  We present a phase field formulation for fracture in functionally graded
materials (FGMs). The model builds upon homogenization theory and accounts for
the spatial variation of elastic and fracture properties. Several paradigmatic
case studies are addressed to demonstrate the potential of the proposed
modelling framework. Specifically, we (i) gain insight into the crack growth
resistance of FGMs by conducting numerical experiments over a wide range of
material gradation profiles and orientations, (ii) accurately reproduce the
crack trajectories observed in graded photodegradable copolymers and
glass-filled epoxy FGMs, (iii) benchmark our predictions with results from
alternative numerical methodologies, and (iv) model complex crack paths and
failure in three dimensional functionally graded solids. The suitability of
phase field fracture methods in capturing the crack deflections intrinsic to
crack tip mode-mixity due to material gradients is demonstrated. Material
gradient profiles that prevent unstable fracture and enhance crack growth
resistance are identified: this provides the foundation for the design of
fracture resistant FGMs. The finite element code developed can be downloaded
from www.empaneda.com/codes.
