boundary condition (9.13) and the boundedness assumption, obtaining the following solutions: wo (5/2) 3) .£[ Yay Suet? (3) } $ Fy 1Y dz, v3) = 0. (9.15) 12,8) (6-2) 2+ x? The corresponding stresses are: we /2) (5/2) ~ . - (2) 2 @-3) | bry (1,14,2)¢ 1-2) ar ae Q-nzl Mby (ny 51.2) 5 2 =dz= (&-3)E : “8 1, Usy-2)? + 22]? " oT) (eyr2) 2+ rt f~ yl? ) pO!) 72) . 2 vy 7 (Myst?) _ bre i (n,>1, z) i: ‘yi i. (E vee Td nce (G2 z) 7427}? (9.16) eee : yy TS ace © oe 7 ‘ 1 106 If this procedure for removing the stress at y = 1 by means of correct- ‘ (3 ions ee is to be of use, the stresses corresponding to oe and p written above in equation (9.16) must be smaller than uce) at 12k y=-l (r=7 2). Before we can estimate the size of (2) > 7) we Ve 12 xyl ? 'yyl must examine the properties of yO ) in more detail. Integratin l,yz & the last equation of (9.16) by parts the following relationship is obtained: (5/2) _ (2) _ ef My pbs ‘yl ot [(2-e,)?+ x2 J? 0 1 A crude estimate shows that a ony Z- 96) ) is 0(ve) uniformly in "4 and & if (n.,1,&) has a uniform bound of O(1) compared with ec. 1 l,y 1 1 /2) From equation (9.9) we can calculate ¥ Cn 1,8) as follows: oy © 51 y 2 Vz mn E,-2 i, = on noire Ee (9.17) oo El. : 2VvE_ =z en - 2} | 601?) (ay 5, BL a Ema) ) az (-1)" =] 11 en — 1 n o Vey (3/2) We have not been successful in determining conditions on b (y,z) (5/2) - such that 4 y is uniformly bounded. It is certainly possible to 3 find examples of interest where y is not uniformly bounded; the tress 1 = 2? ting 1d f han “TE stress yy 74, ain = 1 in general decays no al alt i i S i e ag F. - which predicts that Moy (n, 51, 6) will grow as fast as §) | for large a Fortunately we can deal with the generated stress Syl ol; a) in another way. From equation (9.9) we can easily deduce 1.07 (2). . that t (i.,1,6.) is uniformly 0(1) in comparison with © (remember- yy. 4 1 ing the assumed properties of bL (750). Thus, written as a finetion of the variables (H ,1,%,) is aleo uni- ood L n,t, the sivress + (2) we formly (1) in comparison with e. In chis form it way be used as a part of the nocmal stress at y = 1 associated with the inner approxi- mation developed in section 3. We are in fact proposing (without (2) proof) that the generated stress “yl (nj 5456,) can be divided into two a parts: a pulse near the front at = 0 which stimulates the boundary correction ¢ ; anda more slowly decaying part which can be in-= »Wi 3 Lg” le cluded in the inner approximation. Both the boundary correction and the inner approximation are required as the boundary correction cannot (2) deal with the part of T (1_.,1,6_) which decays slowly away from the yyl L 1 front oo = 0; and the inner approximation cannot deal with the part of (2) T yyl should also be noted that this approach could not be used to deal with (7, 1,8,) which varies too rapidly spatially or wich time. It stresses generated by the dilatation wave front approximation since these effects travel along the rectangle at a faster velocity than the inner approximation allows (the inner approximation predicts that no disturbance cam travel along the rectangle faster than the plate wave 2. a2 2) If 7! / yyt the boundary corrections was explicity known we would now be able to develop p (9/2) (5/2) 12 i2 valid near y = 1 and a similar correction, 9/9? we valid near y = -1. These corrections would in turn génerate shear stresses at the sides y = 1 providing boundary (3) conditicas for the next term in the shear wave front approximation Ye 108 10. Further Approximations. In the course of the development of the results of the preceding section various restrictions, which limit the permissible type of prescribed loads, were obtained. Some of these restrictions were required to avoid consideration of effects variously described by the terminology: corner approximations, initial approximations, Rayleigh wave front approximations, long time approximation and end wave approximations. These effects are not unimportant but so far we have not been able to develop the approximations describing them; therefore we can only demonstrate why they occur and discuss them from an intuitive point of view. First we consider the 'corner approximations'. In sections 8 and $9, devoted ‘to the development of the dilatation wave front approximation and the shear wave front approximation, we found that it was necessary to assume that the rapidly varying part of the prescrib- ed loads at the end x = 1 satisfies the following conditions (cf. (8.9) and (9,11)): aise 0 , bO/?