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Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. (English) Zbl 1427.74029
Summary: We present a new cell-centered Lagrangian scheme on unstructured mesh for hyperelasticity. It is based on the recently proposed Glace scheme [G. Carré et al., ibid. 228, No. 14, 5160–5183 (2009; Zbl 1168.76029)] for compressible gas dynamics. We show how to use the multiplicative decomposition of the gradient of deformation and the entropy property to derive the new scheme. We also prove the compatibility of this discretization with usual calculations of mass. Our motivation is to use hyperelasticity models for the study of finite plasticity, which is an extension of hypoelasticity to finite deformations. Hyperelasticity is a natural choice for extended models in solid mechanics, because of its mathematical structure which is a system of conservation laws with full rotational invariance. We study these properties for the Lagrangian system, and detail the various Eulerian formulations.
We present several test problems, in 1D and 2D planar cases, which shows the capability of the scheme to capture complex shock-waves and to simulate solid-fluid problems. In this article, we use a special equation of state [the authors, Contin. Mech. Thermodyn. 20, No. 3, 173–192 (2008; Zbl 1172.74008)]. Its interest is twofold: we can calculate multi-dimensional plastic phenomenon (such as split shock in 1D uniaxial cases or particular shapes in Taylor test-case), and it gives interesting multi-dimensional test cases for hyperelastic planar schemes.

74B20 Nonlinear elasticity
74S99 Numerical and other methods in solid mechanics
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