Eigendamage: an Eigendeformation model for the variational approximation of cohesive fracture in antiplane shear deformations

1. Introduction

• Ductile fracture involves non-trivial interaction between bulk and surface contributions, complicating both analysis and computation.Beyond elastic deformation, micro-voids and damage zones can produce diffuse-singular strain contributions and plastic deformation.Source: §1 · p. 1
• The work develops an eigendeformation-based model for ductile materials in scalar antiplane shear, intended to capture elastic, plastic, and fracture phenomena.Source: §1 · p. 2
• The two-field model couples displacement and eigendeformation through a limit with non-trivial interaction between plastic deformation and crack opening.Damage is associated with a transition from elastic to inelastic eigendeformation; the stored energy grows linearly for large bulk terms and small inelastic terms.Source: §1 · p. 2
• The multidimensional analysis is technically more challenging than the previously studied one-dimensional beam case.Source: §1 · p. 2

Outline.

• The proof develops compactness and relaxation results, establishes the Γ-liminf and Γ-limsup inequalities, and derives a corollary for energies with an optimized second variable.Source: Outline · p. 2

2. Setting of the problem and main results

• The approximate energy combines an elastic mismatch penalty for ∇u−g with a nonlocal penalty on the eigendeformation γ=gLd.The displacement is approximated in W 1,1(Ω), with γ absolutely continuous in the approximate model.Source: §2 · p. 3
• The limit charges Cantor variation at rate c0 and assigns jumps a direction-independent cost θ([u]), whose growth is comparable to min{|t|,1}.Cantor variation and jump openings contribute differently to the limit energy.Source: §2 · p. 3
• Finite limit energy requires γs=Dsu, while the absolutely continuous parts gLd and ∇uLd need only satisfy ∇u−g∈L2(Ω).The singular parts coincide; the elastic mismatch of the regular parts is subject to an integrability condition.Source: §2 · p. 4
• As ε→0, Eε Γ-converges to E for strong L1 convergence of displacement and flat convergence of eigendeformation.The theorem establishes both the lim inf and lim sup inequalities.Source: §2 · p. 4
• The convergence also holds with W −1,q convergence for the measure component when 1<q<2*, and the energies minimized over γ Γ-converge in strong L1.The minimized energies are defined by taking the infimum over the eigendeformation variable.Source: §2 · pp. 4–5

Preliminaries

• The jump set Ju records approximate jump points, while SBV functions are BV functions whose Cantor part vanishes.SBV 2 additionally requires an L2 approximate gradient and finite jump-set measure.Source: §3 · p. 6
• For BV functions, the derivative splits into absolutely continuous and singular parts, with the singular part further decomposed into jump and Cantor contributions.The jump derivative is represented by the opening [u] and normal νu on Ju.Source: §3 · p. 6
• Convergence in W −1,q implies flat convergence, and weak-* convergence implies flat convergence for measures.Flat convergence with uniformly bounded measure variation implies weak-* convergence.Source: §3 · p. 7

4. Compactness

• The compactness result, combined with the Γ-convergence theorem, implies convergence of almost minimizers.Source: §4 · p. 7
• A technical proposition replaces finite-energy configurations with close SBV configurations whose jump-set energy and L1 bounds are controlled.The proof identifies the passage from disjoint squares to overlapping balls in the nonlocal energy as its main technical difficulty.Source: §4 · pp. 7–8
• Bounded-energy sequences admit subsequences whose displacements converge in L1 to a BV function and whose eigendeformations converge in the flat norm to a measure with γ_s = D_su and ∇u − g in L2.The eigendeformation convergence also holds in W^-1,q for all q < 2*.Source: §4 · pp. 10–11

5. A relaxation result in BV (Ω)

• The relaxation concerns free-discontinuity functionals with linear growth, whose bulk density depends on the approximate gradient and a continuous perturbation.Source: §5 · p. 11
• The main result gives an upper estimate for the L1-relaxed functional on the whole domain for every BV function.Source: §5 · pp. 11, 17
• For every BV function, the set function of the relaxed functional is the restriction of a regular Borel measure.Source: §5 · pp. 12, 14
• Adding a δ-small total-variation term supplies a suitable lower bound needed to apply the integral-representation result.Source: §5 · p. 15
• The relaxation estimate is adapted for Section 7 by additionally imposing an L∞ bound.Source: §5 · pp. 17–18

