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Gamma convergence for a phase-field cohesive energy

Eleonora Maggiorelli, Matteo Negri, Francesco Vicentini, Laura De Lorenzis

TL;DR

The paper develops a Gamma-convergence framework for a cohesive phase-field energy featuring an internal variable to model inelastic response, enabling independent tuning of the strength surface within a variational setting. It rigorously derives the Gamma-limit to a sharp cohesive energy in both 1D and 2D anti-plane geometries, revealing a coupling between elastic, fracture, and inelastic terms that contrasts with classical decoupled models. Optimal-profile analysis yields explicit expressions for the jump-energy φ(j) and the corresponding profile z_j, while the limsup and liminf inequalities establish convergence for discrete finite-element approximations with mesh-dependent regularization length ε_h. Numerical experiments confirm isotropy of the discrete energy and demonstrate robustness to mesh geometry, supporting practical applicability for multiaxial fracture simulations with controlled strength envelopes.

Abstract

Reproducing the key features of fracture behavior under multiaxial stress states is essential for accurate modeling. Experimental evidence indicates that three intrinsic material properties govern fracture nucleation in elastic materials: elasticity, strength, and fracture toughness. Among these, strength remains the most often misunderstood, as it is not a single scalar quantity but rather a full surface in stress space. The flexibility in defining this strength envelope in phase-field models poses significant challenges, especially under complex loading conditions. Existing models in the literature often fail to capture both the qualitative shape and the quantitative fit of experimentally observed strength surfaces. To address this limitation, recent work introduces a new energy functional within a cohesive phase-field framework, specifically designed to control the shape of elastic domains. This model introduces an internal variable to describe the inelastic response. Notably, the strength is decoupled from the internal length, that is not interpreted as a material length scale, as often done in literature, but rather as a purely variational tool. The proposed functional allows for a rigorous variational framework, enabling the use of tools from the calculus of variations. We investigate the Gamma-convergence of the model to a sharp cohesive fracture energy in the one- and two-dimensional (anti-plane) setting, using a finite element discrete formulation and exploiting the strong localization of the damage variable. Notably, unlike classical models where the elastic and fracture energies converge independently, this model exhibits a coupling of all energy terms. We also present numerical simulations exploring the sensitivity of the model to mesh anisotropy, offering insight into both its theoretical robustness and its practical implementation.

Gamma convergence for a phase-field cohesive energy

TL;DR

The paper develops a Gamma-convergence framework for a cohesive phase-field energy featuring an internal variable to model inelastic response, enabling independent tuning of the strength surface within a variational setting. It rigorously derives the Gamma-limit to a sharp cohesive energy in both 1D and 2D anti-plane geometries, revealing a coupling between elastic, fracture, and inelastic terms that contrasts with classical decoupled models. Optimal-profile analysis yields explicit expressions for the jump-energy φ(j) and the corresponding profile z_j, while the limsup and liminf inequalities establish convergence for discrete finite-element approximations with mesh-dependent regularization length ε_h. Numerical experiments confirm isotropy of the discrete energy and demonstrate robustness to mesh geometry, supporting practical applicability for multiaxial fracture simulations with controlled strength envelopes.

Abstract

Reproducing the key features of fracture behavior under multiaxial stress states is essential for accurate modeling. Experimental evidence indicates that three intrinsic material properties govern fracture nucleation in elastic materials: elasticity, strength, and fracture toughness. Among these, strength remains the most often misunderstood, as it is not a single scalar quantity but rather a full surface in stress space. The flexibility in defining this strength envelope in phase-field models poses significant challenges, especially under complex loading conditions. Existing models in the literature often fail to capture both the qualitative shape and the quantitative fit of experimentally observed strength surfaces. To address this limitation, recent work introduces a new energy functional within a cohesive phase-field framework, specifically designed to control the shape of elastic domains. This model introduces an internal variable to describe the inelastic response. Notably, the strength is decoupled from the internal length, that is not interpreted as a material length scale, as often done in literature, but rather as a purely variational tool. The proposed functional allows for a rigorous variational framework, enabling the use of tools from the calculus of variations. We investigate the Gamma-convergence of the model to a sharp cohesive fracture energy in the one- and two-dimensional (anti-plane) setting, using a finite element discrete formulation and exploiting the strong localization of the damage variable. Notably, unlike classical models where the elastic and fracture energies converge independently, this model exhibits a coupling of all energy terms. We also present numerical simulations exploring the sensitivity of the model to mesh anisotropy, offering insight into both its theoretical robustness and its practical implementation.

Paper Structure

This paper contains 12 sections, 11 theorems, 145 equations, 12 figures.

Table of Contents

  1. Introduction
  2. $\Gamma$-convergence for the 1D model
  3. $\Gamma$-convergence for the 2D antiplane model
  4. Some related $\Gamma$-convergence results
  5. Optimal profile
  6. Limsup-inequality
  7. Liminf-inequality
  8. Numerical results
  9. Density
  10. Lower semi-continuity of the limit
  11. Properties of energy density $f$
  12. Numerical surface energy density for the 1D model

Key Result

Theorem 2.1

As $h \to 0$, the functionals $\widetilde{\mathcal{F}}_h$$\Gamma$-converge to $\widetilde{\mathcal{F}}: L^1(I)\times L^1(I)\to [0,+\infty]$ defined as follows: where and the functions $W$ and $\phi$ are given by:

Figures (12)

  • Figure 1: Analytical vs. numerical surface energy density $\phi$.
  • Figure 2: Shape of the elastic domain. Points P and Q are the stress states at which the material starts fracturing for the two tests discussed below.
  • Figure 3: The two different mesh discretizations.
  • Figure 4: Set-up of the numerical simulations.
  • Figure 5: $(U_x,U_y)=(0.5,-0.45)$: loading configuration with multiple solutions. Phase fields for mesh $\mathcal{T}_A$ (left) and mesh $\mathcal{T}_B$ (left) at $(U_x,U_y)= (0.01,0.009)$ . In both cases it is localized on a strip of width $\epsilon_h=0.025.$
  • ...and 7 more figures

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 6.1
  • Remark 6.2
  • Proposition 7.1
  • Remark 8.1
  • Lemma A.1
  • ...and 8 more