A constitutive law for ferroelectric and ferroelastic piezoceramics is implemented in ABAQUS Standard using the subroutine user element. A linear solid element is defined: it is an eight-node hexahedron having the mechanical displacement components and the electric potential as degrees of freedom for each node. The element is formulated for static analysis and it needs the definition of the contribution of this element to the Jacobian (stiffness) and the definition of an array containing the contributions of this element to the right-hand-side vectors of the overall system of equations The subroutine is called for each element that is of a user-defined element type each time element calculations are required. As an example, the element is used for the simulation of a multilayer actuator made of piezoceramics. In this case, the piezoelectric equations are not valid since the electric loading induces non linear phenomena, which are captured through the constitutive law implemented in the user element.
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Elhadrouz, M., Zineb, T.B. & Patoor, E. Finite Element Analysis of the non Linear Behavior of a Multilayer Piezoelectric Actuator. MRS Online Proceedings Library 881, 411 (2005). https://doi.org/10.1557/PROC-881-CC4.11