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

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 I 1 and I 2 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 I i dependence of ∂ W /∂ I j ( i,j = 1,2) was obtained from the biaxial stress−strain data using the Rivlin−Saunders method. In the 3-dimensional plots of ∂ W /∂ I i ( i = 1,2) against both the ( I 1 − 3)- and ( I 2 − 3)-axes, the data points of each derivative at large deformations appear to fall on a plane inclining against the ( I 1, I 2 ) plane, which suggests that both the derivatives linearly depend on each of I 1 and I 2 . The formula of W is reasonably deduced from such linear dependence of ∂ W /∂ I i on I j ( i,j = 1,2) as W = C 10 ( I 1 − 3) + C 01 ( I 2 − 3) + C 11 ( I 1 − 3)( I 2 − 3) + C 20 ( I 1 − 3) 2 + C 02 ( I 2 − 3) 2 . Each of the numerical coefficients C ij is assigned to each of the intercepts at I 1 = I 2 = 3 and the gradients of the two fitted planes in the ( I 1, I 2, ∂ W /∂ I i ) 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 I 1 and I 2 does not even qualitatively reproduce the biaxial stress−strain data.