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Author: Heinz, Sebastian ×

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    Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Heinz, Sebastian; Mielke, Alexander
    We revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in [MTL02]. This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.
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    On the structure of the quasiconvex hull in planar elasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Heinz, Sebastian
    Let K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rank-one convex hull in planar elasticity.
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    Computations of quasiconvex hulls of isotropic sets
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Heinz, Sebastian; Kružik, Martin
    We design an algorithm for computations of quasiconvex hulls of isotropic compact sets in in the space of 2x2 real matrices. Our approach uses a recent result by the first author [Adv. Calc. Var. (2014), DOI: 10.1515acv-2012-0008] on quasiconvex hulls of isotropic compact sets in the space of 2x2 real matrices. We show that our algorithm has the time complexity of O(N log N ) where N is the number of orbits of the set. We show some applications of our results to relaxation of L∞ variational problems.
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    A model for the evolution of laminates in finite-strain elastoplasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Hackl, Klaus; Heinz, Sebastian; Mielke, Alexander
    We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented.
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    Quasiconvexity equals rank-one convexity for isotropic sets of 2 x 2 matrices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Heinz, Sebastian
    Let K be a given compact set of real 2x2 matrices that is isotropic, meaning invariant under the left and right action of the special orthogonal group. Then we show that the quasiconvex hull of K coincides with the rank-one convex hull (and even with the lamination convex hull of order 2). In particular, there is no difference between quasiconvexity and rank-one convexity for K. This is a generalization of a known result for connected sets.