S. Conti and G. Dolzmann An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers J. Mech. Phys. Solids, 113: 126-143 2018 10.1016/j.jmps.2018.02.001
2015
Sergio Conti and Georg Dolzmann On the theory of relaxation in nonlinear elasticity with constraints on the determinant Arch. Rat. Mech. Anal., 217(2): 413-437 2015 http://dx.doi.org/10.1007/s00205-014-0835-9
Abstract: We consider vectorial variational problems in nonlinear elasticity of the form I[u]=∫W(Du)dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional ∫Wqc(Du)dx is an upper bound on the relaxation of I, and coincides with the relaxation if the quasiconvex envelope W qc of W is polyconvex and has p-growth from below with p≧n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.
2014
Sergio Conti, Georg Dolzmann and Stefan Müller Korn's second inequality and geometric rigidity with mixed growth conditions Calc. Var., 50: 437-454 2014 http://dx.doi.org/10.1007/s00526-013-0641-5
Abstract: Geometric rigidity states that a gradient field which is \( L^p\) -close to the set of proper rotations is necessarily \( L^p\) -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in \( L^p+L^q\) and in \( L^p,q\) interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces.
Sergio Conti and Geog Dolzmann Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions Math. Models. Metods App. Sci., 24(14): 2929-2942 2014 http://dx.doi.org/10.1142/S0218202514500419
Abstract: We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.