Abstract
Finite element methods for solving engineering problems are used since decades in industrial applications. This market is still growing and the underlying methodologies, formulations, and algorithms seem to be settled. But still there are open questions and problems when applying the finite element method to situations where finite strains occur. Another problem area is the incorporation of constraints into the formulations, such as incompressibility, contact, and directional constraints needed to formulate anisotropic material behavior. In this section, we present the basic continuum formulation and different discretization techniques that can be used to overcome the problems mentioned above. Additionally, a set of test problems is presented that can be applied to test new finite element formulations.
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- 1.
Small letters are used for indices of vectors and tensors which are related to the basis \(\mathbf{e }_i\) of the current or spatial configuration. The quantities \(x_i\) are the spatial coordinates of X.
- 2.
Note that this split represents physically a different strain energy function than (35).
- 3.
The construction of such principle has advantages. One of them is that the development of efficient algorithms for the solution of the nonlinear equations can be based on optimization strategies.
- 4.
This result corresponds to the variation \(\delta \mathbf{E }\), defined already (49). The partial derivative of W with respect to \(\mathbf{C }\) leads to the second Piola-Kirchhoff stress tensor \(\mathbf{S }\), see (30): \(\mathbf{S } = 2\,\partial W/\partial \mathbf{C }\). Hence Eq. (57) is equivalent to the weak form (50) for a hyperelastic material.
- 5.
The user can overrule the automatic selection of the integration rule, but this is only necessary when special shape functions are used.
- 6.
All data are provided as dimensionless constants, it is assumed that the dimensions match real physical data.
- 7.
Here the variable p is the stress component related to the constraint, e.g., the stress in direction of \(\mathbf{a }\). It has to be scaled in order to yield the correct stress.
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Wriggers, P. (2016). Discretization Methods for Solids Undergoing Finite Deformations. In: Schröder, J., Wriggers, P. (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences, vol 566. Springer, Cham. https://doi.org/10.1007/978-3-319-31925-4_2
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