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The phase-field model of fracture incorporating Mohr-Coulomb, Mogi-Coulomb, and Hoek-Brown strength surfaces

S Chockalingam, Adrian Buganza Tepole, Aditya Kumar

TL;DR

The paper advances phase-field fracture by embedding arbitrary strength surfaces into the driving-force framework, enabling simultaneous prediction of strength-controlled nucleation and Griffith-type crack growth. It provides explicit driving-force construction c_e for Mohr-Coulomb, Hoek-Brown, and Mogi-Coulomb surfaces, including compression-correction and calibration via delta^epsilon to preserve toughness. Through SENT, SENB, and DCBT analyses in Indiana Limestone, the method reproduces strength surfaces exactly at calibration points, converges to the true strength surface as the regularization length vanishes, and demonstrates smooth transitions between strength- and Griffith-dominated regimes. The results highlight the method’s generality, robustness, and potential to extend phase-field fracture to a broad class of brittle materials with arbitrary strength criteria, while maintaining a practical universality of the calibration parameter delta^epsilon.

Abstract

Classical phase-field theories of brittle fracture capture toughness-controlled crack growth but do not account for the material's strength surface, which governs fracture nucleation in the absence of cracks. The phase-field formulation of Kumar et al. (2020) proposed a blueprint for incorporating the strength surface while preserving toughness-controlled propagation by introducing a nucleation driving force and presented results for the Drucker-Prager surface. Following this blueprint, Chockalingam (2025) recently derived a general driving-force expression that incorporates arbitrary strength surfaces. The present work implements this driving force within a finite-element framework and incorporates representative strength surfaces that span diverse mathematical and physical characteristics-the Mohr-Coulomb, 3D Hoek-Brown, and Mogi-Coulomb surfaces. Through simulations of canonical fracture problems, the formulation is comprehensively validated across fracture regimes, capturing (i) nucleation under uniform stress, (ii) crack growth from large pre-existing flaws, and (iii) fracture governed jointly by strength and toughness. While the strength surfaces examined here already encompass a broad range of brittle materials, the results demonstrate the generality and robustness of the proposed driving-force construction for materials governed by arbitrary strength surfaces.

The phase-field model of fracture incorporating Mohr-Coulomb, Mogi-Coulomb, and Hoek-Brown strength surfaces

TL;DR

The paper advances phase-field fracture by embedding arbitrary strength surfaces into the driving-force framework, enabling simultaneous prediction of strength-controlled nucleation and Griffith-type crack growth. It provides explicit driving-force construction c_e for Mohr-Coulomb, Hoek-Brown, and Mogi-Coulomb surfaces, including compression-correction and calibration via delta^epsilon to preserve toughness. Through SENT, SENB, and DCBT analyses in Indiana Limestone, the method reproduces strength surfaces exactly at calibration points, converges to the true strength surface as the regularization length vanishes, and demonstrates smooth transitions between strength- and Griffith-dominated regimes. The results highlight the method’s generality, robustness, and potential to extend phase-field fracture to a broad class of brittle materials with arbitrary strength criteria, while maintaining a practical universality of the calibration parameter delta^epsilon.

Abstract

Classical phase-field theories of brittle fracture capture toughness-controlled crack growth but do not account for the material's strength surface, which governs fracture nucleation in the absence of cracks. The phase-field formulation of Kumar et al. (2020) proposed a blueprint for incorporating the strength surface while preserving toughness-controlled propagation by introducing a nucleation driving force and presented results for the Drucker-Prager surface. Following this blueprint, Chockalingam (2025) recently derived a general driving-force expression that incorporates arbitrary strength surfaces. The present work implements this driving force within a finite-element framework and incorporates representative strength surfaces that span diverse mathematical and physical characteristics-the Mohr-Coulomb, 3D Hoek-Brown, and Mogi-Coulomb surfaces. Through simulations of canonical fracture problems, the formulation is comprehensively validated across fracture regimes, capturing (i) nucleation under uniform stress, (ii) crack growth from large pre-existing flaws, and (iii) fracture governed jointly by strength and toughness. While the strength surfaces examined here already encompass a broad range of brittle materials, the results demonstrate the generality and robustness of the proposed driving-force construction for materials governed by arbitrary strength surfaces.

