Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Plane Transient Waves in Anisotropic Layer, Ray Expansion Approach

Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_104-1

Definitions

The ray method is used to solve the boundary value problems that lead to the propagation of two-dimensional shock waves in anisotropic plates of constant thickness, taking into account rotatory inertia and transverse shear deformations, as well as the coupling of extensional and transverse vibrational modes.

Introduction

Transient wave propagation in an anisotropic plate was the main topic discussed in Cohen and Thomas (1985). The dynamical behavior of elastic anisotropic plates taking into account rotatory inertia and transverse shear deformations was considered in Mindlin and Spencer (1967), Kaczkowski (1960), and Krasyukov (1966). In particular, differential equations were derived in Mindlin and Spencer (1967) for the coupled extensional and transverse vibrational modes of an anisotropic plate of constant thickness, which have the form of two subsystems: the equations of a generalized plane strained state and equations of the Timoshenko type. However, to solve problems of the vibrations of such plates, simplifying assumptions are usually introduced to uncouple the two subsystems. Fig. 1Wave front pattern

The Ray Method

The stressed-strained state of a thin AT-cut quartz plate considering transverse shear deformations and rotatory inertia, as well as the coupling of extensional and transverse vibrational modes, is described by the following set of equations (Mindlin and Spencer 1967):
$$\begin{array}{l}{N}_{1,1}+{N}_{5,3}= h\rho {\ddot{u}}_1,\\ {}{N}_{5,1}+{N}_{3,3}= h\rho {\ddot{u}}_3,\end{array}}$$
(1)
$$\begin{array}{l}{Q}_{1,1}+{Q}_{3,3}= h\rho {\ddot{u}}_2,\\ {}{M}_{1,1}+{M}_{5,3}-{Q}_1=\frac{1}{12}{h}^3\rho\;{\dot{\Phi}}_1,\\ {}{M}_{5,1}+{M}_{3,3}-{Q}_3=\frac{1}{12}{h}^3\rho\;{\dot{\Phi}}_3,\end{array}}$$
(2)
where N1, N2, and N3 are the forces acting in the plane of the plate, Q1 and Q3 are the shear forces, M1 and M3 are the bending moments, M5 is the twisting moment, 2b is the plate thickness, ρ is the density, uj (j = 1,2,3) are the displacement vector components, Φ1 and Φ3 are the rotatory angular velocities of the normal to the middle surface of the plate, x1 and x3 are coordinates of the middle surface, x2 is the coordinate normal to the plate’s middle surface, and an index after a comma indicates differentiation with respect to the appropriate variable.

Let us assume that some dynamical excitation applied at the boundary of the plate y = x1 ν1 + x3 ν3 = 0, whereν1 =  cos φ and ν3 =  sin φ are the components of the normal vector to the boundary, induces the formation of strong surfaces of discontinuity in the plate, each a cylindrical surface S(t) with the directrix L(t) in the x1, x3-plane and generators parallel to the x2-axis (Fig. 1).

