## Abstract

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.

## 1. Introduction

An asymptotic approach is a powerful tool for constructing two-dimensional mathematical theories for thin structures starting from the equations of three-dimensional elasticity. The majority of previous considerations on the subject deal with the derivation of approximate differential equations, governing the static behaviour (see, for example, the most fundamental papers by Green (1962*a*) and Goldenveizer (1963)). At the same time, only a limited number of publications were oriented to asymptotic justification and refinement for the boundary conditions to the above-mentioned two-dimensional governing equations. Among these, we mention the pioneering works by Reiss (1960), Friedrichs & Dressler (1961), Green (1962*b*) and Goldenveizer (1969), and more recent papers by Gregory & Wan (1985) and Wan (2003), see also references therein. The related treatments operate with boundary layers localized near a structure's edge and involve generalized formulation of the Saint-Venant principle.

A dynamic asymptotic analysis takes into consideration a considerably wider range of phenomena than its static counterpart. In particular, the three-dimensional dynamic equations of elasticity require a number of different approximate forms to be fully described asymptotically. Apart from low-frequency long-wave approximation generalizing the static theory, high-frequency approximations are also revealed (e.g. Kaplunov *et al*. 1998; Le 1999). In addition, dynamic boundary conditions should start from the dynamic analogue of the Saint-Venant principle. In the simplest case it involves a low-frequency corrector, expressing the effect of self-equilibrated edge loads (see Babenkova & Kaplunov 2004).

Another specific feature of dynamic asymptotic analysis is associated with the problem of formulating asymptotically consistent initial conditions. This problem appears to be novel within asymptotic structural theories. At the same time, its major importance follows immediately from well-known anomalies with initial conditions in commonly used approximate models. For example, the classical Kirchhoff theory of plate bending assumes only two initial conditions imposed on transverse mid-plane displacement and transverse mid-plane velocity, whereas the original three-dimensional equations in elasticity operate with six initial conditions associated with all the three displacement and velocity components. In addition, the three-dimensional initial conditions allow arbitrary distribution along the plate thickness that usually cannot be approximated by uniform transverse displacement and velocity fields characteristic of the Kirchhoff theory. Similar anomalies are also characteristic of the classical two-dimensional theory of generalized plane stress governing long-wave low-frequency motion associated with plate extension.

In this paper, we develop a general asymptotic treatment of the initial-value problem for a thin elastic plate in the case of long-wavelength initial data, for which a typical length-scale along the mid-plane is assumed to be much greater than the plate thickness. In doing so, we will examine the case of initial data which have arbitrary distribution along the thickness. Composite asymptotic expansions are utilized, taking into consideration all possible long-wave asymptotic dynamic models. Apart from low-frequency models, which coincide in the leading order with the classical theories of plate bending and extension, these include high-frequency theories governing long-wave motions near the so-called thickness stretch and shear resonance frequencies, these coinciding with the natural frequencies of an infinitesimally thin transverse fibre of the plate. The high-frequency theories assume sinusoidal distribution along the thickness. As a result, together with the above-mentioned low-frequency models, they allow us to approximate the arbitrary three-dimensional initial data by expanding them in Fourier series along the thickness.

The paper is organized as follows. The formulation of the problem is presented in §2, including basic relations of long-wave models for plates. In §§3 and 4, asymptotic processes are developed for various sets of three-dimensional initial data. The six most important cases are selected for a separate consideration with respect to the symmetry of initial conditions as well as their uniform or sinusoidal distribution along the thickness. For each of the cases, leading order two-dimensional initial conditions are proposed for all long-wave components. In addition, for a uniform initial field, we also derive refined two-dimensional initial conditions that are adequate for the second-order asymptotic theories for plate bending and extension. In §5, a simple example is discussed for the plane symmetric problem in the case of initial data with uniform distribution along the thickness and sinusoidal variation along the mid-plane. Analysis of the two-term asymptotic behaviour of the exact solution for this model example confirms the consistency of the proposed approximate initial conditions. The Fourier expansions exploited in the paper are given in appendix A.

