Eric Amar

Hörmander's solution of the ¯∂ -equation with compact support

Eric

1 Introduction.

Let ϕ be a strictly plurisubharmonic function of class C 2 in a domain Ω in C n . Let c ϕ (z) be the smallest eigenvalue of i∂ ∂ϕ(z), then ∀z ∈ Ω, c ϕ (z) > 0. We denote by L 2,c p,q (Ω, e ϕ ) the currents in L 2 p,q (Ω, e ϕ ) with compact support in Ω and H p (Ω, e -ϕ ) the space of all (p, 0), ∂ closed forms in L 2 (Ω, e -ϕ ).

In the case n = 1, H. Hedenmalm [START_REF] Hedenmalm | On Hörmander solution of the ∂-equation[END_REF] proved the following theorem Theorem 1.1 Let ω ∈ L 2 0,1 (C, e ϕ ) then there is u ∈ L 2 (C, c ϕ e ϕ ) such that ∂u = ω, and

u L 2 (C,cϕe ϕ ) ≤ C ω L 2 (C,e ϕ ) , provided that ω ⊥ H 0 (C, e -ϕ ).
As he prognosticated in his remark 1.3 in [START_REF] Hedenmalm | On Hörmander solution of the ∂-equation[END_REF] this theorem was generalised to Stein manifolds in [START_REF] Amar | Serre duality and Hörmander's solution of the ∂ -equation[END_REF], and for domains in C n we got :

Theorem 1.2 Let Ω be a pseudo convex domain in C n ; if ω ∈ L 2,c
p,q (Ω, e ϕ ) with ∂ω = 0 if q < n and ω ∈ L 2 p,q (Ω, e ϕ ) with ω ⊥ H n-p (Ω, e -ϕ ) if q = n, then there is u ∈ L 2 p,q-1 (Ω, c ϕ e ϕ ) such that ∂u = ω, and

u L 2 (Ω,cϕe ϕ ) ≤ C ω L 2 (Ω,e ϕ ) .
Still pushed by H. Hedenmalm in several e-mails, the aim is to improve this theorem in the special case of C n by removing the assumption on the compact support for ω. Precisely we shall prove the following Theorem 1.3 Let ω ∈ L 2 p,q (C n , e ϕ ) with ∂ω = 0 if q < n and with ω ⊥ H n-p (Ω, e -ϕ ) if q = n, then there is u ∈ L 2 p,q-1 (C n , c ϕ e ϕ ) such that ∂u = ω, and

u L 2 (C n ,cϕe ϕ ) ≤ C ω L 2 (C n ,e ϕ ) .
I don't know if this is still true for a proper pseudo convex domain Ω of C n ; the method I have works only for C n .

The proof is strongly based on theorem 1.2 via the theorem 2.1, which can be seen as an auto improvement of theorem 1.2.

Solutions with compact support.

The first aim is to prove the following theorem which is an improvement of theorem 1.2, because here the solution is also with compact support.

Theorem 2.1 Let Ω be a pseudo convex domain in C n and ϕ be a strictly plurisubharmonic

function of class C 2 in Ω. Let c ϕ (z) be the smallest eigenvalue of ∂ ∂ϕ(z). There is a constant C > 0 such that if q < n, ∀ω ∈ L 2,c p,q (Ω, e ϕ ), ∂ω = 0, ∃u ∈ L 2,c p,q-1 (Ω, c ϕ e ϕ ), ∂u = ω, u ≤ C ω . If q = n
, we make the hypothesis that ω ⊥ H n-p (Ω, e -ϕ ) and we still get :

∀ω ∈ L 2,c p,n (Ω, e ϕ ), ∃u ∈ L 2,c p,n-1 (Ω, c ϕ e ϕ ), ∂u = ω, u ≤ C ω .
Proof. The idea is the same we used in [START_REF] Amar | An Andreotti-Grauert theorem with L r estimates[END_REF]: we add a weight to force the solution to be small outside a fixed compact set, then we take a limit.

Let ω ∈ L 2,c p,q (Ω, e ϕ ) with ∂ω = 0 if q < n and ω ∈ L 2 p,q (Ω, e ϕ ) with ω ⊥ H n-p (Ω, e -ϕ ) if q = n. Take a p.c. domain D such that D ⊂ Ω, containing the support of ω and defined by a p.s.h. smooth function ρ, i.e. D := {z ∈ Ω :: ρ(z) < 0}. This is always possible because Ω is Stein. Fix ǫ > 0 and set D ǫ := {z ∈ Ω :: ρ(z) < ǫ}, then D ⊂ D ǫ . Let χ ∈ C ∞ (R) such that χ is increasing and convex and such that : ∀t ≤ 0, χ(t) = 0 ; ∀t > 0, χ ′ (t) > 0, χ"(t) > 0. The function σ := χ(ρ) is also p.s.h. and smooth. We have that σ > χ(ǫ) > 0 outside Dǫ so, with ψ k := ϕ + kσ we have that ∂ ∂ψ k = ∂ ∂ϕ + k∂ ∂σ is still positive definite, so ψ k is s.p.s.h. and "big" outside Dǫ . Moreover in D we have ψ k = ϕ. So we apply theorem 1.2 with ψ k and we get the existence of a solution

u k ∈ L 2 p,q-1 (Ω, c ψ k e ψ k ) of ∂u k = ω, with u k L 2 (Ω,c ψ k e ψ k ) ≤ C ω L 2 (Ω,e ϕ ) , (2.1)
because on Supp ω we have ∀k ∈ N, ψ k = ϕ. But, because σ is p.s.h., we have that c ψ k ≥ c ϕ and ∀z / ∈ D, e ψ k (z) = e kσ(z) e ϕ(z) with σ(z) > 0, this means that u k must be small in Dc ǫ .

Now we proceed as in [START_REF] Amar | An Andreotti-Grauert theorem with L r estimates[END_REF]. We have by (2.1) :

Ω |u k | 2 c ϕ e ψ k dm ≤ C ω L 2 (Ω,e ϕ )
because c ψ k ≥ c ϕ . Then, by e ψ k = e kσ e ϕ ≥ e kχ(ǫ) e ϕ on D c ǫ , we get that e kχ(ǫ)

Ω\Dǫ |u k | 2 c ϕ e ϕ dm ≤ Ω\Dǫ |u k | 2 c ϕ e ϕ e kσ dm ≤ C ω 2 L 2 (Ω,e ϕ ) hence Ω\Dǫ |u k | 2 c ϕ e ϕ dm ≤ Ce -kχ(ǫ) ω 2 L 2 (Ω,e ϕ ) . (2.2)
On the other hand we have

Ω |u k | 2 c ϕ e ϕ dm ≤ C ω L 2 (Ω,e ϕ )
hence the sequence {u k } k∈N is uniformly bounded in L 2 (p,q-1) (Ω, c ϕ e ϕ ). So there is a subsequence, still denoted {u k } k∈N , converging weakly to u ∈ L 2 (p,q-1) (Ω, c ϕ e ϕ ), i.e. ∀f ∈ L 2 n-p,n-q+1 (Ω, c ϕ e ϕ ), u k , f → u, f . To see that this form u is 0 a.e. on Ω\D ǫ let us take a component u I,J of it ; it is the weak limit of the sequence of functions {u k,I,J } which means, with the notations v := u I,J , v k := u k,I,J , dµ := c ϕ e ϕ dm,

∀g ∈ L 2 (Ω, c ϕ e ϕ ), Ω vgdµ = lim k→∞ Ω v k gdµ.
As usual take g :

= v |v| 1 E where E := {|v| > 0} ∩ (Ω\D ǫ ) then we get Ω vgdµ = E |v| dµ = lim k→∞ Ω v k gdµ = lim k→∞ E v k v |v| dµ. (2.3)
Now we have by Cauchy-Schwarz

E v k v |v| dµ ≤ v k L 2 (E,dµ) 1 E L 2 (E,dµ) . But v k 2 L 2 (E,dµ) ≤ Ω\Dǫ |u k | 2 dµ ≤ Ce -kχ(ǫ) ω 2 L 2 (Ω,e ϕ ) by (2.2) because χ(ǫ) > 0. Hence, by (2.3), E |v| dµ 2 ≤ C 1 E L 2 (E,dµ) lim k→∞ e -kχ(ǫ) ω 2 L 2 (Ω,e ϕ ) = 0 which implies µ(E) = m(E) = 0 because on E, |v| > 0.
This being true for all components of u, we get that the form u is 0 a.e. on Ω\D ǫ i.e. Supp u ⊂ D ǫ . So we get ∀ϕ ∈ D n-p,n-q (Ω), (-1) p+q-1 ω, ϕ = u k , ∂ϕ → u, ∂ϕ ⇒ u, ∂ϕ = (-1) p+q-1 ω, ϕ hence ∂u = ω as distributions.

We proved that : ∃C > 0, ∀ǫ > 0, ∃u ǫ ∈ L 2 (p,q-1) (Ω, c ϕ e ϕ ) :: ∂u ǫ = ω, u ǫ L 2 (Ω,cϕe ϕ ) ≤ C ω 2 L 2 (Ω,e ϕ ) , and Supp u ǫ ⊂ D ǫ . Because C is independent of ǫ, we can take again a subsequence to get finally that

B c k |ω| 2 e ϕ dm ≤ 1/k + 1. (3.6) Take χ k ∈ C ∞ c (B k ) such that χ k = 1 in B k-1 .
Consider the form ω k := χ k ω ; we have: ω -ω k ≤ 1/k by (3.6). And ∂ω k = ∂χ k ∧ ω. Clearly we can choose χ k such that ∂χ k ∞ ≤ 1 and this is the place where

C n is required. Hence ∂ω k 2 = B k \B k-1 ∂χ k 2 |ω| 2 e ϕ ≤ B k \B k-1 |ω| 2 e ϕ ≤ 1 k .
By (3.5) we have that there exists

u k ∈ L 2,c p,q (C n , c ϕ e ϕ ), ∂u k = ∂ω k with u k ≤ C/k. Now set: µ k := ω k -u k ; we have ∂µ k = 0 and ω -µ k ≤ 1 + C k because ω -µ k = ω -ω k + u k .
Moreover we have that µ k is compactly supported. So there is

v k ∈ L 2,c p,q-1 (C n , c ϕ e ϕ ), ∂v k = µ k and v k ≤ C µ k ≤ C( ω + 1 + C k ).
Because the norm of v k in L 2 p,q-1 (C n , c ϕ e ϕ ) is uniformly bounded, we have a weakly converging sub sequence :

∃v ∈ L 2 p,q-1 (C n , c ϕ e ϕ ) :: ∀f ∈ L 2 n-p,n-q+1 (C n , c ϕ e ϕ ), lim v k , f = v, f . Now choose f := ∂ϕ, with ϕ ∈ D n-p,n-q (C n ) then v k , ∂ϕ = (-1) p+q ∂v k , ϕ = (-1) p+q µ k , ϕ → v, ∂ϕ = (-1) p+q ∂v, ϕ .

