Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials

Abstract

The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. The work examines dielectric materials without piezoelectric coupling or nonlinear ferroelectric switching and considers the inverse flexoelectric effect. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work, we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elastodynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging microstructural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. Furthermore, we investigated the subsonic and the supersonic steady state crack rupture and showed that the Mach cones depend on the microstructural as well as the micro-inertial lengths. An important finding of this work is the existence of surface waves of Bleustein–Gulyaev type that do not appear in classic elastodynamics, but have been found in piezoelectric materials. The case of dielectric metamaterials with negative electric susceptibility is examined for the first time. The results can be useful for other dispersive materials, provided we identify the pertinent microstructural and micro-inertial lengths in accord with the behavior of the material at high frequencies.

Appendix

In order to explain the analogy between the antiplane steady state couple stress problem with the prestretched orthotropic plate problem, we will provide a short account of the plate problem. We will also give the correspondence of the plate’s parameters with the parameters of the couple stress problem, connecting them with the finite element procedure that solves the plate problem. The starting point is that a plate with Kirchhoff-type kinematic assumption ends up with one basic unknown, the out-of-plane displacement w, which is also the prime unknown for the antiplane couple-stress problem. The space of solution is a plane described by the Cartesian coordinates ($\xi , \eta )$ attached to the crack tip and is the same in both plate and antiplane cases. Due to the constant tip velocity that carries also the loading with the same speed, the problem in both cases appears as “static.” Following [26] the orthotropic bending stiffness parameters of the plate are:

$$\begin{aligned} D_{\xi } =\frac{E_{\xi } h^{3}}{12(1-\nu _{\xi } \nu _{\eta } )},\,\,\,\,\,\,\,\,\,\,\,D_{\eta } =\frac{E_{\eta } h^{3}}{12(1-\nu _{\xi } \nu _{\eta } )},\,\,\,\,\,\,\,\,\,\,\,\,2D_{\xi \eta } =\frac{2Gh^{3}}{12} \end{aligned}$$

(A.1)

where $E_{\xi }, E_{\eta }, \nu _{\xi }, \nu _{\eta }$ are the elastic and Poisson’s ratio constants in the $\xi$ and $\eta$ directions, G is the in-plane shear modulus, and h is the plate’s (constant) thickness. If we apply to the plate homogeneous prestressing $N_{\xi }$ and $N_{\eta }$ as line forces in the $\xi$ and $\eta$ directions, we obtain the fourth-order equilibrium equation in terms of w:

$$\begin{aligned} D_{\xi } \frac{\partial ^{4}w}{\partial \xi ^{4}}+\left( {D_{\xi } \nu _{\eta } +D_{\eta } \nu _{\xi } +4D_{\xi \eta } } \right) \frac{\partial ^{4}w}{\partial \xi ^{2}\partial \eta ^{2}}+D_{\eta } \frac{\partial ^{4}w}{\partial \eta ^{4}}=q_\mathrm{plate} +N_{\xi } \frac{\partial ^{2}w}{\partial \xi ^{2}}+N_{\eta } \frac{\partial ^{2}w}{\partial \eta ^{2}} \end{aligned}$$

(A.2)

where $q_\mathrm{plate}$ is the distributed surface load of the plate. Note that all the parameters entering the plate problem can be selected by the finite element code which will be asked to solve Eq. (A.2) and obtain $w\,(\xi ,\eta )$ numerically.

To complete the plate problem, we have to introduce the boundary conditions. For simplicity, we will state the boundary conditions along the $\xi$ and $\eta$ directions. More general boundary conditions can be found in the work of Giannakopoulos and Zisis [26]. Two types of boundary conditions are involved in the plate problem, just as in the antiplane couple stress elasticity problem: the bending moments $M_{\xi }, M_{\eta }, M_{\xi \eta }$ and the shear forces $Q_{\xi }, Q_{\eta }$ (per unit length):

$$\begin{aligned}&\begin{array}{l} M_{\xi } =-D_{\xi } \left( {\frac{\partial ^{2}w}{\partial \xi ^{2}}+\nu _{\xi } \frac{\partial ^{2}w}{\partial \eta ^{2}}} \right) \\ \\ M_{\eta } =-D_{\eta } \left( {\frac{\partial ^{2}w}{\partial \eta ^{2}}+\nu _{\eta } \frac{\partial ^{2}w}{\partial \xi ^{2}}} \right) \\ \\ M_{\xi \eta } =-M_{\eta \xi } =-2D_{\xi \eta } \frac{\partial ^{2}w}{\partial \xi \,\partial \eta } \\ \end{array} \end{aligned}$$

