Modeling Nonlinear Acoustic Wave Propagation With Third‐Order Elastic Constants Using the Finite Element Method

ABSTRACT Due to the widespread usage of specialized piezoelectric actuators in various applications, accurate simulations have become key components of modern design processes. Although simulation tools for piezoelectric devices are widely available, nonlinear effects, which occur when exciting near resonance at elevated amplitudes, are often neglected or linearized. At those resonance frequencies the piezoelectric systems exhibit characteristics of highly dynamical nonlinear systems, showing typical nonlinear effects such as frequency shifts, jump phenomena, and bifurcations. To correctly reproduce this behavior, suitable material models must be identified and incorporated into the simulation environment. Therefore, a finite element formulation for nonlinear piezoelectric systems that exhibit a quadratic elastic nonlinearity for the strain–stress relation in the mechanical field is presented in this work. This results in a third‐order mechanical material tensor, which is composed of third‐order elastic constants. For the electric field and the piezoelectric coupling linear behavior is assumed. An axisymmetric piezoelectric ring is modeled, and the simulation is performed in time‐domain with harmonic excitation. Existing methods for incorporating such quadratic nonlinearities simplify or linearize the nonlinear stiffness matrix. To enable the use of the full set of nonlinear material parameters, the finite element formulation must be derived using tensor notation. This results in tensor‐valued element‐matrices as well as a tensor‐valued nonlinear stiffness matrix, which is unrolled for the simulation process in this work. Simulation results are presented, which show higher order harmonics for the mechanical field in the response spectrum.