Reflection of plane waves from the boundary of an initially stressed incompressible half-space

Abstract

In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on plane waves in an incompressible material. For this problem, the initial stress is not associated with a finite elastic deformation and the material is assumed to be isotropic in the absence of the initial stress. The theory of superposition of infinitesimal deformations on finite deformation is applied to a problem of plane incremental motions in an initially stressed incompressible homogeneous elastic half-space. The general formulation of the problem is presented first and then specialized using a prototype strain energy function. Homogeneous plane waves are considered and the analysis is carried out for incompressible materials in both the deformed and the undeformed reference configurations. In addition to this, problems for the reflection of small amplitude homogeneous waves from the plane boundary of an initially stressed half-space are also considered and graphical results are included, which show the effect of initial stress on reflection. It is noted that the reflection coefficients in this case behave in a similar fashion when the initial stress is a pre-stress.

Keywords

Nonlinear elasticity initial stress plane waves wave propagation reflection reflection coefficients

1. Introduction

The effect of initial stress on the propagation of waves in elastic materials was initially studied by Biot [1, 2]. Biot included initial stress terms in the formulation of the problem as additional terms; vanishing of these terms reduced the results to those presented in the classical linear theory. The theory developed by Biot was later used by Dey and Addy [3, 4] to study the effect of initial stress on plane waves. Later, the same theory was applied for various studies on the reflection and transmission of P and SV waves in isotropic materials [5 –8] and for anisotropic materials [9]. More recently, various problems using classical linear theory with initial stress and other additional conditions have been studied [10 –13]. Singh and Arora [14] have shown that, for an initially stressed rotating orthotropic dissipative medium, two quasi-planar waves exist. Othman and Song [15] have studied the effects of thermal parameters on the reflection of plane waves from a solid half-space. The effect of initial stress on a torsional wave is discussed by Dhua et al. [16], whereas Chatterjee et al. [17] have studied in detail the effect on reflected quasi-P and quasi-SV waves. Very recently, Majhi et al. [18] investigated the reflection and transmission of plane SH-waves in two semi-infinite anisotropic magnetoelastic media. Kundu et al. [19] present a study of the affected behavior of SH-wave propagation in a layer between an anisotropic porous layer and an initially stressed isotropic half-space.

Here, the concept of the strain energy function (which depends on the finite deformation and initial stress) is used to develop the basic equations required to carry out the analysis of plane wave propagation when the material is initially stressed, irrespective of the cause that develops this initial stress. This concept of initial stress was initially used by Hoger [20] and later applied extensively to study various problems [21 –27]. The initial stress considered here is different from the pre-stress, which is generally accompanied by some finite deformation. The effect of pre-stress on wave propagation was examined for compressible and incompressible materials by Ogden and Sotiropoulos [28, 29]. The reader is also referred to the work of Hussain and Ogden [30 –32] on reflection and transmission of plane waves, and the references therein.

The basic theory used in this paper is the nonlinear theory, along with the initial stress tensor embedded in the fourth-order elastic moduli [33] and through the invariants of the right Cauchy–Green deformation tensor. Most of the existing literature is on surface wave propagation in initially stressed media [34 –36], following the same theory. The purpose of this paper is to study plane wave propagation in a homogeneous initially stressed incompressible material in both its deformed and undeformed states. Later, the theory is applied to the problem of the reflection of a plane wave in an initially stressed incompressible material. The effect of homogeneous initial stress is elaborated through graphic illustrations.

2. Plane incremental motions in an initially stressed incompressibleelastic half-space—basic equations

In this section, the reader is generally referred to Shams and Ogden [34] for the kinematics, equilibrium, and invariant formulation of the constitutive law for an elastic solid with initial stress and incremental equations.

Consider a material body that is initially stressed in its reference configuration. Let this reference configuration be denoted $B_{r}$ , and its boundary $\partial B_{r}$ . The initial stress in $B_{r}$ is denoted $τ$ and is not necessarily associated with an elastic deformation. It may be noted that the origin of this stress is not of primary concern in this problem. In the absence of body forces, the initial stress is in equilibrium in $B_{r}$ and hence $Div τ = 0$ , where $Div$ is the divergence in the reference configuration. For the problem under discussion, without loss of generality, the initial stress $τ$ is assumed to be symmetric, homogeneous, and uniform. By contrast, when the initial stresses are residual stresses, they are inhomogeneous and dependence on position then needs to be accounted for. Let $T$ be the Cauchy stress tensor in the configuration $B$ and $S$ be the corresponding nominal stress tensor relative to $B_{r}$ .

The strain energy function W is invariant under a rotation in the reference configuration if it depends on nine independent invariants of the two tensors $C = F^{T} F$ (the right Cauchy–Green deformation tensor) and $τ$ . Here, $F$ is the deformation gradient tensor. A possible set of independent invariants is

\begin{matrix} I_{1} = tr (C), & I_{2} = \frac{1}{2} [I_{1}^{2} - tr (C^{2})], & I_{4} & \equiv {I_{41}, I_{42}, I_{43}}, \\ I_{41} = tr (τ), & I_{42} = tr (τ^{2}), & I_{43} = tr (τ^{3}), \\ I_{5} = tr (C τ), & I_{6} = tr (C^{2} τ), \\ I_{7} = tr (C τ^{2}), & I_{8} = tr (C^{2} τ^{2}) . \end{matrix}

(1)

In the reference configuration, from equation (1), the invariants dependent on $C$ reduce to

I_{1} = I_{2} = 3, I_{5} = I_{6} = I_{41}, I_{7} = I_{8} = I_{42},

(2)

where $T = τ$ , $F = I$ , and $I$ is the identity tensor.

Using the chain rule, one can write

\begin{matrix} \frac{\partial W}{\partial F} = \sum_{r \in I} W_{r} \frac{\partial I_{r}}{\partial F}, \end{matrix}

(3)

where $W_{r} = \partial W ∕ \partial I_{r}$ and $I$ is the index set ${1, 2, 5, 6, 7, 8}$ .

For the purpose of studying the nonlinear effect of initial stress and to avoid unnecessary calculations, the restricted expression of the updated elasticity tensor in its component form is given as

\begin{matrix} A_{0 piqj} = 2 W_{1} B_{pq} δ_{ij} + 2 W_{2} [I_{1} B_{pq} δ_{ij} - B_{iq} B_{jp} + 2 B_{pi} B_{qj} - δ_{ij} {(B^{2})}_{pq} - B_{pq} B_{ij}] \\ + 2 W_{5} Σ_{pq} δ_{ij} + 2 W_{6} [Σ_{pq} B_{ij} + {(Σ B)}_{pq} δ_{ij} + {(B Σ)}_{pq} δ_{ij} + Σ_{ij} B_{pq} + Σ_{pj} B_{iq} + Σ_{qi} B_{jp}] \\ + 4 W_{11} B_{ip} B_{jq} + 4 W_{22} B_{ip}^{*} B_{jq}^{*} + 4 W_{12} [2 I_{1} B_{ip} B_{jq} - B_{ip} {(B^{2})}_{jq} - B_{jq} {(B^{2})}_{ip}] \\ + 4 W_{15} (B_{ip} Σ_{jq} + B_{jq} Σ_{ip}) + 4 W_{16} [B_{ip} {(Σ B + B Σ)}_{jq} + {(Σ B + B Σ)}_{ip} B_{jq}] \\ + 4 W_{25} (B_{ip}^{*} Σ_{jq} + Σ_{ip} B_{jq}^{*}) + 4 W_{26} [B_{ip}^{*} {(Σ B + B Σ)}_{jq} + {(Σ B + B Σ)}_{ip} B_{jq}^{*}] \\ + 4 W_{55} Σ_{ip} Σ_{jq} + 4 W_{56} [Σ_{ip} {(Σ B + B Σ)}_{jq} + {(Σ B + B Σ)}_{ip} Σ_{jq}] \\ + 4 W_{66} {(Σ B + B Σ)}_{ip} {(Σ B + B Σ)}_{jq}, {p, i, q, j} = {1, 2, 3}, \end{matrix}

(4)

where all derivatives of W with respect to I₇ or I₈ are omitted. When $τ = 0$ , equation (4) reduces to the standard isotropic one, as given by Ogden [37] and Hayes and Rivlin [38]. In the reference configuration, equation (4) reduces to the modulus tensor $C$ , which depends on $τ$ . Taking the initial stress to vanish in the reference configuration leads us to the classical expression for $C$ , as noted in linear elasticity.

The conditions on the form of W are [34]

\begin{matrix} 2 W_{1} + 4 W_{2} - p^{(r)} = 0, \\ 2 (W_{5} + 2 W_{6}) = 1, W_{7} + 2 W_{8} = 0, \end{matrix}

(5)

where all the derivatives of W are evaluated in the reference configuration and the invariants assume the values given in equation (2).

Consider an initially stressed incompressible material whose elastic response is characterized by the strain energy function $W (C, τ)$ . Let $λ_{1}, λ_{2}$ , and $λ_{3}$ be the principal stretches corresponding the principal axes $x_{1}, x_{2}$ , and x₃, respectively. Let $τ_{ii}, (i = 1, 2, 3)$ denote the normal initial stress components and $τ_{ij}, i \neq j$ , $(i, j \in {1, 2, 3})$ denote the shear components of the initial stress.

Without loss of generality, it is assumed that $τ_{ij} = 0, i \neq j$ . The principal Cauchy stress components are given by

\begin{matrix} T_{ii} = - p + 2 λ_{i}^{2} W_{1} + 2 λ_{i}^{2} (I_{1} - λ_{i}^{2}) W_{2} + 2 Σ_{ii}^{(r)} W_{7} + 4 λ_{i}^{2} Σ_{ii}^{(r)} W_{8} + 2 λ_{i}^{- 2} Σ_{ii}^{{(r)}^{2}} W_{9} + 4 Σ_{ii}^{{(r)}^{2}} W_{10}, \\ i = {1, 2, 3}, \end{matrix}

(6)

where p is a Lagrange multiplier associated with the constraint.

