Plane waves in anisotropic elastic–plastic material with voids

Abstract

Dispersion equation is derived for the propagation of one-dimensional plane waves in a general linear anisotropic isothermal elastic–plastic material with voids. The plasticity of the considered material is defined through the dislocation of a single slip plane and direction. The derived dispersion equation is then reduced for the relevant wave propagation in particular media, namely, monoclinic, orthotropic, transversely isotropic, and isotropic elastic–plastic material with voids. In general, it is found that there exist four basic waves traveling with distinct speeds in these specific anisotropic elastic–plastic materials with voids. A new wave is found to appear because of the presence of plasticity in the material. Out of the four basic waves traveling in an orthotropic/transversely isotropic material with voids, a wave travels independent of plasticity and void parameters, whereas the remaining three waves depend on plasticity as well as on the presence of voids. The one which is traveling independent of plasticity and voids is nondispersive and nonattenuating, whereas the other waves are dispersive in nature. The speeds of all the existing waves are computed numerically for a specific model, displayed graphically, and discussed.

Keywords

Elastic–plastic anisotropic orthotropic transversely isotropic voids dislocation

1. Introduction

If an elastic material regains its original shape and size after the removal of externally applied forces, then the applied forces must be limited to some range called elastic range. But, whenever the applied forces are delimited to the elastic range, then some of the particles of the material undergo permanent deformation even after removal of the applied forces. Thus, for every elastic material, there exists a domain termed as elastic domain within which there is no permanent deformation, whereas on and outside this domain, the body undergoes permanent deformation. The boundary of this domain is called yield surface or loading surface (Green and Naghdi, 1965). Material possessing such property is a candidate of elastic–plastic materials. Every material occurring in nature or manufactured artificially contains voids, which may be vacuous pores of different shapes and sizes, distributed evenly or unevenly throughout. Such material is called porous material or material with voids. Cowin and Nunziato (1983) and Nunziato and Cowin (1979) developed linear and nonlinear theories of elastic material with voids, whose matrix material is elastic solid and the pores are vacuous pores containing nothing of mechanical of energetic significance. These vacuous pores and the elastic matrix both suffer a change during the process of deformation. The underlying idea of the theory of elastic material with voids is that the bulk density of the material is written as the density of matrix material multiplied by the void volume fraction. The void volume fraction is the ratio of the matrix volume to the bulk volume. During the deformation process, this void volume fraction changes from the reference state and thus introduces a new kinematic variable in the theory corresponding to the change in void volume fraction field, in addition to classical displacement field. Furthermore, the state of stress at a point in the material is that the load across a surface element is transmitted not only through the force stress, but also there exists equilibrated stress associated with the voids. Dey et al. (2004) and Tomar (2007) have nicely reviewed the linear theory of elastic material with voids. Using this theory, many dynamical problems have been attempted by several researchers and they have appeared in the open literature. The worth notables in the last two decades are by Tomar (2005), Tomar et al. (2013), Kaur et al. (2019), Singh and Tochhawng (2019) among several others.

The nonlinear theory of elastic material with voids was later extended by Diaconita (1987) to include plasticity effects, who presented constitutive relations and restrictions on constitutive moduli in view of the thermodynamic stability of the material. Recently, Tomar and Kumar (2020) developed the constitutive relations for an anisotropic elastic–plastic material with voids and obtained field equations for a uniform elastic–plastic material with voids. The plasticity in the material is defined through a single slip plane and direction. They have also explored the propagation of unidirectional plane waves in an infinite elastic–plastic material with voids and found that four basic waves may propagate consisting of three coupled elastic–plastic waves and a lone transverse wave. The transverse wave remains unaffected by the presence of voids and plasticity parameters and propagates with the speed of the classical transverse wave, whereas the remaining coupled elastic–plastic waves are affected significantly by the voids and plasticity parameters present in the medium. The constitutive relations ascribing the behavior of anisotropic materials undergoing elastic–plastic deformations have been developed by Johnson et al. (1970). Zuo (2010) investigated the speeds of plastic waves in the rate-independent elastic–plastic materials with anisotropic elasticity. He proved that the speed of a plastic wave is always less than or equal to the speed of the elastic wave. Some other interesting studies in the pertinent area of research are given by Taylor (1965), Fine (1967), Balaban et al. (1970), Green and Nagdhi (1978), Johnson (1969, 1972), Khan (1980), Tanomoto (1994), Ziegler et al. (1995), Fomin and Kiselev (1997), Tian et al. (2009) and Robert et al. (2018) among others.

Our aim is to explore unidirectional waves in a linear isothermal elastic–plastic material containing voids, which is both elastically and plastically anisotropic. The permanent deformation in the material is considered through the dislocation of a plane passing through the original positions of the particles. Note that there may be many such planes in the material which can dislocate from their original positions after the removal of applied forces. The angle between the plane in the original position and the plane after the dislocation is known as slip-angle and it is different for every dislocated plane. However, in this study, we have considered only a single plane, which dislocates from its position along a particular angle. We developed the equations of motion for the anisotropic elastic–plastic material with voids having a single slip plane and direction. Dispersion equation for the propagation of one-dimensional plane waves is obtained, which further has been reduced for the cases of specific materials such as monoclinic, orthotropic, transversely isotropic, and isotropic elastic–plastic material with voids. For an orthotropic/transversely isotropic material, we found that there may exist four waves propagating in the medium with different speeds. Out of these four waves, one of the waves is arising because of elastic anisotropy, which is independent of plasticity and void parameters as well as nondispersive and nonattenuating, whereas the remaining three waves are because of plastic anisotropy and depend on plasticity and void parameters as well as dispersive too. The disturbance due to plasticity is reduced for elastic anisotropy which results in the disappearance of a wave. For a particular model, numerical computations are carried out to investigate the behavior of propagating waves by taking zinc as an elastic–plastic material with voids.

2. Equations of motion

Following Tomar and Kumar (2020), the constitutive relations and governing equations for linear anisotropic isothermal elastic–plastic material containing void pores are given by

Constitutive relations

S_{i j} = C_{i j k l} E_{k l} + \frac{D_{i j k}}{1 + ϵ} ϕ_{, k} + \frac{B_{i j}}{1 + ϵ} ϕ - \frac{1}{2} \sum_{α} C_{i j k l} a_{m k} a_{n l} (δ_{1 m} δ_{3 n} + δ_{3 m} δ_{1 n}) α_{γ}

(1)

h_{i} = \frac{A_{i j}}{1 + ϵ} ϕ_{, j} + D_{i j k} E_{j k} + \frac{d_{i}}{1 + ϵ} ϕ - \frac{1}{2} \sum_{α} D_{i j k} a_{m j} a_{n k} (δ_{1 m} δ_{3 n} + δ_{3 m} δ_{1 n}) α_{γ}

(2)

g = - B_{i j} E_{i j} - \frac{d_{i}}{1 + ϵ} ϕ_{, i} - \frac{ξ}{1 + ϵ} ϕ + \frac{1}{2} \sum_{α} B_{i j} a_{m i} a_{n j} (δ_{1 m} δ_{3 n} + δ_{3 m} δ_{1 n}) α_{γ}

(3)

where S_ij, h_i, and g are respectively, the force stress tensor, equilibrated stress vector, and equilibrated body force; ϕ is the kinetic variable corresponding to the change in void volume fraction, and E_ij is the elastic strain tensor connected with the displacement vector u_i through the relation given by

2 E_{i j} = u_{i, j} + u_{j, i}

(4)

The quantity α_γ is the plastic strain on αth−slip plane, and C_ijkl, D_ijk, A_ij, B_ij, d_i, and ξ are the constitutive coefficients. The symmetric tensor δ_ij is the Kronecker delta tensor, and a_mk is the cosine of the angle between the slide- plane in undeformed state, and that of in the deformed state, viz, if the slip plane lies initially along x_m-axis in the undeformed state and along x^′_k-axis after the deformation takes place, then a_mk = cos(x_m, x^′_k). Note that the quantity a_mk changes with change of slip plane.