But µ k = ω + e k , with e k ≤ 1 + C k hence µ k , ϕ = ω, ϕ + e k , ϕ → ω, ϕ so ∂v, ϕ = ω, ϕ which means that ∂v = ω in the distributions sense.

∃u ∈ L 2 (p,q-1) (Ω, c ϕ e ϕ ) :: ∂u = ω, u L 2 (Ω,cϕe ϕ ) ≤ C ω 2 L 2 (Ω,e ϕ ) , Supp u ⊂ D, and the theorem is proved.

We have a finer control of the support (see [START_REF] Amar | An Andreotti-Grauert theorem with L r estimates[END_REF], theorem 2.13).

Theorem 2.2 Let Ω be a pseudo convex domain in C n and ϕ be a strictly plurisubharmonic function of class C 2 in Ω. Let c ϕ (z) be the smallest eigenvalue of ∂ ∂ϕ(z).

Let ω ∈ L 2,c p,q (Ω, e ϕ ) with ∂ω = 0 if q < n and ω ⊥ H n-p (Ω, e -ϕ ) if q = n, and with support in Ω\C where C is also a pseudo convex domain, then there is a u ∈ L 2,c p,n-1 (Ω, c ϕ e ϕ ), ∂u = ω, u ≤ C ω with support in Ω\ Ū , where U is any open set relatively compact in C, provided that q ≥ 2.

Proof.

Let ω be a (p, q) form with compact support in Ω\C then ∃v ∈ L 2,c p,q-1 (Ω, c ϕ e ϕ ), ∂v = ω, with compact support in Ω, by theorem 2.1. Because ω has its support outside C we have ω = 0 in C ; this means that ∂v = 0 in C. On the support of v we have, because ϕ is s.p.s.h. and of class C 2 ,

. By Hörmander we have, because C is Stein and bounded, that ∃h ∈ L 2 p,q-2 (C) :: ∂h = v in C. Let U be open and such that Ū ⊂ C. Let χ be a smooth function such that χ = 1 in U and χ = 0 near ∂C ; then set

p,q-1 (Ω) ; moreover u = 0 in Ū because χ = 1 in U hence ∂χ = 0 there. Finally ∂u = ∂v -∂2 (χh) = ω. Because of (2.4) u ∈ L 2 p,q-1 (Ω) implies u ∈ L 2 p,q-1 (Ω, c ϕ e ϕ ) and we are done. If Ω and C are, for instance, pseudo-convex in C n then Ω\C is no longer pseudo-convex in general, so this theorem improves actually the control of the support.

3 Approximation procedure.

p,q (C n , e ϕ ) with ∂ω = 0 if q < n and ω ⊥ H n-p (Ω, e -ϕ ) if q = n, then there is u ∈ L 2 p,q-1 (C n , c ϕ e ϕ ) such that ∂u = ω, and

This is theorem 1.2 of [START_REF] Amar | Serre duality and Hörmander's solution of the ∂ -equation[END_REF] where the assumption of the compact support for ω is removed.

Proof. The idea is to proceed by approximation. So by theorem 2.1 we have the following result:

p,q (C n , e ϕ ), ∃u ∈ L 2,c p,q-1 (C n , c ϕ e ϕ ), ∂u = ω, u ≤ C ω .

(3.5)

Now let ω ∈ L 2 p,q (C n , e ϕ ) then there exists B k = B(0, r k ), r k ≥ r k-1 + 1 → ∞, a sequence of balls in C n such that


possibility of residual strains, which were subsequently accounted for by Ogden and Roxburgh [START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF].

The essence of the theory is that loading is described by an elastic strain-energy function and unloading by a di!erent strain-energy function. The switch between strain-energy functions is e!ected through incorporation of an additional (damage or softening) variable into the strain-energy function, which is then referred to as a pseudo-energy function. Thus, we write =(F, ), where F is the deformation gradient relative to the natural (undeformed, stress-free) con"guration of the material and is the additional variable, which may be either active or inactive. If it is inactive we set it to the constant value unity and write

= (F)"=(F, 1) (1)
for the resulting strain-energy function. When it is active it is taken to depend on the deformation gradient (in general implicitly) and we write " ( (F) and de"ne the strain-energy function w by w(F)"=(F, ( (F)).

(

In [START_REF] Ogden | A pseudo-elastic model for the Mullins e!ect in "lled rubber[END_REF][START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF], was taken to be continuous and to be determined uniquely by F. This contrasts with the theory adopted by Lazopoulos and Ogden [START_REF] Lazopoulos | Nonlinear elasticity theory with discontinuous internal variables[END_REF][START_REF] Lazopoulos | Spherically-symmetric solutions for a spherical shell in "nite pseudo-elasticity[END_REF] in which was related to F implicitly and allowed to depend non-uniquely on F, thus allowing a discontinuity in at certain critical values of F. Such discontinuity is associated with a discontinuity in stress which in turn is related to discontinuous damage and, hence, a discontinuous change in material properties at the critical deformation. This theory was applied in [START_REF] Lazopoulos | Spherically-symmetric solutions for a spherical shell in "nite pseudo-elasticity[END_REF], for example, to describe the in#ation and de#ation cycle for a thick-walled spherical shell; a slightly di!erent pseudo-elastic model for the same problem was used by Ogden [START_REF] Ogden | Elastic and pseudo-elastic instability and bifurcation[END_REF][START_REF] Ogden | Non-smooth changes in elastic material properties under "nite deformation[END_REF].

In the applications of this theory of pseudoelasticity considered so far it has been assumed that the material response remains isotropic. In general, this cannot be the case, particularly if there is residual strain since the response relative to such a residually strained state will be anisotropic. The theory, however, can accommodate this changing material symmetry, as discussed in [START_REF] Ogden | Elastic and pseudo-elastic instability and bifurcation[END_REF], although in the present paper we retain the isotropy assumption for simplicity of illustration. For a general discussion of residual stress in a non-linearly elastic body we refer to Hoger [START_REF] Hoger | On the determination of residual stress in an elastic body[END_REF] and Johnson and Hoger [START_REF] Johnson | The use of strain energy to quantify the e!ect of residual stress on mechanical behaviour[END_REF] and the papers cited therein.

An alternative model for the Mullins e!ect has been developed by Beatty and colleagues in a series of papers and it has some features in common with the theory of pseudo-elasticity discussed here. For uniaxial deformations details can be found in [START_REF] Johnson | The Mullins e!ect in uniaxial extension and its in#uence on the transverse vibration of a rubber string[END_REF][START_REF] Johnson | A constitutive equation for the Mullins e!ect in stress controlled uniaxial extension experiments[END_REF] and for general deformations in, for example, [START_REF] Beatty | A theory of stress softening in incompressible isotropic elastic materials[END_REF][START_REF] Krishnaswamy | The Mullins e!ect in compressible solids[END_REF], which deal with incompressible and compressible materials, respectively.

The theory of pseudo-elasticity (that with continuous ) required here is summarized in Section 2. We apply the theory, with residual strains allowed for, to a problem involving non-homogeneous deformation. Speci"cally, we consider the azimuthal shear of a thick-walled circular cylindrical tube of incompressible initially isotropic material with restriction to plane strain. We formulate the boundary-value problem for a general incompressible pseudo-elastic material for both loading and unloading. The inner circular boundary is taken to be "xed and the outer boundary rotated relative to it at "xed radius by application of a suitable shearing stress (assumed for loading to be monotonically increasing with the angle of rotation).

In Section 3, we describe the equations for azimuthal shear and apply them "rst for an elastic material. We recall that there is an exact solution for this problem for the neo-Hookean form of strain-energy function. We then apply the theory of pseudo-elasticity and obtain equations for unloading in general form. Since the deformation is inhomogeneous the strain-energy function for unloading depends on the radius via the maximum value of the strain-energy function on loading and is therefore itself inhomogeneous. If is the maximum value of the shearing stress on the outer boundary achieved during loading, then there are two critical values of , denoted \ and > , such that the following results hold. Firstly, for ( \ there is no residual strain after unloading. Secondly, if \ ( ( > then there exists a radius, r say, with a(r (b, where a and b are the inner and outer radii of the tube, respectively, such that there is residual strain for a)r(r but no residual strain for r )r)b. In this latter region there is a (residual) rigid-body rotation. Thirdly, if ' > then there is a residual strain distribution for a)r)b.

By using a speci"c example for the constitutive model we calculate the residual strain explicitly. The resulting non-recoverable energy expended in a loading}unloading cycle is also calculated. For unloading paths associated with di!erent values of , the governing equation are solved numerically in order to determine the shear strain distributions and to illustrate the stress softening e!ect, that is the reduction in the shear stress relative to its value on loading at the same strain.

