Antiplane Stoneley waves propagating at the interface between two couple stress elastic materials

Abstract

We investigate antiplane Stoneley waves, localized at the discontinuity surface between two perfectly bonded half-spaces. Both half-spaces are elastic linear isotropic and possess a microstructure that is described within the theory of couple stress materials with micro-inertia. We show that the microstructure deeply affects wave propagation, which is permitted under broad conditions. This outcome stands in marked contrast to classical elasticity, where antiplane Stoneley waves are not supported and in-plane Stoneley waves exist only under very severe conditions on the material properties of the bonded half-spaces. Besides, Stoneley waves may propagate only beyond a threshold frequency (cuton), for which an explicit expression is provided. For a given frequency above cuton, this expression lends the admissible range of material parameters that allows propagation (passband). In particular, significant contrast between the adjoining materials is possible, provided that Stoneley waves propagate at high enough frequency. Therefore, micro-inertia plays an important role in determining the features of propagation. Considerations concerning existence and uniqueness of antiplane Stoneley waves are given: it is found that evanescent and decaying/exploding modes are also admitted. Results may be especially useful when accounting for the microstructure in non-destructive testing (NDT) and seismic propagation.

Appendices

Appendix

Appendix A: Proof for the number of roots of the Rayleigh function

Fig. 9

Simple curve $\gamma$ (green, solid) whose mapping by the Rayleigh function $R^{\mathrm{\scriptscriptstyle A}}(\gamma )$ is used to determine the number of Rayleigh roots. Branch cuts (red, dashed) and branch points are chosen to warrant depthwise decay of the solution (30) (color figure online)

We determine the number of zeros of (34) in the cut complex plane by the argument principle; see Beardon [3]. To fix ideas, we give the proof for $R^{\mathrm{\scriptscriptstyle A}}(\kappa )$, but the argument easily extends to $R^{\mathrm{\scriptscriptstyle B}}(\kappa )$. We construct the mapping of the simple curve $\gamma$ by the Rayleigh function, $R^{\mathrm{\scriptscriptstyle A}}(\gamma )$, and count its index (or winding number). Looking at Fig. 9, we see that $\gamma$ consists of the circle $\gamma _R$ of arbitrarily large radius R, together with the loops $\gamma _{\pm \delta }$ around the centrally symmetric pair of cuts $[\pm \delta , \pm \delta \mp \imath \infty )$ and the loops $\gamma _{\pm \imath }$ around $[\pm \imath , \pm \imath \infty )$. By the asymptotics (38), we infer that, when moving along $\gamma _R$, the image point makes four complete turns around the origin. We now turn to the loops around the cuts, and, in the light of the central symmetry property, only loops sitting in either half-plane are considered, and the resulting winding number is then doubled. On the loop $\gamma _{-\imath }$, we have $A_1$ and $A_2$ purely imaginary, whence Eq. (34) remains in the same form but now in terms of real numbers. In the limit as this loop shrinks down to the cut, $\gamma _{-\imath }$ is mapped onto an open curve approaching the real line from above, i.e. from positive imaginary numbers. Conversely, the loop $\gamma _\delta$ is mapped onto an S-shaped open curve as in Fig. 11, which intersects the real axis three times, named $d_1<d_2<d_3$. In particular, $d_1<0$ is located to the left of the origin, while $d_2 = R^{\mathrm{\scriptscriptstyle A}}(\delta ) = \delta ^4 \eta ^2 \ge 0$ is always to the right. Together, $D(\gamma _{-\imath }) \cup D(\gamma _{-\imath }\delta )$ form a non-simple curve winding once around the origin, that is closed when including the points at infinity. We conclude that six order 1 roots are expected.

