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

John C. Criscione, Jay D. Humphrey, Andrew S. Douglas, William C. Hunter

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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, K1-3, 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 (∂Ki/∂η) with (∂Kj/∂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 = Ai∂W/∂Ki, where the ∂W/∂Ki are scalar response functions and the Ai are kinematic tensors that are mutually orthogonal.

Original languageEnglish (US)
Pages (from-to)2445-2465
Number of pages21
JournalJournal of the Mechanics and Physics of Solids
Volume48
Issue number12
DOIs
StatePublished - Dec 2000

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

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