Pseudo-elasticity theory

The theory of pseudo-elasticity adopted here is based on the pseudo-energy function =(F, ) described in Section 1. For the relevant background on non-linear elasticity we refer to, for example, [13}15]. In this paper we restrict attention to incompressible materials, so that the constraint det F"1

is satis"ed. The nominal stress tensor associated with the strain energy (1) is denoted by S and is given by

S " *= *F (F)!p F\, (4)
where p is the Lagrange multiplier arising from the constraint (3).

In [START_REF] Lazopoulos | Nonlinear elasticity theory with discontinuous internal variables[END_REF] a variational procedure was used to show that, where is active, and F are related, in general implicitly, through the equation

*= * (F, )"0. ( 5
)
This equation, which identi"es stationary points of =(F, ) with respect to , de"nes a hypersurface in the 10-dimensional (F, )-space to which values of must be restricted, subject to the constraint (3). If (5) de"nes uniquely in terms of F then we may write formally " ( (F), [START_REF] Ogden | Non-smooth changes in elastic material properties under "nite deformation[END_REF] as indicated in Section 1, and we introduce the notation w for the resulting strain-energy function, de"ned by [START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF]. In view of ( 5), the nominal stress tensor S associated with w(F) is simply

S" *w *F (F)!pF\" *= *F (F, )!pF\, (7)
where the right-hand side is evaluated for given by ( 6) and p is the counterpart of p when is active. Note that for an incompressible material the Cauchy stress tensors, denoted by and for inactive and active , respectively, are related to S and S, respectively, by "FS , "FS.

The criterion for switching on or o!, i.e. the criterion for a change in material properties, depends on the particular application and hence, on the speci"c model used with the framework of the dependence of = on . In the application to stress softening associated with the Mullins e!ect [START_REF] Ogden | A pseudo-elastic model for the Mullins e!ect in "lled rubber[END_REF][START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF] is taken to be inactive during loading and to switch on during unloading (with loading and unloading being well de"ned relative to the energy expended during a deformation path). In this case, the energy = (F) changes continuously from its value =(F, 1) to =(F, ) with decreasing from the value 1 as unloading proceeds. The corresponding nominal stress also changes continuously, with S evaluated from [START_REF] Hoger | On the determination of residual stress in an elastic body[END_REF] for "1 equal to S . We require to be an objective scalar variable and that =(F, ) satis"es the usual objectivity condition =(QF, )"=(F, ) for all proper orthogonal Q.

(9)

The requirement on then ensures that the dependence of on F determined from ( 5) is objective, and objectivity of w in (2) then follows. Note that objectivity of =(F, ) with respect to F in general guarantees that determined from ( 5) is objective with respect to F by the implicit function theorem since F may be replaced by QF in (5) through [START_REF] Johnson | The Mullins e!ect in uniaxial extension and its in#uence on the transverse vibration of a rubber string[END_REF]. By contrast, objectivity of = (F) does not guarantee that of =(F, ). By the same token, any material symmetry ascribed to = (F) is not in general inherited by =(F, ). See [START_REF] Ogden | Elastic and pseudo-elastic instability and bifurcation[END_REF] for further discussion of these points.

Isotropy

For an isotropic material we write the pseudoenergy function in the form

= M ( , , , ), (10)
where , , , the principal stretches of the deformation, are subject to the incompressibility constraint "1.

Eq. ( 5) specializes to

*= M * "0 (12)
and the principal Cauchy stresses are given by

G " G *= M * G !p, i"1, 2, 3. ( 13
)
We now write

=( , , ),= M ( , , \ \ , ) (14)
so that [START_REF] Krishnaswamy | The Mullins e!ect in compressible solids[END_REF] becomes

*= * ( , , )"0. (15)
Accordingly, Eq. ( 6) may be specialized to give

" ( ( , )" ( ( , ), (16)
the symmetry in and being noted. Eq. ( 13) are combined, on elimination of p, to give

! " *= * , ! " *= * . ( 17
)
Note that in order to avoid undue proliferation of notation we are now using the same notation = for the pseudo-energy function as in Section 1, but with the dependence now on the two independent stretches , and . Similarly, the notation for the function ( has been retained.

Appropriate specializations of Eqs. ( 15) and (17) for simple tension and pure shear are discussed in [START_REF] Ogden | A pseudo-elastic model for the Mullins e!ect in "lled rubber[END_REF][START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF] and we refer to these papers for details and for applications to the modelling of the Mullins stresssoftening e!ect for homogeneous deformations.

As in [START_REF] Ogden | A pseudo-elastic model for the Mullins e!ect in "lled rubber[END_REF][START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF] we model a loading path in ( , )space using "1 so that

= ( , ),=( , , 1) (18)
with loading de"ned by = increasing. On unloading is active, = is decreasing, and is given by [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material[END_REF]. Thus, in parallel with (2) we de"ne the modi-"ed strain-energy function w by w( ,

)"=( , , ( ( , )). (19)
For loading, the Cauchy stress di!erences are given by

? ! " ? *= * ? , "1, 2, (20)
and for unloading by

? ! " ? *w * ? " ? *= * ? , "1, 2. ( 21
)

A plane strain example: simple shear

At this point we specialize to plane strain and set "1 and use the notation

" , " \. ( 22
)
We de"ne the pseudo-energy function in terms of by

= K ( , ),=( , \, ), (23)
so that Eqs. ( 17) and [START_REF] Beatty | Topics in "nite elasticity: hyperelasticity of rubber, elastomers and biological tissues * with examples[END_REF], respectively, may be specialized to

! " = K H , = K E "0, (24)
where the subscripts denote partial derivatives.

In preparation for the discussion, in Section 3, of the azimuthal shear deformation (which is locally a simple shear) we now specialize further by considering the simple shear deformation, for which the amount of shear, denoted , is given in terms of by

" ! \, (25)
where, without loss of generality, we have taken *1 to correspond to *0. To represent the pseudo-energy function in terms of we use the notation = I de"ned by

= I ( , ),= K ( , ). (26)
Then, with the direction of shear taken as the X direction in the (1, 2)-plane the (uniform) shear stress has the form

"= I A ( , ) (27)
and we also have the universal relation

! " . ( 28
)
The second equation in (24) becomes

= I E ( , )"0, (29)
which, in general implicitly, gives in terms of when is active, in which case we write " ( ( ) as the specialization of ( 16) for simple shear. The precise form of this function will depend on the way in which the pseudo-energy function is chosen to depend on . A simple example of the form of ( will be discussed in Section 3.

When is inactive we set

= I ( )"= I ( , 1), (30)
= I AA (0)" '0, (33)
where is the shear modulus in the natural con"guration.

Note that when is active it follows from (29) that

= I AE #= I EE d d "0 (34)
with " ( ( ). Thus, d /d '0 ((0) according as = I AE has the opposite sign to (same sign as) = I EE .

A simple model

The speci"c model used in [START_REF] Ogden | A pseudo-elastic model for the Mullins e!ect in "lled rubber[END_REF][START_REF] Ogden | An energy-based model of the Mullins e!ect[END_REF] has the form

=( , , )" = ( , )#( ) (35)
with

(1)"0 (36)

for incompressible isotropic elasticity. Eq. ( 15) then leads to

( )"!= ( , ) (37)
and is taken to satisfy the inequality

( )(0. ( 38
)
This inequality ensures that in a simple tension test or in simple shear, for example, decreases monotonically with the magnitude of the deformation during unloading. From Eqs. ( 20) and ( 21) we obtain

? ! " ( ? ! ) ( 39
)
in respect of (35). Let ( , ) be the values of ( , ) at the start of unloading and let = "= ( , ). Then

(1)"!= , (40)
which, implicitly, requires that the function depends on the point at which unloading begins. This function is referred to as a damage or softening function and provides a measure of the non-recoverable part of the energy in a loading}unloading cycle, as described below.

If is restricted to being positive then the stress vanishes on unloading only when the original natural con"guration is reached. Then, if is the value of at this point, it follows from (37) that (

)"0, which de"nes uniquely. The value (

) is the non-recoverable part of the energy (per unit volume). If, on the other hand, is not so restricted then the stress may vanish on unloading at the point where "0 before the natural con"guration is reached. Let = be the value of = at this point. Since depends on = then so does = and we have (0)"!= . There is then a residual deformation, given by the stretches ( , ) say, such that = ( , )"= and in this case (0) is the non-recoverable part of the energy (per unit volume).

For simple shear the model (35) leads, in the notation of Section 2.2, to

" " = I ( ) , ! " ( ! )" = I ( ). (41)
Hence "0 gives "0, " , but equally "0 leads to the same conclusion and to a residual value of , say, such that

= I ( )"!(0). ( 42
)
For the azimuthal shear problem discussed in Section 3 this is a local condition with depending on the radius.

Azimuthal shear

Here, we describe the (pure) azimuthal shear deformation of a circular cylindrical thick-walled tube under the assumption of plane strain. The tube cross-section is de"ned by

A)R)B, 0) )2 (43)
in polar coordinates (R, ) in the reference con"guration. Let (r, ) be the corresponding polar coordinates in the deformed con"guration. Then, pure azimuthal shear is the isochoric deformation de-

"ned by r"R, " #g(R), (44)
where g(r)"g(R) is obtained by the solution of the equilibrium equations. We use r as the independent variable and set a"A, b"B so that a)r)b.

The deformation is locally a simple shear of amount depending on r according to

"rg(r), (45)
where the prime signi"es di!erentiation with respect to r. The direction of shear is locally the azimuthal ( ) direction and the radial direction is normal to the planes of shear. For full discussion of the background analysis for azimuthal shear we refer to, for example, Ogden [START_REF] Ogden | Non-linear Elastic Deformations[END_REF].

With reference to Eqs. ( 27) and (28) we may identify locally the ( , r) axes with the (1, 2) axes with increasing counterclockwise. Thus, the stress components are PF " , PP " , FF " . Hence,

PF "= I A ( , ), FF ! PP " PF . ( 46
)
For the azimuthal shear problem there are two independent equilibrium equations. The radial equation has the form

r d PP dr " FF ! PP " = I P ( , ), (47)
and the azimuthal equation is

d dr (r PF )"0. ( 48
)
The latter integrates on use of the "rst equation of (46) to give

= I A ( , )" b/r, ( 49
)
where is a constant equal to the value of PF on r"b.

