[Paper Review] A variational formulation of Griffith phase-field fracture with material strength

Christopher J. Larsen, John E. Dolbow+1 more|arXiv (Cornell University)|Jan 25, 2024

Solidification and crystal growth phenomena5 citations

TL;DR

The paper recasts the Griffith phase-field fracture with material strength as a variational problem, showing that the displacement and phase-field pair minimize two separate functionals, akin to alternating minimization in classical phase-field fracture.

ABSTRACT

In this expository Note, it is shown that the Griffith phase-field theory of fracture accounting for material strength originally introduced by Kumar, Francfort, and Lopez-Pamies (J Mech Phys Solids 112, 523--551, 2018) in the form of PDEs can be recast as a variational theory. In particular, the solution pair $( extbf{u},v)$ defined by the PDEs for the displacement field $ extbf{u}$ and the phase field $v$ is shown to correspond to the fields that minimize separately two different functionals, much like the solution pair $( extbf{u},v)$ defined by the original phase-field theory of fracture without material strength implemented in terms of alternating minimization. The merits of formulating a complete theory of fracture nucleation and propagation via such a variational approach -- in terms of the minimization of two different functionals -- are discussed.

Motivation & Objective

• Motivate a complete macroscopic fracture theory that accounts for elasticity, strength, and critical energy release rate.
• Show that the coupled PDE system can be derived from two separate energy functionals.
• Establish a variational principle consistent with alternating minimization in phase-field fracture.
• Discuss the implications for fracture nucleation and propagation under mono-tonic, quasistatic loading.
• Highlight connections to existing Griffith-based and strength-inclusive fracture theories.

Proposed method

• Define a deformation energy functional E_d^ε(u;v) whose Euler-Lagrange equations recover the momentum balance equations.
• Define a fracture energy functional E_f^ε(v;u) whose Euler-Lagrange equations reproduce the phase-field evolution with strength considerations.
• Show that for fixed v, u minimizes E_d^ε over admissible displacements; for fixed u, v minimizes E_f^ε over admissible phase fields.
• Describe the driving force c_e(X,t) and coefficient δ^ε that incorporate the Drucker-Prager strength surface into E_f^ε.
• Explain the regularization via ε and the irreversibility constraint v ∈ [0,1], v ≤ v_{k-1}.
• Explain the alternating minimization interpretation and its relation to original phase-field fracture theory.

Experimental results

Research questions

• RQ1 Can fracture nucleation and propagation in brittle solids be captured by a variational two-functional framework that includes elasticity, strength, and fracture energy?
• RQ2 How does incorporating the Drucker-Prager strength surface affect the phase-field evolution and fracture onset in the variational setting?
• RQ3 Is the phase-field fracture with material strength equivalent, at ε → 0, to a Griffith-type variational formulation accounting for strength?
• RQ4 What are the mathematical and physical implications of viewing the standard phase-field approach as a special case without strength?

Key findings

• The PDE system for displacement and phase-field can be interpreted as the Euler-Lagrange equations of two separate functionals.
• The deformation energy functional determines body deformation under elastostatic equilibrium.
• The fracture functional governs nucleation and growth of cracks by balancing elastic energy, strength, and fracture energy.
• The framework unifies strength-inclusive fracture with classical Griffith theory through a variational lens.
• For uniform loading, fracture nucleation aligns with violations of the strength surface, and large-crack propagation follows Griffith-like criteria when strength is overcome.
• The approach remains compatible with alternating minimization and clarifies the role of each energy term in fracture dynamics.