Uniaxial Extension and Compression in Stress-Strain Relations of Rubber

Abstract

Abstract A comprehensive literature survey shows the general applicability of the generalized normalized Martin, Roth and Stiehler equation to uniaxial stress-strain data in extension and compression on rubber vulcanizates. The equation can be expressed as F/M=(L−1−L−2) expA (L−L−1) where F is the stress on the undeformed section and L the ratio of stressed to unstressed length. The equation contains two constants—M, Young's Modulus, the slope of the stress-strain curve at L=1, and A an empirical constant. The conformity of stress-strain data to the equation can readily be determined by a plot of logF/(L−1−L−2) against (L−L−1). In almost every case a straight line is obtained, from the slope and intercept of which both the constants can be determined. The range of validity of the equation usually begins near L=0.5 (in the compression region) and continuing through the region of low deformations often extends to the region of rupture in extension. If uniaxial compression data are available the modulus can thus be obtained by interpolation through the region of low deformations, where experimental data are often somewhat unreliable. The value of the modulus M varies with the nature of the rubber, the extent of vulcanization, and the time and temperature of creep or stress relaxation. The value of the constant A is near 0.4 for pure-gum vulcanizates, increasing to values near 1.0 with increasing filler content, and showing an abrupt increase when crystallization occurs. Direct experimental observations where the deformation of a single specimen is varied continuously from compressive to tensile deformation, are cited to show that M, defined as the limit of the ratio of stress to strain, is independent of the direction of approach to the limit at L=0.5. The normalized Mooney-Rivlin plots show F/[2M (L−L−2)] against L−1. These graphs have only limited regions of linearity corresponding to constant values of the coefficients C1 and C2. Since these regions do not include the undeformed state the Mooney-Rivlin equation cannot be used at low elongations or in compression. The values of C1 and C2 show very wide fluctuations for the Mooney-Rivlin plots of experimental data, which are themselves usually well represented by the Martin, Roth, and Stiehler equation with different values of the constant A. In view of all these considerations the conclusion of the present study confirms that of Treloar in his recent publications in failing to find much utility in making Mooney-Rivlin plots. The failure to represent the experimental data at low elongations and the inability to correlate the constants with theoretical predictions based on strain energy or statistical theory considerations are the most serious objections.