In principle the azimuthal equation determines and hence the function g, while the radial equation serves to determine PP (or, equivalently, the Lagrange multiplier p) and hence the stress distribution. We impose on g the boundary conditions

g(a)"0, g(b)" , (50)
so that the inner boundary of the tube is "xed and the outer boundary rotated through a prescribed angle . We assume that is a monotonic increasing function of and we use as the control parameter rather that .

Loading

For loading we take to be inactive so that Eq. (49) specializes to

= I ( )" b/r. ( 51
)
To ensure that this equation yields a unique solution for as a function of r for all *0 we assume that = I ( )'0, = I ( )PR as PR.

For incompressible and compressible isotropic elastic materials the azimuthal shear problem has been considered by several authors and we refer to the recent paper by Jiang and Ogden [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material[END_REF] for full discussion and references. In the incompressible case there is a well-known exact solution for the neo-Hookean material, which, for the deformation considered here, has the form.

= I ( )" , (53)
where is as de"ned in (33).

Eq. ( 51) with ( 53) and (45) gives the solution

" b r , g(r)" b 2 1 a ! 1 r . ( 54
)
The angle of rotation of the outer surface relative to the inner one is given by the linear relationship

"g(b)" b 2 1 a ! 1 b ( 55
)
between and . The total energy, E say, in the material due to the deformation (per unit length normal to the considered plane) is given by

E" @ ? r dr" b 2 1 a ! 1 b . ( 56
)
In this paper we assume for simplicity that loading is governed by the neo-Hookean strain energy so that the explicit solution (54) is available. Let be the maximum value of reached on loading and let be the corresponding value of , so that

= I ( )"b /r. (57)
We set

= I "= I ( ), (58)
which, of course, depends on r and . For the neo-Hookean material we have

= I " , "b / r. ( 59
)

Unloading

We now consider unloading with active so that the identity

= I E ( , )"0 (60)
gives in terms of . The azimuthal equation ( 49) is written as

= I A ( , )"b /r, ) , (61)
which matches (51) at " , "1 (for all r). The radial equation is (47).

When " ( ( ) is determined from (60) the solution of (61) for describes the unloading deforma-tion path as reduces from to 0. When "0 we must have

= I A ( , )"0, (62)
which is solved locally with (60) to give a pair of values ( , ) at each radius r through the tube. As is illustrated below this may yield the trivial solution "0 for all r, in which case there is no residual strain, or "0 for some range of values of r, in which case there is a residual value of for the remaining values of r, or O0 for all r, so that there is residual strain for all r.

The equation

= I A ( , ( ( ))"0 (63)
then yields the residual value of (which depends on r through = I

). Thus, either "0 or " , the latter depending on = I . We now illustrate the results for the model (35), which, when specialized for azimuthal shear, gives

= I ( , )" = I ( )#( ), (64)
and we have

( )"!= I ( ), (1)
In view of the properties (31) and (32) it follows that when "0 either "0 or "0. When "0, the "rst equation of (65) gives

= I ( )"!(0), (67)
which (locally) determines uniquely the residual value of . Since , and hence ( ( ), depends on r through = I the resulting unloading strain energy obtained from ( 64) is inhomogeneous because it depends explicitly on r as well as on .

To illustrate the theory further we now select a speci"c form of . For simplicity, we take to be quadratic in so that (65) and (38) are satis"ed. Thus,

( )"! ( !1)!= I ( !1), (68)
where is a positive (dimensionless) material constant whose interpretation will be discussed below.

The "rst equation of (65) then gives ( !1)"= I ( )

!= I . ( 69
)
For unloading, the pseudo-energy =( , ) becomes the strain energy = I ( , ( ( )),= I ( )#(= I ( )!= I )/ (70) as a function of , and this shows explicitly its dependence on = I and hence on r. For the neo-Hookean material Eq. (69) reduces to

" ( ( ),1#( ! )/ , (71)
and Eq. ( 66) becomes

[1#( ! )/ ] "b / r, ) . ( 72
)

Residual strains

When "0 we deduce from (69) that

= I ( )"= I ! , (73)
and from the properties of = I ( ) it follows that for there to be a solution of this equation (locally) for we must have

= I ' . ( 74
)
If the inequality (74) does not hold then there is no residual strain and the solution "0 applies when "0.

Henceforth, for illustration, we restrict attention to the neo-Hookean strain energy so that is given by (71) on unloading and by (72). Then, by setting "0 and " in (71), we obtain " ! .

This has a real solution for O0 only if ' locally. The resulting residual strain distribution can then be calculated and the corresponding stresses are PF "0, PP " FF . We may think of as a critical value of . If ) for all r then "0 is the only solution when "0, in which case there is no residual strain anywhere in the material, but there is a stress softening e!ect on unloading since (1. This will be discussed later. (77)

We consider separately three cases.

(i) (a)) and ) a/b, \ ; this de-"nes the notation \ . In this case there is no residual strain, "0 for all r, and g(r)"0.

(ii) (b)) ) (a). Then \ ) ) , >
, which de"nes the notation > . In this case let r (a)r )b) be the value of r such that (r )" . Then,

"b / r " r/r , (78)
and hence, ) for r )r)b, * for a)r)r .

(79) (iii) (b)* and * > . In this case there is a residual strain for all r.

In (ii) the solution for corresponding to "0 is

"0 for r )r)b, " "( ! for a)r)r , ( 80
)
and from (78) we see that is given in terms of r by " (r !r/r.

On the use of (45) the solution for the residual value of g(r), denoted g (r), is then found to be g (r)"c for r )r)b, (82)

g (r)"c# 2 cos\ r r ! 1 r (r !r for a)r)r , ( 83
)
where c is a constant. With the boundary conditions (43) it follows from (83) that c" 2

1 a (r !a!cos\ a r , ( 84
)
and this is precisely the residual value of . The local value of the damage function after unloading is given as follows. Where "0 we have " "1! / , and then, from (68), we obtain

( )" /4 . ( 87
)
On the other hand, where "0, we have

(0)" (2 ! )/4. ( 88
)
The total residual energy, denoted E , is calculated (per unit length in the direction normal to the considered plane) as

It is easy to check in each case that dE /d '0 for the relevant range of values of , bearing in mind that for case (ii) we have the connection (78) between r and . The expressions for E in Eqs. (92)}(94) may be compared with the energy for loading given by (56) with " .

Numerical results

Fig. 1 shows a plot of the function

f ( / )"[1#( ! )/ ] / , (95)
which corresponds to the left-hand side of Eq. (72) divided by the factor , for each of the cases

Fig. 1 .

 1 Fig. 1. Plot of the function f ( / ) given by (95) for / "0.7, 1, 1.2 (unloading) and the line f ( / )" / (loading).

For

Acknowledgements

This work was initiated during a visit to the Division of Mechanics at the National Technical University of Athens in March 1999. Discussions with Prof. K.A. Lazopoulos on the subject of pseudo-elasticity are gratefully acknowledged. The visit to Greece was supported by the Royal Society of London and the National Hellenic Research Foundation under the European Science Exchange Programme, and by the Carnegie Trust for the Universities of Scotland.

. Also shown is the straight line f( / )" / corresponding to loading. Note that the middle curve separates the region where there is no (non-zero) residual value of from that where the intercept of the curve for ' with the horizontal axis gives the residual value of identi"ed by (75).

For a representative value of the tube wall thickness (such that b/a"2) we show, in Fig. 2, a plot of the distribution through the wall of the ratio / for three di!erent values of the ratio / and in each case the result for loading is compared with that for unloading at the intermediate point at which " /2. The upper curves correspond to unloading so that it is clear that a larger deformation is associated with a given applied stress on unloading as compared with loading. Note that the nature of the results is governed by the ratio / and not by the speci"c value of . The three cases (a)}(c) in Fig. 2 correspond, respectively, to cases (i)}(iii) identi"ed in Section 3.3. It should be observed that the strain decreases monotonically as the radius increases for both loading and unloading. In Fig. 3 the stress-softening e!ect is illustrated for three di!erent values of / , the shear stress on r"b, PF for loading (the straight lines) and PF for unloading, each scaled by division by and plotted against / (again for b/a"2). It is clear that the stress softening on unloading increases signi"cantly with the extent of loading. It can be shown from ( 71) and (72) that increases monotonically with r for "xed so that the shear stress-softening e!ect shown in Fig. 3 increases as r decreases and is maximal at the inner boundary r"a, where the strain is greatest. Note that (a) and (b) in Fig. 3 correspond to case (ii) in Section 3.3 and (c) to (iii). An illustration of case (i) is not included since the stress softening on r"b is not signi"cant. Indeed, even for (a) the e!ect if quite small.