For an elastic–plastic material containing void pores, we assume that the quantities u_i, ϕ, and E_ij are given by

u_{i} = u_{i}^{e} + u_{i}^{p}, ϕ = ϕ^{e} + ϕ^{p}, E_{i j} = E_{i j}^{e} + E_{i j}^{p}, (i, j = 1,2,3)

(5)

where the superscripts “e” and “p” respectively correspond to the elastic part and plastic part of the corresponding quantity. The elastic and plastic parts of ϕ is assumed to be proportional as

ϕ^{p} = ϵ ϕ^{e}

where ϵ is a constant of proportionality.

Equations of motion

S_{j i, j} + ρ f_{i} = ρ {\ddot{u}}_{i}

(6)

h_{i, i} + g + ρ l = \frac{ρ κ \ddot{ϕ}}{1 + ϵ}

(7)

where f_i is the body force per unit mass, and l is the extrinsic body force density. The quantity ρ is the density, κ is the equilibrated inertia and superposed dot represents the partial time derivative. Using (1)–(3) into (6) and (7), we obtain (after neglecting the body force densities) as

\frac{1}{2} C_{i j k l} [u_{k, l j} + u_{l, k j} - (a_{1 k} a_{3 l} + a_{3 k} a_{1 l}) γ_{, j}] + \frac{1}{1 + ϵ} (D_{i j k} ϕ_{, k j} + B_{i j} ϕ_{, j}) = ρ {\ddot{u}}_{i}

(8)

\begin{matrix} \frac{1}{2} [D_{i j k} {u_{j, k i} + u_{k, j i} - (a_{1 j} a_{3 k} + a_{3 j} a_{1 k}) γ_{, i}} \\ - B_{i j} {u_{i, j} + u_{j, i} - (a_{1 i} a_{3 j} + a_{3 i} a_{1 j}) γ}] + \frac{1}{1 + ϵ} (A_{i j} ϕ_{, j i} - ξ ϕ) = \frac{ρ κ}{1 + ϵ} \ddot{ϕ} \end{matrix}

(9)

where the symbol γ represents the plastic shear strain of the considered slip plane. Equations (8) and (9) are the equations of motion for anisotropic elastic–plastic material with voids having single slip plane and direction.

The relation between the plastic shear stress τ and the force stress S_kl is given by (Johnson, 1972; Johnson et al., 1970)

τ = \frac{1}{2} a_{m k} a_{n l} (δ_{1 m} δ_{3 n} + δ_{3 m} δ_{1 n}) S_{k l}

(10)

where the various symbols have been defined earlier. Plugging (1) into (10), we obtain

τ = \frac{1}{2} (a_{1 i} a_{3 j} + a_{3 i} a_{1 j}) [C_{i j k l} {E_{k l} - \frac{1}{2} (a_{1 k} a_{3 l} + a_{3 k} a_{1 l}) γ} + \frac{1}{1 + ϵ} (D_{i j k} ϕ_{, k} + B_{i j} ϕ)]

(11)

Now, since the plastic shear stress τ remains constant on the yield surface, consequently, the gradient of τ vanishes. When all the flow variables are function of x₁ component only, then the expression of γ can be obtained from (11) as given by

γ = \frac{1}{α} \int {\sum_{r = 1}^{3} α_{r} u_{r, 11} + \frac{1}{1 + ϵ} (β_{1} ϕ_{, 11} + β_{2} ϕ_{, 1})} d x_{1}

(12)

where

\begin{array}{l} α = \frac{1}{4} (C_{1111} + C_{3333} - 2 C_{1133}) \sin^{2} 2 θ + \frac{1}{2} (C_{1113} - C_{1333}) \sin 4 θ + C_{1313} \cos^{2} 2 θ \\ α_{r} = \frac{1}{2} (C_{r 111} - C_{r 133}) \sin 2 θ + C_{r 113} \cos 2 θ \\ β_{1} = \frac{1}{2} (D_{111} - D_{331}) \sin 2 θ + D_{311} \cos 2 θ \\ β_{2} = \frac{1}{2} (B_{11} - B_{33}) \sin 2 θ + B_{13} \cos 2 θ \end{array}

Making use of (12) and the direction cosine matrix given in Johnson (1972) into equations (8) and (9), we see that these equations become as

\sum_{r = 1}^{3} (C_{i 11 r} - \frac{α_{i} α_{r}}{α}) u_{r, 11} + \frac{1}{1 + ϵ} [(D_{i 11} - \frac{α_{i} β_{1}}{α}) ϕ_{, 11} + (B_{i 1} - \frac{α_{i} β_{2}}{α}) ϕ_{, 1}] = ρ {\ddot{u}}_{i}

(13)

\sum_{r = 1}^{3} [(D_{r 11} - \frac{β_{1} α_{r}}{α}) u_{r, 11} - B_{1 r} u_{r, 1}] + \frac{1}{1 + ϵ} [(A_{11} - \frac{β_{1}^{2}}{α}) ϕ_{, 11} - \frac{β_{1} β_{2}}{α} ϕ_{, 1} - ξ ϕ] - β_{2} γ = \frac{ρ κ}{1 + ϵ} \ddot{ϕ}

(14)

where i = 1, 2, 3.

3. Wave propagation

We shall explore the possibility of uni-directional plane waves in an anisotropic elastic–plastic material with voids of infinite extent. Thus, we consider the relevant displacements and the change in void volume fraction in the form of waves advancing in the positive direction of x₁−axis with speed v, as

{u_{i}, ϕ} = {A_{i}, B_{1}} \exp {ι k (x_{1} - v t)}

(15)

where the quantities A_i, (i = 1, 2, 3) and B₁ are the constant amplitudes; k is the wavenumber and connected with the speed v through the relation ω = k v; and ω being the angular frequency. Inserting (15) into (12), we obtain

γ = - \frac{k^{2}}{α} [\sum_{r = 1}^{3} α_{r} A_{r} + \frac{1}{1 + ϵ} (β_{1} - \frac{ι}{k} β_{2}) A_{4}] \exp {ι k (x_{1} - v t)} + \tilde{C}

(16)

where

\tilde{C}

is a constant of integration. Plugging (15) and (16) into (13) and (14) and setting the constant

\tilde{C}

to zero without loss of generality, we obtain

\sum_{r = 1}^{3} (C_{i 11 r} - \frac{α_{i} α_{r}}{α} - ρ v^{2} δ_{i r}) A_{r} + \frac{1}{1 + ϵ} [D_{i 11} - \frac{α_{i} β_{1}}{α} - \frac{ι}{k} (B_{i 1} - \frac{α_{i} β_{2}}{α})] B_{1} = 0

(17)

\sum_{r = 1}^{3} [ι k (D_{11 r} - \frac{β_{1} α_{r}}{α}) - B_{1 r} - \frac{α_{r} β_{2}}{α}] A_{r} + \frac{1}{1 + ϵ} [ι k (A_{11} - \frac{β_{1}^{2}}{α} + \frac{ξ}{k^{2}} - ρ κ v^{2}) - \frac{β_{2}}{α} (2 β_{1} - \frac{ι β_{2}}{k})] B_{1} = 0

(18)

Equations (17) and (18) constitute a set of four homogeneous equations in four unknowns, namely, A₁, A₂, A₃, and B₁. For a nontrivial solution of these equations, the determinant of the coefficient matrix of these equations must vanish, which yields

\sum_{j = 1}^{9} A_{j} v^{9 - j} = 0

(19)

where the expressions of various A_j’s are given in Appendix 1. Equation (19) is the dispersion equation for general anisotropic elastic–plastic material with voids having a single slip plane and direction, which enables us to provide the speeds of the possible propagating waves. We note that this equation is of degree eight in v, therefore, it is expected to provide eight roots, say, v_i, (i = 1(1)8) and each root will correspond to the speed of a possible propagating wave in the medium depending upon the nature of the root. Note that the coefficients of equation (19) depend on elastic moduli, plasticity parameters, and void parameters, hence, in general, the speeds of the possible propagating waves will also depend on them. Moreover, since the coefficients of equation (19) are complex-valued and frequency-dependent, therefore, the propagating waves will be dispersive and attenuating in nature. However, the roots of this equation are not easy to determine analytically, therefore, to facilitate the study, we shall reduce the equation for specific anisotropic material in the following section.

