Three expressions for the strain energy release rate for three-dimensional singular and non-singular finite elements are derived based on Irwin's virtual crack-closure method. The strain energy release rate for the three modes of fracture mechanics can be expressed in terms of the nodal forces ahead of the crack front and the opening displacements behind it. The material is assumed to be isotropic. The formulae are derived for three different element types, namely, 20-node singular element, eight-node and 20-node non-singular elements. The validity of the derived formulae is checked by comparing their results to those from the literature for mode I and mixed mode I and II problems.
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