Beyond the surface S(t), the desired functions Z(x1, x3, t) are expanded in terms of ray series (Achenbach and Reddy 1967; Rossikhin and Shitikova 1995)
$$Z\left({x}_1,{x}_3,t\right)=\sum \limits_{k=0}^{\infty}\frac{1}{k!}\left[Z{,}_{(k)}\right]\left|{}_{t={yG}^{-1}}\right.{\left(t-\frac{y}{G}\right)}^k\nonumber \times H(t-\frac{y}{G}),$$
(3)
where [Z,(k)] are the jumps of the kth-order time-derivative of the function Z(x1, x3, t) across the shock wave front, G is the normal velocity of the wave, y is the distance measured from the plate boundary along the normal to the boundary, and H(t) is the Heaviside unit step function.
To determine coefficients of the ray series (3), the equations of motion (1) and (2) should be differentiated k times with respect to time t, while the relationships between the generalized forces and displacements (Mindlin and Spencer 1967) should be differentiated k + 1 times. Taking their difference across the wave surface S(t) yields
$$\begin{array}{l}\left[{N}_{1,1(k)}\right]+\left[{N}_{5,3(k)}\right]=2 b\rho \left[{v}_{1,\left(k+1\right)}\right],\\[10pt] {}\left[{N}_{5,1(k)}\right]+\left[{N}_{3,3(k)}\right]=2 b\rho \left[{v}_{3,\left(k+1\right)}\right],\end{array}}$$
$$\begin{array}{l}\left[{Q}_{1,1(k)}\right]\!{+}\!\left[{Q}_{3,3(k)}\right]{=}2 b\rho \left[{v}_{2,(k+1)}\right],\\[10pt] \left[{M}_{1,1(k)}\right]\!{+}\!\left[{M}_{5,3(k)}\right]\!{-}\!\left[{Q}_{1,(k)}\right]{=}\frac{2}{3}{b}^3\rho\!\left[{\Phi}_{1,(k+1)}\!\right]\!,\\[10pt] \left[{M}_{5,1(k)}\right]\!{+}\!\left[{M}_{3,3(k)}\right]\!{-}\!\left[{Q}_{3,(k)}\right]{=}\frac{2}{3}{b}^3\rho\!\left[{\Phi}_{3,(k+1)}\!\right]\!,\end{array}}$$
{\begin{aligned}\left[{N}_{\alpha, \left(k+1\right)}\right]&=2b\left\{{\overline{\mathrm{c}}}_{1a}\left[{v}_{1,1(k)}\right]+{\overline{\mathrm{c}}}_{a3}\left[{v}_{3,2(k)}\right]+{K}_{\raisebox{1ex}{\alpha }\!\left/ \!\raisebox{-1ex}{2}\right.+\raisebox{1ex}{5}\!\left/ \!\raisebox{-1ex}{2}\right.}{\overline{\mathrm{c}}}_{a4}\left(\left[{v}_{2,3(k)}\right]+\left[{\Phi}_{3,(k)}\right]\right)\right\},\\ {}\left[{N}_{5,\left(k+1\right)}\right]&=2b\left\{{c}_{55}\left(\left[{v}_{3,1(k)}\right]+\left[{v}_{1,3(k)}\right]\right)+{K}_1{c}_{56}\left(\left[{v}_{2,1(k)}\right]+\left[{\Phi}_{1,(k)}\right]\right)\right\},\\ {}\left[{Q}_{1,\left(k+1\right)}\right]&=2b\left\{{K}_1{c}_{56}\left(\left[{v}_{3,1(k)}\right]+\left[{v}_{1,3(k)}\right]\right)+{K_1}^2{c}_{66}\left(\left[{v}_{2,1(k)}\right]+\left[{\Phi}_{1,(k)}\right]\right)\right\},\\ {}\left[{Q}_{3,\left(k+1\right)}\right]&=2b\left\{{K}_3{\overline{\mathrm{c}}}_{14}\left[{v}_{1,1(k)}\right]+{K}_3{\overline{\mathrm{c}}}_{34}\left[{v}_{3,3(k)}\right]+{K_3}^{2}{\overline{\mathrm{c}}}_{44}\left(\left[{v}_{2,3(k)}\right]+\left[{\Phi}_{3,(k)}\right]\right)\right\},\\ {}\left[{M}_{\alpha, \left(k+1\right)}\right]&=\raisebox{1ex}{2}\!\left/ \!\raisebox{-1ex}{3}\right.{b}^3\left({\gamma}_{1\alpha}\left[{\Phi}_{1,1(k)}\right]+{\gamma}_{\alpha 3}\left[{\Phi}_{3,3(k)}\right]\right),\\{}\left[{M}_{5,\left(k+1\right)}\right]&=\raisebox{1ex}{2}\!\left/ \!\raisebox{-1ex}{3}\right.{b}^3{\gamma}_{55}\left(\left[{\Phi}_{3,1(k)}\right]+\left[{\Phi}_{1,3(k)}\right]\right),\end{aligned}}
(4)
where α = 1, 3, v1, v2, and v3 are the displacement velocities and cpq are the moduli of a quartz crystal,
\begin{aligned} {K_1}^2&=\frac{\pi^2}{12},{K_3}^2=\frac{\pi^2}{24{\overline{c}}_{44}}\Big\{{c}_{22}+{c}_{44}\\&-{\left[{\left({c}_{22}-{c}_{44}\right)}^2+4{c_{24}}^2\right]}^{1/2}\Big\},\\{\overline{\mathrm{c}}}_{pq}&={c}_{pq}-\frac{c_{2p}{c}_{q2}}{c_{22}},\\{\gamma}_{pq}&={\overline{\mathrm{c}}}_{pq}-\frac{{\overline{\mathrm{c}}}_{4p}{\overline{\mathrm{c}}}_{q4}}{{\overline{c}}_{44}},\\{\gamma}_{55}&={c}_{55}-\frac{{c_{56}}^2}{c_{66}}. \end{aligned}

Further the summation convention with respect to repeated indices not in parentheses is adopted; the Latin indices take on the values 1, 2, and 3, and Greek indices take on the values 1 and 2.

Using the compatibility condition for discontinuities in the kth-order derivative of a certain function Z(x1, x3, t) (Thomas 1961; Rossikhin and Shitikova 1995),

\begin{aligned} G\left[{Z}_{,\alpha (k)}\right]&=-\left[{Z}_{,\left(k+1\right)}\right]{\nu}_{\alpha }+\frac{d\left[{Z}_{,(k)}\right]}{dt}{\nu}_{\alpha }\\&+G{\left[{Z}_{,(k)}\right]}_{,s}{\tau}_{\alpha}\left(\alpha =1,3\right) \end{aligned}

