[Paper Review] Morphogenesis and propagation of complex cracks induced by thermal shocks

Blaise Bourdin, Jean‐Jacques Marigo+2 more|arXiv (Cornell University)|Oct 1, 2013

Numerical methods in engineering31 references324 citations

TL;DR

This paper proposes a rate-independent gradient damage model to simulate crack nucleation and propagation in brittle materials under thermal shock. By coupling a material internal length scale with fracture toughness, the model predicts complex crack patterns—such as periodic parallel cracks and hexagonal columnar joints—without assuming crack geometry, achieving excellent agreement with experiments and revealing that crack morphogenesis depends on the internal length while propagation follows Griffith's criterion.

ABSTRACT

We study the genesis and the selective propagation of complex crack networks induced by thermal shock or drying of brittle materials. We use a quasi-static gradient damage model to perform large scale numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion and depends only on the fracture toughness, while crack morphogenesis is driven by the material's internal length. Our numerical simulations feature networks of parallel cracks and selective arrest in two dimensions and hexagonal columnar joints in three dimensions, without any hypotheses on cracks geometry and are in good agreement with available experimental results.

Motivation & Objective

• To understand the mechanisms behind complex crack pattern formation (morphogenesis) and selective propagation in brittle materials subjected to thermal shock.
• To develop a predictive numerical model that captures both crack nucleation and propagation without relying on ad hoc assumptions about crack geometry or initial flaws.
• To bridge the gap between material-level microstructure (internal length) and structural-scale fracture patterns using a unified variational framework.
• To validate the model against experimental data on ceramic slabs and 3D columnar joint formation, demonstrating quantitative agreement across multiple length scales.

Proposed method

• A quasi-static, rate-independent gradient damage model is used, with energy functional E_t(u, α) incorporating damage field α, strain energy, and a regularization term with internal length ℓ and fracture toughness G_c.
• The damage field α evolves via time-discrete minimization of E_t(u, α), enforcing irreversibility and allowing for distributed damage before crack formation.
• The model uses s(α) = 1/(1−α)² and w(α) = α for numerical convenience and stress-softening behavior, ensuring convergence to a Griffith-type energy functional as ℓ→0.
• Thermal shock is modeled via a time-dependent temperature field T_t solving the heat equation, inducing thermal strain ε^th_t = β(T_t − T_0)I.
• Numerical simulations use finite element discretization and post-processing to extract crack spacing and pattern evolution, with no adjustable parameters.
• The model is extended to 3D by straightforward application of the same energy minimization principle, enabling simulation of complex 3D crack networks.

Experimental results

Research questions

• RQ1 How do complex crack patterns such as periodic parallel cracks and hexagonal columnar joints emerge from thermal shock in brittle materials?
• RQ2 What role does the material’s internal length scale ℓ play in determining crack nucleation and initial pattern formation?
• RQ3 Does crack propagation in complex networks follow Griffith’s criterion, and can this be predicted without explicit crack tracking?
• RQ4 Can a single variational damage model simultaneously capture both crack nucleation and propagation across multiple length scales?
• RQ5 Is the scaling law for crack spacing in 2D and 3D identical, and can it be derived from energy minimization alone?

Key findings

• The model reproduces experimentally observed periodic crack patterns in 2D ceramic slabs (e.g., 1 mm × 9.8 mm) with quantitative agreement on crack spacing d as a function of depth a, matching a semi-analytical scaling law from [9].
• The average crack spacing d scales with depth a in a manner consistent with a period-doubling condition in a Griffith model, with ℓ = 46 µm and ℓ₀ = 14 µm derived from material properties.
• In 3D simulations, a disordered initial pattern of small cells evolves into a regular honeycomb structure through energy minimization, indicating a self-organized selection mechanism.
• The final crack pattern’s coarsening and regularity emerge naturally from the energy minimization process, without requiring non-equilibrium or viscous effects.
• While initial crack nucleation depends strongly on the internal length ℓ, the propagation of fully developed cracks is governed solely by fracture toughness G_c and follows Griffith’s criterion.
• The model’s predictions are independent of thermal diffusivity kc, confirming its rate-independent nature and validity for quasi-static simulations.