Phase field modelling of crack propagation in functionally graded materials (1904.08749v1)

Abstract: 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.

• The paper introduces an advanced phase field model that effectively incorporates spatially variable properties to analyze crack propagation in functionally graded materials (FGMs).
• The model was validated against experimental data for graded materials, showing close matches with empirical crack paths and initiation angles.
• Numerical results reveal how specific material gradient orientations correlate to fracture performance, providing insights for designing FGMs with enhanced resistance.

An Overview of "Phase Field Modelling of Crack Propagation in Functionally Graded Materials"

The paper presents an advanced phase field model for analyzing fracture in functionally graded materials (FGMs). This innovative approach leverages the underlying principles of homogenization theory to account for spatial variations in elastic and fracture properties within FGMs. The phase field methodology offers a robust framework capable of capturing the complexity of crack propagation paths influenced by material gradients, a notable challenge within the field of fracture mechanics. Key insights from multiple numerical case studies are systematically addressed, showcasing the potential of phase field modelling in optimizing FGM design for enhanced fracture resistance.

Key Contributions

The paper systematically extends the applicability of phase field methods to FGMs and demonstrates several significant developments:

1. Comprehensive Phase Field Formulation: The paper introduces a phase field model that effectively incorporates spatially variable properties, thus providing a more accurate representation of FGMs. The governing equations characterize crack growth resistance via a strain energy density function and a spatially varying critical energy release rate, $G_c(\bm{x})$.
2. Homogenization and Numerical Implementation: By utilizing a Mori-Tanaka homogenization scheme, the model captures the effective properties from volume fractions of constituent materials, enabling the analysis of FGMs with complex gradations. The implementation utilizes finite element analysis to solve the coupled system of equations, effectively modeling crack initiation and growth without remeshing.
3. Benchmarking and Validation: The paper validates the model through benchmark comparisons with experimental data for graded materials such as photodegradable copolymers and glass-filled epoxy composites. The model demonstrates a close match with empirical crack propagation paths and initiation angles, reinforcing the efficacy of phase field approaches in heterogeneous materials.
4. Complex Crack Path Predictions: Through simulations of conventionally challenging crack paths, such as cracks interacting with multiple holes or varying through the thickness of a plate, the model exhibits prowess in capturing abrupt crack deflections induced by material gradients, even in three-dimensional configurations.
5. Influence of Material Gradients: Numerical results exhibit how specific material gradient orientations correlate to fracture performance, with findings indicating scenarios that optimize crack growth resistance and mitigate unstable fracture. This directional insight holds substantial implications for designing FGMs with superior mechanical properties.

Theoretical and Practical Implications

The findings have profound theoretical and practical implications:

• Theoretical Insight: The paper enhances our understanding of crack tip behavior under varying material properties, contributing to the fundamental fracture mechanics of inhomogeneous materials. The phase field model's adaptability underscores its potential in simulating other complex materials systems.
• Industry Application: Practically, the model's insights can directly inform the design of FGMs in sectors such as aerospace, biomedical, and more, by identifying optimal gradation profiles that enhance durability and reduce the risk of catastrophic failure.

Future Directions

The work alludes to further potential explorations, such as extending the model to metal-based elastic-plastic FGMs and functionally graded composites incorporating advanced materials like carbon nanotubes. Future investigations could focus on integrating phase field formulations with adaptive meshing techniques to enhance computational efficiency further.

Overall, the paper lays a critical foundation for future explorations in FGM design and analysis, positioning phase field methods as highly effective tools for advancing material science in varied applications.