We study the effective behavior of heterogeneous energies arising in the modeling of material voids in geometrically linear elastic materials. Specifically, we consider functionals featuring bulk terms depending on the symmetrized gradient of the displacement and terms comparable to the surface area of the material voids inside the material. Under suitable growth conditions for the bulk and surface densities we prove that, as the microscale $\varepsilon$ tends to zero, the $Γ$-limit admits an integral representation that contains an additional surface term expressed by jump discontinuities of the displacement outside of the void region. This term is related to the phenomenon of collapsing of voids in the effective limit. Under a continuity assumption of the surface density at the $\varepsilon$-scale, we show that the limiting density related to jumps is twice the energy density for voids.
The derivation of homogenization and general Γ-convergence results for integral functionals has been a thriving field over the last decades. We refer to [11] for an overview in the Sobolev setting and mention, without claiming exhaustiveness, various extensions to BV -type frameworks featuring bulk and surface terms [6,12,14,15,16,17,23,24,25,34,37,41]. The analysis generally relies on the localization method for Γ-convergence [21], combined with a global method [7,8] that allows to represent limits as integral functionals. The goal of the present article is to extend this approach to energies in elasticity with material voids.
The formation of material voids inside elastically stressed solids can be formulated in the context of stress-driven rearrangement instabilities (SDRI), see [5,38,39,46]. Energy functionals describing SDRI are characterized by the competition between stored elastic bulk energy and surface contributions. Problems of this type have been intensively studied in recent years using variational methods, including results on existence, regularity, and relaxation [9,18,27,40,44], as well as on linearization [32,30], and dimension reduction [31,33,42,43]. Despite this recent progress, homogenization in this setting remains largely unexplored and appears to be limited to a periodic homogenization result for a scalar model [45].
Setting and background: We now describe in more detail the energy functionals modeling material voids in elastically stressed solids. In the setting of linearized elasticity, a prototypical energy takes the form
where Ω ⊂ R d denotes the reference domain, E is the (sufficiently smooth) void set, and u ∈ H 1 (Ω \ E; R d ) denotes the displacement field. The first term corresponds to the elastic bulk energy, which is quadratic in the symmetrized gradient e(u) := ∇u+∇u T 2 and depends on the fourth-order, positive semi-definite elasticity tensor C. The surface term accounts for the presence of the void set E, where g is a norm and ν E denotes the outer unit normal to ∂E. If u is of the form u = (0, . . . , 0, v) for a scalar function v, then (1.1) reduces to a scalar problem.
Functionals of the type (1.1) are separately lower semicontinuous: for fixed E with Lipschitz boundary, the map u → I(u, E) is weakly lower semicontinuous in H 1 , while for fixed u the map E → I(u, E) can be regarded as a lower semicontinuous functional on sets of finite perimeter. However, as pointed out by Braides, Chambolle, and Solci [9], the functional (1.1) is not lower semicontinuous with respect to the pair (u, E). More precisely, Crismale and the third author showed in [18] that the lower semicontinuous envelope of (1.1) coincides with
We refer to [9] for analogous results in the scalar case and in nonlinear elasticity. In the relaxed formulation, regular void sets are replaced by sets of finite perimeter with essential boundary ∂ * E.
The interaction between the displacement field and the void set gives rise to an additional surface term involving the jump set J u with normal ν u . Indeed, during relaxation, voids may collapse into discontinuities of u, and such collapsed interfaces contribute twice to the surface energy, yielding the density 2g. Note that (1.2) is technically more involved than its scalar or nonlinear counterparts, since u is no longer a (G)SBV function [4], but rather belongs to the space of generalized special functions of bounded deformation (GSBD) [22]. A key achievement of [18] lies in overcoming the lack of Korn’s inequality and dealing with the resulting analytical difficulties in this setting.
In the scalar framework, the result of [9] was extended by Solci [45] to heterogeneous materials with periodic microstructure. Specifically, energies of the form
are considered, where ε > 0 represents the size of the microstructure, and f and g are densities periodic in x and satisfying suitable growth conditions. Under the additional assumption that g is a norm in the ν-variable, the energies (1.3) Γ-converge to ˆΩ\E f hom (∇y) dx + ˆΩ∩∂ * E g hom (ν E ) dH d-1 + ˆJu\∂ * E 2g hom (ν u ) dH d-1 , for suitable homogenized densities f hom and g hom which are independent of the x-variable. The aim of this work is to derive effective models for heterogeneous materials with material voids in the setting of linearized elasticity. In this way, we generalize [45] to (a) the vectorial, geometrically linear setting and to (b) materials with possibly nonperiodic microstructures. In this sense, our work can be understood as a counterpart of [15,34] where we consider material voids in place of fracture sets. The main novelty of our contribution lies in the analysis of collapsing material voids along the homogenization process. Specifically, we study the asymptotic behavior of energy functionals of the form
for suitable bulk and surface densities f ε and g ε , see (f 1 )-(f 3 ) and (g 1 )-(g 3 ) in Section 2 for the precise assumptions. We emphasize that (1.4) allows for more gener
This content is AI-processed based on open access ArXiv data.