Equations (4) are reduced to
{\begin{aligned}&\rho \left({G_{(n)}}^2-{G}^2\right){X_{\left(k+1\right)}}^{(n)}=2\rho {G_{(n)}}^2\frac{{dX_{(k)}}^{(n)}}{dt}+{Gp}_{i j}{l_i}^{(n)}\sum \limits_{f=1}^3{X_{(k)}}^{(f)}{,}_sl{}_j{}^{(f)}\\&\quad+{Gd}_{i\alpha}{l_i}^{(n)}\sum \limits_{\alpha =1}^2{Y_{(k)}}^{\left(\upsilon \right)}{l_{\alpha}}^{\left(\upsilon^{\ast} \right)}-{F}_{i\left(k-1\right)}{l_i}^{(n)},\\ &\rho \left({G_{\left(\upsilon^{\ast} \right)}}^2-{G}^2\right){Y}_{\left(k+1\right)}^{\left(\upsilon \right)}=2\rho {G}_{\left(\upsilon^{\ast} \right)}^2\frac{dY_{(k)}^{\left(\upsilon \right)}}{dt}+{Gp}_{\alpha \beta}^{\ast}{l}_{\beta}^{\left(\upsilon^{\ast} \right)}\sum \limits_{\delta =1}^2{Y}_{(k)}^{\left(\delta \right)}{,}_s{l}_{\alpha}^{\left(\delta^{\ast} \right)}\\&\quad+{Gd}_{\alpha i}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}\sum \limits_{j=1}^3{X_{(k)}}^{(f)}{l_i}^{(f)}-{U}_{\alpha \left(k-1\right)}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)},\\ \end{aligned}}
{\begin{aligned}&{F}_{i\left(k-1\right)}{l}_i^{(n)}=\rho {G}_{(n)}^2\frac{d^2{X}_{\left(k-1\right)}^{(n)}}{dt^2}+{p}_{i j}{l}_i^{(n)}G\sum \limits_{f=1}^3\frac{dX_{\left(k-1\right),s}^{(f)}}{dt}{l}_j^{(f)}\\&\quad+{G}^2{\overline{s}}_{i j}{l}_i^{(n)}\sum \limits_{f=1}^3{X}_{\left(k-1\right)}^{(f)}{,}_{ss}{l}_j^{(f)}+{d}_{i\alpha}{l}_i^{(n)}G\sum \limits_{j=1}^2\frac{dY_{\left(k-1\right)}^{\left(\upsilon \right)}}{dt}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}\\&\quad+{G}^2{\overline{d}}_{i\alpha}{l}_i^{(n)}\sum \limits_{\upsilon =1}^2{Y}_{\left(k-1\right)}^{\left(\upsilon \right)}{,}_s{l}_{\alpha}^{\left(\upsilon^{\ast} \right)},\\ &{U}_{\alpha \left(k-1\right)}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}=\rho {G}_{\left(\upsilon^{\ast} \right)}^2\frac{d^2{Y}_{\left(k-1\right)}^{\left(\upsilon \right)}}{dt^2}+{p}_{\alpha \beta}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}G\sum \limits_{\delta =1}^2\frac{dY_{\left(k-1\right)}^{\left(\delta \right)}{,}_s}{dt}{l}_{\beta}^{\left(\delta^{\ast} \right)}\\&\quad+{\overline{s}}_{\alpha \beta}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}{G}^2\sum \limits_{\delta =1}^2{Y}_{\left(k-1\right)}^{\left(\delta \right)},{{}_{ss}l}_{\beta}^{\left(\delta^{\ast} \right)}+{d}_{\alpha i}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}G\sum \limits_{f=1}^3\frac{dX_{\left(k-1\right)}^{(f)}}{dt}{l}_i^{(f)}\\&\quad+{\overline{d}}_{\alpha i}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}{G}^2\sum \limits_{f=1}^3{X}_{\left(k-1\right)}^{(f)}{,}_s{l}_i^{(f)},\end{aligned}}
(5)
where
{\begin{aligned}{X}_{(k)}^{(f)}&={X}_{i(k)}{l}_i^{(f)},{X}_{1(k)}=\left[{v}_{1,(k)}\right],{X}_{2(k)}=\left[{v}_{3,(k)}\right],\\{X}_{3(k)}&=\left[{v}_{2,(k)}\right],{Y}_{(k)}^{\left(\upsilon \right)}={Y}_{\alpha (k)}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)},{Y}_{1(k)}=\left[{\Phi}_{1,(k)}\right],{Y}_{2(k)}=\left[{\Phi}_{3,(k)}\right],\\ {}{p}_{11}&=2\left({\overline{\mathrm{c}}}_{11}{\tau}_1{\nu}_1+{c}_{55}{\tau}_3{\nu}_3\right),{p}_{12}={p}_{21}=\left({\overline{\mathrm{c}}}_{13}+{c}_{55}\right)\left({\tau}_1{\nu}_3+{\tau}_3{\nu}_1\right),\\{p}_{22}&=2\left({\overline{\mathrm{c}}}_{33}{\tau}_3{\nu}_3+{c}_{55}{\tau}_1{\nu}_1\right), {p}_{13}={p}_{31}=\left({K}_3{\overline{\mathrm{c}}}_{14}+{K}_1{c}_{56}\right)\left({\tau}_1{\nu}_3+{\tau}_3{\nu}_1\right),\\{p}_{23}&={p}_{32}=2\left({K}_1{c}_{56}{\tau}_1{\nu}_1+{K}_3{\overline{\mathrm{c}}}_{34}{\tau}_3{\nu}_3\right),{p}_{33}=2\left({K}_1^2{c}_{66}{\tau}_1{\nu}_1+{K}_3^2{\overline{\mathrm{c}}}_{44}{\tau}_3{\nu}_3\right),\\ {}{d}_{11}&={K}_1{c}_{56}{\nu}_3,{d}_{21}={K}_1^2{c}_{56}{\nu}_1,{d}_{31}={K}_1^2{c}_{66}{\nu}_1,{d}_{12}={K}_3{\overline{\mathrm{c}}}_{14}{\nu}_1,\\{d}_{22}&={K}_3{\overline{\mathrm{c}}}_{34}{\nu}_3,{d}_{32}={K}_3^2{\overline{\mathrm{c}}}_{44}{\nu}_3,{d}_{11}^{\ast}=-3{b}^{-2}{K}_1{c}_{56}{\nu}_1,{d}_{12}^{\ast}=-3{b}^{-2}{K}_1{c}_{56}{\nu}_3\end{aligned}}

