On Finite Element Implementation of Polyconvex Incompressible Hyperelasticity: Theory, Coding and Applications

Abstract

This work concerns mainly the finite element (FE) implementation of polyconvex incompressible hyperelastic models. A user material subroutine (UMAT) has been developed and can be utilized to define the aforementioned material behaviors in the FE system ABAQUS. The subroutine is written using a novel strategy in order to maximally simplify the relations for the analytical material Jacobian (MJ). The UMAT code is attached in the appendix. The developed subroutine allows to significantly decrease the time of computations and to avoid possible convergence difficulties. The structure of the code enables modifications which may lead to a rheological, damage or growth models, for instance.

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

“…where F is a deformation gradient and 'Cof' stands for cofactor. In the special case of isotropy [10], polyconvex potentials of hyperelasticity may be constructed as a sum of convex functionsŴ…”

“…The function (8), defined with a help of (5) and (10), is polyconvex because of the fact that it is constructed as a sum of the convex functions as follows:…”

Polyconvex hyperelastic modeling of rubberlike materials

“…In the case of material full incompressibility, for the uniaxial tension/compression (UT/UC) ̄1 =̄− 1∕2 1 and thus where S 1 is the first P-K stress tensor component. It follows from the relationships given in [26] that in the case of equibiaxial tension (BT) ̄1 =̄2 and the following relations for the components of first P-K stress tensor hold:…”

“…Thus, the experimental measurements obtained in Fig. 5 Isochoric deformation invariants as functions of stretch ratio: a uniaxial tension/compression and pure shear ( Ī 1 =Ī 2 ), b equibiaxial tension/compression [26] the PS test are insufficient for determining how the stored energy function depends on Ī 1 or Ī 2 . The PS data play a secondary role during the material parameter evaluation process.…”

“…Other mathematical conditions such as the stored energy's positivity or the condition of convexity with respect to the principal stretches are either insufficient or too restrictive, cf [3,14,27]. The specific limitations regarding material parameter values can be derived from the polyconvexity conditions, e.g., [26]. Thus, the nonuniqueness of material parameter identification is reduced when the energy polyconvexity conditions are applied.…”