Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials
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O R I G I NA L
Antonios E. Giannakopoulos · Thanasis Zisis
Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials
Received: 11 May 2020 / Accepted: 9 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Flexoelectricity · Dynamic fracture · Dynamic energy release rate · Mach lines · Surface waves · Metamaterials
1 Introduction Flexoelectricity is the ability of materials to convert mechanical strain gradients to electric polarization and vice versa. This implies that dielectric materials and ferroelectrics in paraelectric phase, under inhomogeneous mechanical strain, can produce polarization, and so they can classify as flexoelectrics (see for example [17,34,38,55,81]). Many rocks that consist of earth’s mantle exhibit flexoelectricity, often combined with piezoelectricity (in case of anisotropy). An excellent recent perspective of this unusual electromechanical coupling with emphasis on applications in e
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