Piezoelectric Simulations With Marc

8
Apr

Piezoelectric sensor-actuator technology is often used in high-tech industries like medical, mechanical, automotive engineering and semiconductor markets. Applications of piezo-based ultrasonic sensors range from fuel injectors, flow-rate metering, ultrasonic welding, and high-resolution material testing to medical diagnostics.

A piezoelectric finite element analysis involves computation of coupled stress and electric field in certain crystalline materials. In such materials, mechanical strain generates electric polarization which changes direction with it (direct piezoelectric effect used in sensors) and vice versa (reverse effect used in actuators). This effect is observed in some non-centrosymmetric ionic crystals, whose lattices of positive and negative ions deform differently under the action of external forces, leading to a separation of negative and positive charges.

This electro-mechanical coupling is based on governing differential equations and electro-elastic constitutive material behavior of the piezo-electric continua. In Marc implementation, an electrical charge is brought to a piezoelectric material layer by applying an equivalent force to the corresponding node and the derivatives of the shape functions are taken at the Gauss points. Piezoelectric coupling and capacitance contributions to the element stiffness matrix are computed at each Gauss point.

Marc simultaneously solves for nodal displacements and electric potential, depending on governing equilibrium equations. Such analyses maybe geometrically non-linear, but are materially linear. The material definition for a piezoelectric analysis consists of a mechanical part (same as an elastic stress analysis), an electrostatic part (permittivity, etc.) and coefficients for piezoelectric coupling (either stress based or strain based). Both structural and electrostatic material definition can be isotropic, orthotropic or anisotropic.

Currently, Marc supports linear elements in 2D (4-node plane stress, 4-node plane strain, 4-node axisymmetric quadrilateral) and in 3D (8-node brick and 4-node tetrahedron). The first two or three degrees of freedom (in 2D or 3D respectively) for these elements are available for displacement components (u, v, w) and the last (3rd or 4th) degree of freedom is available for electric potential (voltage ϕ). For complex geometries, tetrahedral and hexahedral elements are most robust and commonly used. These elements can be used in static, transient dynamic, harmonic and buckling analyses.

Existing Marc piezoelectric elements have equivalent heat transfer elements that can be used in coupled thermal-piezoelectric analysis. Such analyses are weakly coupled and solved in a staggered way. During contact analysis, one node of a piezoelectric element touches a segment of another element, and a multi-point constraint relation is set up for nodal displacements and electric potential.

Both mechanical loads and electrostatic loads can be applied as boundary conditions. These loads can have time variation if a transient analysis is performed, or harmonic excitation can also be applied. Marc solves for stresses, strains, electric displacement and electric field intensity at integration points.

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