Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

Abstract

We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of Γ-convergence. This complements the study in [39] by providing a linearization result in arbitrary space dimensions.2010 Mathematics Subject Classification. 74R10, 49J45, 70G75. Key words and phrases. Brittle materials, variational fracture, nonsimple materials, free discontinuity problems, Griffith energies, Γ-convergence, functions of bounded variation and deformation.as n → ∞. We point out that J ∇vn⊂ J vn since v n ∈ W 2,∞ (Ω \ J vn ; R d ).Using v n ∈ W 2,∞ (Ω \ J vn ; R d ) we can choose a sequence (η n ) n with η n → 0 such that z n := y + η n v n ∈ GSBV 2 2 (Ω; R d ) satisfies J zn= J y ∪ J vn and there holds z n → y in measure on Ω. By (4.2), the continuity of W , J zn= J y ∪ J vn , and J ∇zn⊂ J ∇y ∪ J vn we get lim sup n→∞ E ε (z n , Ω) ≤ E ε (y, Ω).(4.3)As J zn= J y ∪ J vn , J ∇y= J v , and J ∇vn⊂ J vn , we also get J ∇zn \ J zn⊂ (J ∇y ∪ J ∇vn ) \ (J y ∪ J vn )⊂ J v \ J vn .(4.4)

Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

“…A first key step in this direction was achieved by Chambolle, Giacomini, and Ponsiglione [18] showing a Liouville-type result for brittle materials storing no elastic energy. To the best of our knowledge, to date counterparts of the quantitative estimate (1.1) are limited to dimension two [46] or, in general dimensions, to a model for nonsimple materials [44] where the elastic energy depends additionally on the second gradient of the deformation, cf. [84].…”

“…[84]. The latter results have been employed successfully to identify linearized models in the small-strain limit [43,44], and to perform dimension reduction [82].…”

“…The authors performed a nonlinear-tolinear analysis in terms of suitably rescaled displacement fields and proved the convergence of minimizers for corresponding boundary value problems. Their study has been extended into various directions, ranging from models for incompressible materials [57,67], from atomistic models [11,81], to multiwell energies [1,80], plasticity [68], viscoelasticity [45], or fracture [43,44]. In all of these results, the rigidity estimate (1.1) or one of its variants plays a key role in establishing compactness.…”

Two-well linearization for solid-solid phase transitions

“…To clarify this, consider the following two-dimensional example. (For related examples, we refer to [38,Example 2.5] or [37,Example 2.4]). Let…”

“…An extension to the case of weaker growth conditions has been the subject of [2]. We further refer to related studies on atomistic systems [17,63], homogenization [43,59], viscoelasticity [39], plasticity [55], or fracture [37,38,60].…”