Rank-one convexity implies polyconvexity in isotropic planar   incompressible elasticity
  We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$ implies the polyconvexity of $W$. 
Ionel-Dumitrel Ghiba and Robert J. Martin and Patrizio Neff
math.CA