## 2. Statement of the problem

Consider a linearly isotropic elastic plate of uniform thickness 2*h*, and of infinite lateral extent, and assume that the plate faces are traction free. A Cartesian coordinate system of axes *Ox*_{1}*x*_{2}*x*_{3} is employed, with origin *O* in the mid-plane of the plate and *Ox*_{3} normal to the plate. We shall let *ρ* denote the mass density, *ν* Poisson's ratio and *E* Young's modulus. The three-dimensional equations of motion can be written as(2.1)where *v*_{j}=*v*_{j}(*x*_{1},*x*_{2},*x*_{3},*t*), *j*=1, 2, 3, are the displacement components, *t* is time and is the distortion wave speed.

The boundary conditions on the traction-free faces may be presented in terms of displacements as(2.2)The problem (2.1) and (2.2) is to be considered together with initial conditions(2.3)where the functions *f*_{j}(*x*_{1},*x*_{2},*x*_{3}) and *g*_{j}(*x*_{1},*x*_{2},*x*_{3}), *j*=1, 2, 3, are prescribed initial data.

The formulated problem is symmetric about the mid-plane, both in terms of geometry and boundary conditions. In view of this, it may be decomposed into symmetric and antisymmetric sub-problems. For the symmetric problem, the tangential displacements *v*_{1} and *v*_{2} are even functions of the thickness coordinate *x*_{3} and the transverse displacement *v*_{3} is odd, whereas in the antisymmetric case these quantities are of opposite parity. Then, the initial data can be represented aswhere the suffices ‘s’ and ‘a’ relate to symmetric and antisymmetric motion, respectively; in doing so, the functions are even functions of *x*_{3} and the others, i.e. , are odd. We can expand all the functions *F*_{e} and *F*_{o} as Fourier series. The even functions may be presented as(2.4)whereas the odd functions can be expanded as(2.5)see appendix A for more details.

In this paper, our specific concern is long-wave motion, i.e. motion for which the typical wavelength *L* considerably exceeds the plate half-thickness *h*; i.e. *η*=*h*/*L*≪1. To this end the functions *f*_{j}(*x*_{1}, *x*_{2}, *x*_{3}) and *g*_{j}(*x*_{1}, *x*_{2}, *x*_{3}) must slowly vary along the mid-plane of the plate. Therefore, we may expect that the original initial-value three-dimensional problem (2.1)–(2.3) is approximated by two-dimensional ones arising from long-wave approximate plate models. Among these long-wave models, low-frequency approximations are usually regarded as the most important. They correspond to the theories of plate extension and plate bending, associated with tangential ((*v*_{1}, *v*_{2})≫*v*_{3}) and transverse (*v*_{3}≫(*v*_{1}, *v*_{2})) motion, respectively.

Let us briefly review the basic asymptotic relations in the aforementioned theories, taking into account terms up to and including *O*(*η*^{2}). First, introduce the scaling(2.6)Throughout the paper Greek indices take the values 1 and 2. We remark that differentiation with respect to *ξ*_{α} and *ζ* will not change the asymptotic order. However, differentiation with respect to *τ* may affect the order. In the variables (2.6), the asymptotic form of displacement components for *plate extension* (symmetric low-frequency motion) becomes (e.g. Goldenveizer *et al*. 1993; Kaplunov *et al*. 1998)(2.7)with tangential displacements of the mid-plane *u*_{α}(*ξ*_{1}, *ξ*_{2}, *τ*) satisfying the equations(2.8)where, here and below, . In addition, to distinguish between differentiation with respect to scaled and original variables, we use the notation (.)_{,α} to indicate differentiation with respect to *ξ*_{α} (*α*=1, 2) and a dot to indicate differentiation with respect to *τ*. We also remark that throughout this paper *v*_{j} (*j*=1, 2, 3) is used exclusively to denote displacements associated with the three-dimensional problem, whereas *u*_{α} and *w* are used, with appropriate superscripts where applicable, to denote the various two-dimensional approximations.