(A.3)

$$\begin{aligned}&\begin{array}{l} Q_{\xi } =-\frac{\partial }{\partial \xi }\left( {D_{\xi } \frac{\partial ^{2}w}{\partial \xi ^{2}}+(D_{\xi } \nu _{\eta } +2D_{\xi \eta } )\frac{\partial ^{2}w}{\partial \eta ^{2}}} \right) +N_{\xi } \frac{\partial w}{\partial \xi } \\ \\ Q_{\eta } =-\frac{\partial }{\partial \eta }\left( {(D_{\eta } \nu _{\xi } +2D_{\xi \eta } )\frac{\partial ^{2}w}{\partial \xi ^{2}}+D_{\eta } \frac{\partial ^{2}w}{\partial \eta ^{2}}} \right) +N_{\eta } \frac{\partial w}{\partial \eta } \\ \end{array} \end{aligned}$$

(A.4)

To complete the analogy, we compare the above plate equations with the steady-state, antiplane couple stress equations assuming that the antiplane displacement and the out of plane plate displacement are the same. Indeed, comparing (A.2) with (5.3), the two equations become identical provided that the following conditions hold:

$$\begin{aligned} \frac{N_{\xi } }{N_{\eta } }= & {} 1-\frac{V^{2}}{c_\mathrm{s}^{2} } \end{aligned}$$

(A.5)

$$\begin{aligned} \frac{q_\mathrm{plate} }{N_{\eta } }= & {} \frac{X_{z} }{\mu } \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{D_{\xi } }{N_{\eta } }= & {} \frac{\ell ^{2}}{2}\left( {1-\frac{V^{2}H^{2}}{6\ell c_\mathrm{s}^{2} }} \right) \end{aligned}$$

(A.7)

$$\begin{aligned} \frac{D_{\eta } }{N_{\eta } }= & {} \frac{\ell ^{2}}{2} \end{aligned}$$

(A.8)

$$\begin{aligned} \frac{D_{\xi } \nu _{\eta } +D_{\eta } \nu _{\xi } +4D_{\xi \eta } }{N_{\eta } }= & {} \frac{\ell ^{2}}{2}\left( {2-\frac{V^{2}H^{2}}{6\ell ^{2}c_\mathrm{s}^{2} }} \right) \end{aligned}$$

(A.9)

The meaning of the above equations is the following: the left-hand side are plate parameters and the right-hand side of these equations are couple stress parameters. Equation (A.5) enters the normalized steady state velocity $V/c_\mathrm{s}$ through the ratio of the pre-stress line loads $N_{\xi } /N_{\eta }$. Equation (A.6) enters the normalized body force $X_{z} /\mu$ through the normalized plate surface load $q_\mathrm{plate} /N_{\eta }$ which interestingly suggests an inverse length. Accordingly, equation (A.8) enters the microstructural length $\ell /\sqrt{2}$ though the ratio $D_{\eta } /N_{\eta }$. Equation (A.7) enters the micro-inertial length $H/\sqrt{12}$ through the ratio $D_{\xi } /N_{\eta }$ (that is plate’s orthotropy dictates the micro-inertial length). Equation (A.9) serves as a consistency equation and essentially fixes the value of $D_{\xi \eta } /N_{\eta }$. The Poisson’s ratio $\nu _{\xi }$ and $\nu _{\eta }$ will have to be selected in accord to the moment type of boundary conditions (A.3). For the problem under investigation, we can select:

$$\begin{aligned} \nu _{\xi }= & {} \nu _{\eta } =0 \end{aligned}$$

(A.10)

$$\begin{aligned} \frac{4D_{\xi \eta } }{N_{\eta } }= & {} \frac{\ell ^{2}}{2}\left( {2-\frac{V^{2}H^{2}}{6\ell ^{2}c_\mathrm{s}^{2} }} \right) \end{aligned}$$

(A.11)

Then, Eq. (5.4) compared to (A.3b) gives $w=0$, in front of the crack tip and $M_{\eta } =0$ at the crack surface.

Equation (5.5) compared to (A.4b) suggests that:

$$\begin{aligned} \frac{Q_{\eta } }{N_{\eta } }=-\frac{t_{3} }{\mu } \end{aligned}$$

(A.12)

This last equation enters the normalized traction $t_{3} /\mu$ of the couple stress problem with the normalized shear load $Q_{\eta } /N_{\eta }$ of the plate problem at the crack surface. Finally, let us examine the case of $\nu _{\xi } \ne 0$ and $\nu _{\eta } \ne 0$. This will lead to a plate boundary condition $\frac{\partial ^{2}w}{\partial \eta ^{2}}+\nu _{\eta } \frac{\partial ^{2}w}{\partial \xi ^{2}}=0$ which would be different that the couple stress boundary condition $\frac{\partial ^{2}w}{\partial \eta ^{2}}=0$.