An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

Abstract

A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration. The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K 1–3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-of-distortion. Distortion is that part of the deformation that does not dilate. Moreover, pure dilatation ( K 2=0), pure shear ( K 3=0), uniaxial extension ( K 3=1), and uniaxial contraction ( K 3=−1) are tests which hold a strain invariant constant. Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy. Albeit useful for large and small strains, distinct advantage is shown for moderate strains (e.g. 2–25%). Central to this work is the orthogonal nature of the invariant basis. If η represents natural strain, then { K 1, K 2, K 3} are such that the tensorial contraction of ( ∂K i / ∂ η) with ( ∂K j / ∂ η) vanishes when i≠ j. This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal. In particular (summation implied) t= A i∂W / ∂K i , where the ∂W/ ∂K i are scalar response functions and the A i are kinematic tensors that are mutually orthogonal.