Considering W to be independent of $I_{5}, I_{9}$ , and $I_{10}$ , the principal Cauchy stress components are

\begin{matrix} T_{11} = - p + 2 λ_{1}^{2} W_{1} + 2 λ_{1}^{2} (λ_{2}^{2} + λ_{3}^{2}) W_{2} + 2 λ_{1}^{2} τ_{11} W_{7} + 4 λ_{1}^{4} τ_{11} W_{8}, \end{matrix}

(7)

\begin{matrix} T_{22} = - p + 2 λ_{2}^{2} W_{1} + 2 λ_{2}^{2} (λ_{1}^{2} + λ_{3}^{2}) W_{2} + 2 λ_{2}^{2} τ_{22} W_{7} + 4 λ_{2}^{4} τ_{22} W_{8}, \end{matrix}

(8)

\begin{matrix} T_{33} = - p + 2 λ_{3}^{2} W_{1} + 2 λ_{3}^{2} (λ_{1}^{2} + λ_{2}^{2}) W_{2} + 2 λ_{3}^{2} τ_{33} W_{7} + 4 λ_{3}^{4} τ_{33} W_{8} . \end{matrix}

(9)

Consider plane incremental motions in the $(x_{1}, x_{2})$ plane with incremental displacement $u$ , having components

\begin{matrix} u_{1} (x_{1}, x_{2}, t), u_{2} (x_{1}, x_{2}, t), u_{3} = 0 . \end{matrix}

(10)

The updated equation of motion [34] in this case is

\begin{matrix} A_{01111} u_{1, 11} + 2 A_{02111} u_{1, 12} + A_{02121} u_{1, 22} + A_{01112} u_{2, 11} \\ + (A_{01122} + A_{02112}) u_{2, 12} + A_{02221} u_{2, 22} - ṗ_{, 1} = ρ u_{1, tt}, \end{matrix}

(11)

\begin{matrix} A_{01112} u_{1, 11} + (A_{01122} + A_{02112}) u_{1, 12} + A_{02221} u_{1, 22} \\ + A_{01212} u_{2, 11} + 2 A_{01222} u_{2, 12} + A_{02222} u_{2, 22} - ṗ_{, 2} = ρ u_{2, tt}, \end{matrix}

(12)

and the incompressibility condition is

\begin{matrix} u_{1, 1} + u_{2, 2} = 0 . \end{matrix}

(13)

From equation (13) it can be deduced that a scalar stream-like function $ψ (x_{1}, x_{2}, t)$ exists, such that

\begin{matrix} u_{1} = ψ_{, 2}, u_{2} = - ψ_{, 1} . \end{matrix}

(14)

Substituting equation (14) into equations (11) and (12), $ṗ$ can be eliminated by cross differentiation followed by subtraction. As a result of this, an equation for $ψ$ is obtained, namely

\begin{matrix} α_{1} ψ_{, 1111} + 2 α_{2} ψ_{, 2112} + 2 α_{3} ψ_{, 1222} + 2 α_{4} ψ_{, 2111} + α_{5} ψ_{, 2222} = ρ (ψ_{, 11 tt} + ψ_{, 22 tt}), \end{matrix}

(15)

where

\begin{matrix} α_{1} = A_{01212}, 2 α_{2} = A_{01111} - 2 A_{01122} - 2 A_{02112} + A_{02222}, \\ α_{3} = A_{02111} - A_{02221}, α_{4} = A_{01222} - A_{01112}, α_{5} = A_{02121} . \end{matrix}

(16)

Also, from equation (4), we have

\begin{matrix} α_{1} = 2 W_{1} λ_{1}^{2} + 2 W_{2} (I_{1} λ_{1}^{2} - λ_{1}^{2} λ_{2}^{2} - λ_{1}^{4}) + 2 W_{7} Σ_{11} + 2 W_{8} (λ_{2}^{2} Σ_{11} + 2 λ_{1}^{2} Σ_{11} + λ_{1}^{2} Σ_{22}) \\ + 4 [W_{77} + 4 W_{78} λ_{1}^{2} + 4 W_{88} λ_{1}^{4}] Σ_{12}^{2}, \end{matrix}

(17)

\begin{matrix} 2 α_{2} = 2 W_{1} (λ_{1}^{2} + λ_{2}^{2}) + 2 W_{2} [λ_{1}^{2} (I_{1} - λ_{1}^{2}) + λ_{2}^{2} (I_{1} - λ_{2}^{2}) - 2 λ_{1}^{2} λ_{2}^{2}] \\ + 2 W_{7} (Σ_{11} + Σ_{22}) + 4 W_{8} [λ_{1}^{2} (3 Σ_{11} - Σ_{22}) + λ_{2}^{2} (3 Σ_{22} - Σ_{11})] \\ + 4 W_{11} {(λ_{1}^{2} - λ_{2}^{2})}^{2} + 4 W_{22} {[λ_{1}^{2} (I_{1} - λ_{1}^{2}) - λ_{2}^{2} (I_{1} - λ_{2}^{2})]}^{2} \\ + 8 W_{12} [λ_{1}^{4} (I_{1} - λ_{1}^{2}) + λ_{2}^{4} (I_{1} - λ_{2}^{2}) - λ_{1}^{2} λ_{2}^{2} (2 I_{1} - λ_{1}^{2} - λ_{2}^{2})] \\ + 16 W_{18} [λ_{1}^{4} Σ_{11} + λ_{2}^{4} Σ_{22} - λ_{1}^{2} λ_{2}^{2} (Σ_{11} + Σ_{22})] \\ + 8 W_{27} (Σ_{11} - Σ_{22}) [λ_{1}^{2} (I_{1} - λ_{1}^{2}) - λ_{2}^{2} (I_{1} - λ_{2}^{2})] + 16 W_{28} [λ_{1}^{4} Σ_{11} (I_{1} - λ_{1}^{2}) \\ + λ_{2}^{4} Σ_{22} (I_{1} - λ_{2}^{2}) - λ_{1}^{2} λ_{2}^{2} (Σ_{22} (I_{1} - λ_{1}^{2}) + Σ_{11} (I_{1} - λ_{1}^{2})] \\ + 4 W_{77} [{(Σ_{11} - Σ_{22})}^{2} - 2 Σ_{12}^{2}] + 16 W_{78} [{(Σ_{11} - Σ_{22})}^{2} - 2 λ_{1}^{2} Σ_{12}^{2}] \\ + 16 W_{88} [{(Σ_{11} - Σ_{22})}^{2} - 2 λ_{1}^{2} λ_{2}^{2} Σ_{12}^{2}], \end{matrix}

(18)

\begin{matrix} α_{3} = 2 [W_{7} + W_{8} (3 λ_{1}^{2} - λ_{2}^{2}) + 2 W_{17} (λ_{1}^{2} - λ_{2}^{2}) + 2 W_{18} (λ_{1}^{2} + λ_{2}^{2}) (λ_{1}^{2} - λ_{2}^{2}) \\ + 2 W_{27} [λ_{1}^{2} (I_{1} - λ_{1}^{2}) - λ_{2}^{2} (I_{1} - λ_{2}^{2})] + 2 W_{28} [λ_{1}^{2} (I_{1} - λ_{1}^{2}) \\ - λ_{2}^{2} (I_{1} - λ_{2}^{2})] (λ_{1}^{2} + λ_{2}^{2})] Σ_{12} + 4 [W_{77} (Σ_{11} - Σ_{22}) + 4 W_{78} [(3 λ_{1}^{2} + λ_{2}^{2}) Σ_{11} \\ - (3 λ_{2}^{2} + λ_{1}^{2}) Σ_{22}] + 2 W_{88} [λ_{1}^{2} Σ_{11} - λ_{2}^{2} Σ_{22}] (λ_{1}^{2} + λ_{2}^{2})] Σ_{12}, \end{matrix}

(19)

\begin{matrix} α_{4} = 2 [W_{7} + W_{8} (3 λ_{2}^{2} - λ_{1}^{2}) + 2 W_{17} (λ_{2}^{2} - λ_{1}^{2}) + 2 W_{18} (λ_{2}^{2} + λ_{1}^{2}) (λ_{2}^{2} - λ_{1}^{2}) \\ + 2 W_{27} [λ_{2}^{2} (I_{1} - λ_{2}^{2}) - λ_{1}^{2} (I_{1} - λ_{1}^{2})] + 2 W_{28} [λ_{2}^{2} (I_{1} - λ_{2}^{2}) \\ - λ_{1}^{2} (I_{1} - λ_{1}^{2})] (λ_{1}^{2} + λ_{2}^{2})] Σ_{12} + 4 [W_{77} (Σ_{22} - Σ_{11}) + 4 W_{78} [(3 λ_{2}^{2} + λ_{1}^{2}) Σ_{22} \\ - (3 λ_{1}^{2} + λ_{2}^{2}) Σ_{11}] + 2 W_{88} [λ_{2}^{2} Σ_{22} - λ_{1}^{2} Σ_{11}] (λ_{1}^{2} + λ_{2}^{2})] Σ_{12}, \end{matrix}

(20)

\begin{matrix} α_{5} = 2 W_{1} λ_{2}^{2} + 2 W_{2} (I_{1} λ_{2}^{2} - λ_{1}^{2} λ_{2}^{2} - λ_{2}^{4}) + 2 W_{7} Σ_{22} + 2 W_{8} (λ_{1}^{2} Σ_{22} + 2 λ_{2} Σ_{22} \\ + λ_{2}^{2} Σ_{11}) + 4 [W_{77} + 4 W_{78} λ_{2}^{2} + 4 W_{88} λ_{2}^{4}] Σ_{12}^{2} . \end{matrix}

(21)

2.1. The strong ellipticity condition

For an incompressible material, equation (15) is strongly elliptic if the coefficients $A_{0 piqj}$ satisfy [34]

\begin{matrix} A_{0 piqj} n_{p} n_{q} m_{i} m_{j} > 0, {p, i, q, j} = {1, 2, 3}, \end{matrix}

(22)

for all nonzero $m, n$ . In addition to these, the incompressibility constraint requires $m \cdot n = 0$ . Here, $m_{i}$ and $n_{i}, i = 1, 2, 3$ , represent the components of $m$ and $n$ , respectively.

For the in-plane motions under consideration, let $m = (m_{1}, m_{2}, 0)$ and $n = (n_{1}, n_{2}, 0)$ be two unit vectors and let $m_{1} = n_{2}$ and $m_{2} = - n_{1}$ so that the incompressibility condition (equation (13)) is satisfied without loss of generality. The strong ellipticity condition (equation (22)) in two dimensions, for this special case of an incompressible material, is given by

\begin{matrix} α_{1} n_{1}^{4} + 2 α_{4} n_{1}^{3} n_{2} + 2 α_{2} n_{1}^{2} n_{2}^{2} + 2 α_{3} n_{1} n_{2}^{3} + α_{5} n_{2}^{4} > 0 . \end{matrix}

(23)

Without loss of generality, it can be assumed that $n_{1} = \cos θ$ and $n_{2} = \sin θ$ . Therefore, inequality (23), after some calculations, gives

\begin{matrix} α_{5} m^{4} + 2 α_{3} m^{3} + 2 α_{2} m^{2} + 2 α_{4} m + α_{1} > 0, \end{matrix}

(24)

where $m = \tan θ$ .

