An intermediately-homogenized peridynamics approach to failure of microstructually disordered materials

TL;DR

The paper develops an intermediately homogenized peridynamics framework to model fracture in microstructure-disordered materials by describing local properties with random fields derived from a Gaussian auxiliary field $\phi$ and mapped to density $\rho$ via $\rho(\phi)=F^{-1}(G(\phi))$, enabling correlated heterogeneity while maintaining $m$-convergence. Bond-based peridynamics with horizon $\delta$ uses a continuum-matching modulus $c_0$ and a fracture criterion involving a critical stretch $s_c$, linked to fracture energy $G$, and the random fields are propagated to bond properties through $c(\boldsymbol{x},\boldsymbol{x}')=\tfrac{c(\boldsymbol{x})+c(\boldsymbol{x}')}{2}$; disorder is implemented with a log-normal density, Gibson–Ashby scalings for $\kappa$ and $s_c$, and an exponential correlation length $\lambda$, typically set to $\lambda=\delta$. Simulations on a snow-like porous material reveal that disorder lowers elastic modulus and peak strength, while $m$-convergence is achieved for $m>3$; damage patterns transition from mode I to branching and distributed damage as disorder increases; the size effect shows a logarithmic strength decrease in unnotched samples and a McClintock–Irwin-type scaling for notched samples with a process-zone size $a_0$ that grows with disorder and length scales. Across systematic tests varying $L$, $a$, $\delta$, and $\lambda$, the fracture toughness $K_c$ is largely independent of disorder and internal scales, whereas $a_0$ depends on both, with a near-linear dependence on $\lambda$ and complex $\delta$-dependence depending on the fixed quantity, indicating a rich, multi-scale interplay between microstructure, damage, and failure. The work provides a principled route for screening failure statistics in disordered materials and informs design of metamaterials and alloys with tailored fracture pathways.

Abstract

Peridynamics provides a versatile tool for fracture modelling in materials where fracture pathways cannot be predicted beforehand, but must be envisaged as an emergent features of the deformation process. One class of materials where this is surely the case are materials with strong microstructural disorder such as random composites, random porous materials or disordered metamaterials. For this class of materials we propose an intermediately-homogenized peridynamic modelling approach where the disordered microstructure is not resolved in full spatial detail but described in terms of random order parameter fields which retain essential information about the local heterogeneity and spatial correlations of material properties.

Figures (10)

• Figure 1: Stress-strain curves of uniaxial tensile simulations for different values of $m$ and $\alpha$. Rows: $\alpha=0.1$(top) and $\alpha=1.0$(bottom); columns: $m= 3.0,5.0,12.0$(left to right). Different colours indicate samples with different random fields.
• Figure 2: Effective elastic modulus and peak stress as functions of $m$ for different degrees of disorder. The square points represent mean values evaluated from 10 samples and the error bars show the corresponding standard deviation.
• Figure 3: Comparison of damage patterns, system size $400 \times 400$ mm, $\delta = \xi = 4$mm, initial crack length $a=16$ mm, left 4 samples: $\alpha = 0.5$, right 4 samples: $\alpha = 1$.
• Figure 4: Elastic response and peak stress of unnotched samples with different sizes and different degrees of disorder; left: apparent elastic modulus (initial slope of stress-strain curve), right: peak stress.
• Figure 5: Failure stresses for varying system size $L$ and initial notch length $a$, for other parameters see text; the fit curves were obtained from \ref{['eq:Irwin']} using the parameters of statistical model H3 in \ref{['tab:fitsAlphaL']}.