4. Specific materials

To reduce the general dispersion equation for specific anisotropic elastic–plastic material with voids, we shall set the values of material constants into (19) such that it reduces to corresponding dispersion relation for specific materials and then the roots of the so reduced dispersion equation will be investigated analytically.

4.1. Monoclinic material

An elastic solid material having one-plane material symmetry is known as monoclinic material. Here, we shall consider the material symmetry about the x₁ x₂-plane. Because of this plane symmetry, some of the constitutive coefficients will vanish, which are given by

C_{1113} = C_{1213} = C_{1333} = D_{311} = B_{31} = 0

Inserting these values of constitutive coefficients in the dispersion equation (19), we obtain

\sum_{j = 1}^{9} A_{j}^{'} v^{9 - j} = 0

(20)

where

\begin{matrix} A_{1}^{'} = ι ρ^{3} (α ξ_{11}^{'} - β_{2}^{2}), A_{2}^{'} = 2 ρ^{3} ω β_{1} β_{2} \\ A_{3}^{'} = - ι ρ^{2} [α (B_{11}^{2} + B_{21}^{2}) + ξ_{12}^{'} (α ξ_{11}^{'} - β_{2}^{2}) - ξ_{11}^{'} ξ_{13}^{'} + ρ ω^{2} (A_{11} α - β_{1}^{2})] \\ A_{4}^{'} = - 2 ω ρ^{2} β_{2} (β_{1} ξ_{12}^{'} - α_{1} D_{111} - α_{2} D_{211}) \\ A_{5}^{'} = - ι ρ [B_{11}^{2} {α_{2}^{2} + α_{3}^{2} - α (C_{1212} + C_{1313})} + B_{21}^{2} {α_{1}^{2} + α_{3}^{2} - α (C_{1111} + C_{1313})} \\ - 2 B_{11} B_{21} (α_{1} α_{2} - α C_{1112}) + β_{2}^{2} ξ_{14}^{'} + ρ ω^{2} {A_{11} (ξ_{13}^{'} - α ξ_{12}^{'}) + β_{1}^{2} ξ_{12}^{'} + α (D_{111}^{2} + D_{211}^{2}) \\ - 2 β_{1} (α_{1} D_{111} + α_{2} D_{211})} + ξ_{11}^{'} {α_{3}^{2} (C_{1111} + C_{1212}) + C_{1313} (α_{1}^{2} + α_{2}^{2}) + ξ_{25}^{'} - α ξ_{14}^{'}}] \\ A_{6}^{'} = - 2 ρ ω β_{2} [- β_{1} ξ_{14}^{'} + α_{1} (D_{111} (C_{1212} + C_{1313}) - C_{1112} D_{211}) + α_{2} (D_{211} (C_{1313} + C_{1111}) - C_{1112} D_{111})] \\ A_{7}^{'} = ι [B_{11}^{2} ξ_{35}^{'} + B_{21}^{2} ξ_{45}^{'} - 2 B_{11} B_{21} (α_{3}^{2} C_{1112} + α_{1} α_{2} C_{1313} - α C_{1112} C_{1313}) \\ - β_{2}^{2} C_{1313} ξ_{15}^{'} + ρ ω^{2} {α C_{1313} (D_{111}^{2} + D_{211}^{2}) + A_{11} (α ξ_{15}^{'} + ξ_{25}^{'} + ξ_{35}^{'} + ξ_{45}^{'}) \\ + 2 β_{1} [D_{111} {α_{2} C_{1112} - α_{1} (C_{1212} + C_{1313})} + D_{211} {α_{1} C_{1112} - α_{2} (C_{1111} + C_{1313})}] + β_{1}^{2} ξ_{14}^{'} \\ + 2 D_{111} D_{211} (α_{1} α_{2} - α C_{1112}) - D_{111}^{2} (α_{2}^{2} + α_{3}^{2} - α C_{1212}) - D_{211}^{2} (α_{1}^{2} + α_{3}^{2} - α C_{1111})} \\ + ξ_{11}^{'} {C_{1313} ξ_{25}^{'} + ξ_{15}^{'} (C_{1313} α - α_{3}^{2})}], A_{8}^{'} = 2 ω β_{2} C_{1313} (β_{1} ξ_{15}^{'} + α_{2} ξ_{16}^{'} + α_{1} ξ_{17}^{'}) \\ A_{9}^{'} = ι ω^{2} [C_{1313} {{(α_{1} D_{211} - α_{2} D_{111})}^{2} + A_{11} {- ξ_{25}^{'} + α ξ_{15}^{'} + 2 β_{1} (α_{1} ξ_{17}^{'} + α_{2} ξ_{16}^{'})} \\ + β_{1}^{2} ξ_{15}^{'} - α (D_{111} ξ_{17}^{'} + D_{211} ξ_{16}^{'})} + α_{3}^{2} (A_{11} ξ_{15}^{'} + D_{111} ξ_{17}^{'} + D_{211} ξ_{16}^{'})] \\ ξ_{11}^{'} = κ ρ ω^{2} - ξ, ξ_{13}^{'} = α_{1}^{2} + α_{2}^{2} + α_{3}^{2}, ξ_{12}^{'} = C_{1111} + C_{1212} + C_{1313} \\ ξ_{14}^{'} = C_{1111} C_{1313} + C_{1212} C_{1313} - ξ_{15}^{'}, ξ_{15}^{'} = C_{1112}^{2} - C_{1111} C_{1212} \\ ξ_{16}^{'} = C_{1111} D_{211} - C_{1112} D_{111}, ξ_{17}^{'} = C_{1212} D_{111} - C_{1112} D_{211} \\ ξ_{25}^{'} = α_{2}^{2} C_{1111} + α_{1}^{2} C_{1212} - 2 α_{1} α_{2} C_{1112}, ξ_{35}^{'} = α_{3}^{2} C_{1212} + α_{2}^{2} C_{1313} - α C_{1212} C_{1313} \\ ξ_{45}^{'} = α_{3}^{2} C_{1111} + α_{1}^{2} C_{1313} - α C_{1111} C_{1313} \end{matrix}

Equation (20) is the dispersion equation for one-dimensional plane wave propagation in monoclinic elastic–plastic material with voids having a single slip plane and direction. This dispersion equation is new and has not been determined hitherto. It can be seen that this equation is also of degree eight in v and hence expected to yield eight roots. Each root will correspond to the speed of a possible wave, but the nature of the roots will inform whether the wave is progressive or nonprogressive, attenuating or nonattenuating, and other characteristics.

