PAMM · Proc. Appl. Math. Mech. 17, 711 – 712 (2017) / DOI 10.1002/pamm.201710324 Structured deformations and applications Marco Morandotti 1, * 1 Fakultät für Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching b. München, Germany The scope of this contribution is to present an overview of the theory of structured deformations of continua and two appli- cations, all of which involve using this rather new formulation of mechanics problems in contexts that are different from one another, thus showing the power and versatility of the theory. c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The necessity of a theoretical apparatus that allows to incorporate multiple scales in the modelling of mechanical deformations became more and more evident as new insight on the microscopic behaviour of material deformations were available, both at the theoretical and at the experimental level. Structured deformations [6] respond to this need by providing a multiscale geom- etry that captures the contributions at the macroscopic level of both smooth geometrical changes and non-smooth geometrical changes at submacroscopic levels. These non-smooth geometrical changes, which are called disarrangements, encode the presence of cracks and defects in the continuum. Structured deformations have been successfully applied in many contexts to model plastic deformations and cracks [4,7–9]; the theory has been extended to more general contexts, especially by defining second-order structured deformations [12], which permit the inclusion of bending effects in the energy functional [3, 11,14]. Also relevant are the works [2,13,16], which focus on interfacial energies, relevant, among other things, for the study of granular and composite materials (see [10] in this context), as well as [15], where a more general functional setting is investigated. In Section 2, we present the general functional and energetic setting. In Section 3, we present the results for two problems. 2 Functional setup and relaxation of energies Let Ω ⊂ R N be a bounded open subset, which we take as the reference configuration of the body. Definition 2.1 (see [6]) A structured deformation is a triple (κ, g, G), where κ is a surface-like subset of Ω, and the injective and piecewise differentiable map g :Ω → R N the piecewise continuous tensor field G :Ω → R N×N are such that 0 <C< det G(x) ≤ det ∇g(x) at each point x ∈ Ω. The crucial result in the theory of Del Piero and Owen [6] is the following approximation theorem, stating that each structured deformation can be seen as the limit, in the sense of L ∞ convergence, of simple deformations. Theorem 2.2 (see [6, Theorem 5.8]) For each structured deformation (κ, g, G) there exists a sequence of injective, piecewise smooth deformations f n and a sequence of surface-like subsets κ n of the body such that g = lim n→∞ f n , G = lim n→∞ ∇f n , and κ = ∪ ∞ n=1 ∩ ∞ j=n κ j . By the convergences in Theorem 2.2, the tensor field G is not influenced by any discontinuities associated with the f n ’s. The non-smooth parts of the approximation f n determine the disarrangements tensor M , in such a way that the following relationship holds: ∇g = G + M. This additive decomposition justifies the names deformation without disarrangements and deformation due to disarrangements for G and M , respectively. In [5] the theory has been cast in a variational framework, thus making it suitable to treat problems involving energy minimisation. The definition of structured deformation can be given in weaker functions spaces, namely the space of special functions of bounded variations SBV (Ω; R d ) and the space of integrable matrix-valued functions L 1 (Ω; R d×N ). This has the advantage of formalising the notion of discontinuity of a function u and to properly define its jump set S(u) (see [1]). Definition 2.3 (see [5]) The space of structured deformations is SD(Ω) := {(g,G): g ∈ SBV (Ω; R d ),G ∈ L 1 (Ω; R d×N )}. In view of Definition 2.3, Theorem 2.2 has the following counterpart. Theorem 2.4 (see [5, Theorem 2.12]) Let (g,G) ∈ SD(Ω). Then there exist u n ∈ SBV (Ω; R d ) such that u n → g in L 1 (Ω; R d ), and ∇u n * ⇀G in M(Ω; R d×N ). Let now u ∈ SBV (Ω; R d ) and consider an initial energy functional defined by E(u) :=  Ω W (∇u(x)) dx +  S(u)∩Ω Ψ([u],ν (u)) dH N-1 , (1) ∗ Corresponding author: e-mail marco.morandotti@ma.tum.de, phone +49 (0)89 289 17989, fax +00 999 999 999 c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim