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Asymptotic analysis for heterogeneous elastic energies with material voids

Stefano Almi, Antonio Flavio Donnarumma, Manuel Friedrich

Abstract

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.

Asymptotic analysis for heterogeneous elastic energies with material voids

Abstract

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 tends to zero, the -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 -scale, we show that the limiting density related to jumps is twice the energy density for voids.
Paper Structure (15 sections, 21 theorems, 257 equations, 4 figures)

This paper contains 15 sections, 21 theorems, 257 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Integral representation and compactness of $\Gamma$-convergence: Proof of Theorem \ref{['first gamma convergence result']}
  4. Integral representation for a class of functionals defined on pairs of function-set
  5. Proof of Proposition \ref{['fundestE0']} and Lemma \ref{['LEMMA 4.3 CRIFRIESOL']}
  6. Fundamental estimate and compactness for the sequence $\mathcal{E}_{\varepsilon}$
  7. Bulk and surface densities: Proof of Proposition \ref{['limsupliminf']}
  8. Identification of the $\Gamma$-limit: Proof of Theorem \ref{['Identification of Gamma-limit']}
  9. An approximation result for void sets
  10. Surface part: Proof of \ref{['identification gamma limit voids']} and \ref{['h0 pc']}
  11. Proof of Theorem \ref{['relaxation']}
  12. Remaining proofs for the bulk density
  13. Definition and properties of $GBD$-functions
  14. G(S)BD functions
  15. Korn's inequality in $GSBD$

Key Result

Theorem 2.1

Let $\Omega \in \mathcal{A}_0$. Let $(f_\varepsilon)_\varepsilon$, $(g_\varepsilon)_\varepsilon$ be collections of functions satisfying $(f_1)$--$(f_3)$ and $(g_1)$--$(g_3)$, respectively. Let $\mathcal{E}_\varepsilon \colon L^0(\Omega;\mathbb{R}^d)\times \mathcal{M}(\Omega)\times \mathcal{A}(\Omega for all $A \in \mathcal{A}(\Omega)$. Moreover, for all $A \in \mathcal{A}(\Omega)$ and for every $(

Figures (4)

  • Figure 1: An example of a competitor in $\mathcal{D}^{e_2}_{\varepsilon}(B_1)$.
  • Figure 2: Consider $A^\prime=B_{1-2\delta}$, $A=B_{1-\delta}$ and $B=B_1 \setminus B_{1-2\delta}$. In blue the void $E$, in red the void $F$, and in purple the void $D$. In light blue $S_A^+ \setminus F$ and $S^-_A \setminus F$, in orange $S^+_B\setminus E$ and $S^-_B\setminus E$. Notice in particular that $\mathcal{L}^2(S^-_A\cap S_B^+)>0$.
  • Figure 3: An example in which shifting a curve by $\theta_\varepsilon$ creates an additional perimeter of $\color{black} 8 \color{black} \theta_\varepsilon$.
  • Figure 4: Example of the construction of the void $D_{n,\theta}^\delta$.

Theorems & Definitions (47)

  • Theorem 2.1: Compactness and Integral representation
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4: Identification of the $\Gamma$-limit
  • Remark 2.5
  • Theorem 2.6: Homogenization
  • proof
  • Remark 2.7: Periodic homogenization
  • Theorem 2.8: Relaxation
  • Theorem 3.1: Integral representation
  • ...and 37 more