2.2. Boundary conditions

Consider the half-space $x_{2} < 0$ bounded by $x_{2} = 0$ . The updated linearized incremental nominal stress tensor is denoted by ${\overset{°}{S}}_{0}$ and is given in its component form as

Ṡ_{0 pi} = A_{0 piqj} u_{j, q} - ṗ δ_{pi} + p u_{p, i},

(25)

where $ṗ$ is the linearized incremental form of p. The incremental traction per unit area of the boundary is ${\overset{°}{S}}_{0}^{T} ν$ , where $ν$ is the unit outward normal to the boundary. The component form of ${\overset{°}{S}}_{0}^{T} ν$ in this case follows from equation (25) as

\begin{matrix} Ṡ_{0 ji} ν_{j} = (A_{0 jilk} + p δ_{jk} δ_{il}) u_{k, l} ν_{j} - ṗ ν_{i} . \end{matrix}

(26)

Since $ν = (0, 1, 0)$ in this case, the only non-vanishing components of ${\overset{°}{S}}_{0}^{T} ν$ are $Ṡ_{021}$ and $Ṡ_{022}$ . These are given by

\begin{matrix} Ṡ_{021} = A_{02111} u_{1, 1} + A_{02121} u_{1, 2} + (A_{02112} + p) u_{2, 1} + A_{02122} u_{2, 2}, \end{matrix}

(27)

\begin{matrix} Ṡ_{022} = A_{01122} u_{1, 1} + A_{02221} u_{1, 2} + A_{02212} u_{2, 1} + (A_{02222} + p) u_{2, 2} - ṗ . \end{matrix}

(28)

Using equation (14) in equations (27) and (28) gives

\begin{matrix} Ṡ_{021} = - (α_{5} - T_{22}) ψ_{, 11} + α_{3} ψ_{, 12} + α_{5} ψ_{, 22}, \end{matrix}

(29)

\begin{matrix} Ṡ_{022} = - A_{02212} ψ_{, 11} + (A_{01122} - A_{02222} - p) ψ_{, 12} + A_{02221} ψ_{, 22} - ṗ, \end{matrix}

(30)

respectively. As a result of taking the derivative of $Ṡ_{022}$ in equation (30) with respect to x₁, the term $ṗ_{, 1}$ appears, which can be eliminated through equation (11). Therefore,

\begin{matrix} Ṡ_{022, 1} = ρ ψ_{, 2 tt} - α_{4} ψ_{, 111} - (2 α_{2} + α_{5} - T_{22}) ψ_{, 112} - 2 α_{3} ψ_{, 122} - α_{5} ψ_{, 222} . \end{matrix}

(31)

Here, the connection

\begin{matrix} A_{0 ijij} - A_{0 ijji} = T_{ii} + p, i \neq j, \end{matrix}

(32)

is being used.

3. Homogeneous plane waves in an initially stressed incompressiblehalf-space in the deformed configuration

The foregoing theory is now applied to study wave propagation in an initially stressed incompressible homogeneous half-space ( $x_{2} < 0$ ) in the deformed configuration for the special model given by

\begin{matrix} W = \frac{μ}{2} (I_{1} - 3) + \frac{\bar{μ}}{2} {(I_{7} - I_{4})}^{2} + \frac{1}{2} (I_{7} - I_{41}), \end{matrix}

(33)

where $μ$ and $\bar{μ}$ are material constants. The material constant $μ$ has the same dimensions as stress and $\bar{μ}$ has dimensions of ${stress}^{- 1}$ . The invariants $I_{1}, I_{41}$ , and I₇ are given by equation (1). This simple model is chosen to illustrate the combined effect of finite deformation and initial stress. Equation (33) can be rewritten in terms of the principal stretches $λ_{1}, λ_{2}$ , and $λ_{3}$ and the principal initial stresses $τ_{1}, τ_{2}$ , and $τ_{3}$ as

\begin{matrix} W = \frac{μ}{2} (λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2} - 3) + \frac{\bar{μ}}{2} {[(λ_{1}^{2} - 1) τ_{1} + (λ_{2}^{2} - 1) τ_{2} + (λ_{3}^{2} - 1) τ_{3}]}^{2} \\ + \frac{1}{2} [(λ_{1}^{2} - 1) τ_{1} + (λ_{2}^{2} - 1) τ_{2} + (λ_{3}^{2} - 1) τ_{3}] . \end{matrix}

(34)

For the incompressible material (equation (33)) in the deformed configuration, we have

\begin{matrix} W_{1} = \frac{μ}{2}, W_{4} = - \bar{μ} I_{7}, W_{7} = \bar{μ} (I_{7} - I_{41}) + \frac{1}{2}, W_{77} = \bar{μ}, \end{matrix}

(35)

which in the reference configuration reduce to

\begin{matrix} W_{1} = \frac{μ}{2}, W_{4} = - \bar{μ} I_{41}, W_{7} = \frac{1}{2}, W_{77} = \bar{μ} . \end{matrix}

(36)

All other $W_{r}, W_{rs} = 0$ .

For this specific strain energy function, equations (17) to (21) give

\begin{matrix} α_{1} = μ λ_{1}^{2} + [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{11} + 4 \bar{μ} Σ_{12}^{2}, \end{matrix}

(37)

\begin{matrix} 2 α_{2} = μ (λ_{1}^{2} + λ_{2}^{2}) + [2 \bar{μ} (I_{7} - I_{41}) + 1] (Σ_{11} + Σ_{22}) + 4 \bar{μ} [{(Σ_{11} - Σ_{22})}^{2} - 2 Σ_{12}^{2}], \end{matrix}

(38)

\begin{matrix} α_{3} = [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{12} + 4 \bar{μ} [(Σ_{11} - Σ_{22}) Σ_{12}, \end{matrix}

(39)

\begin{matrix} α_{4} = [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{12} - 4 \bar{μ} [(Σ_{11} - Σ_{22}) Σ_{12}, \end{matrix}

(40)

\begin{matrix} α_{5} = μ λ_{2}^{2} + [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{22} + 4 \bar{μ} Σ_{12}^{2} . \end{matrix}

(41)

Note that since $Σ$ is symmetric, one can choose axes that correspond to the principal axes of $Σ$ , and therefore, $Σ_{12} = 0$ and $Σ_{11}$ and $Σ_{22}$ are the principal values. Therefore, from equations (37) to (41),

\begin{matrix} α_{1} = μ λ_{1}^{2} + [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{11}, \end{matrix}

(42)

\begin{matrix} 2 α_{2} = μ (λ_{1}^{2} + λ_{2}^{2}) + [2 \bar{μ} (I_{7} - I_{41}) + 1] (Σ_{11} + Σ_{22}) + 4 \bar{μ} {(Σ_{11} - Σ_{22})}^{2}, \end{matrix}

(43)

\begin{matrix} α_{3} = 0, \end{matrix}

(44)

\begin{matrix} α_{4} = 0, \end{matrix}

(45)

\begin{matrix} α_{5} = μ λ_{2}^{2} + [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{22} . \end{matrix}

(46)

Consider an isochoric (homogeneous plane strain) deformation, such that $λ_{3} = 1$ . Here, the updated incompressibility condition is given by [34]

\begin{matrix} λ_{1} λ_{2} λ_{3} = 1, \end{matrix}

(47)

where $λ_{1} = λ$ and $λ_{2} = λ^{- 1}$ .

In this case, using equations (27) and (31), the nonzero traction components are given by

\begin{matrix} Ṡ_{021} = - (α_{5} - T_{22}) ψ_{, 11} + α_{5} ψ_{, 22}, \end{matrix}

(48)

\begin{matrix} Ṡ_{022, 1} = ρ ψ_{, 2 tt} - (2 α_{2} + α_{5} - T_{22}) ψ_{, 112} - α_{5} ψ_{, 222} . \end{matrix}

(49)

The strong ellipticity condition (equation (24)) in this case is given by

\begin{matrix} α_{5} m^{4} + 2 α_{2} m^{2} + α_{1} > 0 . \end{matrix}

(50)

For this inequality to hold $\forall t$ , the necessary and sufficient conditions are

\begin{matrix} α_{1} > 0, α_{5} > 0, α_{2} > - \sqrt{α_{1} α_{5}} . \end{matrix}

(51)

Using equations (42) to (46), equation (15) reduces to

\begin{matrix} α_{1} ψ_{, 1111} + 2 α_{2} ψ_{, 2112} + α_{5} ψ_{, 2222} = ρ (ψ_{, 11 tt} + ψ_{, 22 tt}) . \end{matrix}

(52)

The stress-free incremental boundary conditions on $x_{2} = 0$ follow from equations (48) and (49) as

\begin{matrix} ψ_{, 22} - ψ_{, 11} = 0, \end{matrix}

(53)

\begin{matrix} ρ ψ_{, 2 tt} - (2 α_{2} + α_{5}) ψ_{, 112} - α_{5} ψ_{, 222} = 0, \end{matrix}

(54)

respectively.

Consider now plane waves in an initially stressed incompressible material in the reference configuration. Let us assume that $ψ$ is of the form

\begin{matrix} ψ = f [k (n_{1} x_{1} + n_{2} x_{2} - ct)], \end{matrix}

(55)

where c is the wave speed, k is the wave number, t is time,and f is a four times continuously differentiable function. Since $n_{1} = \cos θ$ and $n_{2} = \sin θ$ , substitution of this expression into equation (52) leads to

\begin{matrix} (α_{1} + α_{5} - 2 α_{2}) \cos^{4} θ + 2 (α_{2} - α_{5}) \cos^{2} θ + α_{5} = ρ c^{2} . \end{matrix}

(56)

This determines the wave speed for any given direction of propagation in the $(x_{1}, x_{2})$ plane and it is easily shown that $ρ c^{2} > 0$ follows from the strong ellipticity conditions (equation (51)). Alternatively, equation (56) determines possible directions in which waves may propagate for given wave speed, material properties, and principal initial stresses. In special cases, it is possible for equation (56) to yield two pairs of distinct directions of propagation.

In the classical theory of incompressible isotropic elasticity, we have $α_{1} = α_{2} = α_{5} = μ$ , where $μ$ is the shear modulus. When $Σ$ vanishes, the material is isotropic and equation (56) thus reduces to $ρ c^{2} = μ$ , independently of the direction of propagation. This gives the speed of a classical shear wave.

Two cases corresponding to different values of $α_{1}, α_{2}$ , and $α_{5}$ can be considered.

