Quasiconvexity is the key mathematical notion in the study of macroscopic mechanical behavior of materials undergoing symmetry-breaking phase transitions. Computing the quasiconvex envelope of a given multi-well energy delivers a coarse-grained description of the material, with energetically optimal fine-scale oscillations correctly accounted for in the energetics, but averaged out in the kinematics.

Explicit formulas for quasiconvex envelopes of physically relevant energies are rare. Nematic elastomers provide an example in which such a formula has been found. This permits to set up finite element simulations of experimentally accessible geometries.

The animation shows a numerical experiment in which a sheet of nematic elastomer is stretched. Only one quarter of the sample is displayed (the rest can be obtained by symmetry). The colors identify the regions where microstructure occurs (blue). The sticks in the insets represent the orientation of the nematic director in a few representative points.
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Details

• The physics of nematic elastomers
• The BTW energy and its quasiconvexification
• Experiments and geometry
• Results

  Selected publications

  Soft elastic response of stretched sheets of nematic elastomers: a numerical study, Sergio Conti, Antonio DeSimone and Georg Dolzmann, J. Mech. Phys. Sol.

  Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Antonio DeSimone and Georg Dolzmann, Arch. Rat. Mech. Anal.