Source-linked AI summary
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
TL;DR
Current phase field models cannot capture brittle compressive-shear fractures in rock-like materials or the effects of cohesion and internal friction angle. This paper introduces a compressive-strain-based driving force with a hybrid formulation, and numerical tests agree well with experimental observations.
Problem
Current phase field models cannot capture compressive-shear fractures in rock-like materials or describe cohesion and internal friction angle effects on load-displacement behavior.
Method
The model uses strain spectral decomposition, compressive strain components, cohesion, and internal friction angle in a new phase-field driving force with a hybrid formulation.
Results
Numerical results for intact specimens and specimens with single or two parallel inclined flaws agree well with experimental results.
Takeaways & Limitations
The proposed phase field model is feasible and practical for simulating brittle compressive-shear fractures in rock-like materials.
Takeaways & Limitations
Typical anisotropic and hybrid phase field methods cannot model compressive-shear fractures because compressive elastic energy is excluded from their driving force.
Abstract
from arXiv · showhide
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
Rock-like materials commonly undergo complex compressive-shear failure, but existing phase field models mainly address tensile-dominated fractures and omit key geomaterial effects. The paper proposes a new driving force and hybrid formulation to model these fractures under compression.
- Motivation: Fracture prediction in rock-like solids remains challenging because geomaterials contain intrinsic flaws and exhibit complex failure patterns.These flaws include microcracks, voids, and soft minerals.
- Motivation: Numerical methods complement experiments by providing physical insights into fracture mechanisms and quantities that are difficult to measure directly.Stress and strain near crack tips are cited as examples of quantities that cannot be directly measured.
- Existing phase field methods: Phase field methods represent sharp fractures as diffuse damage-like zones and naturally track fracture paths through coupled phase and mechanical fields.The zone width is controlled by a length-scale parameter, and fracture propagation follows from solving partial differential equations.
- Existing phase field methods: Existing phase field applications in rock-like materials are relatively limited and largely address mode I tensile-dominated fractures.Examples include Brazilian, notched semi-circular bend, direct tension, and hydraulic-fracture tests.
- Research gap: Current phase field models cannot predict compressive-shear fractures because compressive strains and compressive elastic energy are excluded from phase-field evolution.They also do not account for cohesion and internal friction angle in fracture propagation or load-displacement behavior.
- Proposed approach: The paper introduces a new driving force Hp based on strain spectral decomposition, using compressive strain components while incorporating cohesion and internal friction angle.A hybrid formulation is developed for implementation and tested on intact specimens and specimens containing one or two parallel inclined flaws.
2 Anisotropic phase field model of fracture
The anisotropic phase field model regularizes fracture by replacing sharp discontinuities with a phase field and decomposing strain into tensile and compressive parts. Its variational formulation yields a standard multifield problem suitable for conventional finite element implementation.
- Variational approach and regularization: The fracture formulation minimizes an energy functional combining stored elastic energy, fracture energy, and external work.The elastic energy density is ψε, the critical energy release rate is Gc, and Wext denotes external work.
- Variational approach and regularization: Phase field regularization smears the crack surface across the domain, with ϕ=0 for intact material and ϕ=1 for fully cracked material.The length-scale parameter l0 controls the transition-region width and indirectly reflects crack width.
- Anisotropic strain decomposition: Strain spectral decomposition separates tensile and compressive strain tensors using principal strains and their directions.The positive and negative-part operators are defined as ⟨·⟩±=(·±|·|)/2.
- Anisotropic strain decomposition: The model uses only the tensile energy contribution to capture fractures under tension, while compressive fractures are inhibited in recent PFMs.This motivates distinguishing the tensile and compressive parts of the elastic energy.
- Governing equations: Taking the first variation of the regularized functional produces governing equations for displacement and phase field, transforming strong discontinuities into a standard multifield problem.Conventional finite element methods can then be applied, although the resulting constitutive relationship is anisotropic.
- Governing equations: Irreversibility is enforced by replacing the positive energy with a history field, ensuring a monotonically increasing phase field and preventing fracture healing.A small stability parameter k avoids numerical singularities as ϕ approaches 1.
3 Modified phase field model for compressive-shear frac-
Current phase field methods struggle to model compressive-shear fractures in rock-like solids because compressive elastic energy, cohesion, and internal friction are not adequately represented. The proposed hybrid formulation introduces a new driving force based on compressive strain and these material effects.
