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Phase field modeling of brittle compressive-shear fractures in rock-like materials: A new driving force and a hybrid formulation

Shuwei Zhou, Xiaoying Zhuang, Timon Rabczuk

arXiv:2308.05748v1cs.CEmath.NA

TL;DR

Existing phase field models do not capture compressive-shear fractures in rock-like materials. This paper introduces a model with a compression-based driving force that accounts for cohesion and internal friction angle, and reports numerical results in good agreement with experiments.

  • Problem

    Existing phase field models cannot capture compressive-shear fractures in rock-like materials, while improvements to PFMs for these materials remain limited.

    Source: §1 · p. 4, §3.1 · p. 12

  • Method

    The paper proposes a phase field model with a driving force based on the negative strain parts from spectral decomposition and accounting for cohesion and internal friction angle.

    Source: §1 · p. 5

  • Results

    Numerical results for brittle compressive-shear fractures are in good agreement with experimental results.

    Source: §6 · p. 41

  • Takeaways & Limitations

    The authors conclude that the proposed model is feasible and practicable for simulating brittle compressive-shear fractures.

    Source: §6 · p. 41

    Typical anisotropic and hybrid phase field methods cannot model compressive-shear fractures because their driving force excludes the compressive part of elastic energy.

    Source: §3.1 · p. 12

Abstract

from arXiv · show

Compressive-shear fracture is commonly observed in rock-like materials. However, this fracture type cannot be captured by current phase field models (PFMs), which have been proven an effective tool for modeling fracture initiation, propagation, coalescence, and branching in solids. The existing PFMs also cannot describe the influence of cohesion and internal friction angle on load-displacement curve during compression tests. Therefore, to develop a new phase field model that can simulate well compressive-shear fractures in rock-like materials, we construct a new driving force in the evolution equation of phase field. Strain spectral decomposition is applied and only the compressive part of the strain is used in the new driving force with consideration of the influence of cohesion and internal friction angle. For ease of implementation, a hybrid formulation is established for the phase field modeling. Then, we test the brittle compressive-shear fractures in uniaxial compression tests on intact rock-like specimens as well as those with a single or two parallel inclined flaws. All numerical results are in good agreement with the experimental observation, validating the feasibility and practicability of the proposed PFM for simulating brittle compressive-shear fractures.

1 Introduction

The introduction identifies gaps in phase-field models for rock-like materials and presents a new hybrid formulation intended to capture brittle compressive-shear fractures and their dependence on cohesion and internal friction angle.

1 Introduction

  • Predicting fracture patterns in rock-like solids is challenging, especially for complex patterns, because intrinsic flaws such as microcracks, voids, and soft minerals complicate propagation.These flaws are described as increasing the complexity of fracture propagation.Source: §1 · p. 2
  • Experimental tests are limited by loading conditions and observations of fracture-pattern evolution, and some quantities, including crack-tip stress and strain, cannot be measured directly.The introduction presents numerical approaches as complementary tools for understanding fracture mechanisms.Source: §1 · pp. 2–3
  • Current phase-field models cannot predict compressive-shear fractures in rock-like solids or account for cohesion and internal friction angle in fracture propagation and load-displacement behavior.The models discussed are largely limited to mode-I tensile-dominated cases.Source: §1 · pp. 4–5
  • The proposed model adds an Hp driving force based on spectral decomposition of negative strain, incorporates cohesion and internal friction angle, and uses a hybrid formulation.These features are introduced to model brittle compressive-shear fractures in rock-like materials.Source: §1 · p. 5
  • The model is validated against uniaxial compression tests on intact specimens and specimens containing a single inclined flaw or two parallel inclined flaws.The introduction states that the model reproduces the observed compressive-shear fractures in these tests.Source: §1 · p. 5

2 Anisotropic phase field model of fracture

The section develops the mechanics-community anisotropic phase field model of Miehe et al., framed as an extension of Griffith’s fracture theory, through variational regularization and governing equations.

