Multiaxial Deformations of End-Linked Poly(dimethylsiloxane) Networks. 1. Phenomenological Approach to Strain Energy Density Function

Abstract

The phenomenological strain energy density function (W) for the elastomeric networks of end-linked poly(dimethylsiloxane) (PDMS) has been investigated as a function of the first and second invariants I1 and I2 of the Green's deformation tensor on the basis of the quasi-equilibrium stress−strain relationships of general biaxial deformations varying independently each of two principal strains. The Ii dependence of ∂W/∂Ij (i,j = 1,2) was obtained from the biaxial stress−strain data using the Rivlin−Saunders method. In the 3-dimensional plots of ∂W/∂Ii (i = 1,2) against both the (I1 − 3)- and (I2 − 3)-axes, the data points of each derivative at large deformations appear to fall on a plane inclining against the (I1,I2) plane, which suggests that both the derivatives linearly depend on each of I1 and I2. The formula of W is reasonably deduced from such linear dependence of ∂W/∂Ii on Ij (i,j = 1,2) as W = C10(I1 − 3) + C01(I2 − 3) + C11(I1 − 3)(I2 − 3) + C20(I1 − 3)2 + C02(I2 − 3)2. Each of the numerical coefficients Cij is assigned to each of the intercepts at I1 = I2 = 3 and the gradients of the two fitted planes in the (I1, I2, ∂W/∂Ii) plots. The estimated W satisfactorily reproduces not only the original biaxial stress−strain data but also the data of uniaxial, equibiaxial elongation, and uniaxial compression none of which were used for the original estimation of W. It is also demonstrated that the familiar Mooney−Rivlin type of W composed of only two linear terms of each of I1 and I2 does not even qualitatively reproduce the biaxial stress−strain data.