A quasi‐incompressible and quasi‐inextensible element formulation for transversely isotropic materials

Abstract

The contribution presents a new finite element formulation for quasiinextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu-type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit. KEYWORDSanisotropy, hyperelasticity, mixed finite element design, mixed variational principles, quasi-incompressiblity, quasi-inextensibility

Finite element modeling of electro‐viscoelasticity in fiber reinforced electro‐active polymers

“…In addition to the displacement field, he defined four additional field variables, which are treated on the finite element level as averagese. 53 Stability of mixed finite elements is assessed by the inf-sup condition, among others. The inf-sup condition can be alternatively referred to as Babuška-Brezzi (BB) condition 55,56 or as the LBB condition, including additionally the name of Ladyzhenskaya.…”

“…Nevertheless, the aforementioned finite element has demonstrated relatively satisfying robustness for a wide range of practical applications, considering quasi-incompressible materials that exhibit finite deformations. 53,58,59 DEA materials are regarded as strongly soft in general, however, the fibers are relatively stiff. Therefore, in this work, we propose a coupled and mixed electro-mechanical Q1P0F0 finite element, based on the mechanical counterpart.…”

“…Moreover, the nearly inextensible response along the stiffer preferred direction can lead to a locking phenomenon or poor convergence behavior. In order to tackle those problems, formulations of mixed finite elements for anisotropic hyperelastic materials are proposed in References 53,54, among others. Dal 53 has proposed an extended Hu–Washizu mixed finite element formulation, referred to it as “ Q 1 P 0 F 0” element and performed a comprehensive study to examine the performance of the newly proposed element.…”

“…In order to tackle those problems, formulations of mixed finite elements for anisotropic hyperelastic materials are proposed in References 53,54, among others. Dal 53 has proposed an extended Hu–Washizu mixed finite element formulation, referred to it as “ Q 1 P 0 F 0” element and performed a comprehensive study to examine the performance of the newly proposed element. In addition to the displacement field, he defined four additional field variables, which are treated on the finite element level as averagese 53 .…”

“…Dal 53 has proposed an extended Hu–Washizu mixed finite element formulation, referred to it as “ Q 1 P 0 F 0” element and performed a comprehensive study to examine the performance of the newly proposed element. In addition to the displacement field, he defined four additional field variables, which are treated on the finite element level as averagese 53 . Stability of mixed finite elements is assessed by the inf‐sup condition, among others.…”