[Paper Review] Thermomechanics of damage and fatigue by a phase field model

Giovambattista Amendola, Mauro Fabrizio|arXiv (Cornell University)|Oct 26, 2014

Solidification and crystal growth phenomena17 references4 citations

TL;DR

This paper proposes a thermomechanically consistent phase field model for damage and fatigue using the Ginzburg-Landau equation to describe the evolution of a phase field variable φ representing material damage. The model couples mechanical stress reduction via (1−φ)² scaling, fatigue defined as accumulated structural work, and thermodynamic consistency, with extension to non-isothermal conditions showing thermal effects accelerate damage. Key results include a maximum principle for φ and a thermodynamically coherent framework for fatigue life under thermal cycling.

ABSTRACT

In the paper we present an isothermal model for describing damage and fatigue by the use of the Ginzburg-Landau (G-L) equation. Fatigue produces progressive damage, which is related with a variation of the internal structure of the material. The G-L equation studies the evolution of the order parameter, which describes the constitutive arrangement of the system and, in this framework, the evolution of damage. The thermodynamic coherence of the model is proved. In the last part of the work, we extend the results of the paper to a non-isothermal system, where fatigue contains thermal effects, which increase the damage of materials.

Motivation & Objective

• To develop a thermodynamically consistent model for progressive damage and fatigue in materials under cyclic loading.
• To describe damage evolution via a phase field φ governed by the Ginzburg-Landau equation, representing internal structural changes.
• To extend the isothermal model to non-isothermal conditions, incorporating thermal effects on fatigue and damage.
• To prove mathematical properties such as maximum principle and uniqueness for the phase field φ.
• To define fatigue as a cumulative structural work integral, accounting for mechanical and thermal contributions.

Proposed method

• The phase field φ evolves according to a Ginzburg-Landau-type PDE: ρ∂ₜφ = ∇·(1/κ)∇φ − ℱF′(φ) − ℱ₀G′(φ), with φ ∈ [0,1] representing damage level.
• Stress is reduced via (1−φ)² scaling of the virgin material stress tensor T̃, leading to fracture when φ→1.
• Fatigue ℱ(x,t) is defined as the time integral of [1−φ(τ)] times mechanical work rate, incorporating stress and velocity gradients.
• Thermodynamic consistency is ensured by deriving the evolution equation from a Gibbs free energy functional W(φ,∇φ) involving energy, fatigue, and damage potentials.
• The model is extended to non-isothermal conditions by coupling the Ginzburg-Landau equation with the heat equation and energy balance, including thermal gradients and heat flux.
• The structural power ℙₛⁱ is derived from the phase field dynamics and included in the energy balance, with thermal effects entering through θ⁻¹ terms in the fatigue integral.

Experimental results

Research questions

• RQ1 How can damage and fatigue be consistently described using a phase field approach within a thermomechanical framework?
• RQ2 What is the role of thermal effects in accelerating damage accumulation during cyclic loading?
• RQ3 How does the phase field φ evolve under coupled mechanical and thermal loading, and does it remain bounded in [0,1]?
• RQ4 Can a maximum principle be proven for the phase field φ to ensure physical consistency?
• RQ5 How is fatigue defined in a non-isothermal setting, and what is its dependence on temperature gradients and heat flux?

Key findings

• The phase field φ satisfies a maximum principle, ensuring it remains in [0,1] for all time, which guarantees physical consistency and prevents unphysical damage states.
• The model is thermodynamically coherent: the evolution equation for φ is derived from a free energy functional W(φ,∇φ), ensuring energy dissipation.
• Fatigue ℱ is defined as a cumulative integral of structural work, with a closed-form expression under cyclic loading that includes thermal contributions via θ⁻¹ and ∇θ⁻¹ terms.
• In the non-isothermal case, thermal gradients and heat flux contribute directly to fatigue through the term ∫[1−φ]q·∇θ⁻¹ dτ, showing thermal shocks accelerate damage.
• The structural power ℙₛⁱ is derived as ρφ̇² + (1/κ)∇φ·∇φ̇ + ℱF′(φ) + ℱ₀G′(φ), which appears in the energy balance and ensures consistency with thermodynamics.
• The model reduces to a heat equation with source terms involving φ̇² and ℱF′(φ), showing that damage evolution directly affects thermal energy balance.