The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on January 23, 2008
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbm028
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NONLINEAR TRANSVERSELY ISOTROPIC ELASTIC SOLIDS: AN ALTERNATIVE REPRESENTATION
( Etisalat University College, PO Box 573, Sharjah, United Arab Emirates )
shariff{at}ece.ac.ae
Received 27 February 2007.
Revise 30 October 2007. Revise 27 November 2007.
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Abstract |
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A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis–Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.