2.1 Variational approach and regularization

  • The variational approach treats fracture propagation as energy minimization and regularizes the crack using a phase field that ranges from intact to fully cracked.The length scale parameter l0 controls the transition region and indirectly reflects crack width; as l0 tends to zero, the phase field represents a sharp crack.Source: §2.1 · pp. 6, 8

2.2 Governing equations

  • Strain spectral decomposition separates tensile and compressive parts, while replacing positive elastic energy with a history variable prevents fracture healing and ensures crack irreversibility.This formulation is anisotropic because its stress-strain relationship follows from the split elasticity tensor.Source: §2.2 · pp. 8, 10
  • Regularization transforms the fracture problem with strong discontinuities into a standard multi-field problem that conventional finite element methods can solve.This facilitates implementation of the regularized phase field method.Source: §2.2 · p. 11

3 Modified phase field model for compressive-shear frac-

Existing phase field models struggle to capture realistic compressive-shear fractures in rock-like materials or account for cohesion and internal friction angle. The proposed hybrid model introduces a driving force based on compressive strain that incorporates both material properties.

3.1 Comparison on the current PFMs and their limitations

  • Anisotropic and hybrid models miss compressive-shear fractures because their phase-field driving force excludes compressive energy, while isotropic models produce unrealistic compression fractures.Earlier models also omit cohesion and internal friction angle, which affect fracture propagation and load-displacement curves.Source: §3.1 · p. 12

rock-like solids

  • A direct Mohr-Coulomb criterion is unsuitable for fracture initiation and propagation because crack-tip stress singularities can exceed its strength threshold under small external loads.The criterion is stress-based, whereas the phase field method is energy-based.Source: rock-like solids · p. 15
  • The proposed driving energy uses only the compressive part of the principal strain and incorporates cohesion and internal friction angle.A positive-part operator is introduced to prevent unrealistic fractures from small shear stress, and the driving force is defined to ensure fracture irreversibility.Source: rock-like solids · pp. 15–16
  • Replacing H in Ambati et al. with the new driving force Hp yields a hybrid phase field model that accounts for cohesion and internal friction angle in fracture initiation, propagation, and load-displacement curves.The model uses the critical energy release rate of mode II, Gc.Source: rock-like solids · pp. 16–17

4 Numerical implementation

The governing equations are discretized in weak form using nodal vector-matrix notation and solved with a staggered scheme. The simulations use implicit Generalized-α time integration and Anderson acceleration.

4 Numerical implementation

  • The weak governing equations are expressed in standard vector-matrix notation using nodal displacement and phase-field vectors, with test functions assigned the same discretization.The equations are subsequently written in discrete weak form.Source: §4 · pp. 17–19
  • The displacement and phase fields are solved sequentially with iterations between them, using a staggered scheme selected for greater flexibility and stability than the monolithic scheme.The scheme solves the discrete weak-form equations for the two fields in turn.Source: §4 · p. 19
  • Time integration uses the implicit Generalized-α method, and Anderson acceleration is applied to increase convergence rate in the simulations.The passage cites for the acceleration technology.Source: §4 · p. 19

5 Numerical examples

The numerical examples test the proposed phase field model on intact and flawed rock-like specimens under uniaxial compression, examining fracture patterns and load response across flaw configurations and variations in cohesion, friction angle, and eccentricity.

5.1 An intact specimen under uniaxial compression

  • For intact specimens, 2D and 3D simulations reproduce the experimentally observed V-shaped compressive-shear fracture and a close stress–strain curve, with 3D agreeing better with the test.The 2D curve deviates slightly, which the paper attributes to the added vertical stiffness from the plane-strain displacement restriction.Source: §5.1 · pp. 22–23, Figure 4 · p. 22, Figure 5 · p. 23