3.1. Case A: $α_{1} + α_{5} = 2 α_{2}$

This includes the special case of $α_{1} = α_{2} = α_{5} = μ$ when evaluated in the deformed configuration, which entails $λ_{1} = λ_{2}$ and $Σ_{11} = Σ_{22}$ . This gives $ρ c^{2} = μ$ independently of the direction of propagation in the $(x_{1}, x_{2})$ plane. This also incorporates the classical theory, with $λ_{1} = λ_{2} = 1$ and the stress assumed to be zero.

With $α_{1} + α_{5} = 2 α_{2}$ , equation (56) can be rewritten as

\begin{matrix} ρ c^{2} = α_{1} \cos^{2} θ + α_{5} \sin^{2} θ . \end{matrix}

(57)

For either equation (56) or equation (57), a shear wave can propagate along the principal axis x₁, with $ρ c^{2} = α_{1}$ or along the principal axis x₂, with $ρ c^{2} = α_{5}$ .

If $α_{1} \neq α_{5}$ , equation (57) gives

\begin{matrix} \cos^{2} θ = \frac{α_{5} - ρ c^{2}}{α_{5} - α_{1}} . \end{matrix}

(58)

For this to yield real values of $\cos θ$ , we must have

\begin{matrix} either α_{5} \leq ρ c^{2} \leq α_{1}, or α_{1} \leq ρ c^{2} \leq α_{5} . \end{matrix}

(59)

Equation (58) can be rewritten in the alternative form

\begin{matrix} \tan^{2} θ = \frac{ρ c^{2} - α_{1}}{α_{5} - ρ c^{2}} . \end{matrix}

(60)

For a given wave speed subject to equation (59), equation (60) yields two (in general distinct) directions, symmetric with respect to the axes.

3.2. Case B: $α_{1} + α_{5} \neq 2 α_{2}$

For a given wave speed, the solutions of equation (56) may be written

\begin{matrix} \cos^{2} θ = \frac{α_{5} - α_{2} \pm {[α_{2}^{2} - α_{1} α_{5} + ρ c^{2} (α_{1} + α_{5} - 2 α_{2})]}^{\frac{1}{2}}}{α_{1} + α_{5} - 2 α_{2}} . \end{matrix}

(61)

Considering $0 \leq \cos^{2} θ \leq 1$ , for real solutions from equation (61), either

\begin{matrix} α_{1} + α_{5} - 2 α_{2} > 0, \frac{α_{1} α_{5} - α_{2}^{2}}{α_{1} + α_{5} - 2 α_{2}} \leq ρ c^{2} \leq Min {α_{1}, α_{5}}, \end{matrix}

(62)

\begin{matrix} α_{1} + α_{5} - 2 α_{2} < 0, Max {α_{1}, α_{5}} \leq ρ c^{2} \leq \frac{α_{1} α_{5} - α_{2}^{2}}{α_{1} + α_{5} - 2 α_{2}} . \end{matrix}

(63)

Equal roots arise when

\begin{matrix} ρ c^{2} = \frac{α_{1} α_{5} - α_{2}^{2}}{α_{1} + α_{5} - 2 α_{2}}, \cos^{2} θ = \frac{α_{1} - α_{5}}{α_{1} α_{5} - α_{2}^{2}} . \end{matrix}

(64)

The directions of propagation are given by

\begin{matrix} \tan^{2} θ = \frac{ρ c^{2} - α_{2} \pm {[α_{2}^{2} - α_{1} α_{5} + ρ c^{2} (α_{1} + α_{5} - 2 α_{2})]}^{\frac{1}{2}}}{α_{5} - ρ c^{2}} . \end{matrix}

(65)

Thus, for any given wave speed within the allowed range, there are, in general, four possible distinct directions in which a plane shear wave may propagate. In a special case, these degenerate to two when equation (64) holds.

For the special value $ρ c^{2} = α_{5}$ , the wave propagates either along the x₂ axis (as in case A) or in the direction given by

\begin{matrix} \cos^{2} θ = \frac{2 (α_{5} - α_{2})}{α_{1} + α_{5} - 2 α_{2}}, \end{matrix}

(66)

in which case either $α_{1} < α_{5} < α_{2}$ or $α_{1} > α_{5} > α_{2}$ must hold. Similarly, the special value $ρ c^{2} = α_{1}$ means that the wave is either propagating along the x₁ direction or in the direction given by

\begin{matrix} \cos^{2} θ = \frac{α_{5} - α_{1}}{α_{1} + α_{5} - 2 α_{2}}, \end{matrix}

(67)

which requires that either $α_{2} < α_{1} < α_{5}$ or $α_{2} > α_{1} > α_{5}$ holds.

4. Wave reflection from a plane boundary in the deformed configuration

Consider a plane wave of the form given by equation (55) incident on the boundary $x_{2} = 0$ of the half-space $x_{2} < 0$ . The boundary $x_{2} = 0$ is taken to be free of incremental traction but subject to the normal initial stress in the x₁ direction. The incremental traction-free boundary conditions on $x_{2} = 0$ are given by equations (53) and (54).

Let the direction of propagation of this wave be $n = (n_{1}, n_{2}) = (\cos θ, \sin θ)$ and c be its speed. As a result of this incidence, depending on the material properties and the state of deformation, one or two reflected waves or a surface wave are generated. The general solution for $ψ$ consisting of the incident and two reflected waves in the form is assumed as

\begin{matrix} ψ = f [k (n_{1} x_{1} + n_{2} x_{2} - ct)] + Rf [k (n_{1} x_{1} - n_{2} x_{2} - ct)] + R^{'} f [k^{'} (n_{1}^{'} x_{1} - n_{2}^{'} x_{2} - c^{'} t)], \end{matrix}

(68)

where R and $R^{'}$ are the reflection coefficients. Also, $k^{'}$ and $c^{'}$ are the wave number and wave speed of the second reflected wave. The first reflected wave has the same speed as the incident wave and is reflected at an angle $θ$ to the boundary, while the angle of reflection of the second wave is $θ^{'}$ . Let $n_{1}^{'} = \cos θ^{'}, n_{2}^{'} = \sin θ^{'}$ . For the compatibility of the three waves, they should have the same frequency. For this we must set

\begin{matrix} kc = k^{'} c^{'} . \end{matrix}

(69)

Using equation (68) in equations (53) and (54), it is found that $k n_{1} = k^{'} n_{1}^{'}$ and hence

\begin{matrix} c^{'} n_{1} = c n_{1}^{'}, \end{matrix}

(70)

which is a statement of Snell’s law.

4.1. Case A: $α_{1} + α_{5} = 2 α_{2}$

In this case, from equations (42) to (46), it is implied that $Σ_{11} = Σ_{22} = 0$ , which also implies that $τ_{11} = τ_{22} = 0$ and hence the case refers to vanishing of the initial stress. The wave speed in this case is given by

\begin{matrix} ρ c^{2} = α_{1} n_{1}^{2} + α_{5} n_{2}^{2} . \end{matrix}

(71)

Writing equation (71) for the second reflected wave, we have

\begin{matrix} ρ {c^{'}}^{2} = α_{1} {n_{1}^{'}}^{2} + α_{5} {n_{2}^{'}}^{2} . \end{matrix}

(72)

Using equation (70) to eliminate c and $c^{'}$ between equations (71) and (72), we have ${n_{1}^{'}}^{2} = n_{1}^{2}$ for $α_{1} + α_{5} = 2 α_{2}$ . Thus, the two reflected waves coincide and and there is only one distinct reflected wave. Thus, without the loss of generality, we take $R^{'} = 0$ . This behavior is the same as is found for plane waves in homogeneous isotropic solids in the classical linearized theory [39].

Using equations (68) and (71) in the boundary conditions given by equations (53) and (54) gives

\begin{matrix} (1 + R) [1 - 2 n_{1}^{2}] = 0, \end{matrix}

(73)

\begin{matrix} α_{5} (1 - R) n_{2} n_{1}^{2} = 0, \end{matrix}

(74)

respectively. Considering the possibility $R \neq \pm 1$ , equations (73) and (74) lead to

\begin{matrix} n_{1}^{2} = \frac{1}{2} and n_{2} n_{1}^{2} = 0, \end{matrix}

(75)

and it is impossible for these to occur together. Therefore, the value of R must be either 1 or $- 1$ .

$R = 1$ . This case is possible only when a wave is incident at an angle of $θ = π ∕ 4$ , which results in a unique reflected wave at the same angle. The wave speed in this case is given by

\begin{matrix} ρ c^{2} = (α_{1} + α_{5}) ∕ 2 . \end{matrix}

(76)

In this case, the nonzero displacement component on the boundary $x_{2} = 0$ is

\begin{matrix} u_{2} = - \sqrt{2} kf [k (\frac{x_{1}}{\sqrt{2}} - ct)], \end{matrix}

(77)

which means that there is no displacement along the boundary in the x₁ direction. Since $β_{1} = β_{5} = μ λ^{2}$ in this case, the classical result is recovered for the speed of the shear wave in a deformed isotropic material i.e., $ρ c^{2} ∕ μ = λ^{2}$ .

$R = - 1$ . In this case, either the normal incidence ( $n_{1} = 0$ ) or the grazing incidence ( $n_{2} = 0$ ) can occur. The grazing incidence in this case is not possible for $n_{2} = 0$ , as in that case $ψ \equiv 0$ . The normal incidence results in a wave traveling along the vertical $x_{2}$ axis. From equation (71) when $n_{1} = 0$ , the wave speed is given by $ρ c^{2} ∕ μ = λ^{2}$ .

4.2. Case B: $α_{5} + α_{1} \neq 2 α_{2}$

In the case of an incident wave, equation (55) can be rewritten as

\begin{matrix} (α_{1} + α_{5} - 2 α_{2}) n_{1}^{4} + 2 (α_{2} - α_{5}) n_{1}^{2} + α_{5} = ρ c^{2}, \end{matrix}

(78)

and when there is a reflected wave, we also have

\begin{matrix} (α_{1} + α_{5} - 2 α_{2}) {n_{1}^{'}}^{4} + 2 (α_{2} - α_{5}) {n_{1}^{'}}^{2} + α_{5} = ρ {c^{'}}^{2}, \end{matrix}

(79)

together with Snell’s law, given by equation (70).

From equations (78) and (79), we find that either ${n_{1}^{'}}^{2} = n_{1}^{2}$ or

\begin{matrix} n_{1}^{2} {n_{1}^{'}}^{2} = α_{5} ∕ (α_{1} + α_{5} - 2 α_{2}), \end{matrix}

(80)

and, these two possibilities occur together when

\begin{matrix} n_{1}^{4} = α_{5} ∕ (α_{1} + α_{5} - 2 α_{2}), \end{matrix}

(81)

which defines the transitional angle, say $θ_{0}$ . Thus, when equation (68) is applicable, a given incident wave generates two reflected waves in general. One of these waves is reflected at the same angle as the incident wave; the angle of reflection of the second wave is given by equation (80) with $n_{1}^{'} = \cos θ^{'}$ .

