The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity

• Hencky strain.
• Polyconvex energy.
• Extension of previous results.
• Existence of minimizers.

In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencky energies defined in the first part of this series, i.e.WeH(F)={μkek‖devnlogU‖2+κ2k^ek^[(logdetU)]2ifdetF>0,+∞ifdetF≤0,where F=∇φF=∇φ is the gradient of deformation, U=FTF is the right stretch tensor and devnlogU is the deviatoric part of the strain tensor logU. The main result in this paper is that in plane elastostatics, i.e. for n  =2, the energies of this family are polyconvex for k≥14, k^≥18, extending a previous result which proves polyconvexity for k≥13, k^≥18. This leads immediately to an extension of the existence result.