# Plane Transient Waves in Anisotropic Layer, Ray Expansion Approach

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_104-1

## Synonyms

## 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

## The Ray Method

*N*

_{1},

*N*

_{2}, and

*N*

_{3}are the forces acting in the plane of the plate,

*Q*

_{1}and

*Q*

_{3}are the shear forces,

*M*

_{1}and

*M*

_{3}are the bending moments,

*M*

_{5}is the twisting moment, 2

*b*is the plate thickness,

*ρ*is the density,

*u*

_{j}(

*j*= 1,2,3) are the displacement vector components, Φ

_{1}and Φ

_{3}are the rotatory angular velocities of the normal to the middle surface of the plate,

*x*

_{1}and

*x*

_{3}are coordinates of the middle surface,

*x*

_{2}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* = *x*_{1} *ν*_{1} + *x*_{3} *ν*_{3} = 0, where*ν*_{1} = cos *φ* and *ν*_{3} = 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 *x*_{1}, *x*_{3}-plane and generators parallel to the *x*_{2}-axis (Fig. 1).

*S*(

*t*), the desired functions

*Z*(

*x*

_{1},

*x*

_{3},

*t*) are expanded in terms of ray series (Achenbach and Reddy 1967; Rossikhin and Shitikova 1995)

*Z*,

_{(k)}] are the jumps of the

*k*th-order time-derivative of the function

*Z*(

*x*

_{1},

*x*

_{3},

*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.

*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

*α*= 1, 3,

*v*

_{1},

*v*

_{2}, and

*v*

_{3}are the displacement velocities and

*c*

_{pq}are the moduli of a quartz crystal,

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 *k*th-order derivative of a certain function *Z*(*x*_{1}, *x*_{3}, *t*) (Thomas 1961; Rossikhin and Shitikova 1995),

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

The quantities \( {p}_{\alpha \beta}^{\ast} \) are derived from *p*_{αβ} by replacing \( {\overline{c}}_{mn} \) (*m*, *n* = 1, 3) in the latter with *γ*_{mn} and *c*_{55} with *γ*_{55}, *s* = *x*_{1}*τ*_{1} + *x*_{3} *τ*_{3} (*τ*_{1} = − sin *φ*, *τ*_{3} = 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 *s*_{ij} and \( {s}_{\alpha \beta}^{\ast} \) by replacing *ν*_{δ} in the latter by *τ*_{δ} (*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 −3*b*^{−2}.

*k*= 0, 1, 2 in Eqs. (5) yield

^{∗}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

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.

*t*and coordinate

*s*, i.e.,

*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 *N*_{n} and *N*_{nτ} in the plane of the plate, the shear force *Q*_{n}, and the bending *M*_{n} and twisting *M*_{nτ} 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

*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.

All velocities *G*_{(i)} and \( {{G}}_{\left(\alpha^{\ast} \right)} \) are given in units of the least velocity along the *x*_{1}-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. 3a–e 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 *v*_{1} experiences a discontinuity, but Φ_{3} is continuous together with its first derivative, while the second derivative of Φ_{3} with respect to *x*_{1} also has a jump. On the second wave, the value Φ_{1} experiences a discontinuity, as do the first derivative of *v*_{1} and the second derivatives of *v*_{2} and *v*_{3}. On the third wave, the first derivatives of Φ_{1}, *v*_{2}, *v*_{3} and the second derivative of *v*_{1} are discontinuous. On the fourth wave, the first derivatives of *v*_{1} and Φ_{3}, and on the fifth wave the first derivatives of Φ_{1}, *v*_{2}, *v*_{3} and the second derivative of *v*_{1} 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.

*X*

_{0}and the latter in units of

*Y*

_{0}) for an AT-cut quartz plate of thickness 2b = 2 mm at

*τ*=

*tY*

_{0}= 1 plotted against \( \xi =\frac{x_1}{G_{(1)}}{Y}_0 \). Curves 1–5 represent \( {\overline{v}}_1, \) Φ

_{1}, \( {\overline{v}}_3, \) Φ

_{3}, 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).

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 *v*_{1} and second derivatives of Φ_{1} and Φ_{3} experience discontinuities, on the second wave, the first derivative of Φ_{1} and second derivatives of *v*_{2} and *v*_{3} are discontinuous, on the third wave, *v*_{2} and *v*_{3} and the first derivative of Φ_{1}, in the fourth wave the value of Φ_{3}, the first derivative of *v*_{1} and the second derivative of Φ_{1}, and on the fifth wave the values of *v*_{2} and *v*_{3} and the first derivative of Φ_{1} 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.

*G*

_{L(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

*φ*and considering that \( {s}_{ij}{l}_i^{(n)}{l}_j^{(m)}=0 \) if

*n*≠

*m*yield

Since \( {s}_{ij}={\left.\frac{1}{2}{p}_{ij}\right|}_{\tau_{\sigma }={\nu}_{\sigma }} \)and *s*_{ij, φ} = *p*_{ij}, 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} = \( {{g}}_{\left(\alpha^{\ast} \right)} \).

*φ*should be eliminated from the system of equations

The equality *g* = *G*_{,φ} results in *F*_{1} (*x*_{1}, *x*_{3}, *φ*) = *F*_{,φ} (*x*_{1}, *x*_{3}, *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).

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 *x*_{1}-axis, and the fifth has six lacunae, two of which lie on the *x*_{3}-axis and four are symmetrically located with respect to the origin. On straight lines at angles +35^{°}15^{'} to the *x*_{1}-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 *x*_{1}-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 *x*_{1}−and *x*_{3}−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.

## Cross-References

## References

- Achenbach JD, Reddy DP (1967) Note on wave propagation in linearly viscoelastic media. ZAMP 18(1):141–144zbMATHGoogle Scholar
- Cohen H, Thomas RS (1985) Transient waves in inhomogeneous anisotropic elastic plates. Acta Mech 58(12):41–57MathSciNetzbMATHGoogle Scholar
- Kaczkowski Z (1960) The influence of the shear forces and the rotatory inertia on the vibration of an anisotropic plate. Arch Mech Stosowanej 12(4):531–552zbMATHGoogle Scholar
- Krasyukov VP (1966) Vibrations of anisotropic plates taking into account rotatory inertia and shear deformation. Nauchn Trudy Saratov Politechn Inst 23:107–110Google Scholar
- Mindlin RD, Spencer WJ (1967) Anharmonic, thickness-twist overtones of thickness-shear and flexural vibrations of rectangular, AT-cut quartz plates. J Acoust Soc Am 42(6):1268–1277CrossRefGoogle Scholar
- Musgrave MJP (1970) Crystal acoustics. Introduction to the study of elastic waves and vibrations in crystals. Holden-Day, San FranciscozbMATHGoogle Scholar
- Rossikhin YA, Shitikova MV (1991) Ray method for investigating transient wave processes in a thin elastic anisotropic layer. J Appl Math Mech 55(5):724–732. https://doi.org/10.1016/0021-8928(91)90120-1 MathSciNetCrossRefzbMATHGoogle Scholar
- Rossikhin YA, Shitikova MV (1995) Ray method for solving boundary dynamic problems connected with propagation of wave surfaces of strong and weak discontinuities. Appl Mech Rev 48(1):1–39. https://doi.org/10.1115/1.3005096 CrossRefGoogle Scholar
- Thomas T (1961) Plastic flow and fracture in solids. Academic Press, New YorkzbMATHGoogle Scholar