For a given angle of incidence $θ$ , $θ^{'}$ is calculated from equation (80), $ρ c^{2}$ from equation (78), and $ρ {c^{'}}^{2}$ from equation (79). The reflection coefficients R and $R^{'}$ are calculated using the boundary conditions.

For a given $θ$ , necessary and sufficient conditions for equation (80) to yield a real angle $θ^{'}$ are

\begin{matrix} α_{1} & > 2 α_{2}, \end{matrix}

(82)

\begin{matrix} \tan^{2} θ & \leq \frac{(α_{1} - 2 α_{2})}{α_{5}} \equiv \tan^{2} θ_{c}, \end{matrix}

(83)

where $θ_{c}$ is defined as the critical angle given by the right-hand identity in equation (83).

Through substitution of equation (68) in boundary conditions (53) and (54), after using the propagation condition (78) and equation (80), we get, after some calculations

\begin{matrix} α_{5} (1 + R + R^{'}) - (α_{1} + α_{5} - 2 α_{2}) [(1 + R) {n_{1}^{'}}^{2} n_{2}^{2} + R^{'} n_{1}^{2} {n_{2}^{'}}^{2}] = 0, \end{matrix}

(84)

\begin{matrix} 2 α_{5} [(1 - R) n_{1}^{'} n_{2} + R^{'} n_{1} n_{2}^{'}] - (α_{1} + α_{5} - 2 α_{2}) [(1 - R) n_{1} n_{2} + R^{'} n_{1}^{'} n_{2}^{'}] n_{1} n_{1}^{'} = 0 . \end{matrix}

(85)

Explicit expressions for R and $R^{'}$ are given by

\begin{matrix} R = \frac{{(1 - 2 {n_{1}^{'}}^{2})}^{2} n_{1}^{3} n_{2} - {(1 - 2 n_{1}^{2})}^{2} {n_{1}^{'}}^{3} n_{2}^{'}}{{(1 - 2 {n_{1}^{'}}^{2})}^{2} n_{1}^{3} n_{2} + {(1 - 2 n_{1}^{2})}^{2} {n_{1}^{'}}^{3} n_{2}^{'}}, \end{matrix}

(86)

\begin{matrix} R^{'} = \frac{- 2 (1 - 2 n_{1}^{2}) (1 - 2 {n_{1}^{'}}^{2}) {n_{1}^{'}}^{2} n_{1} n_{2}}{{(1 - 2 {n_{1}^{'}}^{2})}^{2} n_{1}^{3} n_{2} + {(1 - 2 n_{1}^{2})}^{2} {n_{1}^{'}}^{3} n_{2}^{'}} . \end{matrix}

(87)

From equation (87) it is obvious that $R^{'}$ vanishes for $n_{1} = 0$ , which means the angle of incidence $θ = π ∕ 2$ (normal incidence).

We now consider three nontrivial cases, where $R^{'}$ can possibly vanish.

$R^{'} = 0, R \neq \pm 1$ . In this case, equations (84) and (85) yield either $n_{2} = 0$ , i.e., grazing incidence, which is not possible since then $ψ = 0$ , or

\begin{matrix} n_{1}^{2} = \frac{1}{2} = \frac{2 α_{5}}{α_{1} + α_{5} - 2 α_{2}} = {n_{1}^{'}}^{2} . \end{matrix}

(88)

For equation (88) to yield a real angle, inequality (82) along with the stability conditions (equation (51)) must hold. The wave speed in this case is given by

\begin{matrix} ρ c^{2} = α_{2} + α_{5} . \end{matrix}

(89)

$R^{'} = 0, R = 1$ . In this case, we have

\begin{matrix} n_{1}^{2} = \frac{1}{2} . \end{matrix}

(90)

This means that an incident wave at an angle $θ = π ∕ 4$ results in a unique wave reflected at the same angle. The wave speed in this case is given by

\begin{matrix} ρ c^{2} = (α_{1} + α_{5} + 2 α_{2}) ∕ 4 . \end{matrix}

(91)

In this case, the nonzero displacement component on the boundary $x_{2} = 0$ is

\begin{matrix} u_{2} = - \sqrt{2} kf [k (\frac{x_{1}}{\sqrt{2}} - ct)], \end{matrix}

(92)

which means that there is no displacement along the boundary in the x₁ direction.

$R^{'} = 0, R = - 1$ . In this case, we have

\begin{matrix} n_{1}^{2} = \frac{2 α_{5}}{α_{1} + α_{5} - 2 α_{2}} and {n_{1}^{'}}^{2} = \frac{1}{2} . \end{matrix}

(93)

For equation (93)₁ to yield a real angle, inequalities (82) and (51) must hold. The wave speed in this case is given by

\begin{matrix} ρ c^{2} = \frac{α_{5} (α_{1} + α_{5} + 2 α_{2})}{α_{1} + α_{5} - 2 α_{2}} . \end{matrix}

(94)

In this case, the nonzero displacement component on the boundary $x_{2} = 0$ is

\begin{matrix} u_{1} = 2 k {(\frac{2 α_{5}}{α_{1} + α_{5} - 2 α_{2}})}^{1 ∕ 2} f [k {(\frac{2 α_{5}}{α_{1} + α_{5} - 2 α_{2}})}^{1 ∕ 2} x_{1} - ct], \end{matrix}

(95)

which means that there is no displacement normal to the boundary in the x₂ direction.

When equations (82) and (83) are not satisfied, a pair of reflected waves is not possible and an alternate expression for $ψ$ should be used. Therefore, in equation (68), we have $n_{1}^{'} = 1$ and $n_{2}^{'} = - is$ where $s > 0$ so that the latter term in equation (68) decays as $x_{2} \to \infty$ . In this case, equations (69) and (70) change such that

\begin{matrix} k n_{1} = k^{'}, c^{'} n_{1} = c, \end{matrix}

(96)

where $c^{'}$ now represents the speed of the surface wave, whereas c is the speed of the incident wave and can be calculated using equation (78). The reflection coefficients in this case are given by explicit expressions for R and $R^{'}$ , as

\begin{matrix} R = \frac{{(1 + s^{2})}^{2} n_{1}^{3} n_{2} + is {(1 - 2 n_{1}^{2})}^{2}}{{(1 + s^{2})}^{2} n_{1}^{3} n_{2} - is {(1 - 2 n_{1}^{2})}^{2}}, \end{matrix}

(97)

\begin{matrix} R^{'} = \frac{- 2 (1 + s^{2}) (2 n_{1}^{2} - 1) n_{1} n_{2}}{{(1 + s^{2})}^{2} n_{1}^{3} n_{2} - is {(1 - 2 n_{1}^{2})}^{2}} . \end{matrix}

(98)

Using equation (52), we have the propagation conditions for the incident wave and the surface wave,

\begin{matrix} β_{1} n_{1}^{4} + 2 β_{2} n_{1}^{2} n_{2}^{2} + β_{5} n_{2}^{4} = ρ c^{2}, \end{matrix}

(99)

\begin{matrix} β_{1} + 2 β_{2} s^{2} + β_{5} s^{4} = ρ {c^{'}}^{2} (1 - s^{2}), \end{matrix}

(100)

respectively. Using equation (96)₂ and equation (99) to eliminate $ρ {c^{'}}^{2}$ from equation (100) we have the counterpart of equation (80),

\begin{matrix} β_{5} s^{2} = β_{5} - (β_{1} + β_{5} - 2 β_{2}) n_{1}^{2}, \end{matrix}

(101)

where $s > 0$ . The right-hand side of equation (101) is always positive for $β_{1} - 2 β_{2} \leq 0$ . In the case when $β_{1} - 2 β_{2} > 0$ , the angle of incidence should be such that

\begin{matrix} \tan^{2} θ > (β_{1} - 2 β_{2}) ∕ β_{5} . \end{matrix}

(102)

It may be noted that in contrast to the upper bound [40] or lower bound in certain circumstances [41] on the surface wave speed in the case of pre-stressed incompressible materials, there is no restriction observed here when the surface wave is generated by an incident wave. Numerical results illustrating the behavior of the incident and two reflected waves or one reflected wave accompanied by a surface wave for angles of incidence greater than the critical angle are presented in the following section for a special model.

4.3. Analysis

We now apply the foregoing theory to the specific model given by equation (33) for an initially stressed homogeneous incompressible half-space subject to homogeneous plane strain deformation.

We consider the boundary $x_{2} = 0$ of the half-space to be stress-free. Therefore, $T_{22} = 0 = Σ_{22}$ . After a few calculations using the definitions of $I_{41}$ and I₇ from equation (1), equations (42) to (46) reduce to

\begin{matrix} α_{1} = (μ + τ_{11}) λ^{2} + 2 \bar{μ} λ^{2} (λ^{2} - 1) τ_{11}^{2}, \end{matrix}

(103)

\begin{matrix} 2 α_{2} = (μ + τ_{11}) λ^{2} + μ λ^{- 2} + 2 \bar{μ} λ^{2} (3 λ^{2} - 1) τ_{11}^{2}, \end{matrix}

(104)

\begin{matrix} α_{5} = μ λ^{- 2} . \end{matrix}

(105)

The nonzero principal Cauchy stresses in this case are given by

\begin{matrix} T_{11} = [μ + [2 \bar{μ} (I_{7} - I_{41}) + 1] Σ_{11}] λ^{2} - p, \end{matrix}

(106)

\begin{matrix} T_{22} = μ λ^{- 2} - p . \end{matrix}

(107)

Therefore, the stress-free boundary conditions lead to

\begin{matrix} p = α_{5} = μ λ^{- 2}, \end{matrix}

(108)

which holds everywhere in $x_{2} < 0$ , as the underlying state of the material is considered to be uniform.