Introduction

In two recent papers [START_REF] Ogden | The eect of pre-stress on the propagation and re¯ection of plane waves in incompressible elastic solids[END_REF][START_REF] Ogden | Re¯ection of plane waves from the boundary of a pre-stressed compressible elastic half-space[END_REF], for incompressible and compressible isotropic elastic solids, respectively, the in¯uence of pre-stress and pure homogeneous strain on the re¯ection of homogeneous plane waves from the plane boundary of a half-space has been studied in detail. In the present paper, we extend the results in [START_REF] Ogden | The eect of pre-stress on the propagation and re¯ection of plane waves in incompressible elastic solids[END_REF] by considering the eect of a pure homogeneous prestrain on the re¯ection and transmission of homogeneous plane waves at the boundary between two half-spaces of incompressible isotropic elastic material. In general, we take the two halfspaces to consist of dierent materials and to be subject to dierent deformations with their principal axes aligned (one normal to the interface). For pre-stressed materials, this problem has not apparently been considered previously, either for incompressible or compressible materials, except in the paper by Dey and Addy [START_REF] Dey | Re¯ection and refraction of plane waves under initial stresses at an interface[END_REF], which, as pointed out by Norris [START_REF] Norris | Propagation of plane waves in a pre-stressed elastic medium[END_REF], contains fundamental errors. In the case of an incompressible material the analogous problem in which the two half-spaces consist of the same material but are subject to equal and opposite simple shears has been considered recently by Hussain and Ogden [START_REF] Hussain | Re¯ection and transmission of plane waves at a shear-twin interface[END_REF] (see also the related paper [START_REF] Hussain | On the re¯ection of plane waves at the boundary of an elastic half-space subject to simple shear[END_REF]). The problem of interfacial waves (i.e., Stoneley waves) has been considered by a number of authors in respect of pre-stressed incompressible materials (see, for example [7±9]) and interfacial waves and stability by Dowaikh and Ogden [START_REF] Dowaikh | Interfacial waves and deformations in pre-stressed elastic media[END_REF]. For discussion of re¯ection and transmission in the context of the anisotropic theory without pre-strain, the reader is referred to [START_REF] Auld | Acoustic Fields and Waves in Solids[END_REF].

The required equations and notations are summarized in Section 2. In Section 3 the propagation of plane harmonic waves is discussed with reference to the slowness curves for the two materials in respect of two distinct classes of strain-energy functions, exempli®ed by the neo-Hookean and Varga functions.

For the case in which the two half-spaces correspond to (dierent) neo-Hookean materials, the amplitudes of the re¯ected, transmitted and interfacial waves are calculated explicitly in Section 4.1. For each angle of incidence a single re¯ected wave, with angle of re¯ection equal to the angle of incidence, is generated when a homogeneous plane (SV) wave is incident on the boundary from one half-space, and it is accompanied by an interfacial wave (whose amplitude may vanish for certain discrete angles of incidence, which depend on the material parameters and the deformations in the two half-spaces). It is shown that under a certain restriction on the material parameters and deformations a transmitted (homogeneous plane SV) wave and an interfacial wave are generated for all angles of incidence. When this restriction does not hold there exists a critical angle (measured from parallel to the interface) above which there is again a transmitted and an interfacial wave but below which there are two interfacial waves (and no transmitted wave). The interfacial waves have the same speed but decay at dierent rates with distance from the interface.

When the half-spaces consist of (dierent) Varga materials corresponding results are given in Section 4.2. In this case, additional possibilities may arise for certain combinations of angle of incidence, material parameters and deformations. Speci®cally, two re¯ected (homogeneous plane SV) waves may be generated instead of one re¯ected and one interfacial wave and (not in general for the same angles of incidence, material parameters and deformations) two transmitted (homogeneous plane SV) waves may emerge. The ranges of values of the material parameters, deformations and angles of incidence for which the various possibilities arise are identi®ed in Section 4.2.

For each class of materials the conditions under which the incident wave is not refracted at the interface are determined and the angle of incidence for which this happens is obtained explicitly in terms of the deformations and material properties. The theory in Section 4 is illustrated in Section 5 using graphical results to show the dependence of the amplitudes of the waves on the angle of incidence for representative values of the material and deformation parameters.

Basic equations

We consider an incompressible isotropic elastic material subject to pure homogeneous strain. Let k 1 ; k 2 ; k 3 denote the principal stretches of the deformation. Then, the incompressibility condition is expressed as

k 1 k 2 k 3 1: 2:1
Since the material is isotropic, the principal Cauchy stresses are given by r i k i oW ok i À p; i P f1; 2; 3g; 2:2

where W , the strain-energy function per unit volume, is symmetric in k 1 ; k 2 ; k 3 ; and p is a Lagrange multiplier.

We consider incremental motions in the x 1 ; x 2 -plane superimposed on the pure homogeneous deformation such that the displacement vector v has components v 1 x 1 ; x 2 ; t; v 2 x 1 ; x 2 ; t; v 3 0:

The incremental incompressibility condition div v 0 is then used to deduce that v 1 w ;2 ; v 2 Àw ;1 ; 2:3 where wx 1 ; x 2 ; t is a scalar function and ;i denotes o=ox i ; i P f1; 2g:

The incremental nominal stress tensor referred to the homogeneously deformed con®guration is denoted by R; and its components are given by

R ji A 0jilk v k;l pv j;i À pd ij ; 2:4
where p is the increment in p and A 0jilk are the components of the fourth-order tensor A 0 of instantaneous elastic moduli. The components of A 0 in terms of the derivatives of the strainenergy function W are given by

A 0iijj k i k j W ij ; 2:5 A 0ijij k i W i À k j W j k 2 i k 2 i À k 2 j ; i T j; k i T k j ; 2:6 A 0ijij 1 2 A 0iiii À A 0iijj k i W i ; i T j; k i k j ; 2:7 A 0ijji A 0jiij A 0ijij À k i W i ; i T j; 2:8
where W i oW =ok i ; W ij o 2 W =ok i ok j and there is no summation over the repeated indices. The components A 0jilk of A 0 are constants since the deformation is homogeneous. For details see, for example, [START_REF] Ogden | Non-linear Elastic Deformations[END_REF]. The equation of motion is given by A 0jilk v k;jl À p ;i q v i ; i P f1; 2g; 2:9

where q is the material density, a superposed dot indicates the material time derivative and summation is over repeated indices in f1; 2g. Since the motion is restricted to the x 1 ; x 2 -plane we have from (2.9) explicitly in the latter of which the incremental hydrostatic pressure p has been eliminated by dierentiating R 22 with respect to x 1 and then using (2.10).

A 01111 À A

Plane waves

We consider time harmonic plane waves of the form w A exp ikx 1 cos h x 2 sin h À ct; 3:1 where A; c and k are constants, c being the wave speed and k the wave number, both being taken to be positive. Also, h is the angle that the direction of propagation cos h; sin h of the wave in the x 1 ; x 2 -plane makes with the x 1 direction.

Substitution of (3.1) into (2.12) yields a cos 4 h 2b sin 2 h cos 2 h c sin 4 h qc 2 : 3:2

Eq. (3.2) is the propagation condition, which identi®es the wave speed associated with any given angle h.

For qc 2 > 0, it is necessary and sucient that a; b and c satisfy the strong ellipticity inequalities a > 0; c > 0; b > À ac p :

3:3

For an inhomogeneous plane wave of the form

w A expik H x 1 À imx 2 À c H t 3:4
Eq. (2.12) gives a À 2bm 2 cm 4 q1 À m 2 c H 2 3:5 in place of (3.2). For a given wave speed c H , Eq. (3.5) determines m. The wave decays exponentially as x 2 3 ÀII provided m has positive (negative) real part.

We now consider two half-spaces of dierent incompressible isotropic elastic materials. The half-spaces are subjected to pure homogeneous strain and then bonded along their common (plane) boundary in such a way that the principal directions of strain are aligned, one direction being normal to the interface. In rectangular Cartesian coordinates we take the interface to be x 2 0.

Let k 1 ; k 2 ; k 3 be the stretches associated with the half-space x 2 < 0, with strain-energy function W, and material constants a; b; c de®ned by (2.13) with (2.5)±(2.8). Similarly, let k à 1 ; k à 2 ; k à 3 ; W à ; a à ; b à ; c à be the corresponding quantities for the half-space x 2 > 0. For simplicity, we take the deformation to correspond to plane strain with k 3 k à 3 1 so that, with reference to the incompressibility condition (2.1), we introduce the notations k; k à such that

k 1 k À1 2 k; k à 1 k ÃÀ1 2 k à : 3:6
Two distinct strain-energy functions are now examined since these exemplify the range of possible behaviour encountered. For these either 2b a c or 2b T a c in x 2 < 0 and similarly for x 2 > 0.

Case

A: 2b a c; 2b à a à c Ã
In this case (3.2) and (3.5) reduce, respectively, to a cos 2 h c sin 2 h qc 2 3:7

and 1 À m 2 a À À cm 2 À qc H 2 Á 0: 3:8
In terms of the slowness vector s 1 ; s 2 , de®ned by s 1 ; s 2 cos h; sin h=c; 3:9

Eq. (3.7) becomes the slowness curve k 4 s 2 1 s 2 2 q; x 2 < 0; 3:10 in s 1 ; s 2 -space, where q is de®ned by q q=c; 3:11 and a=c k 4 follows from (2.6) and (2.13). By using the dimensionless notation s 1 ; s 2 de®ned by

s 1 ; s 2 s 1 ; s 2 = q p ; 3:12
we can write (3.10) as

k 4 s 2 1 s 2 2 1; x 2 < 0: 3:13
Let h à ; c à ; q à be the counterparts of h; c; q for x 2 > 0 and let q à be de®ned by q à q à =c à 3:14 analogously to (3.11). We use the same non-dimensionalization (3.12) for the slowness vector

s à 1 ; s à 2 cos h à ; sin h à =c à 3:15 and de®ne s à 1 ; s à 2 s à 1 ; s à 2 = q p : 3:16
The counterpart of (3.13) in x 2 > 0 is then written

k Ã4 s Ã2 1 s Ã2 2 D; x 2 > 0; 3:17
where D is de®ned by

D q à =q q à c=qc à : 3:18
Under the considered plane strain conditions the specialization 2b a c forces the strain-energy function to have the neo-Hookean form

W 1 2 lk 2 1 k 2 2 k 2 3 À 3 3:19
(with k 3 1) in x 2 < 0, where l> 0 is the shear modulus, and similarly for x 2 > 0 with shear modulus l à .

In this case the value of D can be obtained from (2.6), (2.13) and (3.19) and is given by D dfk à =kg 2 ; 3:20 where d lq à =ql à : 3:21

Slowness curves for x 2 < 0 and x 2 > 0 are shown superimposed in Fig. 1 for illustrative values of d; k; k à .