Roots should be sought among the zeros of the bi-quartic polynomial

$$\begin{aligned} \eta ^4 \kappa ^8+2(1+ \eta ) \kappa ^6 +\left( \delta _2^2-\delta _1^2\right) \left( 2 \eta +1\right) \kappa ^4 - \left( 2 \eta \delta _2^2 \delta _1^2 +\delta _1^4+\delta _2^4\right) \kappa ^2 +\delta _1^2 \delta _2^2 \left( \delta _2^2-\delta _1^2\right) = 0. \end{aligned}$$

Basically, this is a singularly perturbed polynomial equation inasmuch as $\eta$ is assumed to be small. In this context, we observe that for the case $\eta =0$, corresponding to the strain gradient theory, Rayleigh waves collapse into bulk waves as Eq. (34) reduces to

$$\begin{aligned} \lambda _1 \lambda _2 \left( \lambda _2^2 - \lambda _1^2 \right) , \end{aligned}$$

whose real roots correspond to bulk waves $\lambda _{1,2}=0$. In fact, Rayleigh roots are generally perturbations around either bulk wave speed; see Nobili et al. [22]. It is observed that in Nobili et al. [22] a different choice is made for the cuts, according to which the complex-conjugated pair of zeros may fall outside the Riemann sheet. In fact, with our choice for the cuts, existence of all roots is always warranted.

Appendix B: Proof for the number of roots of the Stoneley frequency function

Fig. 10

Simple curve $\gamma$ (green, solid) whose mapping by the Stoneley frequency equation $D_0(\gamma )$ is used to determine existence and uniqueness of antiplane Stoneley roots. Here, to fix ideas, we have assumed $\delta _1 < \delta$ and $\delta _2 <1$ (color figure online)

To this aim, we enlarge our viewpoint and think of $D_0$ as a function of the complex variable s. Then, $D_0(s)$ appears centrally symmetric, i.e. $D_0(s)=D_0(-s)$. We determine the number of zeros of $D_0(s)$ in the cut complex plane by the argument principle. Accordingly, we determine the index (winding number) of the curve $D_0(\gamma )$, where $\gamma = \gamma _R \cup \gamma _{\pm \delta } \cup \gamma _{\pm \delta _1} \cup \gamma _{\pm \imath }$ is the simple curve shown in Fig. 10. Here, to fix ideas, we assume $\delta _1<\delta$ and $\delta _2 <1$.

When $\varGamma$ is small enough, the following analysis resembles that given for the Rayleigh function. By the asymptotics (44), as the point $\kappa$ moves on the curve $\gamma _R$, its mapping $D_0(\kappa )$ makes four complete turns about the origin, whence the index is 4.

Fig. 11

Image by $D_0(s)$ of the loop $\gamma _\delta$ for $\varOmega <\varOmega _{\mathrm{cuton}}$ (left, index $-\tfrac{1}{2}$) and $\varOmega >\varOmega _{\mathrm{cuton}}$ (right, index $\tfrac{1}{2}$). The dashed line is the common asymptote for the curve at infinity

As in Fig. 11, $\gamma _\delta$ is mapped into an open loop having three intersections with the real axis, $d_1 < 0$, $d_2$ and $d_3>0$, with $d_2 = D_0(\delta )$. The explicit expression for $d_2$ is given in the Appendix. In contrast, $d_1$ and $d_3$ may be found numerically imposing the condition $\mathfrak {I}[D_0(\delta \mp \varepsilon -\imath y)]=0$, respectively, with $\varepsilon \rightarrow 0^+$ and $y>0$. When $\varGamma$ is small enough, this loop looks just like the S-shaped curves encountered in the Rayleigh case, but, unlike there, its intersection $d_2$ with the real axis is not necessarily positive. Indeed, this loop has index $-\tfrac{1}{2}$ inasmuch as $d_2 < 0$, that occurs for small values of $\varOmega$. In this situation, $D_0(s)$ possesses two pairs of roots: a complex-conjugated pair and a purely imaginary pair. Upon increasing $\varOmega$, the cuton frequency $\varOmega _{\mathrm{cuton}}$ is reached such that $d_2 = 0$ and the real root $\kappa _S$ is located precisely at the bulk wave number $\delta$. In consideration of the fact that $\delta$ is a monotonically increasing function of $\varOmega$ and so is $D_0(\varOmega )$, for $\varOmega >\varOmega _{\mathrm{cuton}}$ we have that $D_0(\gamma _\delta )$ winds around the origin as in Fig. 11b. Thus, we find three pairs of roots: a complex-conjugated pair, a purely imaginary pair, and a real pair.