Recasting equations (103) to (105) in the dimensionless form, we have

\begin{matrix} β_{1} ∕ μ = λ^{2} ϵ, 2 β_{2} ∕ μ = λ^{2} ϵ + λ^{- 2} + 4 b_{0} λ^{4} {\bar{τ}}^{2}, β_{5} ∕ μ = λ^{- 2}, \end{matrix}

(109)

where

\begin{matrix} b_{0} = μ \bar{μ}, \bar{τ} = τ_{11} ∕ μ, ϵ = 2 b_{0} (λ^{2} - 1) {\bar{τ}}^{2} + \bar{τ} + 1 . \end{matrix}

(110)

From these expressions, we have

\begin{matrix} (α_{1} + α_{5} - 2 α_{2}) ∕ μ = - 4 b_{0} λ^{4} {\bar{τ}}^{2}, (α_{1} - 2 α_{2}) ∕ μ = - λ^{- 2} - 4 b_{0} λ^{4} {\bar{τ}}^{2}, \end{matrix}

(111)

\begin{matrix} 2 (α_{2} - α_{5}) ∕ μ = λ^{2} ϵ - λ^{- 2} + 4 b_{0} λ^{4} {\bar{τ}}^{2} . \end{matrix}

(112)

We therefore have from equations (82) and (83)

\begin{matrix} - 4 b_{0} {\bar{τ}}^{2} λ^{6} > 1, \end{matrix}

(113)

and

\begin{matrix} \sec^{2} θ \leq - 4 b_{0} {\bar{τ}}^{2} λ^{6} \equiv \sec^{2} θ_{c} . \end{matrix}

(114)

Since $0 < \sec^{2} θ$ , we require $\bar{μ} < 0$ and hence $b_{0} < 0$ .

Provided that equations (113) and (114) hold, we have from equation (80)

\begin{matrix} n_{1} n_{1}^{'} = 1 ∕ \sqrt{Ā_{1}}, \end{matrix}

(115)

where we have defined

\begin{matrix} Ā_{1} = - 4 b_{0} {\bar{τ}}^{2} λ^{6} . \end{matrix}

(116)

From equations (97) and (98), we have

\begin{matrix} R = \frac{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ - (\sec^{2} θ - 2) {(Ā_{1} - \sec^{2} θ)}^{1 ∕ 2}}{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ + (\sec^{2} θ - 2) {(Ā_{1} - \sec^{2} θ)}^{1 ∕ 2}}, \end{matrix}

(117)

\begin{matrix} R^{'} = \frac{- 2 (\sec^{2} θ - 2) (Ā_{1} - 2 \sec^{2} θ) \sin θ}{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ + (\sec^{2} θ - 2) {(Ā_{1} - \sec^{2} θ)}^{1 ∕ 2}}, \end{matrix}

(118)

respectively. It might be noted that for an angle of incidence equal to the critical angle then $θ^{'} = 0$ and there is a grazing reflection.

From these expressions, if $\sec^{2} θ > Ā_{1}$ (i.e., angles of incidence greater than the critical angle), we have

\begin{matrix} R = \frac{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ - i (\sec^{2} θ - 2) {(\sec^{2} θ - Ā_{1})}^{1 ∕ 2}}{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ + i (\sec^{2} θ - 2) {(\sec^{2} θ - Ā_{1})}^{1 ∕ 2}}, \end{matrix}

(119)

\begin{matrix} R^{'} = \frac{2 (\sec^{2} θ - 2) (2 \sec^{2} θ - Ā_{1}) \sin θ}{{(Ā_{1} - 2 \sec^{2} θ)}^{2} \cos^{2} θ \sin θ + i (\sec^{2} θ - 2) {(\sec^{2} θ - Ā_{1})}^{1 ∕ 2}}, \end{matrix}

(120)

respectively. These expressions also follow from equations (97) and (98), respectively. The value of $s^{2}$ in this case follows from equation (101) and is given by

\begin{matrix} s^{2} = \cos^{2} θ (\sec^{2} θ - Ā_{1}), \end{matrix}

(121)

which has a positive right-hand side if either $Ā_{1} \leq 1$ or if $Ā_{1} > 1$ and $\sec^{2} θ > Ā_{1}$ .

We can rewrite equation (114) in the form

\begin{matrix} \cos^{2} θ \geq 1 ∕ Ā_{1} . \end{matrix}

(122)

This means, for real $θ^{'}$ , the inequality (122) should hold. For values of $θ$ outside of this range, a surface wave exists whose reflection coefficient is given by equation (120). Figure 1 refers to the values of $θ$ and $Ā_{1}$ where inequality (122) holds.

The strong ellipticity conditions (equation (51)) in this case give the sufficient conditions as

\begin{matrix} λ^{2} ϵ > 0, 4 b_{0} {\bar{τ}}^{2} λ^{6} + ϵ λ^{4} + 2 ϵ^{\frac{1}{2}} λ^{2} + 1 > 0, \end{matrix}

(123)

which further imply that $ϵ > 0$ where $ϵ$ is given by equation (110). This holds $\forall \bar{τ}$ when $8 b_{0} (λ^{2} - 1) > 1$ . For $8 b_{0} (λ^{2} - 1) < 1$ , we have

\begin{matrix} \frac{- 1 + \sqrt{1 - 8 b_{0} (λ^{2} - 1)}}{4 b_{0} (λ^{2} - 1)} < \bar{τ} < \frac{- 1 - \sqrt{1 - 8 b_{0} (λ^{2} - 1)}}{4 b_{0} (λ^{2} - 1)} . \end{matrix}

(124)

Using equations (78) and (111), the dimensionless wave speed of the incident wave in this case is given by

\begin{matrix} ρ c^{2} ∕ β_{5} = (- 4 b_{0} λ^{6} {\bar{τ}}^{2}) \cos^{4} θ + (4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1) \cos^{2} θ + 1, \end{matrix}

(125)

as a function of $θ$ .

Similarly, for the reflected wave, the speed in its dimensionless form is given by

\begin{matrix} ρ {c^{'}}^{2} ∕ β_{5} = {(- 4 b_{0} λ^{6} {\bar{τ}}^{2})}^{- 1} [\sec^{4} θ + (4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1) \sec^{2} θ - 4 b_{0} λ^{6} {\bar{τ}}^{2}], \end{matrix}

(126)

as a function of $θ$ . For a reflected wave to exist, $θ^{'}$ in equation (126) must be real and should fall in the range to satisfy the inequality (122). For angles outside of this range, $c^{'}$ is the speed of a surface wave, which increases indefinitely (and its amplitude vanishes) as the incident wave approaches normal incidence. The behaviors of $ρ c^{2} ∕ β_{5}$ (dashed graph) and $ρ {c^{'}}^{2} ∕ β_{5}$ are shown in Figure 2 for $0 \leq θ \leq θ_{c}$ where $θ_{c} = \sec^{- 1} (\sqrt{Ā_{1}})$ . For angles of incidence greater than the critical angle, $c^{'}$ represents the surface wave and is given by $c^{'} = c ∕ n_{1}$ where c is the speed of the incident wave given by equation (125). The speed of surface wave for $θ > θ_{c}$ is given by

\begin{matrix} ρ {c^{'}}^{2} ∕ β_{5} = \sec^{2} θ + λ^{4} - 1 . \end{matrix}

(127)

For the special value of $ρ c^{2} = β_{5}$ , equation (125) gives

\begin{matrix} \cos^{2} θ = 0, or \cos^{2} θ = (4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1) ∕ 4 b_{0} λ^{6} {\bar{τ}}^{2}, \end{matrix}

(128)

which means that the wave travels either along the $x_{2}$ axis or in the direction given by equation (128)₂. For equation (128)₂ to give real angles, either of the following two should hold:

\begin{matrix} λ^{4} ϵ < 1 < 4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1, λ^{4} ϵ > 1 > 4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1 . \end{matrix}

(129)

For $ρ c^{2} = β_{1}$ , equation (125) gives

\begin{matrix} \cos^{2} θ = 1, or \cos^{2} θ = (λ^{4} ϵ - 1) ∕ 4 b_{0} λ^{6} {\bar{τ}}^{2}, \end{matrix}

(130)

where for real angles either of the following conditions should hold

\begin{matrix} 4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1 < λ^{4} ϵ < 1, 4 b_{0} λ^{6} {\bar{τ}}^{2} + λ^{4} ϵ - 1 > λ^{4} ϵ > 1 . \end{matrix}

(131)

When $\bar{τ} = 0$ in equation (109), that is the case when the initial stress vanishes, the case is equivalent to $β_{1} + β_{5} = 2 β_{2}$ (see Section 4.1) and there is only one reflected wave at the same angle as the incident wave, with the dimensionless wave speed given by

\begin{matrix} ρ c^{2} ∕ β_{5} = (λ^{4} - 1) \cos^{2} θ + 1 . \end{matrix}

(132)

This result reduces to $ρ c^{2} ∕ μ = 1$ when specialized for reference configuration (i.e., $λ = 1$ ). This is the same as that for classical linear theory. The result (equation (125)) is also deducible from equations (58) and (59) and either of the following two inequalities hold:

\begin{matrix} 1 \leq ρ c^{2} ∕ β_{5} \leq λ^{4}, λ^{4} \leq ρ c^{2} ∕ β_{5} \leq 1 . \end{matrix}

(133)

From equation (116), we note that as $\bar{A_{1}} \to 0$ , the case corresponds to infinitesimally small initial stress. In this case, equation (109) gives the expressions independent of the initial stress and the results are therefore comparable to the classical linearized theory for isotropic materials in the deformed configuration. Figure 3 shows the behavior of $| R^{'} |$ and $| R | (= 1)$ when the initial stress has infinitesimally small magnitude. We see that $| R^{'} |$ is bounded and vanishes at various angles of incidence. It is important to note that in the case of infinitely small values of $Ā_{1}$ , the inequality (122) does not hold and the plot of $| R^{'} |$ (for example, Figure 3(a)) refers to a surface wave for all angles of incidence $θ$ . The results are plotted using equations (86) and (87). Similar behavior is shown in the case of pre-stressed materials in Ogden and Sotiropoulos [29] for pure homogeneous strain and in Hussain and Ogden [30] for simple shear.

It is obvious that, for $Ā_{1} \to \infty$ (very large magnitude of initial stress), $| R^{'} |$ tends to vanish and becomes more confined in the band along the normal angle of incidence (see Figure 4(d)), whereas $| R | \to 1$ (see Figure 5(d)). For intermediate values of $Ā_{1}$ , refer to Figures 4 (for $| R^{'} |$ ) and 5 (for $| R |$ ). It may be observed that since the stretches and initial stress occur in a product, a variation in either of the two leads to similar values of $Ā_{1}$ . Also, the plots in Figure 4 refer to the amplitude of a surface wave for the range of values of $θ$ where the inequality (122) does not hold. For instance, in Figure 4(a), where $Ā_{1} = 2.391$ , the inequality (122) does not hold for the range $0.867532 < θ < 2.27406$ and hence in this range the plot refers to a surface wave.

Another aspect is the symmetry of the amplitudes in Figures 4 and 5 about the angle of incidence for all values of stretch. This aspect is also observed by Ogden and Sotiropoulos [29], who discuss a special class of material with pre-stress and find that increasing or decreasing magnitude of the pre-stress (the initial stress in our case) does not hinder the symmetric behavior in the case of $|R|$ . This is in contrast with results obtained for simple shear by Hussain and Ogden [30], where the symmetric behavior is lost with the increase in stretch. However, the symmetric behavior in the present case is lost (see Figure 4(c)) for a range of higher values of $Ā_{1}$ in the case of $| R^{'} |$ . The range in each case is different, depending on the values of $λ$ and b₀.