4.2. Orthotropic material

An elastic material having three-plane material symmetry is called orthotropic material. For an orthotropic material, all the odd order constitutive coefficients vanish and the even order tensors shall take the form given by

C_{1112} = C_{1233} = 0, A_{11} = a_{1}, B_{11} = b_{1}, B_{33} = b_{3}

where a₁, b₁, and b₃ are scalar constants. Using these values in the dispersion equation (20), we obtain

(ρ v^{2} - C_{1212}) ({\bar{A}}_{1} v^{6} + {\bar{A}}_{2} v^{4} + {\bar{A}}_{3} v^{2} + {\bar{A}}_{4}) = 0

(21)

where

\begin{array}{l} {\bar{A}}_{1} = ρ^{2} (β_{2}^{2} + α ξ - α ρ κ ω^{2}), {\bar{A}}_{2} = ρ [α b_{1}^{2} - β_{2}^{2} {\bar{c}}_{2} + ξ {\bar{c}}_{3} + ρ ω^{2} (a_{1} α - κ {\bar{c}}_{3})] \\ {\bar{A}}_{3} = b_{1}^{2} (α_{3}^{2} - α C_{1313}) + β_{2}^{2} C_{1111} C_{1313} + ξ {\bar{c}}_{1} + ρ ω^{2} (a_{1} {\bar{c}}_{3} - {\bar{c}}_{1} κ), {\bar{A}}_{4} = ω^{2} a_{1} \\ {\bar{c}}_{1}, {\bar{c}}_{1} = α C_{1111} C_{1313} - α_{1}^{2} C_{1313} - α_{3}^{2} C_{1111}, {\bar{c}}_{2} = C_{1111} + C_{1313}, {\bar{c}}_{3} = α_{1}^{2} + α_{3}^{2} - α {\bar{c}}_{2} \end{array}

Equation (21) is the dispersion equation for an orthotropic elastic–plastic material with voids having a single slip plane and direction.

4.3. Transversely isotropic material

An elastic material having material symmetry about an axis perpendicular to the plane of isotropy is called transversely isotropic material. For a transversely isotropic material, we shall have the following relations between the various constitutive coefficients given by

C_{1111} \neq 0, C_{1133} = C_{1122}, C_{2222} = C_{3333}, C_{1313} \neq 0, C_{1212} = \frac{1}{2} (C_{1111} - C_{1122}), ξ \neq 0

And the quantities A₁₁, B₁₁, and B₃₃ are the same as given earlier for orthotropic material. Using these relations into (21), we obtain

(ρ v^{2} - C_{1212}) ({\bar{A}}_{1} v^{6} + {\bar{A}}_{2} v^{4} + {\bar{A}}_{3} v^{2} + {\bar{A}}_{4}) = 0

(22)

where the expressions of the quantities

{\bar{A}}_{i}

’s have been defined earlier. Equation (22) is the dispersion equation for the one-dimensional plane waves propagating in the transversely isotropic elastic–plastic material with voids having a single slip plane and direction. Note that equations (21) and (22) are exactly matched with a minor difference that the symbol C₁₂₁₂ in the former is arbitrarily nonzero, whereas in the later it is specifically nonzero. Equation (21) or (22) yields

ρ v^{2} - C_{1212} = 0, {\bar{A}}_{1} v^{6} + {\bar{A}}_{2} v^{4} + {\bar{A}}_{3} v^{2} + {\bar{A}}_{4} = 0

(23)

The first equation of (23) yields $v (\equiv v_{4}) = \sqrt{C_{1212} / ρ}$ and arises because of elastic anisotropy which is real valued and corresponds to the speed of a propagating wave. It is also clear that the corresponding wave depends only on the elastic parameters and remains independent of plasticity and voids present in the medium. Moreover, the wave is nondispersive and nonattenuating. Therefore, we can say that this wave is arising in the medium because of elastic anisotropy. The second equation of (23) is cubic in v² and may give the speeds of three propagating waves. It is clear from the nature of the coefficients of this equation that the speeds of propagating waves shall depend on material parameters and frequency of the propagating waves. Hence, the waves corresponding to the roots of equation (23)₂ are arising because of elasticity, plasticity, and voids present in the medium. These waves are called elastic–plastic waves in the considered anisotropic material and possess quasi-nature (Johnson, 1972).

We conclude that in a transversely isotropic/orthotropic elastic–plastic material with voids having a single slip plane and direction, there may propagate four waves with distinct speeds. Out of these four propagating waves, one of the waves is nondispersive, nonattenuating, and propagates independent of the presence of plasticity and voids in the medium, whereas the remaining three waves are dispersive, depend upon the presence of voids, plasticity, and may be attenuating.

It can be noticed immediately that in the absence of plastic anisotropy from the medium, that is, when the slide angle θ = 0, the coefficient ${\bar{A}}_{4}$ vanishes and the second equation of (23) reduces to

v^{2} (A_{1}^{'} v^{4} + A_{2}^{'} v^{2} + A_{3}^{'}) = 0

(24)

where A^′₁ = ρ(ξ − ρκω²),

A_{2}^{'} = ρ ω^{2} (a_{1} + κ C_{1111}) + b_{1}^{2} - ξ C_{1111}

, and A^′₃ = ω²a₁C₁₁₁₁.

The roots of equation (24) are given by

v_{1}^{2} = 0, v_{2,3}^{2} = \frac{1}{2 A_{1}^{'}} (- A_{2}^{'} \pm \sqrt{A_{2}^{' 2} - 4 A_{1}^{'} A_{3}^{'}})

(25)

We see that one of the roots vanishes, which indicate that the speed of the corresponding wave vanishes and hence does not appear. Thus, we conclude that the plasticity of the medium is responsible to generate a new wave, which disappears in its absence. The nature of the remaining waves can be studied from the formula of $v_{2,3}^{2}$ in (25). Each of these formulas corresponds to the speed of a wave. However, the nature of the root will decide whether the corresponding wave is progressive or nonprogressive. Hence, in general, there exist three elastic waves propagating with distinct speeds in an transversely isotropic/orthotropic elastic medium with voids. Out of which, speeds of the two waves depend on the material parameters, void parameters, and the frequency parameter, whereas the speed of the remaining one wave is independent of these parameters. The waves will face critical frequency if the quantity in the denominator of the formula in (25)₂ vanishes, that is, when $ω = ω_{c} = \sqrt{ξ / (ρ κ)}$ . This means that the corresponding waves are propagating waves for ω > ω_c, whereas in the range 0 < ω ≤ ω_c, they are not the propagating waves but merely represent distance decaying sinusoidal vibrations. An interesting analysis can be conducted for equation (23) by setting v = V⁻¹ to obtain

{\bar{A}}_{4} V^{6} + {\bar{A}}_{3} V^{4} + {\bar{A}}_{2} V^{2} + {\bar{A}}_{1} = 0

(26)

We see that the constant term ${\bar{A}}_{1}$ vanishes when $ω^{2} = (β_{2}^{2} + α ξ) / α ρ κ = (ω_{1}^{2}, say)$ . Hence, at this value of ω, one pair of the roots of equation (26) vanishes, that is, one pair of the roots of equation (23)₂ approaches toward infinity. Thus, the wave corresponding to this pair of roots may face critical frequency depending on the behavior of these roots, when frequency ω <, > ω₁. We shall present further detailed analysis under Section 6.

4.4. Isotropic material

For an isotropic elastic–plastic material with voids, the constitutive coefficients take the form

C_{i j k l} = λ δ_{i j} δ_{k l} + μ (δ_{i k} δ_{j l} + δ_{i l} δ_{j k}), A_{11} = a_{1}, B_{11} = B_{33} = b_{1}

With these values, equation (19) takes the form

(ρ v^{2} - μ) (A v^{6} + B v^{4} + C v^{2} + D) = 0

(27)

where

\begin{array}{l} A = κ (ω^{2} - c_{3}^{2}), B = - (κ ω^{2} c_{4}^{2} + c_{5}^{4} + A c_{1}^{2}), C = [(c_{1}^{2} - c_{2}^{2}) A + c_{5}^{4}] c_{2}^{2} \sin^{2} 2 θ + κ ω^{2} c_{1}^{2} c_{4}^{2} \\ D = κ (c_{2}^{2} - c_{1}^{2}) ω^{2} c_{2}^{2} c_{4}^{2} \sin^{2} 2 θ, c_{1}^{2} = \frac{λ + 2 μ}{ρ}, c_{2}^{2} = \frac{μ}{ρ}, c_{3}^{2} = \frac{ξ}{ρ κ}, c_{4}^{2} = \frac{a_{1}}{ρ κ}, c_{5}^{2} = \frac{b_{1}}{ρ} \end{array}

Equation (27) is the dispersion equation for the isotropic elastic–plastic material with voids, which is the same as obtained by Tomar and Kumar (2020) for the corresponding problem and the investigation has been carried out there.