The quantities $${p}_{\alpha \beta}^{\ast}$$ are derived from pαβ by replacing $${\overline{c}}_{mn}$$ (m, n = 1, 3) in the latter with γmn and c55 with γ55, s = x1τ1 + x3 τ3 (τ1 =  −  sin φ, τ3 =  cos φ) is the distance measured from the origin along the wave surface, $$\rho {G}_{(f)}^2$$ and $$\rho {G}_{\left({\gamma}^{\ast}\right)}^2$$ are the principal values, and $${l}_i^{(f)},{l}_{\alpha}^{\left(\gamma^{\ast} \right)}$$ are the unit principal directions of the symmetric tensors $${s}_{ij}={\left.\frac{1}{2}{p}_{ij}\right|}_{\tau_{\delta }={\nu}_{\delta }}$$ and $${s}_{\alpha \beta}^{\ast}={\left.\frac{1}{2}{p}_{\alpha \beta}^{\ast}\right|}_{\tau_{\delta }={\nu}_{\delta }}$$ (δ = 1, 3), respectively; the tensors $${\overline{s}}_{ij}$$ and $${{\overline{s}}^{\ast}}_{ij}$$are obtained from sij and $${s}_{\alpha \beta}^{\ast}$$ by replacing νδ in the latter by τδ (δ = 1, 2), and, finally, $${d}_{13}^{\ast},$$ $${d}_{23}^{\ast},$$ $${d}_{22}^{\ast},$$ and $${d}_{21}^{\ast}$$ are derived from the corresponding asterisk-free quantities by multiplying the latter by −3b−2.

