A Theory of Large Elastic Deformation

Abstract

It is postulated that (A) the material is isotropic, (B) the volume change and hysteresis are negligible, and (C) the shear is proportional to the traction in simple shear in a plane previously deformed, if at all, only by uniform dilatation or contraction. It is deduced that the general strain‐energy function, W, has the form $$\mathrm{W=}\frac{G}{4}{\displaystyle \sum _{\text{i=1}}^3}{\left({\lambda }_i-\frac{1}{{\lambda }_i}\right)}^2+\frac{H}{4}{\displaystyle \sum _{\text{t=1}}^3}\left({\lambda }_i^2-\frac{1}{{\lambda }_i^2}\right),$$ where the λ_i's are the principal stretches (1+principal extension), G is the modulus of rigidity, and H is a new elastic constant not found in previous theories. The differences between the principal stresses are σ_i[minus]σ_i=λ_i∂ W/∂λ_i[minus]λ_i∂ W/∂λ_i. Calculated forces agree closely with experimental data on soft rubber from 400 percent elongation to 50 percent compression.