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A Novel Approach to Elastodynamics: I. The Two-Dimensional Case

We introduce a new approach to constructing analytic solutions of the linear PDEs describing elastodynamics. This approach is illustrated for the case of a homogeneous isotropic half-plane body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions, was first solved by Lamb using the Fourier-Laplace transform. The solution of the general problem can also be expressed in terms of the Fourier transform, but this representation involves transforms of unknown boundary values. This necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary data.


Introduction
The problem considered in this paper has a long and illustrious history, which begins with the classic works of Sir Horace Lamb in 1904 [1,2,3]. In [1], Lamb treated four basic problems, the so called Lamb's problems, including both the two and the three dimensional cases. These problems have been studied by several authors, see for example [4,5,6,7,8,9,10,11,12]. For the axisymmetric point-load problem, Pekeris [4] obtained an exact expression for the surface motion in the case that the point load is given by a Heaviside function. This result was further extended by Schiel and Protázio [6]. For the three dimensional cases, Payton, using the Betti-Rayleigh reciprocal theorem [5], obtained the displacement and the surface motion. Gakenheimer, in his PhD thesis [7], made a systematic investigation of the travelling normal point load, by using the Cagniard-deHoop method. Bakkeraq et al. in [9] revisited this problem and presented a straightforward implementation of the Cagniard-deHoop method. Several related problems of contact mechanics and wave propagation can be found in the books of Miklowitz [2], of Johnson [3], and of Payton [13].
The extensive study of Lamb's problems is perhaps due to the fact that they arise in a variety of physical situations, as well as to their interesting mathematical structure.
In this paper, we study two-dimensional problems under the usual assumption of plane strain. We consider arbitrary initial conditions and general stress boundary conditions, including normal line load, tangential line load, and mixed line load. We refer to the latter three stress conditions as Lamb's boundary conditions.
Most studies of Lamb's problems are based on the Helmholtz decomposition and on the use of the Laplace transform in time. In particular, Lamb was the first to use the above approach, as well as a certain superposition which is in fact equivalent to the use of the Fourier transform. Miklowitz made use of the Cagniard-deHoop method [2], which itself is based on the Laplace transform. Payton made use of Green's functions and of the Laplace transform.
Helmholtz decomposition has the advantage of decomposing the P-waves and S-waves. However, it has the disadvantage of introducing higher order derivatives to the boundary conditions. The use of Laplace transform in time, essentially restricts the problem to the case of homogeneous initial conditions.
Our new approach is based on the unified approach for solving linear and integrable nonlinear PDEs introduced by one of the authors in [14]. A crucial role in this method is played by the so-called global relations, which are algebraic relations coupling appropriate transforms of the unknown boundary values with transforms of the given data. For linear PDEs this method uses three novel steps [15]- [44]: 1. Derive a representation for the solution in terms of an integral involving a contour in the complex Fourier plane. This representation is not yet effective, because in addition to transforms of the given initial and boundary conditions, it also contains transforms of unknown boundary values. 2. Analyse certain transformations in the complex Fourier plane which leave invariant the transforms of the unknown boundary values. 3. Eliminate the transforms of the unknown boundary values, by combining the results of the first two steps and by employing Cauchy's theorem (or more precisely Jordan's lemma).
This paper is organised as follows. In section 2, we recall the governing equations of elastodynamics and derive the global relations. In section 3 we implement step 1 and in section 4 we implement steps 2 and 3. In section 5 we use the general representations derived in section 4 in order to analyse Lamb's problems. These results are further discussed in section 6.
For certain complicated boundary value problems it seems that it is not possible to eliminate from the integral representation of the solution the transforms of the unknown boundary values. However, for some of these problems, by using the global relations, one can derive expressions for the Laplace transforms of the boundary values in terms of the given data [45]. A summary of this less effective approach is presented in the Appendix. It is interesting that for the particular case of zero initial conditions, the formulae presented in the Appendix reduce to the formulae first derived in the classical works of Lamb.

