  We formulate a phase field crack growth model for mode III fracture in a
Maxwell-type viscoelastic material. To describe viscoelastic relaxation, a
field variable of viscously flowed strain is employed in addition to a
displacement field and damage phase field used in the original elastic model.
Unlike preceding models constructed in the mechanical engineering community,
our model is based only on the generic procedure for driving (uni-directional)
gradient flow system from a physically natural system energy and employ no
additional assumption such as the super-imposed relations for stress and strain
(and their time derivatives) valid only for linear viscoelasticity. Numerical
simulations indicate that the competition between increase in deformation by
applied loading and the viscoelastic relaxation determines whether a distinct
crack propagation has occurred from an initial crack. Furthermore, we consider
the numerical results from an energetic perspective.