Fig. 12

Image by $D_0(s)$ of the loop $\gamma _{-\imath }$ (index $\tfrac{1}{2}$)

Similarly, $\gamma _{\delta _1}$ is mapped onto a loop closed at infinity which never encircles the origin and contributes nothing to the index. Finally, the loop $\gamma _{-\imath }$ is mapped onto the real axis from above (i.e. from the side of positive imaginary part) moving from left to right; see Fig. 12. This curve brings an index $\tfrac{1}{2}$ regardless of $\varOmega$.

Appendix C: Linear approximation to the cuton frequency

The analytic expressions of the coefficients in the linear approximation (46) are

$$\begin{aligned} d_2&= \beta ^2\varGamma \Bigg [ \left( \sqrt{\left( \delta ^2-\delta _1^2\right) \left( \delta ^2+\delta _2^2\right) }-\delta ^2 \eta ^{\mathrm{\scriptscriptstyle B}}\right) ^2 \\&\quad -\sqrt{\left( \delta ^2-\delta _1^2\right) \left( \delta ^2+\delta _2^2\right) } \left( \sqrt{\delta ^2-\delta _1^2}+\sqrt{\delta ^2+\delta _2^2}\right) ^2\Bigg ] \\&\quad +\frac{\delta ^4 (\eta ^{\mathrm{\scriptscriptstyle A}}) ^2}{\beta ^2 \varGamma } -2 \delta ^4 \eta ^{\mathrm{\scriptscriptstyle A}}\eta ^{\mathrm{\scriptscriptstyle B}}+2 \sqrt{\left( \delta ^2-\delta _1^2\right) \left( \delta ^2+\delta _2^2\right) } \delta ^2 \eta ^{\mathrm{\scriptscriptstyle A}}\\&\quad -\sqrt{\left( \delta ^2+1\right) \left( \delta ^2-\delta _1^2\right) \left( \delta ^2+\delta _2^2\right) } \left( \sqrt{\delta ^2-\delta _1^2}+\sqrt{\delta ^2+\delta _2^2}\right) , \end{aligned}$$

and

$$\begin{aligned} a_1&= \frac{\sqrt{2} \delta }{\beta ^2 \varGamma } \Bigg \{ \delta ^2 \left[ -2 \beta ^2 \varGamma \left( \sqrt{\delta ^2-\delta _1^2}+\sqrt{\delta ^2+\delta _2^2}-\eta ^{\mathrm{\scriptscriptstyle B}}\sqrt{\delta ^2+1}\right) -\sqrt{\delta ^2+1} (2 \eta ^{\mathrm{\scriptscriptstyle A}}+1)\right] \\&\quad-\beta ^2 \varGamma \Big [\sqrt{\delta ^2-\delta _1^2} \delta _2^2+\sqrt{\delta ^2-\delta _1^2}-\delta _1^2\sqrt{\delta ^2+\delta _2^2}\\&\quad +\sqrt{\delta ^2+\delta _2^2}+2 \sqrt{\left( \delta ^2+1\right) \left( \delta ^2-\delta _1^2\right) \left( \delta ^2+\delta _2^2\right) }\Big ] -\sqrt{\delta ^2+1} \Bigg \}. \end{aligned}$$

Naturally, in the special case $\varGamma \rightarrow 0$, we retrieve the result already obtained for Rayleigh waves in Nobili et al. [22].