¢4,6)) = 0 ‘We could require that only those prescribed loads which satisfy the (3/2) first of the conditions (10.1) are considered. However since b | (3/2) includes both the prescribed stress b , and stresses generated by the dilatation wave front approximation, it seems very artificial to assume that the second condition of (10.1) is true. The question then arises: what can we do if condition (10.1) is not true? A suitable choice of variable allows the corner x = 0, y = 1 to be examined in 109 detail and a corner approximation investigated. However the equations governing the behavior of this corner approximation turned out to be the full equations of two-dimensional elasticity even at the lowest order, and the relevant domain was an infinite quarter space. This is a classic unsolved problem in elasticity. A careful examination of the equations (7.10) for the boundary corrections to the wave front approximations shows that another choice for c besides c = a and c = 1 is significant. For c = 8, where 8< 1 is the real root of the follow- ing equation (Rayleigh's equation): (2-82)2 -~ 4 /i-g2 Vi-g2/o2 = 0 , (10.2) Solutions % a0 aa) can be found which leave the side r = 0 stress free and which correspond to effects decaying away from the side r = 0, The difficulty is that it is not possibile to decide what conditions should be specified for Pao» Yegat the end x = 0. If it was possible t to develop a ‘corner approximation! this would naturally suggest con- ditions to allow thecomplete determination of the Rayleigh wave front approximation $,,, ¥,,. The corner approximations would also be assoc- $ jated with the phenomenon of ‘end waves’ bouncing back and forth across the end of the rectangle. In the development of the inner approximation and the dif- (2) (2) fusive approximations it is assumed that Ps (x.t), P, (x,t), y (x,t) and a (x,t), the prescribed stresses at the sides y = 1, are all zero initially and heve their first two time derivatives zero initially. Intuitively speaking, this means that at the lowest order there is no overall vibration of the rectangle due to waves bouncing 116 from side to side. It also means that we must assume suitable initial conditions for the inner approximation and diffusive approximations. The development of an ‘initial approximation’ would allow general init- ial conditions to be specified for the elastic plate problem and would also allow a more convincing development of initial conditions for the corresponding inner approximation. This ‘initial approximation" has so far not been sucessfully obtained. Therefore we must assume that the rectangle is initially at rest and restrict the initial growth of the prescribed stresses at y = 1 as described above. In the development of the diffusive approximations it was assumed that the diffusive approximation valid near the end x = 0 has small effect ax tha other end x = 1 (and vice-versa for the diffusive approximation significant near x = 1). In section 6 this assumption was shown to be valid for finite times of smaller magnitude than 0(1/e). This is a completely satisfactory conclusion as we are not concerned with the solution to the problem after the dilatation wave propagating from the end x = 0 reaches the end x = 1, However it is of interest to note that we can develop a long-time approximation by introducing the variables x,y and + = et and the corresponding approximations, say v, Oise t88)s u, Cy, 15 e)- We find that the lowest order term in the . — (0) , expansion for v, can be written vy, (x,t) (wing familiar notaticn) where Ve satisfies the following differential equation: > , vO) 4 4art 7 9 , (10.3) t,tt 3.4 t,xxxx The two diffusive approximations can then be ‘matched’ on to a sclution of (10.3). 111 11. Summary. The most significant and complete result obtained in the preceding work was development of the plate wave equation and the Euler-Bernoulli plate equation from the full equations of dynamic elasticity using a rational perturbation expansion technique. It was found in section 3 that a slowly varying normal stress applied at one end of an elastic rectangle propagates according to the plate wave equation. In section 4 it was found that a slowly varying bending moment and average shear stress applied at one end of an elastic rect- angle diffuse according to the Euler~Bernoulli plate equation. In later sections some results concerning effects arising from rapidly varying applied loads were obtained. A