Keywords: hyperelastoplasticity, Arbitrary Eulerian Lagrangian method, material force, configurational force.

This contribution aims at providing the framework of an Arbitrary Lagrangian Eulerian formulation for hyperelastic as well as hyperelastoplasticity problem classes. This ALE formulation is based on the dual balance of momentum in terms of spatial forces (the well-known Newtonian forces) as well as material forces (also known as configurational forces). The balance of spatial momentum results in the usual equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimise the shape or the finite element mesh. The main difference with traditional ALE approaches is that the combination of the Lagrangian and Eulerian description is no longer arbitrary, in other words the mesh motion is no longer user defined but completely embedded within the mechanical formulation [1,2,3,4].

A new ALE hyperelastic setting will be developed in rate form. We will deal with two systems of partial differential equations: The discretised spatial and the material momentum equation. The spatial equation will then be linearised by taking the material time derivative while the material equation will be linearised by taking the spatial time derivative. The solution defines the optimal spatial and material configuration in the context of energy minimisation in hyperelastic setting.

The hyperelastic setting will provide the platform to extend the formulation to include plasticity. In ALE hyperelastoplastic formulations additional equations are required to update the stresses. The principle of maximum plastic dissipation as well as the consistency conditions in spatial and material setting will introduce the spatial and material plastic parameters and rate form of the stress-strain relations. The solution defines the optimal spatial and material configuration in the context of energy minimisation in hyperelastoplasticity setting.

1

E. Kuhl, H. Askes, and P. Steinmann, "An ALE formulation based on spatial and material settings of continuum mechanics, Part 1: Generic hyperelastic formulation", Computer methods in applied mechanics and engineering, 193, 4207-4222, 2004. doi:10.1016/j.cma.2003.09.030

2

H. Askes, E. Kuhl and P. Steinmann, ""An ALE formulation based on spatial and material settings of continuum mechanics, Part 2: Classification and applications", Computer methods in applied mechanics and engineering, 193, 4223-4245, 2004. doi:10.1016/j.cma.2003.09.031

3

P. Steinmann, "On spatial and material settings of hyperelastodynamics", Acta Mech. 156, 193-218, 2002. doi:10.1007/BF01176756

4

Z. Uthman and H. Askes, "A hyperelastodynamic ALE formulation based on spatial and material forces", 3rd European Confrenece on Computacional Mechanics, eds. C.A. Mota Soares et al., Dordrecht(CD Rom proceedings), 2006. doi:10.1007/1-4020-5370-3_83

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