5.2 Uniaxial compression of a specimen with an inclined flaw

  • At α = 45°, shear zones form at flaw tips and initiate two cracks that widen and propagate obliquely; the modeled fracture patterns agree with experimental results.The fractures ultimately form an inclined pattern as loading increases.Source: Figure 7 · p. 25, §5.2 · p. 26, Figure 8 · p. 26
  • Under the same parameters, the anisotropic model [25] shows no load-displacement drop stage and simulates only wing and anti-wing tensile cracks, unlike the proposed model.The paper concludes that the anisotropic PFM is not suitable for predicting compressive-shear fracture in rock-like materials.Source: Figure 8 · p. 26, §5.2 · p. 26, Figure 12 · p. 28
  • At α = 15°, 30°, 60°, and 75°, modeled fractures are inclined shear; at α = 0°, flaw-tip fractures initiate at u = 0.1378 mm and propagate horizontally as normal shear by u = 0.1400 mm.Source: §5.2 · p. 29
  • As flaw inclination increases in the compressive-shear mode, the maximum sustainable load and the specimen’s maximum displacement increase.The paper attributes the displacement increase to a larger factual load-bearing area at higher inclination.Source: §5.2 · p. 29
  • Changing Gc and l0 does not affect inclined-flaw fracture patterns, although larger l0 produces wider fractures and peak load rises with Gc and falls with l0.The reported load-displacement curves show sensitivity to l0.Source: §5.2 · p. 29, Figure 15 · p. 31, Figure 17 · p. 32
  • For an inclined flaw, increasing cohesion or internal friction angle leaves the fracture pattern unchanged but increases peak load, reflecting greater compressive load-bearing capacity.The reported increase in peak load with cohesion c and friction angle φ is stated to correspond to experimental observations.Source: §5.2 · pp. 31, 34
  • As eccentricity increases, the inclined-flaw fracture propagates at a lower angle to the horizontal, while peak load decreases only negligibly.The text attributes the negligible peak-load decrease to an almost unchanged factual load-bearing area under compression.Source: §5.2 · p. 34, Figure 22 · p. 35

flaws

  • Two-flaw simulations reproduce fracture initiation, coalescence, and boundary-reaching propagation, with quasi-coplanar paths for Type A and non-coplanar S-shaped coalescence for Type B.For Type A, fractures initiate at u = 0.1178 mm, coalesce at u = 0.1184 mm, and reach the specimen boundaries at u = 0.1200 mm; for Type B, initiation is at u = 0.1244 mm and coalescence and boundary-reaching occur at u = 0.1258 mm.Source: flaws · p. 36, Figure 24 · p. 37
  • Non-coplanar two-flaw specimens have a larger peak load than coplanar specimens, attributed to their larger factual load-bearing area.The simulated compressive-shear patterns are reported to align with experimental coalescence observations for multiple inclined flaws [5]; tensile wing cracks are outside this study’s focus.Source: flaws · p. 37, Figure 27 · p. 39, Figure 28 · p. 40

6 Conclusions

The paper proposes a phase field model for brittle compressive-shear fractures in rock-like materials, using a new driving force based on compressive strain that accounts for cohesion and internal friction angle.

6 Conclusions

  • The model reconstructs its phase-field evolution driving force from compressive strain components and accounts for cohesion and internal friction angle in fracture propagation.The authors state that other phase field models cannot account for the influence of cohesion and internal friction angle.Source: §6 · p. 41
  • A hybrid formulation was implemented in COMSOL within the conventional finite element method framework.Source: §6 · p. 41
  • Numerical results agree well with previous experiments on intact specimens and specimens containing one or two parallel inclined flaws, supporting the model’s feasibility for brittle compressive-shear fracture simulation.The tests used specimens under uniaxial compression.Source: §6 · p. 41
  • Fracture initiation, propagation, coalescence, and branching emerge autonomously without external fracture criteria or a prescribed propagation path.The authors identify this as an advantage for modeling complex compressive-shear fractures in rock-like materials.Source: §6 · p. 41
  • Suppressing tensile strain excludes tensile and tension-shear fractures; the authors propose combining this driving force with previously used ones to address mixed-mode fractures in rocks.Source: §6 · p. 41
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