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A variational phase-field model for anisotropic fracture accounting for multiple cohesive lengths

Angela Maria Fajardo Lacave, Francesco Vicentini, Fabian Welschinger, Laura De Lorenzis

TL;DR

This work addresses anisotropic fracture by developing a single-damage-variable variational phase-field model (the multi-cohesive model, MCM) that independently controls crack nucleation and propagation through direction-specific cohesive lengths while decoupling elastic and fracture anisotropy. It builds SM and MDM baselines, derives strength surfaces and directional critical stresses for all three models, and shows that MCM yields a smooth, tunable strength surface with enhanced calibration freedom under multiaxial loading. The authors perform second-order stability analyses and validate the formulations with 2D and 3D simulations, including layered and laminate configurations, demonstrating directional crack nucleation and propagation along prescribed material directions. Overall, the MCM offers computational efficiency of single-variable formulations and the directional flexibility of multi-variable models, making it well-suited for complex anisotropic fracture in composites and other heterogeneous materials.

Abstract

We propose a novel variational phase-field model for fracture in anisotropic materials. The model is specifically designed to allow a more flexible calibration of crack nucleation than existing anisotropic fracture formulations, while avoiding the introduction of multiple damage variables. In addition to the classical components of anisotropic phase-field models based on a single damage variable -- namely, anisotropic elasticity and the extension of the fracture energy density via a second-order structural tensor -- the proposed approach introduces fracture anisotropy through a cohesive degradation function with potentially distinct cohesive lengths along the principal material directions. For this reason, we refer to it as multi-cohesive model. This feature enables independent control of the critical stresses governing crack nucleation in each material direction. We analyze the homogeneous solution and its second-order stability, and we compare the resulting strength surfaces with those of two representative anisotropic phase-field models available in the literature. Finally, numerical simulations in two and three dimensions demonstrate the capability of the proposed model to independently control crack nucleation and propagation in anisotropic fracture problems of increasing complexity.

A variational phase-field model for anisotropic fracture accounting for multiple cohesive lengths

TL;DR

This work addresses anisotropic fracture by developing a single-damage-variable variational phase-field model (the multi-cohesive model, MCM) that independently controls crack nucleation and propagation through direction-specific cohesive lengths while decoupling elastic and fracture anisotropy. It builds SM and MDM baselines, derives strength surfaces and directional critical stresses for all three models, and shows that MCM yields a smooth, tunable strength surface with enhanced calibration freedom under multiaxial loading. The authors perform second-order stability analyses and validate the formulations with 2D and 3D simulations, including layered and laminate configurations, demonstrating directional crack nucleation and propagation along prescribed material directions. Overall, the MCM offers computational efficiency of single-variable formulations and the directional flexibility of multi-variable models, making it well-suited for complex anisotropic fracture in composites and other heterogeneous materials.

Abstract

We propose a novel variational phase-field model for fracture in anisotropic materials. The model is specifically designed to allow a more flexible calibration of crack nucleation than existing anisotropic fracture formulations, while avoiding the introduction of multiple damage variables. In addition to the classical components of anisotropic phase-field models based on a single damage variable -- namely, anisotropic elasticity and the extension of the fracture energy density via a second-order structural tensor -- the proposed approach introduces fracture anisotropy through a cohesive degradation function with potentially distinct cohesive lengths along the principal material directions. For this reason, we refer to it as multi-cohesive model. This feature enables independent control of the critical stresses governing crack nucleation in each material direction. We analyze the homogeneous solution and its second-order stability, and we compare the resulting strength surfaces with those of two representative anisotropic phase-field models available in the literature. Finally, numerical simulations in two and three dimensions demonstrate the capability of the proposed model to independently control crack nucleation and propagation in anisotropic fracture problems of increasing complexity.

Paper Structure

This paper contains 31 sections, 90 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Phase-field models for anisotropic fracture
  3. Energy density functional
  4. Energy density functional of the SM
  5. Energy density functional of the MDM
  6. Energy density functional of the proposed MCM
  7. Time-discrete variational formulation
  8. Irreversibility and non-negativity constraints
  9. Admissible strain and stress domains, strength surfaces
  10. Directional critical stress under uniaxial conditions
  11. Strength surfaces and directional critical stresses of phase-field models for anisotropic fracture
  12. Standard model
  13. Strength surface of the SM
  14. Directional critical stress of the SM
  15. Multi-damage model
  16. ...and 16 more sections

Figures (17)

  • Figure 1: Orientation-dependent Young's modulus for 2D analysis.
  • Figure 2: Strength surface and polar plot of the directional critical stress according to the SM
  • Figure 3: Strength surface and polar plot of the directional critical stress according to the MDM.
  • Figure 4: MCM admissible strain and stress domains for the orthotropic 2D case.
  • Figure 5: Strength surface and polar plot of the directional critical stress according to the MCM.
  • ...and 12 more figures