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On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. (English) Zbl 1129.74009
Summary: This paper discusses the multiplicative decomposition of deformation gradient into its volumetric and isochoric parts and its implications in the case of anisotropy. An analysis is carried out showing that the volumetric-isochoric split of the stored energy function can be justified and systematically derived on the basis of physical assumption that the spherical part of the stress depends on the determinant of deformation gradient without ad hoc introduction of multiplicative split. The analysis shows that care must be exercised in the case of anisotropic material description in order not to violate certain physical requirements. Additive splits of the energy can be justified on the basis of certain physical observations and independently of the multiplicative decomposition of deformation gradient. Specifically, it is shown that a spherical state of stress will cause, even in the incompressible case, a change of shape. In fibre reinforced materials, the split of the stored energy function into a part related to the matrix and a part related to the fibre is considered, showing that the volumetric-isochoric split should be applied to the matrix part only.

74B20 Nonlinear elasticity
74E10 Anisotropy in solid mechanics
Full Text: DOI
[1] Bednarczyk, H.; Sansour, C., On the choice of integrity base of strain invariants for constitutive equations of isotropic materials, (), 26-33
[2] Boehler, J.P., Applications of tensor functions in solid mechanics, (1987), Springer Wien · Zbl 0657.73001
[3] Flory, P.J., Thermodynamic relations for high elastic materials, Trans. Faraday soc., 57, 829-838, (1961)
[4] Lu, S.H.C.; Pister, K.S., Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids, Int. J. solids structures, 11, 927-934, (1975) · Zbl 0315.73018
[5] Maugin, G.A., Material forces: concepts and applications, Appl mech. rev., 48, 213-245, (1995)
[6] Ogden, R.W., Elastic deformation of rubberlike solids, (), 499-537
[7] Ogden, R.W., Nonlinear elastic deformation, (1984), Ellis Horwood Chichester · Zbl 0541.73044
[8] Simo, J.C.; Taylor, R.L.; Pister, K.S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. methods appl. mech. engrg., 51, 177-208, (1985) · Zbl 0554.73036
[9] Smith, G.F., On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J. engrg. sci., 9, 899-916, (1971) · Zbl 0233.15021
[10] Spencer, A.J.M., Theory of invariants, (), 239-353
[11] Spencer, A.J.M., Deformations of fibre-reinforced materials, (1972), Clarendon Press Oxford · Zbl 0238.73001
[12] Wang, C.C., On representations for isotropic functions: part I. isotropic functions for symmetric tensors and vectors, Arch. rational mech. anal., 33, 249-267, (1969) · Zbl 0332.15012
[13] Wang, C.C., Corrigendum to my recent papers on: representations for isotropic functions, Arch. rational mech. anal., 43, 392-395, (1971)
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