In the case of *plate bending* (antisymmetric low-frequency motion), we have(2.9)where *w*(*ξ*_{1}, *ξ*_{2}, *τ*) is the transverse displacement of the mid-plane and(2.10)

By neglecting terms of *O*(*η*^{2}) in the relations (2.7)–(2.10), we arrive at the classical theory of generalized plane stress and the classical Kirchhoff theory of plate bending. It is important for what follows to note that the classical theories support the uniform distribution along the thickness for the principal displacements.

The goal of this paper is to analyse the transition from the six three-dimensional initial conditions (2.3) in the original three-dimensional problem to two or four two-dimensional initial conditions corresponding to the two-dimensional equations of plate bending and extension, respectively. The study will include both the classical and refined theories. We may expect *a priori* that the initial conditions for equations (2.8) and (2.10) can only approximate in the leading order the original initial data given by the first terms (*a*_{0}/2) in the Fourier series (2.4), related to uniform distribution along the thickness.

To satisfy the initial conditions corresponding to the terms with cos(*nπζ*) and sin[(2*m*−1)*πζ*/2] in (2.4) and (2.5), we have to take into consideration another type of long-wave theories. These describe high-frequency motions in the vicinities of thickness stretch and shear resonance frequencies. The consideration below operates, as a rule, with the leading order version of these theories; their basic relations and full details may be found, for example, in the book by Kaplunov *et al*. (1998). In particular, in the vicinities of thickness *stretch* resonance frequencies,(2.11)we have(2.12)with the upper and lower expressions in brackets corresponding to symmetric and antisymmetric motions, respectively, and the two-dimensional long-wave amplitude *w*^{st}(*ξ*_{1}, *ξ*_{2}, *τ*) satisfying the equation(2.13)where for symmetric motion(2.14)and for antisymmetric(2.15)with

For the long-wave theories valid in the vicinities of the thickness *shear* resonance frequencies,(2.16)the basic relations, analogous to (2.12), can be written as(2.17)where does not depend on the thickness variable and(2.18)in which for symmetric motion,(2.19)and for antisymmetric(2.20)The relations (2.11)–(2.15) are associated with high-frequency transverse motion both for the symmetric and antisymmetric cases, whereas the relations (2.16)–(2.20) deal with symmetric and antisymmetric tangential motion. They may be utilized for satisfying, at the leading order, all prescribed sinusoidal distributions along the thickness in the Fourier series expansions (2.4) and (2.5).

Now we are in a position to proceed with a general asymptotic analysis of initial-value problems for the two-dimensional equations (2.8) and (2.10) for plate extension and bending, starting from the original three-dimensional formulation (2.1)–(2.3) and making use of the high-frequency long-wave theories presented in this section. We study here non-homogeneous initial conditions for velocities, assuming that *v*_{j}|_{τ=0}=0 (*j*=1, 2, 3). Without loss of generality, our treatment is focused on six initial-value sub-problems for equations (2.1) and (2.2). The corresponding initial conditions (at *τ*=0) for the velocities are provided by(2.21)(2.22)(2.23)(2.24)(2.25)(2.26)where *M* and *N* are integers.

The initial conditions (2.21)–(2.23) relate to symmetric motion, with (2.24)–(2.26) relating to antisymmetric. The asymptotic processes, developed below for satisfying all the aforementioned conditions, always take into consideration both the low-frequency and high-frequency long-wave models introduced in this section. In respect of the uniform velocity fields (2.21) and (2.24), we justify the traditional initial conditions in the classical plate theories for extension and bending, and , respectively. We also construct *O*(*η*^{2}) corrections to these conditions, the corrected initial conditions being adequate for second-order plate equations (2.8) and (2.10). At the same time, for the velocity fields (2.22), (2.23), (2.25) and (2.26), which are not generally characteristic of low-frequency long-wave theories, we nevertheless extract appropriate uniform initial velocity fields resulting in dynamic extension and bending. In addition, in all the cases (2.21)–(2.26), we also propose initial conditions for all required high-frequency models.

