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

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 (K₂ = 0), pure shear (K₃ = 0), uniaxial extension (K₃ = 1), and uniaxial contraction (K₃ = -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 {K₁,K₂,K₃} 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.