Case

B: 2b T a c; 2b à T a à c Ã
In this case we take the strain-energy functions to satisfy b ac p , which was used by Ogden and Sotiropoulos [START_REF] Ogden | The eect of pre-stress on the propagation and re¯ection of plane waves in incompressible elastic solids[END_REF] and in [START_REF] Hussain | Re¯ection and transmission of plane waves at a shear-twin interface[END_REF][START_REF] Hussain | On the re¯ection of plane waves at the boundary of an elastic half-space subject to simple shear[END_REF], and similarly b

à a à c à p . Then (3.2) takes the form a p cos 2 h È c p sin 2 h É 2 qc 2 3:22 and (3.5) becomes a p À À c p m 2 Á 2 q 1 À À m 2 Á c H 2 : 3:23
The slowness curve corresponding to (3.22) is given by

k 2 s 2 1 È s 2 2 É 2 s 2 1 s 2 2 ; x 2 < 0 3:24
in dimensionless form with the notation (3.12) and q de®ned by (3.11). Similarly, for x 2 > 0 the corresponding equation of the slowness curve is fd; k; k à g f10; 2; 1:2g, inner (outer) ellipse for x 2 < 0> 0, (c) fd; k; k à g f3; 1:2; 3g, short (tall) ellipse for

x 2 < 0> 0. k Ã2 s Ã2 1 È s Ã2 2 É 2 s Ã2 1 È s Ã2 2 É D; 3:25
where D is again de®ned by (3.18).

As in Case A we illustrate the slowness curves for particular values of k and k à in the two halfspaces in Fig. 2. For this we use the Varga strain-energy function, which is given by

W 2lk 1 k 2 k 3 À 3 3:26
(with k 3 1 here) for x 2 < 0, where l is the shear modulus. Similarly for x 2 > 0 with shear modulus l à . For this material the value of D is obtained from (3.26) and (2.6) as

D dk à k Ã2

Re¯ection and transmission at the interface

For continuity of traction at the interface associated with the underlying deformation we require r à 2 r 2 . The boundary conditions corresponding to continuous displacement are

v 1 v à 1 ; v 2 v à 2 on x 2 0, where v 1 ; v 2 are the displacement components in x 2 < 0 and v à 1 ; v à 2 are those in x 2 > 0. From (2.
3) these boundary conditions can be written in terms of the scalar functions w and w à as We now consider a wave incident on the boundary x 2 0 from the region x 2 < 0 with direction of propagation cos h; sin h in the x 1 ; x 2 -plane and speed c. Because of the symmetry with respect to the normal direction to the interface we henceforth, without loss of generality, restrict attention to values of h in the interval 0; p=2. We write the solution comprising the incident wave, a re-¯ected wave (with angle of re¯ection equal to the angle of incidence) and an interfacial wave in x 2 < 0 as

w A exp ikx 1 cos h x 2 sin h À ct AR exp ikx 1 cos h Àx 2 sin h À ct AR H exp ik H x 1  À imx 2 À c H t à ; 4:5
where R is the re¯ection coecient and R H measures the amplitude of the interfacial wave. The notations k H ; m; c H are as used in (3.4) and m has positive real part.

In the half-space x 2 > 0 we write the solution comprising a transmitted and an interfacial wave in the form Eq. (4.7) states, in particular, that the ®rst components of the slowness vectors for each homogeneous plane wave interacting at the boundary x 2 0 are equal, in this case s 1 s à 1 .

w à AR à expik à x 1 cos h à x 2 sin h à À c à t AR ÃH expik ÃH x 1 im à x 2 À c
Thus, by reference to the slowness curves for x 2 < 0 and x 2 > 0, superimposed as exempli®ed in Figs. 1 and2, the range of angles of incidence for which a transmitted wave exists can be iden-ti®ed. In Fig. 1(b), for example, since the inner curve corresponds to x 2 < 0 there is, for every angle of incidence (i.e. for every s 1 associated with the curve) a point on the outer curve for which s à 1 s 1 , and hence a transmitted wave. In Fig. 1(a), on the other hand, there are values of s 1 on the slowness curve for x 2 < 0 for which there are no corresponding values of s à 1 on the x 2 > 0 slowness curve, and therefore a range of angles of incidence for which no transmitted wave exists. This will be discussed further in Section 4.1. The situation in respect of Fig. 2 is less straightforward and will be examined in Section 4.2.

We now consider again Cases A and B separately. We do not examine here the mixed cases in which 2b a c; 2b à T a à c à or 2b T a c; 2b à a à c à since these do not introduce features that are qualitatively distinct from those arising in Case B.

Case

A: 2b a c; 2b à a à c Ã
In this case we see from Eq. (3.8) that m AE1, which yields an interfacial wave in the half-space x 2 < 0 for m 1. The zeros of the other quadratic factor in (3.8) correspond to m i tan h and m Ài tan h, which are associated, respectively, with the incident and re¯ected waves in x 2 < 0. Thus, in x 2 < 0 the solution (4.5) applies with m 1.

In x 2 > 0 the solution consisting of a transmitted wave and an interfacial wave may be written as (4.6) subject to (4.7) and with m à 1. The latter arises from the ®rst factor in the analogue of Eq. (3.8) for x 2 > 0. The second factor gives

m Ã2 a à À q à c ÃH 2 =c à : 4:8
Using the fact that a à =c à k Ã4 together with (4.7) and de®nitions (3.11), (3.14) and (3.18), Eq. (4.8) may be written as

m Ã2 k Ã4 À D k 4 À tan 2 h Á : 4:9
If this corresponds to a transmitted wave then we must have

m à Ài tan h à ; 4:10
where h à is a real angle given by

tan 2 h à D k 4 À tan 2 h Á À k Ã4 : 4:11 If Dk 4 > k Ã4 , i.e., from (3.20), dk 2 > k Ã2
, then the right-hand side of (4.11) is positive for all h and a transmitted wave exists for all angles of incidence (including h 0, which corresponds to grazing incidence). If, on the other hand, Dk 4 < k Ã4 , then there is a critical angle, h h c say, such that

tan 2 h c k Ã4 =D À k 4 : 4:12
In this case there is a transmitted wave for angles of incidence such that h c < h 6 p=2, but for 0 6 h < h c there is no transmitted wave and there are two interfacial waves, one with m à 1, and the other with m à given as the positive solution of (4.9). These interfacial waves have the same speed c H according to (4.7) but in general the value of m à given by (4.9) is not equal to unity. In fact, it can be unity (for some h such that 0 6 h < h c ) only if k Ã4 À Dk 4 P 1. At the critical angle h h c we have h à 0 and the ®rst term in (4.6) corresponds to a plane shear (body) wave propagating parallel to the boundary in x 2 > 0 (grazing transmission). At this point it is convenient to introduce the notations de®ned by t tan h; t à tan h à ; t c tan h c ; 4:13 noting that h à is a real angle when the right-hand side of (4.11) is non-negative and that t; t à and t c are all non-negative. We now examine conditions under which there is no refraction, i.e. a transmitted wave has the same direction of propagation as the incident wave (h à h). For this to be the case we must have

t à t,
1 À t 2 1 R 2R H d1 À t Ã2 R à 2R ÃH ; 4:20 2t1 À R À i1 À t 2 R H d2t à R à i1 À t Ã2
F t 1 3it t 2 À it 3 d2t À it à À i À it t à t it à i d 2 1 3it à t Ã2 À it Ã3 ; 4:24 G H 21 À t 2 À id3 À t 2 t à À i d 2 1 3it à t Ã2 À it Ã3 ; 4:25 G à À1 t 2 d1 t Ã2 ; G ÃH 1 À t 2 2it à dt à À i 2 ;
4:26 and F Àt is obtained from F t by replacing t by Àt without changing t à . In these equations t à is given by (4.11) and is replaced by im à with m à > 0 when the right-hand side of (4.11) is negative. In Section 5.1 some graphical results for the absolute values of R; R H ; R à and R ÃH are given for illustration. Note that the results for k k à 1, when neither half-space is subject to pre-strain, are included here as a special case, and are illustrated in Figs. 3 and4 in Section 5.1.

Case B: b ac

p ; b à a à c à p
In this case, from (3.23), after using a=c k 4 and Snell's law cos h=c 1=c H , we have

1 t 2 k 2 À m 2 2 1 À m 2 k 2 t 2 2 ;
which can be reorganized as

m 2 t 2 m 2 1 t 2 À t 2 k 2 k 2 À 2 0: 4:27
Note that m it and m Àit are the solutions of (4.27) corresponding to the incident and re¯ected waves respectively. The other solutions are

m AE 1 À k 2 À 1 2 =1 t 2 q : 4:28 If k 6 2 p
then m is real for all h and the positive solution of (4.28) corresponds to an interfacial wave in

x 2 < 0. If k > 2 p
then there is a critical value of h; h c say, for which m 0 and this is given by t 2 c k 2 k 2 À 2;

4:29 where the notation t c tan h c is used. It follows that m is real for h c 6 h 6 p=2. For h c < h 6 p=2 there is a re¯ected wave accompanied by an interfacial wave and for h h c the interfacial wave becomes a plane shear (body) wave propagating parallel to the boundary in x 2 < 0 (grazing re-¯ection). When 0 < h < h c the interfacial wave is replaced by a second re¯ected wave with angle of re¯ection, h H say, obtained from (4.28) by replacing m by Ài tan h H to give

t H 2 k 2 k 2 È À 2 À t 2 É 1 t 2 ; 4:30
where t H tan h H . The speed c H of the second re¯ected wave is obtained from Snell's law in the form c H c cos h H = cos h together with (4.30). Note that Eq. ( 4.30) may be rewritten as cos 2 h cos 2 h H 1=k 2 À 1 2 ; 4:31 which is the same as Eq. ( 5.5) given in [START_REF] Ogden | The eect of pre-stress on the propagation and re¯ection of plane waves in incompressible elastic solids[END_REF] for the Varga material and does not depend on the presence of the half-space x 2 > 0 (it requires only the use of Snell's law and the propagation condition). The two re¯ected waves coincide when the angle of incidence, h 0 say, is given by t 2 0 k 2 À 2; 4:32 where t 0 tan h 0 . Clearly, this gives a non-trivial real angle only if k > 2 p . For the half-space x 2 > 0, we obtain an equation similar to (4.27), namely

m Ã2 t Ã2 m Ã2 1 Â t Ã2 À t Ã2 k Ã2 k Ã2 À 2 Ã 0: 4:33
The solution m à it à corresponds to a transmitted wave provided t à is real and positive. The other relevant solution is

m à AE 1 À k Ã2 À 1 2 =1 t Ã2 q 4:34
with the plus sign when m à is real. The nature of m à in (4.34) depends on that of t à , which is obtained by using the propagation condition (3.22) and its counterpart for x 2 > 0 together with Snell's law (4.7). This gives a quadratic for t Ã2 , which we write as

t Ã4 À D Ã À À 2k Ã2 Á t Ã2 k Ã4 À D Ã 0; 4:35
where D Ã is de®ned by

D Ã Dk 2 t 2 2 =1 t 2 k Ã2 t Ã2 2 =1 t Ã2 4:36
and D is given by (3.27). Eq. (4.35) may have real, pure imaginary or complex solutions for t à depending on the values of k; k à and D and on the angle of incidence. We delimit the possibilities as follows.