Governing Equations and Global Relations
The transient problem for two dimensional elastodynamics in the half plane with Lamb's boundary conditions is defined as follows: Let u = u(x, y, t), v = v(x, y, t), denote the displacements of a homogeneous isotropic half space body. The governing equations of motion without external body forces, are the Lamé-Navier equations: where λ, µ are the Lamé constants and ρ denotes the density of the material, which without loss of generality is normalized to unity, i.e., ρ ≡ 1. Let the initial conditions be denoted by u(x, y, 0) = u 0 (x, y), u t (x, y, 0) = u 1 (x, y), Let the stress boundary conditions be denoted by For a tangential line load, the functions g 1 and g 2 are given by for a normal line load, for a moving normal line load with a constant speed C, Here σ 0 is a constant, δ(x) denotes the Dirac-δ function, X(t) and Y (t) are functions which depend only on t.
Notations Hat, "∧", will denote the two-dimensional Fourier transform with respect to x and y, whereas tilde, "∼", will denote the Fourier transform with respect to x. In particular, where C − denotes the lower half complex l-plane.
We emphasise that since x ∈ R, the x−Fourier transform is well-defined only for k ∈ R; on the other hand, since 0 < y < ∞, the y−Fourier transform is well-defined for l in the lower half complex l−plane.
By applying the two-dimensional Fourier transform to equations (2.1) and by using in the resulting equations the boundary conditions (2.3), we obtain the following equations: Introducing the notations where the functions F P (k, l, t) and F Q (k, l, t) are defined as follows: Solving equations (2.12) for {P, Q}, we obtain the following expressions: t 0 e −iω1s F P (k, l, s)ds where the dispersion relations ω 1 and ω 2 are given by 15) and the known functions P 0 (k, l), P 1 (k, l), Q 0 (k, l), Q 1 (k, l) are given in terms of the initial conditions by In what follows, we take The function (k 2 + l 2 ) 1 2 has the branch points ±ik in the complex l−plane; we connect these two branch points by a branch cut and we fix a branch in the cut plane by the requirement that, Let u (j)± , v (j)± , U (j) , V (j) , j = 1, 2, denote the following unknown functions: Similarly, let g (j)± , f (j)± , G (j) , F (j) , j = 1, 2, denote the following known functions: Using the above notations, equations (2.14) become where the known functions N P (k, l, t) and N Q (k, l, t) are defined as follows: (2.21b) We will refer to equations (2.20) as the global relations. The global relations express the two-dimensional Fourier transforms of the solution (u, v) in terms of the given initial and boundary data, as well as in terms of the transforms U (1) , U (2) , V (1) , V (2) of the unknown boundary values. The important observation is that equations (2.20) are valid for all values of l in the lower half complex l−plane. It turns out that using this fact it will be possible to eliminate the unknown transforms.

An Integral Representation Involving the Unknown Transforms
We first observe that for any fixed k ∈ R and any fixed t, Indeed, these functions are all single-valued, thus it only remains to establish the analyticity in the neighbourhood l = ±ik. The definition of U (1) , i.e. equation (2.18c), implies that this function possesses the following expansion for l near ±ik: Solving equations (2.20) we find We observe that the following important transformation is valid in the complex l−plane: l → −l maps ω 1 to − ω 1 , ω 2 to − ω 2 , and leaves U (j) , V (j) , j = 1, 2 invariant.
Applying the inverse Fourier transform formula to these equations, we obtain Denote by H 1 (k, l, t) and H 2 (k, l, t) the following functions appearing in equations (3.5): We observe that for any fixed k ∈ R and fixed t, 0 ≤ t < T , T > 0, the above functions are analytic in the entire complex l−plane. Indeed, equation (3.1) and the analogous equation for U (2) , imply that in the neighbourhood of l = ±ik, the following expansion is valid: Similarly, The restriction y > 0, as well as the analyticity of the functions H 1 (k, l, t) and H 2 (k, l, t), allow us to deform the contour of integration from the real axis to a contour γ k in the upper half l−plane (the particular choice of γ k will be determined in the next section): and The function l 12 has two branch points ±k λ+µ µ , which we connect by a horizontal branch cut; the function l 21 has two branch points ±ik λ+µ λ+2µ , which we connect by a vertical branch cut, see Fig.1. We fix the branches by the requirements that, , as l → ∞.  The following transformations are valid in the cut l-plane: Using in equations (2.20a) and (2.20b) the transformations l → l 12 and l → −l respectively, and then combining the two resulting equations, we obtain the following equations: kû(k, l 12 , t) + l 12v (k, l 12 , t) = (λk 2 + l 2 12 (λ + 2µ))V (2) (k, l, t) + 2µkl 12 U (2) (k, l, t) + N P (k, l 12 , t),  Similarly, using in equations (2.20a) and (2.20b) the transformations l → −l and l → l 21 respectively, and then combining the two resulting equations, we obtain the following equations: (4.5b) Let C 1 (k, l) = λk 2 + l 2 12 (λ + 2µ), C 2 (k, l) = 2µkl 12 , (4.6a) Simplifying the expressions for ∆ 1 and ∆ 2 we find Equations (4.5) imply − D 2 (k, l)(l 21û (k, l 21 , t) − kv(k, l 21 , t) − N Q (l 21 ))], (4.9a) We fix the choice of the contour γ k by requiring that every term in the RHS of (4.8) and (4.9) does not have a pole or a branch point above this contour, see Fig.2.
We summarize the above result in the following proposition: with the initial conditions (2.2) and the stress boundary conditions (2.3). Assume that the given functions have sufficient smoothness and decay. A solution of the above initial-boundary value problem, which decays for large (x, y), is given by equations . c) The contours γ k depicted in Fig.2, are deformations of the real axis and determined by the requirement that the zeros of ∆ j , j = 1, 2 and the branch points of l 21 and l 12 are all below γ k .

