Strain energy function of an uncross-linked butadiene rubber estimated from planar extension

The derivatives of the strain energy function u with respect to the invariants of the strain tensor (I₁ and I₂) are estimated for uncross-linked butadiene rubber by using the BKZ constitutive equation. The derivatives at small deformations show anomalous behavior; namely, an upturn for ∂u/∂I₁ and a downturn for ∂u/∂I₂ take place, as is the case of cross-linked rubbers. At large deformations, u is well described by u = A₁(I₁ −3) + A₂(I₂ −3) with numerical constants A₁ and A₂. This behavior is also quite similar to that for cross-linked rubbers. The non-zero positive constant A₂ for the melt suggests that the non-zero value is due to neither the inhomogeneity in network structure nor high extension of constituent polymer chains.