4.5. Limiting frequency

We note that the various coefficients of equation (19) depend on frequency. Hence, in general, the roots of (19) representing the speeds of possible propagating waves depend on the frequency, and thus the waves are expected to be dispersive. We shall analyze the behavior of the speeds of these waves propagating through the monoclinic, orthotropic, and transversely isotropic elastic–plastic medium with voids under very high and very low frequency cases. This analysis of dispersion equation relevant to the transversely isotropic and relevant to the orthotropic elastic–plastic medium with voids need not to do separately as their corresponding dispersion equation is exactly the same with a minor difference as explained above. Moreover, the analysis of the dispersion equation relevant to the isotropic elastic–plastic material with voids under limiting frequency has already been investigated by Tomar and Kumar (2020) hence not given here.

4.5.1. Limitly high frequency

For very high-frequency waves, the frequency approaches to infinity, that is, ω → ∞. In this case, the dispersion equation (20) relevant to the monoclinic elastic–plastic medium with voids reduces to

B_{1}^{'} v^{8} + B_{2}^{'} v^{6} + B_{3}^{'} v^{4} + B_{4}^{'} v^{2} + B_{5}^{'} = 0

(28)

where

\begin{matrix} B_{1}^{'} = ρ^{4} κ α, B_{2}^{'} = - ρ^{3} {κ (α ξ_{12}^{'} - ξ_{13}^{'}) + A_{11} α - β_{1}^{2}} \\ B_{3}^{'} = - ρ^{2} [A_{11} (ξ_{13}^{'} - α ξ_{12}^{'}) + β_{1}^{2} ξ_{12}^{'} + α (D_{111}^{2} + D_{211}^{2}) - 2 β_{1} (α_{1} D_{111} + α_{2} D_{211}) \\ + κ {α_{3}^{2} (C_{1111} + C_{1212}) + C_{1313} (α_{1}^{2} + α_{2}^{2}) + ξ_{25}^{'} - α ξ_{14}^{'}}] \\ B_{4}^{'} = ρ [α C_{1313} (D_{111}^{2} + D_{211}^{2}) + A_{11} (α ξ_{15}^{'} + ξ_{25}^{'} + ξ_{35}^{'} + ξ_{45}^{'}) \\ + 2 β_{1} [D_{111} {α_{2} C_{1112} - α_{1} (C_{1212} + C_{1313})} + D_{211} {α_{1} C_{1112} - α_{2} (C_{1111} + C_{1313})}] \\ + β_{1}^{2} ξ_{14}^{'} + 2 D_{111} D_{211} (α_{1} α_{2} - α C_{1112}) - D_{111}^{2} (α_{2}^{2} + α_{3}^{2} - α C_{1212}) \\ - D_{211}^{2} (α_{1}^{2} + α_{3}^{2} - α C_{1111}) + κ {C_{1313} ξ_{25}^{'} + ξ_{15}^{'} (C_{1313} α - α_{3}^{2})}] \\ B_{5}^{'} = C_{1313} {{(α_{1} D_{211} - α_{2} D_{111})}^{2} + A_{11} {- ξ_{25}^{'} + α ξ_{15}^{'} + 2 β_{1} (α_{1} ξ_{17}^{'} + α_{2} ξ_{16}^{'})} \\ + β_{1}^{2} ξ_{15}^{'} - α (D_{111} ξ_{17}^{'} + D_{211} ξ_{16}^{'})} + α_{3}^{2} (A_{11} ξ_{15}^{'} + D_{111} ξ_{17}^{'} + D_{211} ξ_{16}^{'}) \end{matrix}

Equation (28) being quartic in v² will provide four roots representing the speeds of four waves. Hence, under very high frequency, there may exist four basic waves in an infinite monoclinic elastic–plastic material with voids propagating with different speeds depending on elastic moduli, voids, and plasticity parameters.

The dispersion equation (21)/(22) relevant to the orthotropic/transversely isotropic elastic–plastic medium with voids leads to (23), where one of the roots is clearly corresponds to a propagating wave with speed $v_{4} = \sqrt{C_{1212} / ρ}$ and the other equation, namely, (23)₂ after making ω → ∞ reduces to

ρ^{3} α κ v^{6} + ρ^{2} (κ {\bar{c}}_{3} - a_{1} α) v^{4} + ρ ({\bar{c}}_{1} κ - a_{1} {\bar{c}}_{3}) v^{2} - a_{1} {\bar{c}}_{1} = 0

Which is a cubic equation in v², whose roots are given by

{\bar{v}}_{1}^{2} = \frac{a_{1}}{ρ κ}, {\bar{v}}_{2,3}^{2} = \frac{1}{2 ρ α} (- {\bar{c}}_{3} \pm \sqrt{{\bar{c}}_{3}^{2} - 4 α {\bar{c}}_{1}})

(29)

Thus, we conclude that in the case of very high frequency, there exist four waves in orthotropic/transversely isotropic elastic–plastic material with voids propagating with distinct speeds. We note that the speed v₄ is independent of voids and plasticity parameters, the speed v₁ depends only on void parameters, whereas the speeds v₂ and v₃ depend significantly on elasticity and plasticity parameters but independent of void parameters.

4.5.2. Limitly low frequency

For very low frequency waves, the frequency approaches to zero. Hence, when ω → 0, the dispersion equation (20) relevant to the monoclinic elastic–plastic medium with voids reduces to

v^{2} (B_{1} v^{6} + B_{2} v^{4} + B_{3} v^{2} + B_{4}) = 0

(30)

where

\begin{matrix} B_{1} = - ρ^{3} (α ξ + β_{2}^{2}), B_{2} = - ρ^{2} [α (B_{11}^{2} + B_{21}^{2}) - ξ_{12}^{'} (α ξ + β_{2}^{2}) + ξ ξ_{13}^{'}] \\ B_{3} = - ρ [B_{11}^{2} {α_{2}^{2} + α_{3}^{2} - α (C_{1212} + C_{1313})} + B_{21}^{2} {α_{1}^{2} + α_{3}^{2} - α (C_{1111} + C_{1313})} \\ - 2 B_{11} B_{21} (α_{1} α_{2} - α C_{1112}) + β_{2}^{2} ξ_{14}^{'} - ξ {ξ_{25}^{'} + α_{3}^{2} (C_{1111} + C_{1212}) + C_{1313} (α_{1}^{2} + α_{2}^{2}) - α ξ_{14}^{'}}] \\ B_{4} = B_{11}^{2} ξ_{35}^{'} + B_{21}^{2} ξ_{45}^{'} - 2 B_{11} B_{21} (α_{3}^{2} C_{1112} + α_{1} α_{2} C_{1313} - α C_{1112} C_{1313}) \\ - β_{2}^{2} C_{1313} ξ_{15}^{'} - ξ {C_{1313} ξ_{25}^{'} + ξ_{15}^{'} (C_{1313} α - α_{3}^{2})} \end{matrix}

It is clear that one of the roots of equation (30) vanishes, which mean that the corresponding wave disappears. Whereas, the other roots are given by

B_{1} v^{6} + B_{2} v^{4} + B_{3} v^{2} + B_{4} = 0

Which is a cubic equation in v² and hence will provide three roots corresponding to the speeds of three waves depending on elastic moduli, voids, and plasticity parameters. Thus, we may conclude that under limitly low frequency, there exist three waves in an infinite monoclinic elastic–plastic material with voids.

