Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function W. We will demonstrate these methods using the planar isotropic rank-one convex function Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin,where λ_max≥ λ_min are the singular values of F, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies W: GL ⁺(2) → R with an additive volumetric-isochoric split of the form W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF)with a concave volumetric part. This example is therefore of particular interest with regard to Morrey’s open question whether or not rank-one convexity implies quasiconvexity in the planar case.

The macroscopic interfacial healing behaviour in a series of urea-urethane networks as function of the hydrogen bonds and disulphides content is presented. The polymers were prepared with different crosslinking densities but with the same amount of dynamic covalent bonds (disulphide linkages). Tensile and fracture measurements were adopted to evaluate the degree of recovery of the mechanical properties after damage. Healing kinetics and healing efficiencies were quantitatively determined as a function of network composition, healing temperature and contact time. Finally, the recovery of mechanical properties was correlated with the viscoelastic response of the networks through rheological measurements and time-temperature superposition principle (TTS). The application of the TTS approach on both fracture healing and DMTA and subsequent mathematical descriptive model led to a better understanding of the influence of polymer architecture and that of the amount of reversible groups on the healing process.