Physics

Geology

Finite element method

Pure mathematics

Phase field models

Mathematics

Mechanics

Structural engineering

Engineering

Paleontology

Geotechnical engineering

Phase boundary

Metallurgy

Materials science

Stress field

Context (archaeology)

Strain energy release rate

Computer science

Phase (matter)

Quantum mechanics

Field (mathematics)

Fracture (geology)

Fracture mechanics

Fracture toughness

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith's energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness-for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimisation - without the need for \textit{ad hoc} criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms.

An assessment of phase field fracture: crack initiation and growth