The dispersion equation (23)₂ relevant to the orthotropic/transversely isotropic elastic–plastic medium with voids under limitly low frequency case reduces to

v^{2} (A_{1}^{″} v^{4} + A_{2}^{″} v^{2} + A_{3}^{″}) = 0

(31)

where

A_{1}^{''} = ρ^{2} (β_{2}^{2} + α ξ), A_{2}^{''} = ρ (α b_{1}^{2} - β_{2}^{2} {\bar{c}}_{2} + ξ {\bar{c}}_{3}), A_{3}^{''} = ξ {\bar{c}}_{1} + β_{2}^{2} C_{1111} C_{1313} + b_{1}^{2} (α_{3}^{2} - α C_{1313})

The roots of equation (31) are given by

v_{1}^{2} = 0, v_{2,3}^{2} = \frac{1}{2 A_{1}^{''}} (- A_{2}^{''} \pm \sqrt{A_{2}^{'' 2} - 4 A_{1}^{''} A_{3}^{''}})

(32)

Here, the zero root represents that the corresponding wave disappears, and the other two roots correspond to the speeds of possible propagating waves, whose nature will depend on the sign of the coefficients $A_{1}^{''}$ , $A_{2}^{''}$ , and $A_{3}^{''}$ . Thus, we can say that under limitly low frequency case, there exist three propagating waves in an infinite orthotropic/transversely isotropic elastic–plastic medium with voids, one of which is not influenced by the presence of plasticity and voids in the medium, whereas the other two waves depend on elastic, plastic, and void parameters of the medium.

5. Characterization of waves

Note that the roots of the dispersion equation for the general anisotropic elastic–plastic material with void pores cannot be obtained analytically; therefore, it is not directly possible to characterize the types of propagating waves. A similar case happens with the dispersion equation relevant to the monoclinic material. However, an understanding about the nature of propagating waves can be envisioned from the roots of dispersion equation relevant to the orthotropic, transversely isotropic, and isotropic elastic–plastic material with voids. We see that the speed v₄ is no different than the speed of transverse waves in the elastic medium, and the remaining three speeds v_i (i = 1, 2, 3) correspond to the sets of three coupled elastic–plastic waves with quasi-nature. Each set of these coupled waves is influenced by the elastic anisotropy, plastic anisotropy, and presence of voids in the medium with dispersive and may be attenuating behaviors.

6. Computational study

To analyze the characteristic behavior of the existing waves in the orthotropic/transversely and isotropic elastic–plastic material with voids, the following values of various relevant material parameters for zinc are given in Table 1 as (Dhaliwal and Singh, 1980).

Table 1.

Numerical values of various parameters.

Symbol	Value	Unit	Symbol	Value	Unit
C ₁₁₁₁	1.628 × 10¹¹	Pa	C ₁₃₁₃	7.70 × 10¹⁰	Pa
C ₃₃₃₃	6.27 × 10¹⁰	Pa	C ₁₁₃₃	5.08 × 10¹⁰	Pa
ρ	7.14 × 10³	kg/m³	ξ	1.2196069 × 10¹¹	Pa
κ	9.5656 × 10⁻¹⁸	m²	b ₁	8.52849 × 10¹⁰	Pa
b ₃	7.41 × 10¹⁰	Pa	a ₁	1.7798 × 10⁻⁴	Pa·m²

Using these values, the dispersion equation (22) has been solved for given values of frequency ω ranging from 0 to 1.8 × 10¹² Hz at a fixed value of slip angle θ = 30^o. Clearly, one of the roots of this equation is real, therefore it corresponds to the speed of a propagating wave in the medium. The other roots are given by equation (23)₂, which is cubic in v² and shall provide three roots. Each root represents the speed of the coupled elastic–plastic wave. To compute these roots, a MATLAB program is developed and the roots of the said equation are computed.

The variation of $v_{i}^{2}, (i = 1,2,3)$ against the angular frequency ω has been depicted graphically through Figure 1. It can be seen from this figure that the waves having speeds v₁ and v₂ are propagating waves for all values of frequency ω, but the wave having speed v₃ is a nonpropagating wave in the frequency range 0 < ω ≤ 1.3393 × 10¹² and becomes propagating wave in the range ω > 1.3393 × 10¹². Therefore, the frequency ω = 1.3393 × 10¹² acts as a critical frequency for the wave having speed v₃. This value of critical frequency exactly matches with the frequency ω₁, which has been discussed under Section 4.3. Note that the speeds $v_{1}^{2}$ and $v_{2}^{2}$ have been magnified with a factor 10⁴ to show the clear comparison between the speeds.

Figure 2(a)–(c) depict the variation of the speeds v_i, (i = 1, 2, 3) against the frequency ω for three different values of the slip angle θ, namely, 0^°, 30^°, and 45^°. From Figure 2(a), it is clear that the wave having speed v₁ is dispersive in nature for the low frequency range 0 < ω < 0.4 × 10¹¹ and nondispersive for the high frequency range ω > 0.4 × 10¹¹. It can also be noticed that this wave disappears in the absence of plasticity, that is, when the slip angle θ = 0^°. Also, the speed of this wave increases with the increase of the slip angle θ and becomes dispersive in the vicinity of zero frequency. Hence, we conclude that the wave having speed v₁ is dispersive for low frequency, nondispersive for high frequency, and disappears when the plasticity effect is neglected from the material. This clearly shows that this wave arises because of the plasticity present in the medium. Figure 2(b) shows that the wave having speed v₂ is again dispersive in the low frequency range 0 < ω < 0.8 × 10¹² and nondispersive in the high frequency range ω > 0.8 × 10¹². Also, the speed of this wave decreases with the increase of the slip angle θ. From Figure 2(c), we conclude that the wave having speed v₃ is a propagating wave only after the critical frequency ω = ω₁, below which it is a nonpropagating wave. This wave is dispersive for all values of frequency beyond ω = ω₁ but only in the vicinity of ω = ω₁ and becomes nondispersive in the high frequency range. It can also be noticed that when the slip angle θ = 0^°, then the value of critical frequency ω = ω₁ is slightly less than the case when slip angle θ ≠ 0^°, in the case of transversely isotropic/orthotropic elastic medium with voids as discussed under Section 4.3. Thus, we may conclude that the wave having speed v₁ is propagating with the slowest speed, whereas the wave having speed v₃ propagates with the fastest speed in their respective range of propagation. The fastest wave faces critical frequency, and the value of critical frequency slightly depends on the plasticity present in the medium.

Figure 3(a)–(c) depict the speeds v_i, (i = 1, 2, 3) against the frequency ω for three different values of the void parameter ξ, namely, 0, 1.21960369 × 10⁸ and 1.21960369 × 10¹¹. Figure 3(a) and (b) depict the dispersion curves of the waves traveling with speeds v₁ and v₂. The speeds of these waves slightly decrease with the decrease of void parameter ξ only in a very low frequency range, but both propagate with the same and constant speeds in the high frequency range irrespective of the presence of void parameter ξ. Figure 3(c) depicts the dispersion curve of the wave having speed v₃, and observe that the critical frequency increases with the increase of void parameter ξ. Thus, we conclude that in the low frequency range, the waves traveling with speeds v₁ and v₂ are poorly affected, whereas the wave having speed v₃ is significantly affected in terms of the value of critical frequency only by the void parameter ξ.