Restricting below by three terms in the ray series (3) for the functions to be found and putting k = 0, 1, 2 in Eqs. (5) yield
{\begin{aligned}{X}_{(0)}^{\left(f,f\right)}&={h}_{(0)}^{(f)}(z),{X}_{(0)}^{\left(n,f\right)}=0\left(n\ne f\right),{X}_{(0)}^{\left(\alpha^{\ast}, f\right)}=0,\\ {}{Y}_{(0)}^{\left(\alpha^{\ast}, \alpha \right)}&={h}_{\left(\alpha \right)}^{\left(\alpha^{\ast} \right)}(z),{Y}_{(0)}^{\left(\alpha^{\ast}, \beta \right)}=0\left(\alpha^{\ast} \ne \beta \right),{Y}_{(0)}^{\left(n,\alpha \right)}=0,\\ {}{X}_{(1)}^{\left(f,f\right)}&={h}_{(1)}^{(f)}(z)+{tH}_{(0)}^{\left(f,f\right)},{X}_{(1)}^{\left(n,f\right)}={a}_{\left(f,n\right)}^{(n)}{h}_{(0)}^{(n)}{,}_z\left(n\ne f\right),\\ {}{X}_{(1)}^{\left(\alpha^{\ast}, f\right)}&={a}_{\left(f,\alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{h}_{(0)}^{\left(\alpha^{\ast} \right)},{Y}_{(1)}^{\left(\alpha^{\ast}, \alpha \right)}={h}_{(1)}^{\left(\alpha^{\ast} \right)}+{tM}_{(0)}^{\left(\alpha^{\ast}, \alpha \right)},\\ {}{Y}_{(1)}^{\left(\alpha^{\ast}, \beta \right)}&={d}_{\left(\beta, \alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{h}_{(0)}^{\left(\alpha^{\ast} \right)}{,}_z\left(\alpha \ne \beta \right),{Y}_{(1)}^{\left(n,\alpha \right)}={d}_{\left(\alpha, n\right)}^{(n)}{h}_{(0)}^{(n)},\\ {}{X}_{(2)}^{\left(f,f\right)}&={h}_{(2)}^{(f)}(z)+{tH}_{(1)}^{\left(f,f\right)}+\frac{1}{2}{t}^2{\chi}_{(1)}^{\left(f,f\right)},{X}_{(2)}^{\left(n,f\right)}={H}_{(1)}^{\left(n,f\right)}+{ta}_{\left(f,n\right)}^{(n)}{H}_{(0),z}^{\left(n,n\right)}\left(n\ne f\right),\\ {}{X}_{(2)}^{\left(\alpha^{\ast}, f\right)}&={H}_{(1)}^{\left(\alpha^{\ast}, f\right)}+{ta}_{\left(f,\alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{M}_{(0)}^{\left(\alpha^{\ast}, \alpha \right)},{Y}_{(2)}^{\left(\alpha^{\ast}, \alpha \right)}={h}_{(2)}^{\left(\alpha^{\ast} \right)}(z)+{tM}_{(1)}^{\left(\alpha^{\ast}, \alpha \right)}+\frac{1}{2}{t}^2{\chi}_{(1)}^{\left(\alpha^{\ast}, \alpha \right)},\\ {}{Y}_{(2)}^{\left(\alpha^{\ast}, \beta \right)}&={M}_{(1)}^{\left(\alpha^{\ast}, \beta \right)}+{td}_{\left(\beta, \alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{M}_{(0),z}^{\left(\alpha^{\ast}, \alpha \right)}\left(\alpha \ne \beta \right),{Y}_{(2)}^{\left(n,\alpha \right)}={M}_{(1)}^{\left(n,\alpha \right)}+{td}_{\left(\alpha, n\right)}^{(n)}{H}_{(0)}^{\left(n,n\right)},\end{aligned}}
(6)
where the first superscript in parentheses indicates the mode number (the first three waves are numbered by 1, 2, and 3 and the fourth and fifth modes by 1 and 2, respectively) and the second indicates the projection number; $${h}_{(k)}^{(1)},$$ $${h}_{(k)}^{(2)},$$ and $${h}_{(k)}^{(3)}$$ (k = 0, 1, 2) are arbitrary functions of the argument z = s − g(f) t; $${h}_{(k)}^{\left(1^{\ast}\right)}$$ and $${h}_{(k)}^{\left(2^{\ast}\right)}$$ (k = 0, 1, 2) are arbitrary functions of the argument z   = s − $${{g}}_{\left(\alpha^{\ast} \right)}$$ t (for convenience, the indices at z have been omitted in the formulas); the functions $${H}_{(k)}^{\left(n,f\right)},{M}_{(k)}^{\left(\alpha^{\ast}, \alpha \right)}$$ (k = 0,1), $${M}_{(1)}^{\left(n,\alpha \right)},$$ $${\chi}_{(1)}^{\left(f,f\right)}$$, and $${\chi}_{(1)}^{\left(\alpha^{\ast}, \alpha \right)}$$, which depend on the aforementioned arbitrary functions and their z-derivatives, are not given here, and
\begin{aligned} {g}_{(n)}&=\frac{p_{ij}{l}_i^{(n)}{l}_j^{(n)}}{2\rho {G}_{(n)}},\\{g}_{\left(\upsilon^{\ast} \right)}&=\frac{p_{\alpha \beta}^{\ast}{l}_{\alpha}^{\left(\upsilon^{\ast} \right)}{l}_{\beta}^{\left(\upsilon^{\ast} \right)}}{2\rho {G}_{\left(\upsilon^{\ast} \right)}}. \end{aligned}
(7)

Using relationships (6) the solution could be constructed in the form of the ray series (3) with the corresponding Heaviside functions for each of the five shock wave modes. As the problem under consideration is linear, the final result could be obtained by simply adding together the ray series constructed. The five sets of arbitrary functions appearing in the solution are determined by five boundary conditions. Following Rossikhin and Shitikova (1991), several types of boundary conditions are considered below.