- Current PFM limitations: Anisotropic and hybrid phase field methods cannot model compressive-shear fractures because compressive elastic energy is excluded from the phase-field driving force.
- Current PFM limitations: Isotropic formulations can model compression fractures, but the resulting fractures are reported to be unrealistic.
- Current PFM limitations: Existing phase field methods omit cohesion and internal friction angle, despite their significant effects on compressive-shear propagation and load-displacement curves.
- Modified driving force: The model reconstructs a driving energy ψp using cohesion, internal friction angle, and a positive operator within a Mohr-Coulomb-based formulation.
- Modified driving force: Only the compressive part of the principal strain is used to construct the energy, while fracture irreversibility is enforced through Hp = max.
- Modified formulation: Replacing H with Hp yields a hybrid phase field model that accounts for cohesion and internal friction angle in fracture initiation, propagation, and load-displacement curves.
4 Numerical implementation
The numerical implementation discretizes the displacement and phase fields with finite elements and solves their coupled weak forms sequentially. A staggered scheme is used for flexibility and stability.
- Finite-element discretization: The phase field model is implemented within conventional finite element methods using discretized displacement and phase-field variables.
- Finite-element discretization: Shape functions and their derivative matrices define the trial and test fields, strains, gradients, and corresponding weak-form equations.
- Discrete equations: The discrete equations include internal and external displacement forces together with the degraded elasticity matrix and phase-field stiffness matrices.
- Solution procedure: A staggered scheme solves the displacement and phase fields sequentially with iterations between them, providing greater flexibility and stability than a monolithic scheme.
- Solution procedure: The implementation additionally uses implicit Generalized-α time integration and Anderson acceleration to increase simulation convergence rate.
5 Numerical examples
Numerical examples test the proposed phase field model on intact specimens and specimens containing one or two inclined flaws. The simulations reproduce observed compressive-shear fracture patterns and associated loading responses.
- Benchmark examples: The benchmark suite includes uniaxial compression tests on intact specimens and specimens with single or double inclined pre-existing flaws.
- Intact specimen: For intact specimens, both 2D and 3D simulations produce V-shaped compressive-shear fractures under increasing axial compression.The V-shaped pattern is frequently observed experimentally.
- Intact specimen: The proposed PFM reproduces the intact specimen’s fracture pattern and stress-strain curve, with the 3D simulation agreeing better with experiment than the 2D plane-strain model.
- Single inclined flaw: For a specimen with one inclined flaw, shear zones develop near flaw tips, fractures initiate there, propagate obliquely, and agree with experimental results.
- Comparison with experiments: Across the flaw configurations, numerical fracture patterns are reported to align with experiments, supporting the model’s capability for complex compressive-shear fractures.
- Parameter effects: Increasing cohesion and internal friction angle increases compressive load-bearing capacity while preserving the modeled fracture pattern.
- Two inclined flaws: Two coplanar flaws produce coalescing quasi-coplanar fractures, whereas non-coplanar flaws produce S-shaped coalescence and a larger peak load.
6 Conclusions
The proposed phase field model combines strain spectral decomposition, a compressive-strain driving force incorporating cohesion and internal friction, and a hybrid formulation to simulate brittle compressive-shear fractures. Numerical examples reproduce experimental observations while autonomously capturing complex fracture evolution, but the model excludes tensile and tension-shear fractures.
- 6 Conclusions: The proposed model uses strain spectral decomposition with emphasis on compressive strain to construct a new phase-field driving force.The driving force is reconstructed in the phase-field evolution equation.
- 6 Conclusions: Cohesion and internal friction angle are incorporated into the driving force to influence fracture initiation and propagation.This addresses effects that previous phase-field models could not consider.
- 6 Conclusions: A hybrid formulation enables implementation of the proposed model within a conventional finite-element framework and COMSOL.The model is applied to uniaxial compression of intact specimens and specimens containing one or two parallel inclined flaws.
- 6 Conclusions: The numerical results are in good agreement with previous experimental results, validating the model for brittle compressive-shear fractures.The examples cover rock-like specimens with inclined flaws as well as intact specimens.
- 6 Conclusions: Fracture initiation, propagation, coalescence, and branching occur autonomously without external fracture criteria or prescribed propagation paths.This supports modeling complex compressive-shear fracture patterns in rock-like materials.
- 6 Conclusions: Because tensile strain is suppressed, the model excludes tensile fractures and fractures under tension-shear mode.The authors identify combining this driving force with previous ones as a future direction for mixed-mode fractures in rocks.