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Phase field modelling of crack propagation in functionally graded materials
Hirshikesh, Sundararajan Natarajan, Ratna K. Annabattula, Emilio Martínez-Pañeda
TL;DR
Fracture in functionally graded materials is difficult to model because gradients deflect cracks and limit conventional crack-path tracking, especially in 3D. The paper develops a homogenization-based phase field formulation with spatially varying elastic and fracture properties, and demonstrates complex crack-path prediction across several graded-material cases. The results identify gradient profiles and orientations associated with enhanced crack-growth resistance and reproduce reported experimental trajectories.
Problem
Crack-path topology in functionally graded materials is difficult to track with conventional numerical techniques, particularly in three-dimensional problems.
Method
The paper develops a phase field fracture formulation that combines homogenization theory with spatially varying elastic and fracture properties.
Results
The formulation models complex crack-propagation paths, identifies volume-fraction profiles and gradient orientations affecting fracture resistance, and accurately reproduces experimental crack trajectories.
Takeaways & Limitations
Material gradient profiles and orientations can be selected to prevent unstable fracture and enhance crack-growth resistance within the modelled FGM settings.
Takeaways & Limitations
The introduction notes that existing techniques have limited ability to track crack-path topology, particularly in 3D; one studied complex-crack-pattern setting leaves the crack trajectory uncertain.
Abstract
from arXiv · showhide
We present a phase field formulation for fracture in functionally graded materials (FGMs). The model builds upon homogenization theory and accounts for the spatial variation of elastic and fracture properties. Several paradigmatic case studies are addressed to demonstrate the potential of the proposed modelling framework. Specifically, we (i) gain insight into the crack growth resistance of FGMs by conducting numerical experiments over a wide range of material gradation profiles and orientations, (ii) accurately reproduce the crack trajectories observed in graded photodegradable copolymers and glass-filled epoxy FGMs, (iii) benchmark our predictions with results from alternative numerical methodologies, and (iv) model complex crack paths and failure in three dimensional functionally graded solids. The suitability of phase field fracture methods in capturing the crack deflections intrinsic to crack tip mode-mixity due to material gradients is demonstrated. Material gradient profiles that prevent unstable fracture and enhance crack growth resistance are identified: this provides the foundation for the design of fracture resistant FGMs. The finite element code developed can be downloaded from www.empaneda.com/codes.
1. Introduction
FGMs offer spatially tailored properties but commonly exhibit brittle fracture, making crack initiation and growth difficult to predict when material gradients deflect cracks and induce mode mixity.
- Material inhomogeneity can deflect cracks from self-similar propagation even under pure mode I loading when the gradient is misaligned.
- Gradient-induced crack-tip mode mixity produces complex trajectories that are difficult for conventional numerical methods to capture.
- Discrete approaches include remeshing, X-FEM, scaled boundary finite elements, and cohesive zone models.
- These techniques are limited in tracking crack-path topology, particularly in three-dimensional problems.
- Variational energy-minimization approaches are presented as promising tools for modelling crack advance in FGMs.
- The paper extends phase field fracture modelling to compositionally graded materials.
2. A phase field fracture formulation for FGMs
The formulation represents graded fracture through spatially varying elastic and fracture properties, a phase field for damage, homogenized constituent behavior, and finite element solution procedures.
- 2.1. Governing balance equations: Elastic and fracture properties vary gradually in space through strain energy density ψ(x) and critical energy release rate Gc(x).
- 2.1. Governing balance equations: The phase field parameter φ approximates a discrete crack, with φ = 0 denoting intact material and φ = 1 completely damaged material.
- 2.1. Governing balance equations: The regularized crack surface uses an intrinsic length scale ℓ, and the total potential energy combines crack-surface and bulk-energy contributions.
- 2.1. Governing balance equations: The bulk formulation uses strain ε, a spatially varying stiffness matrix C(x), displacement kinematics, and coupled field equations involving the Cauchy stress tensor.
- 2.2. Homogenization scheme: Local effective elastic properties are obtained from constituent volume fractions using a Mori-Tanaka homogenization scheme.
- 2.2. Homogenization scheme: The effective bulk and shear moduli determine Young’s modulus and Poisson’s ratio through standard relations.
- 2.2. Homogenization scheme: Fracture response depends on Gc and ℓ; spatially varying fracture resistance is represented through Gc(x), computed from constituent fractions using the rule of mixtures.
- 2.3. Finite element implementation: The coupled equations are discretized with finite elements, solved using a staggered approach, and implemented with graded integration-point properties and a hybrid irreversibility treatment.
3. Results
The phase field framework reproduces graded-material crack paths across numerical and experimental cases, while revealing how gradient orientation and fracture resistance shape crack growth. The results identify gradation profiles that delay initiation, enhance growth resistance, and prevent unstable fracture.
- Experimental validation: Numerical predictions agree closely with experiments on graded photodegradable copolymers and glass-filled epoxy composites.The comparisons span a wide range of functionally graded materials.
- Crack-growth response: Gradation orientation and crack placement strongly alter force–displacement responses, initiation loads, post-initiation drops, and crack-growth resistance.Cracks may deflect toward compliant regions, while increasing fracture resistance along the propagation path produces substantial subcritical growth and later final failure.
- Crack-growth response: FGM profiles can be tailored to increase crack-initiation delay or crack-growth resistance and prevent unstable fracture.The reported mechanisms include spatially increasing Gc along the crack path and mode mixity induced by material gradients.
- Experimental validation: Crack trajectories in graded copolymers closely match experiments and outperform maximum tangential stress criteria in correlation.The model captures initial kinking and subsequent propagation paths caused by gradient-induced mode mixity.
- Crack-growth response: The framework predicts crack deflection toward compliant material and delayed fracture relative to homogeneous specimens under selected gradients.For vertical gradation, the graded sample delays fracture through the local Gc and gradient-induced mode mixity.
- Three-dimensional fracture: In three-dimensional graded specimens, cracks initiate at the edge with smallest Gc, propagate along x and z, and eventually fracture the specimen.The formulation also predicts nucleation, complete trajectories, and failure in complex geometries using the initial mesh.
4. Conclusions
The phase field formulation captures complex fracture in FGMs across two- and three-dimensional problems, including crack initiation, deflection, and unstable propagation. Its predictions reproduce experiments and identify gradient designs associated with improved crack growth resistance.
- Fracture resistance in FGMs is represented through spatial variation of the critical energy release rate, Gc.The formulation can be extended to related material systems and homogenization schemes, including metal-based elastic-plastic FGMs under stated assumptions.
- The method models complex crack paths arising from crack-tip mode mixity induced by material-property variation in two- and three-dimensional problems.The framework addresses both two-dimensional and three-dimensional boundary value problems.
- The formulation predicts crack initiation from arbitrary nucleation sites and crack deflection without re-meshing.It also models unstable crack propagation within an implicit framework.
- Numerical predictions accurately reproduce experiments across different FGM specimens and a wide range of material-gradient profiles.The results provide quantitative insight into how property gradation affects crack propagation response.
- The framework identifies combinations of volume-fraction profiles and gradient orientations that optimize crack growth resistance.These findings support the design of fracture-resistant FGMs, including components exposed to aggressive environments.