Let us assume first that at the edge of the plate, the following vales are given: three velocities of displacements and two velocities of angular rotation of the normal to the middle surface of the plate as functions of the time t and coordinate s, i.e.,
{\begin{aligned}{\left.{v}_n\right|}_{y=0}=\sum \limits_{k=0}^{\infty }{X}_{n(k)}^0(s)\frac{t^k}{k!},\\{\left.{v}_{\tau}\right|}_{y=0}=\sum \limits_{k=0}^{\infty }{X}_{\tau (k)}^0(s)\frac{t^k}{k!},\\{\left.{v}_2\right|}_{y=0}=\sum \limits_{k=0}^{\infty }{X}_{3(k)}^0(s)\frac{t^k}{k!},\\ {}{\left.{\Phi}_n\right|}_{y=0}=\sum \limits_{k=0}^{\infty }{Y}_{n(k)}^0(s)\frac{t^k}{k!},\\{\left.{\Phi}_{\tau}\right|}_{y=0}=\sum \limits_{k=0}^{\infty }{Y}_{\tau (k)}^0(s)\frac{t^k}{k!}.\end{aligned}}
This yields the following relations for determining the arbitrary functions:
{ \begin{aligned}&\sum \limits_{f=1}^3{h}_{(0)}^{(f)}{C}_N^{(f)}={X}_{N(0)}^0,\sum \limits_{\alpha =1}^2{h}_{(0)}^{\left(\alpha^{\ast} \right)}{B}_M^{\left(\alpha^{\ast} \right)}={Y}_{M(0)}^0\\&{}\sum \limits_{f=1}^3{h}_{(1)}^{(f)}{C}_N^{(f)}=-\sum \limits_{f=1}^3\sum \limits_{\left(f\ne n\right)n=1}^3{a}_{\left(f,n\right)}^{(n)}{h}_{(0)}^{(n)},{{}_zC}_N^{(f)}-\sum \limits_{f=1}^3\sum \limits_{\alpha =1}^2{a}_{\left(f,\alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{h}_{(0)}^{\left(\alpha^{\ast} \right)}{C}_N^{(f)}+{X}_{N(1)}^0,\\ &{}\sum \limits_{\alpha =1}^2{h}_{(1)}^{\left(\alpha^{\ast} \right)}{B}_M^{\left(\alpha^{\ast} \right)}=-\sum \limits_{\alpha =1}^2\sum \limits_{\left(\alpha \ne \beta \right)\beta =1}^2{d}_{\left(\beta, \alpha^{\ast} \right)}^{\left(\alpha^{\ast} \right)}{h}_{(0)}^{\left(\alpha^{\ast} \right)}{,}_z{B}_M^{\left(\beta^{\ast} \right)}-\sum \limits_{n=1}^3\sum \limits_{\alpha =1}^2{d}_{\left(\alpha, n\right)}^{(n)}{h}_{(0)}^{(n)}{B}_M^{\left(\alpha^{\ast} \right)}+{Y}_{M(1)}^0,\\ &{}\sum \limits_{f=1}^3{h}_{(2)}^{(f)}{C}_N^f=-\sum \limits_{f=1}^3\sum \limits_{\left(f\ne n\right)n=1}^3{H}_{(1)}^{\left(n,f\right)}{C}_N^{(f)}-\sum \limits_{\alpha =1}^2\sum \limits_{f=1}^3{H}_{(1)}^{\left(\alpha^{\ast}, f\right)}{C}_N^{(f)}+{X}_{N(2)}^0,\\ &{}\sum \limits_{\alpha =1}^2{h}_{(2)}^{\left(\alpha^{\ast} \right)}{B}_M^{\left(\alpha^{\ast} \right)}=-\sum \limits_{\alpha =1}^2\sum \limits_{\left(\alpha \ne \beta \right)\beta =1}^2{M}_{(1)}^{\left(\alpha^{\ast}, \beta \right)}{B}_M^{\left(\beta^{\ast} \right)}-\sum \limits_{n=1}^3\sum \limits_{\alpha =1}^2{M}_{(1)}^{\left(n,\alpha \right)}{B}_M^{\left(\alpha^{\ast} \right)}+{Y}_{M(2)}^0,\end{aligned}}
(8)
where in each system of equations, the index N will take on the values n, τ, 3 and the index M takes on the values n, τ and moreover $${C}_n^{(f)}={l}_1^{(f)}{\nu}_1+{l}_2^{(f)}{\nu}_3,$$ $${C}_{\tau}^{(f)}={l}_1^{(f)}{\tau}_1+{l}_2^{(f)}{\tau}_3,$$ $${C}_3^{(f)}={l}_3^{(f)}$$, $${B}_n^{\left(\alpha^{\ast} \right)}={l}_1^{\left(\alpha^{\ast} \right)}{\nu}_1+{l}_2^{\left(\alpha^{\ast} \right)}{\nu}_3$$, and $${B}_{\tau}^{\left(\alpha^{\ast} \right)}={l}_1^{\left(\alpha^{\ast} \right)}{\tau}_1+{l}_2^{\left(\alpha^{\ast} \right)}{\tau}_3$$.

Considering the case when at the plate edge the following values are given, the forces Nn and N in the plane of the plate, the shear force Qn, and the bending Mn and twisting M moments, then a set of equations similar to (8) for the arbitrary functions could be derived. As this takes place, the right-hand sides of the equations would involve, instead of $${X}_{N(k)}^0$$ and $${Y}_{M(k)}^0$$ (k = 0, 1, 2), the functions $${N}_{N(k)}^0$$ and $${M}_{M(k)}^0$$ defined on the boundary. Here $${N}_{n(k)}^0$$ and $${N}_{\tau (k)}^0$$ are the coefficients of the Maclaurin series for the boundary forces in the pane of the plate, $${N}_{2(k)}^0$$ are the same for the boundary shear force, and $${M}_{n(k)}^0$$ and $${M}_{\tau (k)}^0$$ are the coefficients for the boundary bending and twisting moments, respectively.

Boundary conditions of other types may be treated in an analogous fashion.