Paper Structure

This paper contains 18 sections, 70 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Kinematics and basic definitions
  4. Constitutive behavior of the material
  5. Governing equations of the strength-incorporated phase-field theory
  6. Driving force $c_{\mathtt{e}}$
  7. Choice of strength surfaces
  8. Mohr-Coulomb
  9. Hoek-Brown
  10. Mogi-Coulomb
  11. Analysis and results
  12. The strength limit: fracture nucleation under uniform multi-axial stress
  13. The Griffith limit: fracture nucleation and propagation from a large crack
  14. Strength-Griffith mediated fracture nucleation
  15. Uniform stress in the vanishing crack limit: The single-edge notched tension test
  16. ...and 3 more sections

Figures (12)

  • Figure 1: Illustration of the Mohr-Coulomb criterion. Failure occurs when the shear stress on any plane exceeds the sum of the cohesion $c$ of the material and the frictional resistance arising from the normal compression stress clamping the plane.
  • Figure 2: Experimental strength data for Indiana Limestone from conventional triaxial compression testing and fitted Mohr-Coulomb (\ref{['eq:MC_s1s3']}) and Hoek-Brown (\ref{['eq:HBTC_In']}) strength criteria. The axial compression stress $s_1$ is plotted against the confining pressure $s_3$.
  • Figure 3: Illustration of the boundary value problem used to model strength-controlled failure under uniform multiaxial stress.
  • Figure 4: Comparison of plots of material strength surface and phase-field strength surface for the Mohr-Coulomb criterion in principal stress space under plane stress conditions using material parameters of Indiana Limestone. Solid black line is the exact Mohr-Coulomb material strength surface $\mathcal{F}_{\mathtt{MC} }$ defined in \ref{['eq:generalform_MC']}. (a) Phase-field strength surface ${\mathcal{F}}^\varepsilon_\mathtt{MC}$ defined in \ref{['eq:PFSS_MC']}. (b) Compression corrected phase-field strength surface ${\mathcal{F}}^\varepsilon_\mathtt{MC, cc}$ defined in \ref{['eq:PFSS_MC_cc']}. The critical stresses from finite element analysis are plotted as crosses in the corresponding colors for the different $\varepsilon$. The legend is the same for both plots. The red star-marked points are the calibrated strength locations (uniaxial tensile strength $\sigma_{\mathtt{ts}}^\mathtt{MC}$ and uniaxial compressive strength $\sigma_{\mathtt{cs}}^\mathtt{MC}$) that are exactly captured by the phase-field theory for all $\varepsilon$.
  • Figure 5: Comparison of plots of material strength surface and phase-field strength surface for the 3D Hoek-Brown criterion in principal stress space under plane stress conditions using material parameters of Indiana Limestone. Solid black line is the exact 3D Hoek-Brown material strength surface $\mathcal{F}_{\mathtt{HB} }$ defined in \ref{['eq:generalform_HB']}. (a) Phase-field strength surface ${\mathcal{F}}^\varepsilon_\mathtt{HB}$ defined in \ref{['eq:PFSS_HB']}. (b) Compression corrected phase-field strength surface ${\mathcal{F}}^\varepsilon_\mathtt{HB, cc}$ defined in \ref{['eq:PFSS_HB_cc']}. The critical stresses from finite element analysis are plotted as crosses in the corresponding colors for the different $\varepsilon$. The legend is the same for both plots. The red star-marked points are the calibrated strength locations (uniaxial tensile strength $\sigma_{\mathtt{ts}}^\mathtt{HB}$ and uniaxial compressive strength $\sigma_{\mathtt{cs}}^\mathtt{HB}$) that are exactly captured by the phase-field theory for all $\varepsilon$.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5