Figures 6 and 7 show real and imaginary plots for $R^{'}$ and R, respectively. These plots use varying values of $λ$ for fixed $b_{0}$ and $\bar{τ}$ and the respective values of $Ā_{1}$ are stated. The plots for Real( $R^{'}$ ) show a sharp rise for a particular intermediate range of values of $Ā_{1}$ (see Figure 6(c)) and the amplitude drops as the values of $λ$ increase (or, in other words, for very large values of $Ā_{1}$ ). The changing range of the vertical axis may be noted in Figures 4 and 6. This is a behavior expected from equation (118) as $Ā_{1} \to \infty$ . The loss of symmetric behavior is noted in Figure 6(e).

Figure 7 shows the symmetric and bounded behavior of R; it is worth noting that for even very small increases in the values of $λ$ , the imaginary part of the amplitude vanishes and the real part is such that $| R | \leq 1$ . This is obvious from equation (117) when $Ā_{1} \to \infty$ .

In reference to the discussion in Section 4.2, for vanishing of $R^{'}$ , we note from Figures 3(a), 4, and 6 that $R^{'}$ vanishes as $θ$ approaches $π ∕ 2$ . Also, from equations (90) and (91), we expect $R^{'}$ to vanish at either or both $θ = π ∕ 4$ and

\begin{matrix} \cos^{2} θ = 2 ∕ Ā_{1} = 2 \cos^{2} θ_{c} . \end{matrix}

(134)

This fact is recoverable from equation (118) and is obvious in Figures 6 and 7.

5. Homogeneous plane waves in an initially stressed incompressiblehalf-space in the reference configuration

For an initially stressed incompressible material in the reference configuration, equations (17) to (21) reduce to

\begin{matrix} α_{1} = α_{1} + τ_{11} + α_{3} (τ_{11} + τ_{22}) + α_{5} τ_{12}^{2}, \end{matrix}

(135)

\begin{matrix} 2 α_{2} = 2 α_{1} + τ_{11} + τ_{22} + 2 α_{3} (τ_{11} + τ_{22}) + α_{5} [{(τ_{11} - τ_{22})}^{2} - 2 τ_{12}^{2})], \end{matrix}

(136)

\begin{matrix} α_{3} = τ_{12} + α_{5} τ_{12} (τ_{11} - τ_{22}), \end{matrix}

(137)

\begin{matrix} α_{4} = τ_{12} - α_{5} τ_{12} (τ_{11} - τ_{22}), \end{matrix}

(138)

\begin{matrix} α_{5} = α_{1} + τ_{22} + α_{3} (τ_{11} + τ_{22}) + α_{5} τ_{12}^{2} . \end{matrix}

(139)

Considering the specific model given by equation (33) in the reference configuration, the counterparts of equations (37) to (41) are

\begin{matrix} α_{1} = α = μ + τ_{11}, \end{matrix}

(140)

\begin{matrix} 2 α_{2} = 2 β = 2 μ + τ_{11} + τ_{22} + 4 \bar{μ} {(τ_{11} - τ_{22})}^{2}, \end{matrix}

(141)

\begin{matrix} α_{5} = γ = μ + τ_{22}, \end{matrix}

(142)

where $τ_{12} = 0$ .

Using the strong ellipticity condition (24) for this special case, we have

\begin{matrix} γ m^{4} + 2 β m^{2} + α > 0 . \end{matrix}

(143)

For inequality (143) to hold generally, the necessary and sufficient conditions are simply

\begin{matrix} α > 0, γ > 0, β > - \sqrt{α γ} . \end{matrix}

(144)

These conditions thus ensure positive real values for $ρ c^{2}$ through the strong ellipticity condition (22).

For the model given by equation (33), when $T_{22} = 0$ on $x_{2} = 0$ , we find

\begin{matrix} p_{0} = γ, \end{matrix}

(145)

which holds everywhere, as the underlying state of the material is considered to be uniform. From equation (2)₁, we also have $p_{0} = μ$ and from equation (41), $τ_{22} = 0$ , for consistency.

With stress-free boundary conditions for a plane wave in an initially stressed incompressible material in the reference configuration, the propagation condition is given by

\begin{matrix} (α + γ - 2 β) \cos^{4} θ + 2 (β - γ) \cos^{2} θ + γ = ρ c^{2} . \end{matrix}

(146)

This determines the wave speed for any given direction of propagation in the $(x_{1}, x_{2})$ plane. Also, a shear wave polarized in the $(x_{1}, x_{2})$ plane can propagate in any direction in the same plane, provided the strong ellipticity conditions (144) hold.

In the classical theory of incompressible isotropic elasticity, we have $α = β = γ = μ$ , where $μ$ is the shear modulus. Equation (146) thus reduces to $ρ c^{2} = μ$ , independently of the direction of propagation. This gives the speed of a classical shear wave.

We now consider two cases corresponding to different values of $α, β$ and $γ$ .

5.1. Case A: $α + γ = 2 β$

This includes the special case of $α = β = γ = μ$ when evaluated in the undeformed configuration. This gives $ρ c^{2} = α$ independently of the direction of propagation in the $(x_{1}, x_{2})$ plane. This incorporates the classical theory with initial stress assumed to be zero in the undeformed configuration. Also, from equations (140) to (142), for this special case, we require $τ_{11} = τ_{22} = 0$ , since we have already assumed $τ_{22} = 0$ .

With $α + γ = 2 β$ , equation (146) can be rewritten as

\begin{matrix} ρ c^{2} = α \cos^{2} θ + γ \sin^{2} θ . \end{matrix}

(147)

For either equation (146) or equation (147) a shear wave can propagate along the principal axis x₁, with $ρ c^{2} = α$ or along the principal axis x₂, with $ρ c^{2} = γ$ .

If $α \neq γ$ , equation (147) gives

\begin{matrix} \tan^{2} θ = \frac{ρ c^{2} - α}{γ - ρ c^{2}}, \end{matrix}

(148)

with

\begin{matrix} either γ \leq ρ c^{2} \leq α, or α \leq ρ c^{2} \leq γ . \end{matrix}

(149)

For a given wave speed subject to conditions (149), equation (148) yields two (in general, distinct) directions, symmetric with respect to the axes.

5.2. Case B: $γ + α \neq 2 β$

For a given wave speed, the solutions of equation (146) may be written

\begin{matrix} \cos^{2} θ = \frac{γ - β \pm {[β^{2} - α γ + ρ c^{2} (α + γ - 2 β)]}^{\frac{1}{2}}}{α + γ - 2 β} . \end{matrix}

(150)

Considering $0 \leq \cos^{2} θ \leq 1$ , for real solutions from equation (150), we must have either

\begin{matrix} α + γ - 2 β > 0, \frac{α γ - β^{2}}{α + γ - 2 β} \leq ρ c^{2} \leq Min {α, γ}, \end{matrix}

(151)

\begin{matrix} α + γ - 2 β < 0, Max {α, γ} \leq ρ c^{2} \leq \frac{α γ - β^{2}}{α + γ - 2 β} . \end{matrix}

(152)

Equal roots arise when

\begin{matrix} ρ c^{2} = \frac{α γ - β^{2}}{α + γ - 2 β}, \cos^{2} θ = \frac{α - β}{α + γ - 2 β} . \end{matrix}

(153)

We may write the directions of propagation in the form

\begin{matrix} \tan^{2} θ = \frac{ρ c^{2} - β \pm {[β^{2} - α γ + ρ c^{2} (α + γ - 2 β)]}^{\frac{1}{2}}}{γ - ρ c^{2}} . \end{matrix}

(154)

Thus, for any given wave speed within the allowed range there are, in general, four possible distinct directions in which a plane shear wave may propagate. In a special case, these degenerate to two when condition (153) holds.

For the special value $ρ c^{2} = γ$ , the wave propagates either along the x₂ axis (as in case A) or in the direction given by

\begin{matrix} \cos^{2} θ = \frac{2 (γ - β)}{α + γ - 2 β}, \end{matrix}

(155)

in which case either $α < γ < β$ or $α > γ > β$ must hold. Similarly, the special value $ρ c^{2} = α$ means that the wave is propagating either along the x₁ direction or in the direction given by

\begin{matrix} \cos^{2} θ = \frac{γ - α}{α + γ - 2 β}, \end{matrix}

(156)

which requires that either $β < α < γ$ or $β > α > γ$ hold.

6. Wave reflection from a plane boundary in the reference configuration

For the model given by equation (33), p₀ given by equation (145), $τ_{22} = 0$ , and the stress-free incremental boundary conditions, consider a plane wave of the form given by equation (55) incident on the boundary $x_{2} = 0$ in the half-space $x_{2} < 0$ in the reference configuration. The boundary $x_{2} = 0$ is taken to be free of incremental traction but subject to the normal initial stresses $τ_{11}, τ_{22}$ .

Let the direction of propagation of this wave be $n = (n_{1}, n_{2}) = (\cos θ, \sin θ)$ and c be its speed. We now consider the following cases.

6.1. Case A: $γ + α = 2 β$

In this case, from equations (140) to (142), we require $τ_{11} = τ_{22} = 0$ . This special case therefore corresponds to vanishing of the initial stress, just as in classical mechanics. Also, it follows that $α = γ = β$ in this case. We therefore have

\begin{matrix} ρ c^{2} = α . \end{matrix}

(157)

This shows that for the materials following $γ + α = 2 β$ in the reference configuration, a single wave travels independently of the direction of propagation, with a fixed speed $α$ or $γ$ .

6.2. Case B: $γ + α \neq 2 β$

We are now considering the case of two reflected SV waves. In the case of an incident wave, equation (146) can be rewritten as

\begin{matrix} (α + γ - 2 β) n_{1}^{4} + 2 (β - γ) n_{1}^{2} + γ = ρ c^{2}, \end{matrix}

(158)

and when there is a reflected wave, we also have

\begin{matrix} (α + γ - 2 β) {n_{1}^{'}}^{4} + 2 (β - γ) {n_{1}^{'}}^{2} + γ = ρ {c^{'}}^{2}, \end{matrix}

(159)

together with Snell’s law, given by equation (70).

From equations (158) and (159), we find that either ${n_{1}^{'}}^{2} = n_{1}^{2}$ or

\begin{matrix} n_{1}^{2} {n_{1}^{'}}^{2} = γ ∕ (α + γ - 2 β), \end{matrix}

(160)

and these two possibilities occur in the case when

\begin{matrix} n_{1}^{4} = γ ∕ (α + γ - 2 β), \end{matrix}

(161)

which defines the transitional angle, say, $θ_{0}$ . Thus, a given incident wave generates two reflected waves, in general. One of these waves is reflected at the same angle as the incident wave; the angle of reflection of the second wave is given by equation (160).

