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Volumetric-distortional decomposition of deformation and elasticity tensor. (English) Zbl 1257.74018
Math. Mech. Solids 15, No. 6, 672-690 (2010); erratum ibid. 16, No. 2, 248-249 (2011).
Summary: The deformation gradient admits a multiplicative decomposition into a purely volumetric component and a purely distortional component. For a hyperelastic material, based on this decomposition, the elastic strain energy potential, the stress, and the elasticity tensor can be expressed in general as a function of both the volumetric deformation and the distortional deformation. However, the volumetric-distortional decomposition of deformation has often been employed in a fully decoupled form of the elastic strain energy potential, which is expressed as the sum of a term depending solely on the volumetric deformation and a term depending solely on the distortional deformation. This work has three main objectives. First, to derive the elasticity tensor in the general (non-decoupled) case, in its material, spatial, and linear forms; this is achieved by extensive use of fourth-order tensor algebra, and in particular of the properties of the so-called spherical operator, which is largely used, but very seldom given the dignity of being assigned a symbol and a name, in the literature. Second, to show that a fully decoupled potential gives rise to an elasticity tensor which may be inconsistent with its linearized counterpart, as some components of the linear elasticity tensor in general do not have a corresponding term in nonlinear decoupled elasticity tensor. Third, to obtain the conditions under which a purely hydrostatic stress causes a purely volumetric deformation, by means of the developed theory; the results show that this condition is satisfied if and only if the elastic potential is fully decoupled. While the whole approach is completely independent of the material symmetry, the cases of isotropy and transverse isotropy are shown as an example.

##### MSC:
 74B20 Nonlinear elasticity
Full Text:
##### References:
 [1] Ogden, R.W., Non-linear Elastic Deformations (1997) [2] Flory, P.J., Transactions of the Faraday Society 57 pp 829– (1961) · doi:10.1039/tf9615700829 [3] Ogden, R.W., Journal of Mechanics and Physics of Solids 26 pp 37– (1978) · Zbl 0377.73044 · doi:10.1016/0022-5096(78)90012-1 [4] Weiss, J.A., Computer Methods in Applied Mechanics and Engineering 135 pp 107– (1996) · Zbl 0893.73071 · doi:10.1016/0045-7825(96)01035-3 [5] Miehe, C., International Journal of Numerical Methods in Engineering 37 pp 1981– (1994) · Zbl 0804.73067 · doi:10.1002/nme.1620371202 [6] Holzapfel, G.A., Nonlinear Solid Mechanics. A Continuum Approach for Engineering (2000) · Zbl 0980.74001 [7] Holzapfel, G.A., Journal of Elasticity 61 pp 1– (2000) · Zbl 1023.74033 · doi:10.1023/A:1010835316564 [8] Hill, R., Journal of Mechanics and Physics of Solids 13 pp 89– (1965) · Zbl 0127.15302 · doi:10.1016/0022-5096(65)90023-2 [9] Walpole, L.J., Proceedings of the Royal Society of London A 391 pp 149– (1984) · Zbl 0521.73005 · doi:10.1098/rspa.1984.0008 [10] Quintanilla, R., Journal of Applied Mechanics 74 pp 455– (2007) · Zbl 1111.74605 · doi:10.1115/1.2338053 [11] Ting, T.C.T., Mathematics and Mechanics of Solids 6 pp 235– (2001) · Zbl 1028.74010 · doi:10.1177/108128650100600301 [12] Marsden, J.E., Mathematical Foundations of Elasticity (1994) [13] Curnier, A., Journal of Elasticity 37 pp 1– (1995) · Zbl 0828.73016 · doi:10.1007/BF00043417 [14] Bonet, J., Nonlinear Continuum Mechanics for Finite Element Analysis (Second Edition) (2008) · Zbl 1142.74002 · doi:10.1017/CBO9780511755446 [15] Federico, S., Journal of Mechanics and Physics of Solids 52 pp 2309– (2004) · Zbl 1115.74358 · doi:10.1016/j.jmps.2004.03.010 [16] Federico, S., Meccanica 43 pp 279– (2008) · Zbl 1163.74532 · doi:10.1007/s11012-007-9090-6 [17] Federico, S., International Journal of Engineering Science 46 pp 164– (2008) · Zbl 1213.74056 · doi:10.1016/j.ijengsci.2007.09.005 [18] Walpole, L.J., Advances in Applied Mechanics 21 pp 169– (1981) · Zbl 0512.73056 · doi:10.1016/S0065-2156(08)70332-6 [19] Gasser, T.C., Journal of the Royal Society Interface 3 pp 15– (2006) · doi:10.1098/rsif.2005.0073 [20] Federico, S., Considerations on incompressibility in linear elasticity [21] Spencer, A.J.M., Deformations of Fibre-Reinforced Materials (1972) · Zbl 0238.73001 [22] Kunin, I.A., International Journal of Engineering Science 19 pp 1551– (1981) · Zbl 0479.15023 · doi:10.1016/0020-7225(81)90078-1
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