An approximate analytical solution for a dynamical simple shear deformation problem of an infinite strip including flexoelectric and micro‐inertial effects by Laplace transform

We study the flexoelectric effect induced in an infinite strip of a dielectric material, due to an applied shear deformation at the upper surface, while the lower surface is fixed. The model incorporates dynamic spontaneous polarization, flexocoupling deformation, and microinertia effects to evaluate these effects combined. The variation of the Hamiltonian functional in the medium, and of the external forces yields the governing equations and the boundary conditions. There are many types of applied boundary conditions besides the initial conditions, where the initial conditions are imposed due to the dynamics of many effects in the mathematical modelling formulation. In addition to that, the boundary conditions are divided into two groups, where the first group contains mechanical boundary conditions such as traction boundary conditions and displacement boundary conditions for the classical treatment as in the theory of elasticity while for the strain gradient theory of elasticity, conditions for higher‐order traction vector should be imposed and cannot be neglected. The proposed technique depends on applying Laplace's transform to the field equations to obtain the characteristic equation, the roots of which are expressed in terms of the s$s$‐transform parameter and other parameters related to the elastic and electric properties of the material. Those roots which produce a bounded solution are then approximated using Taylor's expansion, then the boundary conditions are applied to complete the solution. This technique enables to obtain the solution in terms of simple analytical functions that are easily inverted from the s$s$‐domain to the time domain. This technique is applied to a simple shear deformation problem to study the dynamical flexoelectric effect in dielectric materials due to shear displacement and external electric potential. A suitable material, such as Strontium Titanate material, SrTiO3$SrTi{O}_3$, is used for numerical simulation, and the results are analyzed, discussed, and represented graphically.