Numerical Example

Utilizing the above obtained formulas, the angle φ dependence of the propagation velocities of the strong discontinuity surface G(n), $${G}_{\left(\gamma^{\ast}\right)}$$ and of the velocities g(n), $$g_{\left(\gamma^{\ast} \right)}$$ at which the perturbations propagate along the wave surfaces could be investigated. Figures 2 and 3 illustrate dimensionless magnitudes in polar coordinates (curves 1, 2, and 3 in Fig. 2a represent the velocities $${\overline{G}}_{(1)},{\overline{G}}_{(2)},{\overline{G}}_{(3)}$$ and curves 1 and 2 in Fig. 2b the velocities $${\overline{G}}_{\left(1^{\ast}\right)},{\overline{G}}_{\left(2^{\ast}\right)}$$). For reasons of symmetry, Fig. 2 shows only the upper parts of the curves and Fig. 3 the right-hand parts; in so doing after the reflection in the vertical axis, the solid curves in Fig. 3 become dashed curves and vice versa. Fig. 2The φ-dependence of normal dimensionless velocities of propagating waves Fig. 3The φ-dependence of tangential dimensionless velocities of propagating disturbances

All velocities G(i) and $${{G}}_{\left(\alpha^{\ast} \right)}$$ are given in units of the least velocity along the x1-axis direction, and all velocities g(i) and $${{g}}_{\left(\gamma^{\ast} \right)}$$ are given in units of the least velocity in the direction of propagation at φ = 50°.

It is clear from the comparison of the curves that the extremal values of the velocities $${\overline{G}}_{(n)}$$and $${\overline{G}}_{\left(\gamma^{\ast} \right)}$$ correspond to vanishing velocities $${\overline{g}}_{(n)}$$ and $${\overline{g}}_{\left(\gamma^{\ast} \right)}$$; moreover, the values $${\overline{g}}_{(n)},{\overline{g}}_{\left(\gamma^{\ast} \right)}$$ change sign as they go through zero (the solid and dashed curves in Fig. 3ae represent positive and negative values, respectively), i.e., the radial tubes may deviate from the normal in either direction, depending on the direction in which the wave surface is propagating.

As examples of the solution of boundary-value problems, let us consider the propagation of plane waves in a quartz plate in the direction φ = 0°, triggered by a shock impulse of type (8), when all the quantities $${X}_{N(k)}^0$$ and $${Y}_{N(k)}^0$$, except $${X}_{n(0)}^0$$ and $${Y}_{n(0)}^0$$, vanish or when all the quantities $${N}_{N(k)}^0$$ and $${M}_{N(k)}^0$$, except $${N}_{2(0)}^0$$ and $${M}_{\tau (0)}^0$$, vanish.

A shock of the first type in the direction φ = 0° gives rise to five wave modes, propagating with velocities G(1) > $${{G}}_{\left(1^{\ast} \right)}$$ > G(2) > $${{G}}_{\left(2^{\ast} \right)}$$ > G(3) (numbered in decreasing order of magnitude). On the first wave, the value of v1 experiences a discontinuity, but Φ3 is continuous together with its first derivative, while the second derivative of Φ3 with respect to x1 also has a jump. On the second wave, the value Φ1 experiences a discontinuity, as do the first derivative of v1 and the second derivatives of v2 and v3. On the third wave, the first derivatives of Φ1, v2, v3 and the second derivative of v1 are discontinuous. On the fourth wave, the first derivatives of v1 and Φ3, and on the fifth wave the first derivatives of Φ1, v2, v3 and the second derivative of v1 experience discontinuities. Numerical analysis of the solution with boundary conditions of the first type shows that the fourth mode does not make a significant contribution to determining the displacement velocities and its rotatory angular velocities, and thus it may be ignored.

To illustrate the aforesaid, Fig. 4 shows the dimensionless displacement and angular velocities (the former in units of the initial velocity X0 and the latter in units of Y0) for an AT-cut quartz plate of thickness 2b = 2 mm at τ = tY0 = 1 plotted against $$\xi =\frac{x_1}{G_{(1)}}{Y}_0$$. Curves 1–5 represent $${\overline{v}}_1,$$ Φ1, $${\overline{v}}_3,$$ Φ3, and $${\overline{v}}_2$$, respectively. Along the interval from 0 to 0.53, all five modes contribute to the solution, from 0.53 to 0.76, the first four waves; from 0.76 to 0.86, the first three waves; from 0.86 to 0.96, the first two waves; and from 0.96 to 1.00, the first wave only (Fig. 4a). Fig. 4Epure of displacement and rotation angle velocities; (a) shock excitation of the first type, (b) shock excitation of the second type

If the initial shock is of the second type, the nature of each of the five waves changes as follows. On the first wave, the first derivative of v1 and second derivatives of Φ1 and Φ3 experience discontinuities, on the second wave, the first derivative of Φ1 and second derivatives of v2 and v3 are discontinuous, on the third wave, v2 and v3 and the first derivative of Φ1, in the fourth wave the value of Φ3, the first derivative of v1 and the second derivative of Φ1, and on the fifth wave the values of v2 and v3 and the first derivative of Φ1 experience discontinuities. Analysis of the computed data in this case implies that the first and second waves do not significantly affect the values of the displacement velocities and rotary velocities, and their contribution to the solution may be ignored. Curves of the dimensionless values are plotted in Fig. 4b.