(a) There are two positive solutions for t Ã2 and hence two transmitted waves if

k Ã2 > 2; k Ã4 > D Ã P 4 k Ã2 À À 1 Á : 4:37
When the equality holds on the right-hand side the two solutions (and hence the transmitted waves) coincide. The direction of propagation, h à 0 say, is then de®ned by

t Ã2 0 k Ã2 À 2 4:38
similarly to (4.32), where t à 0 tan h à 0 . If k Ã4 D à > 4k Ã2 À 1 then one of the transmitted waves becomes a plane shear (body) wave propagating parallel to the boundary in x 2 > 0. One solution of (4.35) is then t à 0, while the other de®nes a critical angle h à c in x 2 > 0 with t à c tan h à c given by

t Ã2 c k Ã2 k Ã2 À À 2 Á 4:39
analogously to (4.29). (b) There is one positive and one negative solution for t Ã2 , and hence one transmitted and one interfacial wave, if

D Ã > k Ã4 ; 4:40
where k Ã2 may be greater than, equal to or less than 2. If D Ã k Ã4 < 2k Ã2 the transmitted wave becomes a plane shear (body) wave as in (a). (c) There are two negative solutions for t Ã2 , and hence two interfacial waves, if

k Ã2 < 2; k Ã4 > D Ã P 4k Ã2 À 1 4:41
with the solutions equal when equality holds on the right-hand side. In this case the interfacial waves propagate parallel to the boundary. (d) There are two complex conjugate solutions for t Ã2 , and hence two interfacial waves, if

D Ã < 4k Ã2 À 1 6 k Ã4 : 4:42
As for (b) this can hold for k Ã2 greater than, equal to or less than 2. In this case the interfacial waves propagate away from instead of parallel to the boundary. Note that waves of this type are not generated in x 2 < 0 for any values of the parameters.

In connection with determination of which of cases (a)±(d) holds it should be observed that if k 2 6 2 then D Ã , regarded as a function of t P 0 is monotonically increasing with minimum Dk 4 at t 0. On the other hand, if k 2 > 2 then, as t increases from zero, D Ã decreases from Dk 4 to its minimum 4Dk 2 À 1 where t t 0 and thereafter increases monotonically, reaching the value Dk 4 again when t t c .

The equation D Ã k Ã4 identi®es values of t corresponding to transition between regimes (a) and (b) or (b) and (c). If k 2 6 2 and k Ã4 < Dk 4 then this equation is not satis®ed for any real t, while if k Ã4 > Dk 4 there is one positive value of t, which becomes zero if the latter inequality is replaced by equality. If k 2 > 2 and k Ã4 > Dk 4 there is again one positive value of t; t 1 say, satisfying t 1 > t c , while if Dk 4 > k Ã4 P 4Dk 2 À 1 there are two positive values of t; t 1 and t 2 say, distributed according to t 1 < t 0 < t 2 < t c .

Transition between regimes (a) and (d) or (c) and (d) is identi®ed by the equation D Ã 4k Ã2 À 1. This yields one positive value of t if 4k Ã2 À 1 > Dk 4 (greater than t c ) and two if Dk 4 > 4k Ã2 À 1 > 4Dk 2 À 1, with a distribution similar to that noted in the above paragraph in respect of D Ã k Ã4 .

As h increases from 0 through to p=2, ten possible distributions of ranges of values of h associated with the regimes (a)±(d) can occur depending on the various inequalities restricting the parameters k; k à and D. We do not catalogue all of these here but illustrate some of the possibilities. Firstly, for example, if 4Dk 2 À 1 > k Ã4 ; k 2 > 2 then (b) applies for all angles of incidence. Secondly, if k Ã4 > Dk 4 ; Dk 2 À 1 > k Ã2 À 1; k 2 > 2; k Ã2 > 2 then there is a transition from a sector in which (a) holds to one in which (b) holds as h increases. Thirdly, if As in Case A, the solutions w and w à may be written in the forms (4.5) and (4.6) respectively, with m in (4.5) given by the positive solution of (4.28) when real and replaced by Ài tan h H when imaginary. In (4.6), tan h à and t à im à are the relevant solutions of (4.35) appropriate to which of the possibilities (a)±(d) applies.

2k Ã2 P k Ã4 > Dk 4 > 4k Ã2 À 1 > 4Dk 2 À 1; k 2 > 2 the transition is from (c) to (d) to (c) to (b). Finally, for Dk 4 > k Ã4 ; k Ã2 À 1 > Dk 2 À 1; k 2 > 2; k Ã2 > 2
The angle of incidence for which there is no refraction may be found as in Case A, this time by substituting t à t in Eq. (4.35) and making use of the de®nition (4.36). The resulting equation has two solutions for t 2 , namely

t 2 D p k 2 À k Ã2 1 À D p ; t 2 À D p k 2 k Ã2 D p 1 : 4:43
The second of these is negative and therefore does not yield a transmitted wave. The ®rst is positive under exactly the same conditions as its counterpart (

t1 À R À imR H t à R à im à R ÃH ; 4:47 1 À t 2 1 R 1 m 2 R H d1 À t Ã2 R à d1 m Ã2 R ÃH ; 4:48 t1 m 2 1 À R À im1 À t 2 R H t à d1 m Ã2 R à im à d1 À t Ã2
F t 1 2imt m 2 t 2 À imt 3 im 3 t d2mt À im à t à À i À imt m à t à t imt à im à d 2 1 2im à t à m Ã2 t Ã2 À im à t Ã3 im Ã3 t à ; 4:52 G H 1 m 2 À idm à t à À i1 À t 2 À idm à t à À i À im à t à d 2 t à im à 2 ; 4:53 G à m1 À t 2 À m à 1 m 2 dm à 1 À t Ã2 À m1 m Ã2 ; 4:54 G ÃH m1 À t 2 it à 1 m 2 À dit à 1 m Ã2 m1 À

Numerical results

In Sections 5.1 and 5.2 some graphical results are given for Cases A and B, respectively. For discussion of the stability of the pair of half-spaces we refer to Dowaikh and Ogden [START_REF] Dowaikh | Interfacial waves and deformations in pre-stressed elastic media[END_REF]. 1(a), which is the relevant one for Fig. 3, it can be seen that there is no transmitted wave for 0 6 h 6 h c , where h c is the critical value of h given by (4.12), in this case approximately 0:685. From Fig. 3(a) we observe that jRj 1 in this range, while Fig. 3(b) shows that jR H j vanishes at h approximately equal to 0:553. In the case of Fig. 4 there is a transmitted wave for all angles of incidence.

Figs. 5 and6 show the results for when there is pre-strain. With reference to the slowness curve in Fig. 1(b), we see that there is a transmitted wave for every angle of incidence, which is relevant for Fig. 5. In respect of Fig. 6, from Section 4 and from the discussion of the slowness curves in In each of Figs. 3±6 there are no non-trivial results for grazing incidence since jRj 1 and R H R Ã R ÃH 0 when h 0. In each of Figs. 3 and6 there is a critical angle associated with grazing transmission, approximately 0:685 and 0:983, respectively, where one of the interfacial waves changes to a transmitted wave (in x 2 > 0). In x 2 < 0 there is no such transition because there is only one re¯ected wave and one interfacial wave for each angle of incidence.

Case B: b ac

p ; b à a à c à p
For this case graphical results are given in Figs. 7±9. In Fig. 7, with reference to the slowness curves in Fig. 2(a), there is one re¯ected wave accompanied by an interfacial wave in x 2 < 0 and one transmitted wave accompanied by an interfacial wave in x 2 > 0 for each angle of incidence (i.e. (b) applies). As in Case A, there are no non-trivial results for grazing incidence.

In Fig. 8, with reference to the slowness curves in Fig. 2(b), there are three separate regions, namely 0 6 h 6 0:743; 0:743 6 h 6 1:06 and 1:06 6 h 6 p=2 corresponding to (d) to (a) to (b) transition as h increases from zero. In x 2 < 0 there is one re¯ected wave accompanied by an interfacial wave for each value of h since k < 2 p , as discussed in Section 4.2. For 0 6 h 6 0:743, we have two interfacial waves in x 2 > 0, for 0:743 6 h 6 1:06 we have two transmitted waves in x 2 > 0 and ®nally, for 1:06 6 h 6 p=2, one transmitted wave is accompanied by an interfacial wave. In Fig. 9, with reference to the slowness curves in Fig. 2(c) for x 2 > 0, we have one transmitted wave with an interfacial wave in 0 6 h 6 p=2 but for x 2 < 0 we have two intervals, namely 0 6 h 6 h c with two re¯ected waves and h c 6 h 6 p=2 with one re¯ected wave accompanied by an interfacial wave, where h c , approximately 1:38 here, is the critical value identi®ed in Eq. (4.29). Note that for h h 0 , approximately 1:12, the two re¯ected waves coincide and then jR H j 1 and R R Ã R ÃH 0, as can be seen from the ®gure. The vanishing of R; R Ã and R ÃH is associated with vanishing of the common factor t im in (4.50) and (4.51) or, equivalently, t H t.

