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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 69
HyperElastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the SethHill Strain Measures K. Farahani and H. Bahai
Department of System Engineering, Brunel University, Uxbridge, Middlesex, United Kingdom K. Farahani, H. Bahai, "HyperElastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the SethHill Strain Measures", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 69, 2002. doi:10.4203/ccp.75.69
Keywords: constitutive equations, energy conjugate stresses, hyperelastic.
Summary
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
SethHill conjugate stress tensors in the principal axes was derived in [10]. Some
well known relations of the SethHill 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 hypoelastic 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]. Hyperelastic 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, Hyperelastic constitutive equations for the SethHill 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 stressstrain 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. References
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