In conclusion, let us compute the ray velocities GL(n), $${{G}}_{L\left(\alpha^{\ast} \right)}$$ of strong discontinuity waves (the velocities at which the perturbations propagate along the ray), construct the ray velocity curves, compare them with the corresponding phase velocity curves G(n), $${{G}}_{\left(\alpha^{\ast} \right)}$$ (Fig. 2a, b), and, in addition, determine the angles γ(n), $${{\gamma}}_{\left(\alpha^{\ast} \right)}$$ that characterize the deviation of the rays from the wave normal. As already mentioned, the extrema of the phase velocities G(n), $${{G}}_{\left(\alpha^{\ast} \right)}$$ are the zero of the velocities g(n), $${{g}}_{\left(\alpha^{\ast} \right)}$$, at which the perturbations travel along the appropriate wave fronts. This suggests that g(n), $${{g}}_{\left(\alpha^{\ast} \right)}$$ are the derivatives with respect to φ of the phase velocities G(n), G(α∗). Indeed, differentiating the expression Fig. 5Ray velocities Fig. 6Angles characterizing rays divergence from the wave normals
$${s}_{ij}{l}_i^{(n)}{l}_j^{(n)}=\rho {G}_{(n)}^2$$
with respect to φ and considering that $${s}_{ij}{l}_i^{(n)}{l}_j^{(m)}=0$$ if n ≠ m yield
$${G}_{(n),\varphi }={s}_{ij,\varphi }{l}_i^{(n)}{l}_j^{(n)}{\left(2\rho {G}_{(n)}\right)}^{-1}.$$
(9)

Since $${s}_{ij}={\left.\frac{1}{2}{p}_{ij}\right|}_{\tau_{\sigma }={\nu}_{\sigma }}$$and sij, φ = pij, it follows that formula (9) coincides with formula (7) for g(n), i.e., G(n), φ = g(n). Similar reasoning yields $${{G}}_{\left(\alpha^{\ast} \right)}$$, φ = $${{g}}_{\left(\alpha^{\ast} \right)}$$.

To determine the location of the perturbation at any fixed instant of time, the parameter φ should be eliminated from the system of equations
\begin{aligned} F\left({x}_1,{x}_3,\varphi \right)&={x}_1{\nu}_1+{x}_3{\nu}_3- Gt=0,\\{F}_1\left({x}_1,{x}_3,\varphi \right)&={x}_1{\tau}_1+{x}_3{\tau}_3- gt=0, \end{aligned}
(10)
wherein, for convenience, the index of the wave number has been omitted.

The equality g = G,φ results in F1 (x1, x3, φ) = F,φ (x1, x3, φ), i.e., the ray velocity curves are the envelopes of the wave fronts of plane waves radiating at a certain time from a point source placed at the origin (Musgrave 1970).

Equations (10) yield the equations of the ray velocity surfaces in polar form
$$\rho =\sqrt{{x_1}^2+{x_3}^2}={G}_Lt,{G}_L=\sqrt{G^2+{g}^2}$$
(11)
and the expressions for the angles
$$\cos \gamma ={GG}_L^{-1},\sin \gamma ={gG}_L^{-1}$$
(12)

describing the deviation of the rays from the wave normal.

Figure 5 shows the ray velocity curves for five plane waves determined by formula (10) (in view of symmetry, only the upper parts of the curves are shown). It is seen that smooth wave fronts are observed for only two modes: quasi-longitudinal wave (the first mode) and quasi-extensional rotational wave (the second mode). In quasi-transverse (third and fifth) and quasi-transverse rotary (fourth) modes, the wave fronts contain lacunae: the third and fourth modes have lacunae each, symmetrically located with respect to the x1-axis, and the fifth has six lacunae, two of which lie on the x3-axis and four are symmetrically located with respect to the origin. On straight lines at angles +35°15' to the x1-axis, these lacunae are not symmetrical about the straight lines. It could be observed that the AT-cut is also inclined to the crystallographic axis at the angle of 35°15'. The ray L may intersect the wave surface at five, seven, or nine points, i.e., along a ray in the AT-cut quartz plate up to nine elastic modes of different velocities may propagate, two of which are quasi-longitudinal and the others are quasi-transverse.

In Fig. 6, the angles of deviation γ of the rays from the wave normal are plotted against the inclination φ-angle of the wave normal to the x1-axis, as obtained from (12). It is seen that all the curves, except the third one, have two extrema and cut the φ-axis at three points (the third curve has three extrema and four points of intersection with the φ-axis). In other words, the rays deviate most from the normal in two directions (for the fifth mode – in three directions) and coincide with the wave normal in three directions (for the fifth mode – in four), two of which are the directions of the x1−and x3−axes.

Conclusion

The ray method is used to solve the boundary-value problems that lead to the propagation of two-dimensional shock waves in anisotropic plates of constant thickness, taking into account rotary inertia and transverse shear deformations, as well as the coupling of extensional and transverse vibrational modes. The essence of the method is to construct the solution behind the shock wave fronts by using ray series expansions. Transient two-dimensional wave propagation in semi-infinite AT-cut quartz plates has been studied.

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