Figure 4(a)–(c) depict the speeds of the waves having speed v_i, (i = 1, 2, 3) against the frequency ω for three different values of parameter a₁, namely, 0, 1.7798 × 10⁻⁷ and 1.7798 × 10⁻⁴. From these figures, it is clear that the speeds of the existing waves increase with the increase of the parameter a₁. The wave having speed v₁ is a propagating wave only in the frequency ranging from 0 to 0.38783 × 10¹² and disappears beyond the frequency ω = 0.38783 × 10¹², when the parameter a₁ = 0. The waves having speeds v₁ and v₂ become highly dispersive when the value of parameter a₁ decreases. The wave having speed v₂ increases with an increase of angular frequency ω, when void parameter a₁ = 1.7798 × 10⁻⁴, but the speed of this wave decreases with the increase of angular frequency ω for other two values of a₁. The wave traveling with speed v₃ appears only after the critical frequency and is significantly affected by the parameter a₁.

Figure 5 depicts the variation of the speeds v_i, (i = 1, 2, 3) against the frequency ω for two different values of the parameter b₁, namely, 8.52849 × 10⁸ and 8.52849 × 10¹⁰. The speeds of the waves having speeds v₁ and v₂ decrease when the parameter b₁ increases in the low frequency range, but the waves propagate with almost the same and constant speeds in the high frequency range. It can be noticed that the parameter b₁ is found to be responsible to bring down the value of critical frequency for the wave having speed v₃ with an increase of this parameter. Hence, for high frequency range, all the waves propagate with almost constant speeds and independent of the void parameter b₁. The effect of void parameter b₃ is also almost similar as effect of b₁ on the propagating waves. Hence, it is not shown graphically.

Figure 6(a)–(d) depict the effect of anisotropy on the dispersion curves of different propagating waves at a fixed slip angle θ = 30^°. It is clear from these figures that for all the frequencies, the speed v₁ is higher and the speed v₂ is lower in isotropic material than the transversely isotropic material. The speed v₃ does not have much influence of transverse isotropy, and the speed v₄ is highest in isotropic material than those in orthotropic/transversely isotropic material. Note that the speed of the wave having speed v₄ depends on the material constant C₁₂₁₂.

7. Conclusion

Propagation of one-dimensional plane waves has been explored in an anisotropic elastic–plastic material with voids. The effect of plasticity is considered through the slip angle of a dislocation plane form its original position. For the orthotropic, transversely isotropic, and isotropic elastic–plastic material with voids, we found that there may exist four waves propagating in the medium. We conclude that

One of the existing waves depends on the elastic parameter only and arises because of elastic anisotropy, whereas the other three existing waves travel with distinct speeds in the medium and depend on elastic, plastic, and void parameters. These waves are dispersive in the low frequency range, whereas nondispersive in the high frequency range.

The wave having speed v₁ propagates with the slowest speed, whereas the wave having speed v₃ propagates with the fastest speed in their respective propagating range.

In the absence of plasticity, there exist only three propagating waves in the medium. Out of these three waves, one of them is a transverse wave, whereas the other two are longitudinal waves. The wave having speed v₁ arises because of the plasticity present in the medium.

The wave having speed v₃ faces critical frequency, whereas the remaining waves do not face any critical frequency and exist for all frequencies. The value of the critical frequency depends upon the void parameters ξ, b₁, and b₃ and slightly on the plasticity parameter θ.

All the existing waves are found to be influenced by the anisotropy of the medium.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: One of the authors SK is grateful to the Council of Scientific and Industrial Research, New Delhi for providing financial support in the form of Senior Research Fellowship through Grant No. 09/135(0817)/2018-EMR-1.

ORCID iD

Sushil Kumar Tomar

Appendix 1

The expressions of various coefficients occurring in equation (19) are given by

\begin{matrix} A_{1} = ι ρ^{3} (α ξ_{11}^{'} - β_{2}^{2}), A_{2} = 2 ω ρ^{3} β_{1} β_{2} \\ A_{3} = - ι ρ^{2} [α (B_{11}^{2} + B_{21}^{2} + B_{31}^{2}) + ξ_{12}^{'} (α ξ_{11}^{'} - β_{2}^{2}) - ξ_{11}^{'} ξ_{13}^{'} + ρ ω^{2} (A_{11} α - β_{1}^{2})] \\ A_{4} = - 2 ω ρ^{2} β_{2} (β_{1} ξ_{12}^{'} - α_{1} D_{111} - α_{2} D_{211} - α_{3} D_{311}) \\ A_{5} = - ι ρ [B_{11}^{2} {α_{2}^{2} + α_{3}^{2} - α (C_{1212} + C_{1313})} + B_{21}^{2} {α_{1}^{2} + α_{3}^{2} - α (C_{1111} + C_{1313})} \\ + B_{31}^{2} {α_{1}^{2} + α_{2}^{2} - α (C_{1111} + C_{1212})} - β_{2}^{2} (C_{1112}^{2} + C_{1113}^{2} + C_{1213}^{2} \\ + C_{1111} C_{1212} + C_{1111} C_{1313} + C_{1212} C_{1313}) + ξ_{11}^{'} {α (- ξ_{14}^{'} + C_{1113}^{2} + C_{1213}^{2}) + α_{1}^{2} (C_{1212} + C_{1313}) \\ + α_{2}^{2} (C_{1111} + C_{1313}) + α_{3}^{2} (C_{1111} + C_{1212}) - 2 (α_{1} α_{2} C_{1112} + α_{1} α_{3} C_{1113} + α_{2} α_{3} C_{1213})} \\ + 2 α (B_{11} B_{21} C_{1112} + B_{11} B_{31} C_{1113} + B_{21} B_{31} C_{1213}) + ρ ω^{2} {A_{11} (ξ_{13}^{'} - α ξ_{12}^{'}) + β_{1}^{2} ξ_{12}^{'} \\ + α (D_{111}^{2} + D_{211}^{2} + D_{311}^{2}) - 2 β_{1} (α_{1} D_{111} + α_{2} D_{211} + α_{3} D_{311})} - 2 (α_{1} α_{2} B_{11} B_{21} \\ + α_{1} α_{3} B_{11} B_{31} + α_{2} α_{3} B_{21} B_{31})] \\ A_{6} = - 2 ρ ω β_{2} [β_{1} (C_{1113}^{2} + C_{1213}^{2} - ξ_{14}^{'}) + α_{1} (C_{1212} D_{111} - C_{1112} D_{211} - C_{1113} D_{311} \\ + C_{1313} D_{111}) + α_{2} (C_{1313} D_{211} - C_{1112} D_{111} + C_{1111} D_{211} - C_{1213} D_{311}) + α_{3} (C_{1111} D_{311} \\ - C_{1213} D_{211} - C_{1113} D_{111} + C_{1212} D_{311})] \\ A_{7} = ι [B_{11}^{2} {α (C_{1213}^{2} - C_{1212} C_{1313}) + α_{3}^{2} C_{1212} + α_{2}^{2} C_{1313} - 2 α_{2} α_{3} C_{1213}} \\ + B_{21}^{2} {α (C_{1113}^{2} - C_{1111} C_{1313}) + α_{3}^{2} C_{1111} + α_{1}^{2} C_{1313} - 2 α_{1} α_{3} C_{1113}} \\ + B_{31}^{2} {α ξ_{15}^{'} + α_{2}^{2} C_{1111} + α_{1}^{2} C_{1212} - 2 α_{1} α_{2} C_{1112}} + 2 {B_{11} B_{21} (α_{2} α_{3} C_{1113} - α_{3}^{2} C_{1112} \\ + α_{1} α_{3} C_{1213} - α_{1} α_{2} C_{1313}) + B_{11} B_{31} (α_{1} α_{2} C_{1213} - α_{1} α_{3} C_{1212} - α_{2}^{2} C_{1113} + 2 α_{2} α_{3} C_{1112}) \\ + B_{21} B_{31} (α_{1} α_{2} C_{1113} - α_{1}^{2} C_{1213} + 2 α_{1} α_{3} C_{1112} - 2 α_{2} α_{3} C_{1111})} \end{matrix}

