Phase field predictions of microscopic fracture and R-curve behaviour of fibre-reinforced composites

1. Introduction

• Computational micromechanics uses finite-element simulations of an RVE or embedded microstructural cell to vary constituent properties and derive homogenised behaviour for larger-scale simulations.The approach is presented as a route to virtual material optimisation.Source: §1 · p. 2
• Mesh-based CDM and CZM models can exhibit mesh-induced direction bias, while aligning a refined mesh with fibres requires complex generation and high computational cost.Enriched methods such as X-FEM and the Floating Point Method may also fail to track crack coalescence and branching topology.Source: §1 · p. 3
• The phase-field fracture model is presented as an alternative that can simulate complex crack paths, including branching and coalescence in arbitrary geometries and dimensions.It is described as a variational fracture approach based on the Griffith energy balance.Source: §1 · p. 3
• The paper identifies crack-path topology and macroscopic fracture toughness as unresolved phenomena, including effects of fibres, matrix, interfaces, voids, and defects.These include toughening or embrittlement mechanisms such as fibre bridging and crack branching.Source: §1 · p. 3
• The coupled phase-field/cohesive-zone framework models matrix cracking and fibre–matrix debonding, and examines their effects on homogenised fracture toughness.The work also investigates matrix bridging and the influence of matrix toughness, interface properties, and porosity.Source: §1 · p. 3
• The framework is evaluated in single-edge-notched three-point-bending tests, with predictions compared against measured crack paths and load–displacement curves.The introduction also identifies R-curve sensitivity to matrix toughness and interface strength and toughness, and explores microstructures with varying porosity.Source: §1 · p. 3

2. Numerical model

• The framework combines phase-field fracture and cohesive-zone models in ABAQUS user elements to simulate matrix and fibre cracking alongside fibre–matrix debonding.Source: §2 · p. 4

2.1. Phase field fracture model

• The phase field tracks crack interfaces with a damage variable that transitions from 0 in intact regions to 1 inside cracks.Its non-local formulation supports mesh objectivity, and the length-scale parameter governs fracture-process-zone size.Source: §2.1 · p. 4

2.2. Cohesive zone model

• A bi-linear traction–separation law models fibre–matrix debonding using interface modulus, strength, and fracture energy, with mixed-mode damage evolution.Damage onset uses a quadratic nominal-stress criterion, and damage evolution follows the Benzeggagh–Kenane fracture criterion.Source: §2.2 · pp. 5–6

3.1. Singe-edge cracked plate subjected to tension

• With interface decohesion, many interfaces debonded before peak load, producing nonlinear behavior and multistep softening as debonding coalesced with matrix cracks.Source: §3.1 · p. 8
• When matrix cracking initiated first, no interface decohesion occurred; fibre-induced crack deflection instead produced zig-zag softening and increased fracture surface area and macroscopic toughness.Source: §3.1 · p. 8
• Work of fracture was 21.7 J/m2 with interface debonding, 20.4 J/m2 without it, and 14.3 J/m2 for bulk matrix; reducing interface strength was proposed to improve toughness.The passage notes that high interface strength can instead improve composite strength, while reduced interface strength is consistent with toughening approaches in ceramic fibre-reinforced composites.Source: §3.1 · p. 9

3.2. Single-edge notched three-point bending test

• Damage began with fibre–matrix interface debonding; coalescing interface decohesion and matrix cracking formed a continuous crack path, with matrix ligaments bridging the crack.The simulations also captured crack evolution with increasing remote load.Source: §3.2 · p. 10
• The embedded-cell model accurately captured the measured load–CMOD response and softening regime, with maximum load underestimated by around 10%, within experimental scatter.It also reproduced the measured CMOD at maximum load.Source: §3.2 · p. 10, Figure 5 · p. 11
• Predicted TPB R-curves agree closely with experiments, and their rise is attributed to matrix-ligament bridging and softening of the fibre–matrix interface.Source: §3.2 · p. 12

3.2.1. Sensitivity study

• Reducing phase-field length scale ℓ raises the force–displacement response, while increasing matrix fracture energy Gm raises peak load and CMOD at maximum load.Interface toughness has relatively little effect on load–CMOD responses; interface strength affects maximum load, while matrix and interface toughness contribute to softening. Increasing matrix toughness is more effective for enhancing overall fracture toughness, supporting the use of thermoplastics for high-toughness applications.Source: §3.2.1 · pp. 12–13

3.2.2. The effect of porosity

• Circular holes blunt the crack, but the predicted toughening applies to regular matrix-pocket voids, unlike manufacturing-induced voids that can degrade mechanical behaviour.The paper notes that manufacturing voids are often irregular and close to fibre–matrix interfaces, where they can reduce interface and macroscopic fracture toughness.Source: §3.2.2 · p. 15
• At ∆a = 0.8 mm, the 10%-porosity sample has 37% higher fracture toughness than the pore-free sample and exhibits continuously rising, more stable crack growth.Increasing porosity also decreases modulus and strength; strength is reduced by approximately 17% at 10% porosity.Source: §3.2.2 · p. 15, Figure 8 · p. 16

• Predictions for single-edge notched beam bending experiments correlate excellently with experimental results, both qualitatively and quantitatively.Source: §4 · p. 17
• Increasing matrix toughness is suggested to be more effective for toughening than enhancing interface fracture toughness.Source: §4 · p. 17
• Void inclusions in matrix-resin regions can enhance composite fracture toughness through crack blunting.Source: §4 · p. 17
• The coupled phase-field and cohesive-zone framework uses an embedded cell to examine microstructure and constituent-property effects on macroscopic fracture toughness.Source: §4 · p. 17
• The embedded-cell computational framework is presented as a multiscale virtual tool for investigating microstructure and material properties toward more efficient designs for higher fracture toughness.Source: §4 · p. 17