The Normal Line Load with the Homogeneous Initial Conditions
Consider a normal line load suddenly applied to an isotropic elastic half plane body. In this case, where σ 0 is a constant and h(t) is the Heaviside function defined by h(t) = 0, t ≤ 0; h(t) = 1, t > 0.
Similar results can be obtained for the other Lamb's problems.

Conclusions
The main result of this paper is the derivation of equations (4.10). These equations express the displacements (u(x, y, t), v(x, y, t)) in terms of an integral along the real line and integrals along contours γ k of the complex l−plane; these integrals involve transforms of the given initial and boundary data. Indeed, equations (4.10) involve the functions N P (k, l, t) and N Q (k, l, t), which are defined in equations (2.21) in terms of the Fourier transforms P 0 (k, l, t), Q 0 (k, l, t), P 1 (k, l, t), Q 1 (k, l, t) of the initial data (see equations (2.16)), as well as in terms of certain known transforms G (1) (k, l, t), G (2) (k, l, t), F (1) (k, l, t), F (2) (k, l, t) of the boundary data (see equations (2.19)). The starting point of the derivations of equations (4.10) is the derivation of the global relations (2.20). These equations are the direct consequence of the application of the two-dimensional Fourier transform and of the substitution in the resulting equations of the given initial and boundary conditions. Equations (2.20) involve the known functions N P and N Q , as well as the unknown functions U (1) , U (2) , V (1) , V (2) (these functions involve certain transforms of the unknown boundary values u(x, 0, t) and v(x, 0, t), see equations (2.8) and (2.18)). The elimination of the above unknown functions is achieved by the following steps: 1. By employing the transformation l → −l, which leave the unknown functions (U (1) , U (2) , V (1) , V (2) ) invariant, and by utilising the analyticity properties of these functions, we obtain the integral representations (3.9). These representations involve an integral along the real k − axis and an integral along the contour γ k of the complex l−plane. 2. Using the transformations l → l 21 and l → l 12 which map U (2) to U (1) ,V (2) to V (1) and U (1) to U (2) ,V (1) to V (2) respectively, we express the unknown functions (U (1) , U (2) , V (1) , V (2) ) in terms of known functions as well as in terms of certain unknown functions which however are analytic in a certain domain of the complex l−plane, see equations (4.8) and (4.9). 3. Using (4.8) and (4.9) in equations (3.9) and employing Jordan's lemma we obtain equations (4.10).
The main advantages of the new approach are the following: 1. The new method provides an analytic solution of Lamb's problem with arbitrary initial and boundary conditions.
2. This solution is expressed in terms of the given initial and boundary data. The relevant representation is novel even for the particular case of homogeneous initial conditions (this case has been analysed by several authors). An alternative approach using the Laplace transform is briefly discussed in the Appendix. By comparing equations (4.10) and equation (6.1), the advantage of the new formulae becomes clear. Actually, taking into consideration that the initial-boundary value problems of the equations of elastodynamics are well posed for any finite t, the use of the Laplace transform, which requires t → ∞, is clearly inappropriate.
3. The new method can be employed for the solution of several related initialboundary value problems, including the problem of the isotropic half space in the axisymmetric case. Furthermore, it can be extended to problems in three dimensions [46].
In the particular case of the problem of homogeneous initial condition and the normal line load boundary condition, equation (6.4) reduces to the classic Lamb's solution [1] [2].