## 3. Symmetric motion

We shall first consider the symmetric initial-value problem and construct the iteration processes for solving the three-dimensional problem (2.1) and (2.2) for initial conditions (2.21)–(2.23). The sought for solutions to the formulated problems may be written as(3.1)Here, the three-dimensional distributions , and (*j*=1, 2, 3) are associated with plate extension and high-frequency motion in the vicinities of thickness stretch and thickness shear resonance frequencies, respectively. All of these are assumed to be of the same asymptotic order. In this case, the powers of *η* in (3.1) determine the contributions of each of the aforementioned modes. The sought for powers *a*, *b*_{m} and *c*_{n} depend on prescribed initial conditions. In addition, the proposed asymptotic structure of (3.1) takes into account the specific behaviour of the tangential and transverse plate displacements, dictated by the formulae (2.7), (2.12) and (2.17).

The functions , and may be expanded in asymptotic series (e.g. Kaplunov *et al*. 1998),(3.2)Here, *j*=1, 2, 3 and the suffix *q* takes the symbolic value ‘ext’, with the suffix *p* taking the symbolic values ‘st’ or ‘sh’.

The three-dimensional functions in the expansion (3.2) can be expressed in terms of two-dimensional functions introduced in §2. In particular, we have from (2.7) that(3.3)where and denote the coefficients in the expansion of the solution to equation (2.8).

Similarly, we deduce from (2.12) and (2.17) that(3.4)and(3.5)with *w*^{st(m,0)}(*ξ*_{1}, *ξ*_{2}, *τ*) and satisfying equations (2.13) and (2.18), respectively.

Later, we shall also require the two-term expansion . In the same manner as it done in Kaplunov *et al*. (1998) for the antisymmetric case, may be obtained in the form(3.6)In this case, satisfy the higher-order analogue of equation (2.18), taking into account terms of *O*(*η*^{2}).

### (a) Case 1s

Let us begin with the uniform, with respect to thickness, tangential velocity field, specified by the initial conditions (2.21). This type of initial condition gives rise to predominantly extensional motion. Thus, we may set in (3.1) *a*=0, expecting, in general, *b*_{m}>0 and *c*_{n}>0 (*m*, *n*=1, 2, …). At the same time, the theory of plate extension, resulting in the formulae (2.7), creates a discrepancy of order *O*(*η*) in the initial condition for the transverse velocity. This discrepancy is given, as follows from (2.7)_{2}, by a linear variation in *ζ*. To remove this discrepancy, and at the leading order satisfy the condition , we should take into consideration high-frequency stretch modes, for which transverse velocity dominates. To this end, we may now put *b*_{m}=2 (*m*=1, 2, …) in (3.1). As a result, we get a further discrepancy *O*(*η*^{2}) in the initial condition for tangential velocity. It originates both from high-frequency stretch motion (see the corresponding terms in (3.1)_{1}) and from the quadratic term in *ζ* predicted by the theory of plate extension, see (2.7)_{1}. Finally, to balance the above-mentioned discrepancy, we also have to incorporate shear modes, setting in (3.1) *c*_{n}=2 (*n*=1, 2, …), thus obtaining(3.7)We next differentiate the expansions (3.2) with respect to *τ*, taking into account the relations (3.3)–(3.5), and substitute the result into (3.7) and initial conditions (2.21), gathering terms with equal powers of the small parameter *η*, finally establishing that at *τ*=0,(3.8)Now the initial values of the two-dimensional quantities , , and may be defined. To this end we first expand *ζ* in (3.8)_{2} in a Fourier series in sin[(2*m*−1)*πζ*/2], i.e. in , starting from the formula given in appendix A, see (A 8). As a result, we may determine the initial value for in terms of the given quantities *A*_{α}. Then, by using formulae (A 5) and (A 6), we are able to expand the functions *ζ*^{2}, and in (3.8)_{3} as Fourier series expressed in terms of cos(*nπζ*), i.e. in , obtaining the initial values for and . Finally, we have at *τ*=0,(3.9)

