[Paper Review] Mesoscopic theory of microcracks

Christina Papenfuß, P. Ván+1 more|arXiv (Cornell University)|Dec 4, 2002

Elasticity and Wave PropagationEngineering22 citations

TL;DR

This paper develops a mesoscopic theory for microcrack evolution in brittle materials by introducing a distribution function over crack size and orientation, enabling derivation of balance equations and dynamic evolution laws. The key contribution is a systematic framework linking microcrack dynamics to macroscopic damage via fabric-alignment tensors and a damage parameter, derived from thermodynamic principles and validated through Rice-Griffith-type crack growth laws.

ABSTRACT

The mesoscopic concept is a way to deal with complex materials with an internal structure within continuum mechanics. It consists of extending the domain of the balance equations by mesoscopic variables and of introducing a local distribution function of these variables as a statistical element. In our case microcracks are modelled as penny shaped and completely characterized by their diameter and the unit normal to the crack surface. Two examples of crack dynamics are given as well as a possible definition of a damage parameter. Orientational order parameters (fabric-alignment tensors) are defined and balance like dynamic equations for them are derived.

Motivation & Objective

• To develop a mesoscopic framework that captures microcrack dynamics in brittle materials beyond classical continuum mechanics.
• To model microcracks as penny-shaped, fixed, and non-healing entities characterized by radius and orientation.
• To derive macroscopic damage parameters—such as average crack length and fabric-alignment tensors—from a mesoscopic distribution function.
• To establish dynamic evolution equations for these damage parameters based on mesoscopic balance laws and crack growth kinetics.
• To connect the mesoscopic approach to phase field and Landau-type theories by showing analogous equation forms for damage evolution.

Proposed method

• Introduce mesoscopic fields on an extended space ℝ³ₓ × ℝₜ × M, where M = [lₘ, lₘₐₓ] × S² represents crack size and orientation.
• Define a crack distribution function f(l, n, x, t) as the probability density of finding a crack with size l and normal n at position x and time t.
• Derive mesoscopic balance equations for mass, momentum, angular momentum, and energy, including a crack number density N(l, n, x, t).
• Implement a crack growth law based on the Rice-Griffith criterion: ḋl = −α + βσ²l for l ≥ l_c, with σ ∝ (e_z · n)².
• Derive the Fokker–Planck-type equation for f: ∂f/∂t = −(1/l²) ∂/∂l [l²(−α + βv_σ₀²l t² (e_z·n)⁴)] for l ≥ l_c.
• Define macroscopic damage parameters as moments of f, including scalar average crack length and tensorial fabric-alignment tensors.

Experimental results

Research questions

• RQ1 How can microcrack dynamics be systematically described within a continuum mechanics framework that accounts for size and orientation?
• RQ2 What is the appropriate mesoscopic field theory that allows the derivation of macroscopic damage evolution from microscale crack statistics?
• RQ3 How do crack growth laws, such as the Rice-Griffith criterion, translate into evolution equations for the mesoscopic distribution function?
• RQ4 What are the macroscopic damage parameters (e.g., average crack length, fabric tensors) and how do they evolve over time?
• RQ5 To what extent does the derived equation of motion for damage parameters resemble those in phase field or Landau theory of phase transitions?

Key findings

• The mesoscopic distribution function f(l, n, x, t) evolves according to a partial differential equation derived from mesoscopic balance laws and a thermodynamically consistent crack growth law.
• For l ≥ l_c, the distribution function evolves as ∂f/∂t = −(1/l²) ∂/∂l [l²(−α + βv_σ₀²l t² (e_z·n)⁴)], with the growth rate dependent on the fourth power of the cosine of the crack normal relative to the loading direction.
• The macroscopic damage parameter, such as the average crack length, is obtained as a moment integral of f over l and n, and its time evolution is derived from the distribution function dynamics.
• Fabric-alignment tensors, defined as fourth-order moments of f, evolve according to balance-like equations that reflect the anisotropic nature of crack growth under biaxial loading.
• The equation of motion for the damage parameter is derived from first principles (mesoscopic balance equations) rather than postulated, and it shares the same mathematical form as equations in phase field models.
• The model incorporates a critical crack length l_c below which cracks do not grow, consistent with the Griffith criterion, and ensures non-healing by enforcing ḋl = 0 for l < l_c.