Section 6: Material modelling in solid mechanics - GAMM 2012

Section 6: Material modelling in solid mechanics 21
elements should usually be preferred. Therefore, we use a solid-shell element presented by Schwarze
and Reese [2] which combines the advantages of both solid elements and shell elements at the
same time. This element allows for displaying realistically the three-dimensional geometry while
still providing the suitable shape for thin structures.
In addition, the present solid-shell formulation utilizes a reduced integration scheme within the
shell plane using one integration point, whereas a full integration is used in thickness direction.
Thus, an arbitrary number of integration points can be chosen over the shell thickness. Thereby,
different fibre orientations of the layers can be taken into account easily, since the material
parameters can be defined for each integration point separately.
Moreover, the proposed solid-shell formulation removes all potential locking phenomena. In
particular, volumetric locking in case of (nearly) incompressible materials as well as Poisson
thickness locking in bending problems of shell-like structures are eliminated by use of the enhanced
assumed strain (EAS) concept. In addition, to cure the transverse shear locking which is present
in standard eight-node hexahedral elements, the assumed natural strain (ANS) method is applied.
[1] S. Reese. A micromechanically motivated material model for the thermo-viscoelastic material
behaviour of rubber-like polymers, Int J Plast, 19, 909–940, 2003.
[2] M. Schwarze, S. Reese. A reduced integration solid-shell finite element based on the EAS and
the ANS concept - large deformation problems, Int J Numer Methods Engng, 85, 289–329,
2011.
On consistent tangent operator derivation and comparative study of rubber-like material
models at finite strains
Mokarram Hossain, Paul Steinmann (Universität Erlangen-Nürnberg)
The overall micro-structure of rubber-like materials can be idealized by chain-like macromolecules
which are connected to each other at certain points via entanglements or cross-links. Such
special structure leads to a completely random three-dimensional network [2,3]. To model the
mechanical behaviour of such randomly-oriented micro-structure, several phenomenological and
micro-mechanically motivated network models for nearly incompressible hyperelastic polymeric
materials have been proposed in the literature. To implement these models for polymeric material
(undoubtedly with widespread engineering applications) in finite element method, one would
require two important ingredients, e.g. the stress tensor and the consistent fourth-order tangent
operator where the latter is the result of linearization of the former.
In this contribution, an extensive overview on several hyperelastic rubber-like material models
has been presented. Special focus is given particularly to derive the accurate stress tensors and
tangent operators which yield quadratic convergence when the governing nonlinear equations for
a boundary value problem are solved by the Newton-like iterative schemes. A simple but efficient
algorithm will be demonstrated to testify the correctness of the tangent operator locally of a
particular model without going into details of the finite element implementation [1].
[1] Steinmann P, Hossain M, Possart G (2011), Hyperelastic models for rubber-like materials:
Consistent tangent operators and suitability for Treloar’s data. Archive of Applied Mechanics,
In review (2011)
[2] Boyce MC, Arruda EM (2000), Constitutive models of rubber elasticity: a review. Rubber
Chemistry and Technology, 73: 504-523, 2000