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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 75
Edited by: B.H.V. Topping and Z. Bittnar
Paper 69

Hyper-Elastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the Seth-Hill Strain Measures

K. Farahani and H. Bahai

Department of System Engineering, Brunel University, Uxbridge, Middlesex, United Kingdom

Full Bibliographic Reference for this paper
K. Farahani, H. Bahai, "Hyper-Elastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the Seth-Hill Strain Measures", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 69, 2002. doi:10.4203/ccp.75.69
Keywords: constitutive equations, energy conjugate stresses, hyper-elastic.

The concept of energy conjugacy first presented by Hill [1] states that a stress measure is said to be conjugate to a strain measure if represents power or rate of change of internal energy per unit reference volume. Based on this definition, Guo and Man [2] derived explicit tensorial formulations for conjugate stress for , while earlier, the stress measure conjugate to logarithmic strain tensor , had been derived by Hoger [3]. Also, following the Hill's principal axis method and energy conjugacy notion, a relation between the components of two Seth-Hill conjugate stress tensors in the principal axes was derived in [10]. Some well known relations of the Seth-Hill strain measures with their conjugate stresses are givien in [5,6,7]. Also basic relations were derived for the rate of logarithmic strain [4].

Use of hypo-elastic constitutive equations for large strains in nonlinear finite element applications usually needs special considerations [8]. For example, the strain doesn't vanish in some elastic loading and unloading closed cycles, and they need objective rate tensors, and incrementally objective algorithm for numerical application and integration. Some of them may fluctuate under excessive shear deformation [8]. Hyper-elastic constitutive equation in comparison, don't need these considerations. However, their behaviour for large elastic strains is important, and may differ in tension and compression. In the present work, Hyper-elastic constitutive equations for the Seth-Hill strains and their conjugate stresses are explored as a natural generalisation of Hook's law for finite elastic deformations. Based on the uniaxial and simple shear tests, the response of the material for different constitutive equations is examined. Some required kinematic and tensor equations are used from [6,10]. Together with an objective rate model, the effect of different constitutive laws on Cauchy stress components is compared. It is shown that the logarithmic strain and its conjugate stress give answers closer to that of the rate model. In addition, use of Biot stress-strain pairs for a bar element, results in an elastic spring which obeys the hook's law even for large deformations and behaves the same in tension and compression. The volume change of material is another factor which is noticed.

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A. Hoger, D.E. Carlson, "Determination of stretch and rotation in the polar decomposition of deformation gradient", Quart. App. Math. 10, 1984.
K. Farahani, R. Naghdabadi, "Conjugate stresses of the Seth-Hill strain tensors", International Journal of Solids and Structures, 37, 5247-5255, 2000. doi:10.1016/S0020-7683(99)00209-7

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