summary ot the derivation of these results is now presented. In sections 3 - 9 an approximate solution of the original system of equations (2.10), (2.11) was obtained as the sum of several different approximations, each satisfying part of the problem and all interacting with one another. These parts of the overall approximate solution are: the inner approximation, two diffusive approximations, two quasi-static boundary layer approximations, the dilatation wave front approximation and associated boundary correction, the shear wave front approximation and associated boundary correction, the corner approximations and the Rayleigh wave front approximations (these last two were not completely developed). Before any of these approximations can be obtained the pre- seribed stresses at the end x = 0 must be divided into slowly varying 112 parts and rapidly varying parts. We write (cf. (2.11c) and (2.11d)): : Qi) ( Txx syst 3e€,6) = ef “ (y,t) + ca 3/2 (3/2) —E g (y, 1 Uy ,t/s) : (11.1) 3/2 » 63/2) ( Tyy (O.y, tre, 8) = t)te y,t/6) 3/2 Here eDiy ty and = f (yt) are required to have continuous second t-derivatives of 0(1) in comparison with « for O< t . This ; 1 condition is especially important at t = 0 since that ¢‘ dUy.t,) and (3/2) ; . (1) g (y,t) rise smoothly from t = 0. We require that a (y,t/&) (3/2) and b (y,t/§) are zero for t >6 , where 6/e is small. Given certain prescribed stresses we can examine their second time deriva- tives and if these are large only for a short time a scale $ is deter- mined and a division as in (11.1) can be nade. If these conditions are not satisfied the present methods would req ire modification. (for example, the methods developed here cannot be used for a problem where the impulsive part of the load has a duration as long as the time required for a shear wave to propagate across the rectangle; nor “ean prolonged rapidly oscillating loads be considered). We also re- quire the prescribed end stresses to possess continuous second y-deriv- atives. This avoids the propagation of waves across the ends. (2) (2) A similar condition is required for the stresses Py »P5 > (2) (2) 4, 249 prescribed at y= 1. It is required that these given funct- ions are twice continuously differentiable for 0 <t<™, O< x< 1. Having ascertained that the given duta satisfies the above requirements we can now develop an approximate solution for the original system (2.10) (2.11). The first step is to calculate the lowest order term 113 9?) (Stes Ak rare of the dilatation wave front approximation. For 3/2 particular case é=e / this may be explicitly written as follows (cf. (8.6)): 0 0, a * 36?) - *a,E, a & ofl Og Z } 735 é 9 (nq vn, 65,72) 42. sin nny “Jo (Pq Oy 2n-1 . 2 + l wa a & )I of avn _(zZ -2))dz.cos Bod TY,— > V5 n=1 a a a oO atx = ottx (1) (2) 34 where Eo oP >T, 7 T/2 ° 8on (sin any dy, € and a & [is Dy, y Beast ayay The expression (11.2) describes the propagation of the prescribed mormal stress a yy ey 6 from the end into the interior of the rect-— angle (with a velocitya). As the disturbance described by 9? ) pro- pagates along the rectangle it produces higher order shear stresses on the sides y = Ht. To deal with these generated stresses the dilatation wave front corrections are developed. The lowest order terms pO! (attx Inv ot-x da a yor? ) (attx ty at-x) are described by van Ye e3l? Mee ve ST? equation (8.20). They correspond to wavelets propagating across the rectangle and moving with the dilatation wave speed. (These effects are related to the Stoneley waves of seismology). Next we calculate the lowest order term of the shear wave front approximation, yp es, Ys a» according to equation (9.9) vee ait (for the special case $= ez). ig eepacasion describes the pro- pagaliou of the prescribed shear screis het (y,t/>) aid shear stress- es generated py the dilatation wave fie approsimacion (uesevived oy 0 yG/2), yGl2)) at x = 0 into the jinceriar of che rectangle (with a velocity 1). The difficulty at this stage is Lhat it is not possible to decide in advance, for a generai prescribed stress 41) (y,t/§), what (3/2) (3/2) (3/2) the pr i c g T the properties of the generated shear stresses t,0)", Tyan » Txyak?s are at x= 0. Ina particular case we must decide how much of these generated stresses to include with the rapidly varying effects and then consider the remaining part with the slowly varying effects. Assuming that we have successfully and fully determined the (3/2) | (3/2) | (3/2) voperties of the shear stresses T T , t x = 0, we Prop cars xya * xyak ” x ag! ° (3/2) . can calculate the shear wave front approximation Vy . this express- : (2) ion generates higher order normal stresses + 1 at the sides y= 1. vY To deal with these generated stresses we introduce shear wave front : Sis + boundary corrections yO?) (s, Lee “3D and yO? ees, . x, a 31D These 1k Ve ve” wae Ve” corrections describe effects important near each side moving with the (3) shear wave. it is found thac $12 can only accomodate a normal stress at y = 1 which decays rapidly away from the wave front t = x. In gen- 2) ,t4+x t-x . . 