Appl Math Optim (2017) 75:99–116
A Simple Derivation and Classical Representations
of Energy Variations for Curved Cracks
Published online: 12 January 2016
© Springer Science+Business Media New York 2016
Abstract We consider conﬁgurational variations of a homogeneous (anisotropic) lin-
ear elastic material ⊂ R
withacrackK. First, we provide a simple way to
compute conﬁgurational variations of energy by means of a volume integral. Then,
under increasing information on the regularity of the displacement ﬁeld we show how
to obtain classical representations of the energy release due to Eshelby, Rice and Irwin.
A rigorous functional setting for these representations to hold is provided.
Keywords Energy release · Eshelby tensor · J-integral · Stress intensity factors
Mathematics Subject Classiﬁcation 74A45
In fracture mechanics it is common to employ a variational approach in which evolu-
tions are deﬁned in terms of energies and their variations, most often with respect to
variations of the crack set. This approach dates back to Grifﬁth’s criterion  which is
formulated right in terms of energy release G (i.e. variations of potential energy with
respect to variations of crack surface area) and toughness G
(a material parameter).
Actually, the mathematical argument of  was “just” an algebraic estimate of the
energy release in terms of material and geometrical parameters, based on the results
of Inglis  on stress singularities around elliptical holes and cracks.
Financial support provided by ERC grant no. 290888 QuaDynEvoPro and by INdAM GNAMPA.
Department of Mathematics, University of Pavia, Via A. Ferrata 1, 27100 Pavia, Italy