TY - JOUR

T1 - Invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

AU - Criscione, John C.

AU - Humphrey, Jay D.

AU - Douglas, Andrew S.

AU - Hunter, William C.

N1 - Funding Information:
The first author would like to acknowledge that numerous discussions with Dr Joseph C. Criscione, materials scientist, contributed to the foundation of this theory. Also, we acknowledge Dr Alexander Spector for pointing out/interpreting works of Lurie so that we could compare our analysis with his. Financial support from the NIH (grant HL 30552 to WCH) is gratefully acknowledged.

PY - 2000/12

Y1 - 2000/12

N2 - 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.

AB - 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.

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U2 - 10.1016/S0022-5096(00)00023-5

DO - 10.1016/S0022-5096(00)00023-5

M3 - Article

AN - SCOPUS:0034559428

VL - 48

SP - 2445

EP - 2465

JO - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

SN - 0022-5096

IS - 12

ER -