For a given angle of incidence $θ$ , $θ^{'}$ is calculated from equation (160), $ρ c^{2}$ from equation (158), and $ρ {c^{'}}^{2}$ from equation (159). The reflection coefficients R and $R^{'}$ are calculated using the boundary conditions.

For a given $θ$ , necessary and sufficient conditions for equation (160) to yield a real angle $θ^{'}$ are

\begin{matrix} α & > 2 β, \end{matrix}

(162)

\begin{matrix} \tan^{2} θ & \leq \frac{α - 2 β}{γ} \equiv \tan^{2} θ_{c}, \end{matrix}

(163)

where $θ_{c}$ is defined as the critical angle given by the right-hand identity in equation (163).

The explicit expressions for R and $R^{'}$ are given by equation (86) and it is obvious that $R^{'}$ vanishes for the normal incidence. The three nontrivial cases where $R^{'}$ can possibly vanish are considered next:

$R^{'} = 0, R \neq \pm 1$ . In this case, either $n_{2} = 0$ , i.e., grazing incidence, which is not possible, or

\begin{matrix} n_{1}^{2} = \frac{1}{2} = \frac{2 γ}{α + γ - 2 β} = {n_{1}^{'}}^{2} . \end{matrix}

(164)

For equation (164) to yield a real angle, inequality (162) must hold along with equation (144). The wave speed in this case is given by

\begin{matrix} ρ c^{2} = β + γ . \end{matrix}

(165)

$R^{'} = 0, R = 1$ . In this case, we have $n_{1}^{2} = 1 ∕ 2$ and there is no displacement along the boundary in the x₁ direction.

The wave speed is given by

\begin{matrix} ρ c^{2} = (α + γ + 2 β) ∕ 4 . \end{matrix}

(166)

$R^{'} = 0, R = - 1$ . In this case, we have

\begin{matrix} n_{1}^{2} = \frac{2 γ}{α + γ - 2 β} and {n_{1}^{'}}^{2} = \frac{1}{2} . \end{matrix}

(167)

For equation (167)₁ to yield a real angle, inequality (162) must hold along with equation (144). The wave speed in this case is given by

\begin{matrix} ρ c^{2} = \frac{γ (α + γ + 2 β)}{α + γ - 2 β} . \end{matrix}

(168)

In this case, it is found that there is no displacement normal to the boundary in the x₂ direction.

6.3. Analysis

For the specific model given by equation (33), for an initially stressed incompressible material in the reference configuration, we have

\begin{matrix} α = μ + τ_{11}, 2 β = 2 μ + τ_{11} + 4 \bar{μ} τ_{11}^{2}, γ = μ . \end{matrix}

(169)

The case when $α + γ = 2 β$ refers to the case when the initial stress vanishes. The results are therefore equivalent to those for an isotropic linear elastic material in the classical theory. For details, see the discussion in Section 6.1.

For the material model given by equation (33), $α + γ \neq 2 β$ and the general results in Section 6.2 should apply. For this, we define the dimensionless quantity

\begin{matrix} Ā_{2} = (α + γ - 2 β) ∕ β = - 4 b_{0} {\bar{τ}}^{2} . \end{matrix}

(170)

Since $γ > 0$ , owing to strong ellipticity, and $α > 2 β$ from equation (160), we must have $\bar{μ} < 0$ . Also we have from equations (162) and (163) for real angles

\begin{matrix} Ā_{2} < 1, and \cos^{2} θ \geq 1 ∕ Ā_{2}, \end{matrix}

(171)

which gives the range of values for the angle of incidence $θ$ for which a reflected wave exists. For $0 \leq Ā_{2} < 1$ , the second reflected wave is replaced by a surface wave.

The strong ellipticity conditions (51), in this case, give the sufficient conditions as

\begin{matrix} \bar{τ} + 1 > 0, 4 b_{0} {\bar{τ}}^{2} + \bar{τ} + 2 {(\bar{τ} + 1)}^{\frac{1}{2}} + 2 > 0 . \end{matrix}

(172)

Using equations (78) and (111), the dimensionless wave speed of the incident wave in this case is given by

\begin{matrix} ρ c^{2} ∕ γ = (- 4 b_{0} {\bar{τ}}^{2}) \cos^{4} θ + (4 b_{0} {\bar{τ}}^{2} + \bar{τ}) \cos^{2} θ + 1, \end{matrix}

(173)

as a function of $θ$ .

Similarly, for the reflected wave, the speed in its dimensionless form is given by

\begin{matrix} ρ {c^{'}}^{2} ∕ γ = {(- 4 b_{0} {\bar{τ}}^{2})}^{- 1} [\sec^{4} θ + (4 b_{0} {\bar{τ}}^{2} + \bar{τ}) \sec^{2} θ - 4 b_{0} {\bar{τ}}^{2}], \end{matrix}

(174)

as a function of $θ$ . Since, for a reflected wave to exist, $θ^{'}$ in equation (174) must be real and should fall in the range to satisfy the inequality (122). For angles outside of this range, $c^{'}$ is the speed of a surface wave, which increases indefinitely (and its amplitude vanishes) as the incident wave approaches normal incidence. The behaviors of $ρ c^{2} ∕ γ$ (dashed graph) and $ρ {c^{'}}^{2} ∕ γ$ are shown in Figure 8 for $0 \leq θ \leq θ_{c}$ , where $θ_{c} = \sec^{- 1} (\sqrt{Ā_{2}})$ . For angles of incidence greater than the critical angle, $c^{'}$ represents the surface wave and is given by $c^{'} = c ∕ n_{1}$ where c is the speed of the incident wave given by equation (125). The speed of surface wave for $θ > θ_{c}$ is given by

\begin{matrix} ρ {c^{'}}^{2} ∕ γ = \sec^{2} θ . \end{matrix}

(175)

We hence observe that the stability of waves in the reference configuration in this case depends on the magnitude of the initial stress when $b_{0}$ is fixed. Figures 9 and 10 are counterpart plots (for $λ = 1$ ) of Figures 4 and 5. Replacing $Ā_{1}$ by $Ā_{2}$ , the expressions for R and $R^{'}$ follow from equation (117) for a second reflected wave and from equation (119) for a surface wave, respectively. A comparison shows that in the absence of stretches, $R^{'}$ allows a large value of $\bar{τ}$ until it vanishes. The behavior is similar for intermediate values of $Ā_{2}$ , that is, we see a sharp increase in the magnitude of $R^{'}$ for particular choices of $\bar{τ}$ for fixed $b_{0}$ . The behavior in the absence of stretches for very small and very large values is illustrated in Figures 9(a, b) and 9(g, h) for $R^{'}$ , respectively, and in Figure 10(a, b) and Figure 10(g, h) for R, respectively. The real and imaginary parts of $R^{'}$ and R are shown in Figures 11 and 12. In these figures, the symmetry of the curves about the angle of incidence is obvious; however, the loss of symmetric behavior is noted in Figure 11(g) in the case of $R^{'}$ . In the case of each plot for $R^{'}$ , only those values of $θ$ refer to a reflected wave for which the inequality (171) holds. For instance, in Figure 9(a), since $Ā_{2} = 2$ , a reflected wave does not exist for the range $π ∕ 4 < θ < 3 π ∕ 4$ (or equivalently $0.7854 < θ < 2.3562$ ) and rather a surface wave exists.

7. Conclusions

In this paper, a detailed discussion is presented to elaborate the effect of a homogeneous initial stress present in a material body in its reference state on the speed of plane waves.

A prototype strain energy function is used, along with various specialized cases, and a theoretical discussion is presented to study the effect of the initial stress on the speed of plane waves. For a detailed discussion of the reflection of plane waves from the boundary of an incompressible material, expressions for the reflection coefficients are derived; these include parameters representing the presence of the initial stress. It is found that the presence of a homogeneous initial stress has a considerable effect on the wave. In most cases, an incident plane wave is reflected as a plane wave (with the same angle of reflection as the angle of incidence) and an additional plane wave, the angle of reflection of which is in terms of the angle of incidence. An inequality involving a transitional angle is found for a second reflected wave to exist, which depends on the values of material parameters, the initial stress, and the angle of incidence. In the case of violation of this inequality, a surface wave exists, which travels along the boundary of the half-space. The inequality is plotted to show the region of permissible values of the angle of incidence and the initial stress parameter. Also, the reflection coefficient for the incident wave exhibits symmetric behavior about the angle of incidence for any choice of parameters, whereas the behavior in the case of the second reflected wave is mostly symmetric but is lost for a particular range of parametric values.

Footnotes

Acknowledgements

This work formed part of a PhD thesis submitted by the author to the University of Glasgow, UK.

Funding

This work was supported by the faculty development program for the National University of Sciences and Technology through the Higher Education Commission of Pakistan.

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Reflection of plane waves from the boundary of an initially stressed incompressible half-space

Abstract

Keywords

1. Introduction

2. Plane incremental motions in an initially stressed incompressibleelastic half-space—basic equations

2.1. The strong ellipticity condition

2.2. Boundary conditions

3. Homogeneous plane waves in an initially stressed incompressiblehalf-space in the deformed configuration

3.1. Case A: α 1 + α 5 = 2 α 2

3.2. Case B: α 1 + α 5 ≠ 2 α 2

4. Wave reflection from a plane boundary in the deformed configuration

4.1. Case A: α 1 + α 5 = 2 α 2

4.2. Case B: α 5 + α 1 ≠ 2 α 2

4.3. Analysis

5. Homogeneous plane waves in an initially stressed incompressiblehalf-space in the reference configuration

5.1. Case A: α + γ = 2 β

5.2. Case B: γ + α ≠ 2 β

6. Wave reflection from a plane boundary in the reference configuration

6.1. Case A: γ + α = 2 β

6.2. Case B: γ + α ≠ 2 β

6.3. Analysis

7. Conclusions

Footnotes

Acknowledgements

Funding

References

3.1. Case A: $α_{1} + α_{5} = 2 α_{2}$

3.2. Case B: $α_{1} + α_{5} \neq 2 α_{2}$

4.1. Case A: $α_{1} + α_{5} = 2 α_{2}$

4.2. Case B: $α_{5} + α_{1} \neq 2 α_{2}$

5.1. Case A: $α + γ = 2 β$

5.2. Case B: $γ + α \neq 2 β$

6.1. Case A: $γ + α = 2 β$

6.2. Case B: $γ + α \neq 2 β$