Section: Application Domains

Rubber elasticity

At the continuum level, rubber is modelled by an energy $E$ defined as the integral over a domain $D$ of ${ℝ}^d$ of some energy density $W$ depending only locally on the gradient of the deformation $u$: $E\left(u\right)={\int }_{D}W\left(\nabla u\left(x\right)\right)dx$. At the microscopic level (say 100nm), rubber is a network of cross-linked and entangled polymer chains (each chain is made of a sequence of monomers). At this scale the physics of polymer chains is well-understood in terms of statistical mechanics: monomers thermally fluctuate according to the Boltzmann distribution [61] . The associated Hamiltonian of a network is typically given by a contribution of the polymer chains (using self-avoiding random bridges) and a contribution due to steric effects (rubber is packed and monomers are surrounded by an excluded volume). The main challenge is to understand how this statistical physics picture yields rubber elasticity, and how this can be taken advantage of in order to devise physically-based constitutive laws for rubber. A rigorous derivation can be performed using tools of stochastic homogenization. The challenge is now to devise a numerical method to pass from polymer physics to an explicit continuum constitutive law, and to apply this approach to physically relevant models.