Thus, we arrive at three two-dimensional initial-value problems. The first relates to the plate extension governed by the higher-order equations (2.8) and the refined initial conditions(3.10)Two others deal with equations (2.13) and (2.18) of high-frequency long-wave motion in the vicinities of thickness stretch and shear resonance frequencies, respectively. In this case for high-frequency stretch vibration modes (*m*=1, 2, …), equation (2.13) is supplied with the initial conditions(3.11)whereas for shear high-frequency vibration modes (*n*=1, 2, …), equations (2.18) require(3.12)

Note that by neglecting the corrector *O*(*η*^{2}) in the conditions (3.10), we readily recover the classical two-dimensional initial conditions usually imposed on the tangential velocity on the plate mid-plane in the classical theory of generalized plane stress.

### (b) Case 2s

In this case, one of the high-frequency stretch modes, the *m*=*M* mode, with the critical frequency , has the same leading order thickness variation as that prescribed by the initial conditions, cf. (2.12) and (2.22). For this reason, this particular mode will provide a dominant contribution to the velocity field. The contributions of the other modes may be calculated similarly to case 1s. Accordingly, after setting *a*=1, *b*_{M}=1, *b*_{m}=3 (*m*≠*M*) and *c*_{n}=1, the appropriate form of (3.1) becomes(3.13)Taking into account (3.2)–(3.5), we may present the initial conditions (2.22) as(3.14)Then, by expanding all the functions of *ζ* in (3.14)_{2} in Fourier series in terms of cos(*nπζ*), i.e. in , we have at *τ*=0,(3.15)Thus, for the plate extension, in respect of the initial velocity field of the form (2.22), equations (2.8) require the initial conditions(3.16)

### (c) Case 3s

Similarly to the previous case, only high-frequency shear mode with the critical frequency , the *n*=*N* mode, appears in the leading order problem. Now we put in (3.1) *a*=2, *b*_{m}=2, *c*_{N}=0 and *c*_{n}=2 (*n*≠*N*), thus obtaining(3.17)The initial conditions (2.23) then become(3.18)Next we may find consequently at *τ*=0 all the sought for quantities, dictating that(3.19)We do not write down here the expression for the term , corresponding to the correction to the *N*th shear mode inducing the transient motion. Finally, for the case of initial velocity field (2.23), we arrive at the following two-dimensional initial conditions in the refined asymptotic theory of plate extension:(3.20)

## 4. Antisymmetric motion

The consideration below is similar to that presented for the symmetric case in §3. The antisymmetric three-dimensional solution can be written as(4.1)where , and are given by expansions (3.2) with the suffix *q* taking now the symbolic value ‘ben’. Using (2.9), (2.12) and (2.17), it is possible to establish that(4.2)and(4.3)and(4.4)with , and satisfying equations (2.10), (2.13) and (2.18), respectively. Later, we will also require the second term in the expansion (3.2)_{2} for . It may be written as (see Kaplunov *et al*. 1998)(4.5)We remark that the two-term expansion satisfies the second-order asymptotic analogue of equation (2.13).

### (a) Case 1a

First, consider the uniform transverse velocity field specified by the initial conditions (2.24). In this case, we may put *a*=0, and () in (4.1), resulting in(4.6)The initial conditions (2.24) may now be represented in the form(4.7)revealing that at *τ*=0,(4.8)As a result, in the case of the uniform initial velocity field (2.24), the initial-value problem for plate bending is governed by the higher-order equation (2.10) and the refined initial conditions(4.9)By neglecting the corrector in (4.9), we arrive at the initial conditions usually imposed on transverse mid-plane velocity in the classical theory of bending.