1s eral the generated stress A at, 7D does not satisfy this con- © (2) dition. We assume that we can consider Tyy1 in two parcs: a rapidly (3) ecaying part producing the correction term ¢ 2 aad a more slowly varying part which can be considered with the prescribed normal stress 2 . . at y=1, Ps ) in the analysis of the inner approximation. This division oft Ex, 1,44) has not been established. yyl Je 3/2 115 It is now necessary to investigate the lowest order term in the corner approximations. These are required to describe the effects produced by non zero prescribed stresses at the corners x = 0, y = tl. These corner approximations have not been developed. Also connected with these corner approximations are the Rayleigh wave front approxi- mations. The existence of a wave front important only near the ‘sides y = +1 and travelling with the Rayleigh wave speed (velocity g in our notation) was established. However this Rayleigh wave front approxi- mation can only be fully determined after the corner approximations have been developed. It may also prove necessary to develop an end wave approximation to account for the normal stresses at x = 0 generat- ed by the dilatation wave front approximation p@) and the accompanying boundary corrections yo, yo!?), It is an obvious deficiency in the present work that the corner approximations and Rayleigh wave front approximations have not been obtained. The results described so far account for the lowest order effects produced by ‘the rapidly varying part of the prescribed stresses. We now develop the lowest order tarms of the inner approximation, wl) x,t) and vO) (x,t). This approximation describes the effects produced by slowly varying prescribed and generated stresses at y =*1, as well as the propagation of the average prescribed and generated slowly varying normal stress from x = 0 into the interior of the ract- angle. The equations for uy and vw, written below, do not include the generated slawly varying stresses. However the modification vequir- ed to do this is minor. The equations obtained determining v9), 0) i i 116 are: (0) _ 402-1) 1(,(2)_ ,@) “itt a Yi 2 7 QD (11.3) (0) 2 fla 0 wine") © Bacay js yseddy , ul Gye) = 0, 1.4) ul) (x,0) =0 , uf?) (x,0) =O , (11.5) wo. =+ (? p{?)) > (11.6) (0) = (0) = v, (x,0) =O , v, 6&9) =¢ ., (11.7) It is of significance that the equation (11.3) can be identified as the plate wave equation. The lowest order terms of the two diffusive approximations, vo Et) and vo?) = (Acx -x Ve moment and the average shear effects produced by the slowly varying —(0) prescribed and generated stresses at each end. The discussion of vg _(1/2) and Va" in sections 4 and 6 does not include the generated (however »t), describe the spreading of the bending ‘the extension of the results to do this is not difficult). The term (0) Va is a solution of the following problem: —(0) ~4o0%1 -() _ Va.tt + 3 a2 d,zzzz ~ 0 >. (11.8) . 1 {0 2 Vay 22 (Ost) — CD | yt (y,t)ay , 2 (11.9) 117 | —(0) —(0) lim vi’ (z,t) =O , limv; ‘(z2,t)=0 , (11.10) d d,z eer) Zo —(0 —(0) WO (2,0) = 0 , Va rt2,0)=0 , (11.11) where z = = Equation (11.8) can be identified as the Fuler-Bernoulli € beam equation (see section 4). Lastly we obtain the lowest order boundary layer approxi- (1) x (1) x (2) 1-x (2) 1-x 2Z Gey.t), Vo Cyst), Uys ( e? — yst,) and voy mations u »¥3t) which describe the quasi-static effects arising from the slowing vary- ing stresses at each end. These effects are important only near each end. In the above summary we have only considered the various approximations obtained for times before the fastest wave (the dilatat- ion wave) reaches the end x = 1. To proceed further we would have to consider the stresses generated at x = 1 by the approximate solution as new data and repeat the whole procedure with the role of the ends x = 0 and x = 1 interchanged. 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