Abstract
Anisotropic material with inextensible fibers introduce constraints in the mathematical formulations. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution methods like the finite element method the presence of constraints—in this case associated to a possible fiber inextensibility compared to a matrix—lead to so called locking-phenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed that can handle anisotropic materials with inextensible fibers that can be relaxed to extensible fiber behaviour. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime.
Highlights
Many different approaches were developed over the last decade to formulate finite elements for anisotropic material with inextensible fibers
Finite elements for large strain anisotropic behaviour were developed in this paper
Special emphasis was put on a formulation that was able to enforce inextensible fiber extensions
Summary
Many different approaches were developed over the last decade to formulate finite elements for anisotropic material with inextensible fibers. Examples Several numerical examples are considered to show the performance of the new formulation for different loading cases In these examples the following discretization schemes are compared: Fig. 2 Part of the AceGen code for the mixed element based on a perturbed Lagrangian formulation for transverely anisotropic material. Tetrahedral elements for the constraint formulation (9), (10), (11) and (12) with quadratic ansatz functions (21) for the deformations and linear ansatz, see (23), for the Lagrangian multiplier σc These elements are labeled T2-A1 in the following. Hexahedral elements for the constraint formulation (9), (10), (11) and (12) with quadratic ansatz functions (20) for the deformations and linear ansatz, see (22), for the Lagrangian multiplier σc These elements are labeled H2-A1 in the following.
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