\begin{matrix} + β_{2}^{2} (- C_{1111} C_{1213}^{2} - C_{1113}^{2} C_{1212} - ξ_{15}^{'} C_{1313} + 2 C_{1112} C_{1113} C_{1213}) \\ + 2 α {B_{11} B_{31} (C_{1113} C_{1212} - C_{1112} C_{1213}) + B_{21} B_{31} (C_{1111} C_{1213} - C_{1112} C_{1113}) \\ + B_{11} B_{21} (C_{1112} C_{1313} - C_{1113} C_{1213})} + ξ_{11}^{'} {α (C_{1111} C_{1213}^{2} + C_{1113}^{2} C_{1212} + C_{1313} ξ_{15}^{'} \\ - 2 C_{1112} C_{1113} C_{1213}) + α_{3}^{2} (C_{1111} C_{1212} - C_{1112}^{2}) + α_{2}^{2} (C_{1111} C_{1313} - C_{1113}^{2}) \\ + α_{1}^{2} (C_{1212} C_{1313} - C_{1213}^{2}) + 2 α_{2} α_{3} (C_{1112} C_{1113} - C_{1111} C_{1213}) + 2 α_{1} α_{2} (C_{1113} C_{1213} - C_{1112} C_{1313}) \\ + 2 α_{1} α_{3} (C_{1112} C_{1213} - C_{1113} C_{1212})} + ρ ω^{2} {A_{11} {α (ξ_{15}^{'} + C_{1113}^{2} - C_{1111} C_{1313} \\ - C_{1212} C_{1313} + C_{1213}^{2}) - 2 (α_{1} α_{2} C_{1112} + α_{1} α_{3} C_{1113} + α_{2} α_{3} C_{1213}) + α_{1}^{2} (C_{1212} + C_{1313}) \\ + α_{2}^{2} (C_{1111} + C_{1313}) + α_{3}^{2} (C_{1212} + C_{1111})} + 2 β_{1} {α_{1} (C_{1112} D_{211} - C_{1212} D_{111} + C_{1113} D_{311} \\ - C_{1313} D_{111}) + α_{2} (C_{1112} D_{111} - C_{1111} D_{211} + C_{1213} D_{311} - C_{1313} D_{211}) + α_{3} (C_{1113} D_{111} \\ - C_{1111} D_{311} + C_{1213} D_{211} - C_{1212} D_{311})} + β_{1}^{2} (C_{1111} C_{1212} + C_{1111} C_{1313} + C_{1212} C_{1313} \\ - C_{1112}^{2} - C_{1113}^{2} - C_{1213}^{2}) + α {D_{211}^{2} (C_{1111} + C_{1313}) + D_{111}^{2} (C_{1212} + C_{1313}) + D_{311}^{2} (C_{1111} + C_{1212}) \\ - 2 (C_{1112} D_{111} D_{211} + C_{1113} D_{111} D_{311} + C_{1213} D_{211} D_{311})} - D_{111}^{2} (α_{2}^{2} + α_{3}^{2}) - D_{211}^{2} (α_{1}^{2} + α_{3}^{2}) \\ - D_{311}^{2} (α_{1}^{2} - α_{2}^{2}) + 2 (α_{1} α_{2} D_{111} D_{211} + α_{1} α_{3} D_{111} D_{311} + α_{2} α_{3} D_{211} D_{311})}] \end{matrix}

\begin{matrix} A_{8} = 2 ω β_{2} [β_{1} {C_{1213} (C_{1111} C_{1213} - C_{1112} C_{1113}) + C_{1113}^{2} C_{1212} + C_{1313} ξ_{15}^{'}} \\ + α_{1} {C_{1213} (C_{1113} D_{211} - C_{1213} D_{111}) + C_{1112} (C_{1213} D_{311} - C_{1313} D_{211}) - C_{1212} (C_{1113} D_{311} + C_{1313} D_{111})} \\ + α_{2} {C_{1113} (C_{1213} D_{111} - C_{1113} D_{211}) + C_{1112} (C_{1113} D_{311} - C_{1313} D_{111}) - C_{1111} (C_{1213} D_{311} - C_{1313} D_{211})} \\ + α_{3} {C_{1112} (C_{1113} D_{211} + C_{1213} D_{111}) - C_{1113} C_{1212} D_{111} - C_{1111} C_{1213} D_{211} - D_{311} ξ_{15}^{'}}] \\ A_{9} = ι ω^{2} [α {{(C_{1112} D_{311} - C_{1113} D_{211})}^{2} + 2 D_{111} (C_{1313} C_{1112} D_{211} - C_{1112} C_{1213} D_{311} \\ - C_{1113} C_{1213} D_{211} + C_{1212} C_{1113} D_{311}) + C_{1213}^{2} D_{111}^{2} + 2 C_{1111} C_{1213} D_{211} D_{311} - C_{1212} C_{1313} D_{111}^{2} \\ - C_{1111} C_{1313} D_{211}^{2} - C_{1111} C_{1212} D_{311}^{2} + A_{11} (C_{1111} C_{1212} C_{1313} - C_{1111} C_{1213}^{2} - C_{1212} C_{1113}^{2} \\ - C_{1313} C_{1112}^{2} + 2 C_{1112} C_{1113} C_{1213})} + α_{1}^{2} (A_{11} (C_{1213}^{2} - C_{1212} C_{1313}) + C_{1313} D_{211}^{2} \\ - 2 C_{1213} D_{211} D_{311} + C_{1212} D_{311}^{2}) + α_{2}^{2} (A_{11} (C_{1113}^{2} - C_{1111} C_{1313}) + C_{1313} D_{111}^{2} \\ - 2 C_{1113} D_{111} D_{311} + C_{1111} D_{311}^{2}) + α_{3}^{2} (A_{11} (C_{1112}^{2} - C_{1111} C_{1212}) + C_{1212} D_{111}^{2} \\ - 2 C_{1112} D_{111} D_{211} + C_{1111} D_{211}^{2}) + 2 α_{1} α_{2} {A_{11} (C_{1313} C_{1112} - C_{1113} C_{1213}) \\ + D_{311} ξ_{18}^{'} - C_{1313} D_{111} C_{211}} + 2 α_{1} α_{3} (A_{11} (C_{1212} C_{1113} - C_{1112} C_{1213}) \\ + D_{211} ξ_{18}^{'} - C_{1212} D_{111} D_{311}) + 2 α_{2} α_{3} {A_{11} (C_{1111} C_{1213} - C_{1112} C_{1113}) \\ + D_{111} ξ_{18}^{'} - C_{1111} D_{211} D_{311}} + 2 β_{1} {α_{1} {D_{311} (C_{1112} C_{1213} - C_{1212} C_{1113}) \\ + D_{211} (C_{1113} C_{1213} - C_{1313} C_{1112}) + D_{111} (C_{1212} C_{1313} - C_{1213}^{2})} + α_{2} {D_{311} (C_{1112} C_{1113} - C_{1111} C_{1213}) \\ + D_{111} (C_{1113} C_{1213} - C_{1313} C_{1112}) + D_{211} (C_{1111} C_{1313} - C_{1113}^{2})} + α_{3} {D_{211} (C_{1112} C_{1113} \\ - C_{1111} C_{1213}) + D_{111} (C_{1112} C_{1213} - C_{1212} C_{1113}) - D_{311} ξ_{15}^{'}}} + β_{1}^{2} (C_{1313} ξ_{15}^{'} - 2 C_{1112} C_{1113} C_{1213} \\ + C_{1212} C_{1113}^{2} + C_{1111} C_{1213}^{2})], ξ_{18}^{'} = C_{1112} D_{311} + C_{1113} D_{211} - C_{1213} D_{111} \end{matrix}

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