6. Estimate from below of the Γ-lower limit

• The liminf estimate splits into surface, volume, and Cantor terms, handled respectively by blow-up, a direct argument, and slicing.The volume estimate is proved directly because it is unclear how to apply slicing or blow-up to the weakly converging γ-dependent term in the flat topology.Source: §6 · p. 18

6.1. Estimate from below of the surface term.

• The surface lower-bound minimization over suitable Radon measures is solved directly using the Hardy–Littlewood–Pólya majorization principle.Source: §6.1 · p. 18
• The surface-term Γ-liminf set function is a regular Borel measure, and rescaling bounds its density with only the non-local term contributing.The density estimate is reduced to a minimization problem on a suitable class of Radon measures.Source: §6.1 · pp. 18, 20
• For a pure-jump function, an approximating sequence converges in displacement and flat measure while matching the limiting energy and remaining unchanged outside an infinitesimal neighborhood of the jump set.The construction maintains a uniform displacement bound and an L2 bound on the difference between the displacement gradient and the density of the measure.Source: §6.1 · pp. 22–23
• The centered measure μ0 minimizes G, and majorization extends its optimality to continuous concave functions h with h(0) = 0.This extends the minimization result beyond the specific function f used in G.Source: §6.1 · pp. 29–30
• The surface-term lower estimate gives the jump density θ([u]) by combining a local estimate with the minimization result for the surface functional.The estimate applies to admissible (u, γ) on open sets A ⊂ Ω.Source: §6.1 · p. 30

6.2. Estimate from below of the volume term.

• The volume-term lower estimate for admissible BV limits is established using an auxiliary lower-semicontinuity lemma and SBV approximations.The lemma treats BV convergence with uniformly bounded jump measure alongside weak L2 convergence of the auxiliary fields.Source: §6.2 · pp. 30, 32, 33

6.3. Estimate from below of the Cantor term.

• For the Cantor-term lower estimate, the authors replace the d-dimensional ball average in the second term with a cube average, then use Fubini to split it into a (d−1)-dimensional cube average and an interval average for slicing and the one-dimensional result.Source: §6.3 · p. 33
• For every direction ξ, the energy satisfies E′(u, γ, A) ≥ c0|D_cu · ξ|(A), yielding the Cantor-part lower bound used in the full estimate.The directional estimate is obtained by slicing; the combined bounds include c0|D_cu|(A).Source: §6.3 · p. 36, §6.4 · p. 36

6.4. Conclusion and proof of Theorem 2.1 (i).

• The Γ-liminf inequality follows by combining lower bounds for the volume, Cantor, and jump contributions on every open subset A of Ω.The volume density is |∇u − g|² + c0|g| away from the jump and Cantor sets; the jump density is θ([u]), and the Cantor contribution is c0|D_cu|.Source: §6.4 · pp. 36–37

• For u in W(Ω), the recovery construction first approximates u energetically optimally, then defines the eigendeformation approximation from that displacement sequence.Source: §7 · p. 37
• The construction sets uε equal to u outside the δ-neighborhood of the jump set and uses linear interpolation inside, with δ = ε².Source: §7 · p. 37
• Defining gε = g − ∇u + ∇uε preserves ∇uε − gε = ∇u − g and yields γε converging to γ in the flat norm.Source: §7 · p. 38
• In the limsup estimate, the inner-integral term yields the cohesive jump-cost expression as ρ decreases to zero.Source: §7 · p. 40
• Density and relaxation extend the upper-bound estimate from W(Ω) to BV, and together with Theorem 6.18 establish Γ-convergence.Source: §7 · p. 41

8. Γ-convergence for the minimal energies with respect to γ

• The section proves Γ-convergence for the minimal energies obtained by minimizing with respect to γ.The result concerns the energies ˜Eε and ˜E.Source: §8 · p. 42
• Bounded minimal energies yield, up to a subsequence, L1 convergence to a BV function bounded by K.The compactness result follows by choosing low-energy γε and applying Theorem 4.2.Source: §8 · p. 42
• The liminf proof selects low-energy eigendeformations and invokes compactness, while the limsup proof uses a recovery sequence.The liminf argument passes to a subsequence with flat-norm convergence of γε; the limsup uses the recovery sequence of (u, γopt).Source: §8 · pp. 42–43