Fig. 1 .

 1 Fig.1. Slowness curves for 2b a c; 2b à a à c à : (a) fd; k; k à g f0:6; 1; 1g, inner (outer) circle for x 2 > 0< 0, (b) fd; k; k à g f10; 2; 1:2g, inner (outer) ellipse for x 2 < 0> 0, (c) fd; k; k à g f3; 1:2; 3g, short (tall) ellipse for x 2 < 0> 0.

Fig. 2 .

 2 Fig. 2. Slowness curves for b ac p ; b à a à c à p : (a) fd; k; k à g f6; 1:4; 2g, inner (outer) curve for x 2 < 0> 0, (b) fd; k; k à g f1:1; 0:7; 3:5g, short (tall) curve for x 2 < 0> 0, (c) fd; k; k à g f3; 2:5; 1:2g, tall (broad) curve for x 2 < 0> 0.

  there are ®ve sectors with transition (b) to (a) to (d) to (a) to (b).

5. 1 .

 1 Case A: 2b a c; 2b à a à c à Graphical results for a selection of values of d; k; k à and l=l à are given in Figs. 3±6. Figs. 3 and 4 are for the case with no pre-strain with d < 1 and d > 1, respectively. With reference to the slowness curve shown in Fig.

Fig. 3 .

 3 Fig. 3. Plots of jRj; jR H j; jR à j and jR ÃH j in (a), (b), (c) and (d), respectively, against h0 6 h 6 p=2 for 2b a c; 2b à a à c à with fd; k; k à g f0:6; 1; 1g and l=l à 1:5.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Plots of jRj; jR H j; jR à j and jR ÃH j in (a), (b), (c) and (d), respectively, against h0 6 h 6 p=2 for 2b a c; 2b à a à c à with fd; k; k à g f3; 1; 1g and l=l à 2.

Fig. 1 (

 1 Fig.1(c) it is apparent that there are two interfacial waves in x 2 > 0 for 0 6 h 6 h c , where here h c is approximately 0:983. A transmitted wave is accompanied by an interfacial wave for h c 6 h 6 p=2. Fig. 6(a) shows that jRj 1 for 0 6 h 6 h c , while from Figs. 6(b) and (c) we see that each of jR H j and jR Ã j vanishes at one value of h. From (4.22) and (4.23) with (4.25) and (4.26) this value is calculated as 0:75 for jR H j and 0:763 for jR Ã j.In each of Figs. 3±6 there are no non-trivial results for grazing incidence since jRj 1 and R H R Ã R ÃH 0 when h 0. In each of Figs.3 and 6there is a critical angle associated with grazing transmission, approximately 0:685 and 0:983, respectively, where one of the interfacial waves changes to a transmitted wave (in x 2 > 0). In x 2 < 0 there is no such transition because there is only one re¯ected wave and one interfacial wave for each angle of incidence.

Fig. 6 .Fig. 8 .

 68 Fig.6. Plots of jRj; jR H j; jR à j and jR ÃH j in (a), (b), (c) and (d), respectively, against h0 6 h 6 p=2 for 2b a c; 2b à a à c à with fd; k; k à g f3; 1:2; 3g and l=l à 2.

Fig. 7 .

 7 Fig. 7. Plots of jRj; jR H j; jR à j and jR ÃH j in (a), (b), (c) and (d), respectively, against h0 6 h 6 p=2 for b ac p ; b à a à c à p with fd; k; k à g f6; 1:4; 2g and l=l à 3.

  01122 pt 1;11 À p ;1 A 02121 t 1;22 A 02121 À r 2 t 2;12 q t 1 ;

				2:10
	A 02222 À A 02211 pt 2;22 À p ;2 A 01212 t 2;11 A 02121 À r 2 t 1;12 q t 2 :	2:11
	By eliminating p from (2.10) and (2.11) and using (2.3) we obtain an equation for w; namely
	aw ;1111 2bw ;1122 cw ;2222 q w ;11	w ;22 ;	2:12
	where the material constants a; b; c are de®ned by
	a A 01212 ; c A 02121 ; 2b A 01111 A 02222 À 2A 01122 À 2A 01221 ;	2:13
	as, for example, in [13].
	From (2.4), on use of (2.3), the shear and normal components of the incremental nominal
	traction R 21 ; R 22 on a plane x 2 constant are expressible in terms of w through
	R 21 cw ;22 À c À r 2 w ;11 ;			2:14
	ÀR 22;1 2b c À r 2 w ;112 cw ;222 À q	w ;2 ;	2:15

  where w à is the counterpart of w for x 2 > 0.The boundary conditions for continuous incremental traction on the interface are where R 21 ; R 22 are the traction components in x 2 < 0 and R à 21 ; R à 22 are those in x 2 > 0. From (2.14), (2.15) and the continuity condition r à 2 r 2 , the boundary condition (4.2) takes the forms in terms of w and w à , where, in order to obtain (4.4), the second equation in (4.2) has been replaced by R 22;1 R à 22;1 and use made of (2.15) and its counterpart for x 2 > 0. Notice that (4.3) and (4.4) are independent of the pre-stress r 2 .

	w ;1 w à ;1 ; w ;2 w à ;2	on x 2 0;		4:1
	R 21 R Ã 21 ; R 22 R Ã 22	on x 2 0;		4:2
	cw ;22 À w ;11 c à w à ;22 À w à ;11 ;		4:3
	2b cw ;112 cw ;222 À q w ;2 2b à c à w à ;112 c à w à ;222 À q Ã	w à ;2 ;	4:4

  à is the transmission coecient and R ÃH is the analogue of R H for x 2 > 0. The transmitted wave has direction of propagation cos h à ; sin h à , wave number k à and speed c à , while k ÃH ; m à ; c ÃH are the counterparts of k H ; m; c H . Note that the interfacial wave decays as x 2 3 I provided m à has positive real part.According to Snell's law we have k cos h k H k à cos h à k ÃH or, equivalently, cos h=c 1=c H cos h à =c à 1=c ÃH :

	4:7

ÃH t; 4:6 where R

  and (4.11) then gives t 2 Dk 4 t 2 À k

	k 4 2w ;112 w ;222 À q	w ;2 d k Ã4 h	2w à ;112 w à ;222 À q Ã	w à ;2	i	;	4:18
	where d is de®ned by
	d c à =c l à k 2 =lk Ã2 :						4:19

Ã4

: 4:14

If D 1 this can only be satis®ed if k à k and it is then satis®ed for all angles of incidence. In this case it follows from (3.20) and (3.21) that d 1 and the shear wave speeds in the two materials are equal. If D T 1, then Eq. (4.14) may be written as

t 2 Dk 4 À À k Ã4 Á 1 À D: 4:15

Then, if D < 1 this yields a real angle of incidence provided Dk 4 > k Ã4 (and hence k à < k) and (4.11) has a real solution for the transmitted angle for all angles of incidence. If, on the other hand, D > 1, then (4.15) yields a real angle provided Dk 4 < k Ã4 (and hence k à > k). In this case there is a critical angle, given by (4.12), the solution of (4.15) being in the required range of values for the existence of a transmitted wave to be assured (i.e. t > t c ). Using (4.5) and (4.6) appropriately specialized in the boundary conditions (4.1), we obtain

1 R R H R à R ÃH ; 4:16 t1 À R À iR H t à R

à iR ÃH : 4:17 When (4.11) is replaced by (4.9), t à is replaced by im à in (4.15) with m à real and positive.With the specialization 2b a c; 2b à a à c à , the boundary condition (4.4) becomes Again with appropriate specialization, substitution of w and w à from (4.5) and (4.6) into (4.3) and (4.16) leads to

  R ÃH ; 4:21 with t à replaced by im à where necessary. Note that when d 0 Eqs. (4.20) and (4.21) reduce to equations given in[START_REF] Ogden | The eect of pre-stress on the propagation and re¯ection of plane waves in incompressible elastic solids[END_REF] for a single half-space with r 2 0.

	The solution of Eqs. (4.16), (4.17), (4.20) and (4.21) may be written
	R	t iF Àt t À iF t	;	R H À	2tG H t À iF t	;	4:22
	R Ã	2itt iG à t à À iF t	;	R ÃH À	2itt iG ÃH t à À iF t	;	4:23
	where F t; G

H ; G Ã ; G ÃH are de®ned by

  4.15) in Case A. The coecients R; R H ; R Ã and R ÃH are determined by using the boundary conditions(4.

	d c à =c	l à kk 2 1 lk à k Ã2 1	:				4:45
	By substituting the values of w and w à from (4.5) and (4.6) in (4.1), (4.3) and (4.44) we obtain
	1 R R H R Ã R ÃH ;					4:46
							1), (4.3)
	and (4.4), with (4.4) taking the form
	2k 2 1w ;112 w ;222 À q	w ;2 d 2k à 2 n	1w à ;112 w à ;222 À q Ã	w à ;2	o	4:44
	in this case, where

  . (4.50)±(4.55) reduce to those in (4.22)±(4.24), (4.254) when both m and m à are set equal to 1. In Section 5.2 graphical results for the absolute values of R; R H ; R à and R ÃH are given for illustration.

	t Ã2 :	4:55

In these equations, for given t; m is obtained from Eq. (4.28) so as to have positive real part, t à from (4.35) and then m à from (4.34). Note that the same notation has been used as in (4.24)±(4.26) and that Eqs

Acknowledgements

The work of W. Hussain was supported by a UK government ORS award and by the University of Glasgow.