### (b) Case 2a

For initial conditions (2.25), we take *a*=1, , and () in (4.1), i.e.(4.10)Then we have at *τ*=0,(4.11)with . By expanding and in (4.11)_{2} as Fourier series expansions in , i.e. in , we obtain for the initial values of the sought for quantities, indicating that at *τ*=0,(4.12)Thus, in the case of the three-dimensional initial conditions (2.25), the problem of plate bending is governed by the equation (2.10) and the following two-dimensional initial conditions:(4.13)

### (c) Case 3a

In this case, we have in (4.1) *a*=2, , for and , i.e.(4.14)with the initial conditions (2.26) rewritten in the form(4.15)where now . Finally, we get at *τ*=0,(4.16)In the case of the initial velocity field (2.26), the appropriate two-dimensional initial conditions for the theory of plate bending are accordingly provided by(4.17)

## 5. Model problem

To verify the proposed asymptotic initial conditions and amplify the methodology developed, we consider a model problem. As a benchmark, we use the exact solution to the original problem (2.1)–(2.3) for the initial data in formula (2.3) given by(5.1)Here and later we make use of the scaled variables (2.6). The initial conditions (5.1) belong to the case 1s according to the classification of §2. We have and in (2.21), where .

We seek a solution of the formulated plane problem in the form(5.2)and apply the Laplace transform technique. Let the functions and be the Laplace transforms of the functions and , respectively, with the Laplace transform parameter *p*, i.e.(5.3)These satisfy the transformed equations of motion,(5.4)whereThe boundary conditions on the faces become(5.5)The solution of the problem (5.4) and (5.5) is(5.6)where denotes the symmetric Rayleigh–Lamb denominator(5.7)and(5.8)with . In the limit, , the equation has two imaginary roots of for which(5.9)and infinitely many large imaginary roots for which(5.10)(5.11)Here, , and , are defined by equations (2.14) and (2.19), respectively. It is clear that the poles (5.9) are associated with low-frequency motion, whereas (5.10) and (5.11) are associated with high-frequency motion in the vicinities of thickness stretch and thickness shear resonance frequencies.

Next we may apply the residue theorem to expand the sought for functions and as(5.12)Omitting the intermediate calculations, we write out the asymptotic behaviour of and keeping the terms of , resulting in(5.13)where(5.14)

We have from (5.13) at *τ*=0,(5.15)Thus, uniform with respect to thickness initial data, (5.1) do induce various long-wave modes.

Now return back to the asymptotic treatment of §3*a*. Then, by substituting and into (3.10) and taking into account (2.7), we arrive at the identical expressions for and to those in (5.15). The same substitution into (3.12), (2.17) and (3.11), (2.12) leads to the identical formulae for , and , , respectively.

It should be also mentioned that the inspection of similar model problems for all the other cases (2.22)–(2.26) justifies the asymptotic nature of the approximate initial conditions developed in §§3 and 4.

## 6. Concluding remarks

In this paper, we have derived asymptotic initial conditions for long-wave models in thin plates corresponding to initial velocity fields arbitrary distributed along the thickness. The case of initial displacement fields may be considered in the same manner. For the latter, however, we should keep in mind additional consistency at *t*=0 of the imposed initial conditions with boundary conditions on traction-free faces.

Among the developed asymptotic conditions, those for low-frequency models corresponding to the widespread theories of plate bending and extension appear to be most important. In fact, when dealing with more realistic problems, that necessarily involve a small structural damping, we will observe that the time decay rate of high-frequency modes will be much greater than that for low frequency. These rates are usually proportional to typical time-scales that are of different asymptotic orders for low-frequency and high-frequency motions, cf. for example cosine terms in formula (5.13). For the same reason, even asymptotically secondary low-frequency components associated with initial velocity data may contribute significantly to related displacement fields, as follows from the above-mentioned formula (5.13).

## Acknowledgments

The research is supported by UK Engineering and Physical Sciences Research Council (GR/S29751). This award is very gratefully acknowledged.

## Footnotes

↵† Present address: Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK.

- Received May 12, 2005.
- Accepted February 2, 2006.

- © 2006 The Royal Society