Uniaxial extension and compression in stress-strain relations of rubber

Abstract

A survey of experimental data from the literature in cases where the deformation of a specimen is varied continuously from uniaxial compression to tensile deformation shows that Young's Modulus M, defined as the limit of stress to strain in the undeformed state, is independent of the direction of approach to the limit. The normalized stress-strain relation of Martin, Roth, and Stiehler (MRS, 1956) is F/M = (L-1 - L-2) exp A (L - L-1) where F is the stress on the undeformed section, L is the extension ratio, and M and A are constants. Values of M and A are obtained from the intercept and slope of a graph of experimental observations of log F/(L-1 - L-2) against (L - L-1) including observations of uniaxial compression if available. They found the value of A to be about 0.38 for pure-gum vulcanizates of natural rubber and several synthetics. In later work several observers have now found that the equation is also valid for vulcanizates containing a filler, but A is higher, reaching a value of about 1 for large amounts of filler. In extreme cases A is not constant at low deformations. The range of applicability in many cases now is found to extend from the compressive region where L = 0.5 up to the point of tensile rupture or to a point where A increases abruptly because of crystallization. Taking A as a constant parameter in the range 0.36 to 1, graphs are presented showing calculated values of (1) F/M as a function of L and (2) the normalized Mooney-Rivlin plot of F/[2M(L - L-2)] against L-1. Each of the latter graphs has only a limited region of linearity corresponding to constant values of the Mooney-Rivlin coefficients C1 and C2. Since this region does not include the undeformed state, where L = 1, or any of the compression region, the utility of the Mooney-Rivlin equation is extremely limited, since it can not be used at low elongations. The coefficients are dramatically altered for rubbers showing different values of the MRS constant A. For rubbers showing the higher values of A, the coefficients are radically altered and the region of approximate linearity is drastically reduced.