Energy integral in fracture mechanics (J-integral) and gauss-bonnet theorem

The J-integral (a path-independent energy integral) formalism is the standard method of analyzing nonlinear fracture mechanics. It is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J-integral for dislocation-disclination fields and gives the force on dislocation-disclination fields as a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of the J-integral. This geometric interpretation gives that the J-integral is an alternative expression of the well-known theorem in differential geometry, i.e., the Gauss-Bonnet theorem (with genus = 1). The geometrical expression of the J-integral shows that the Eshelby's energy-momentum (the physical quantity of the material space) is closely related to the Einstein 3-form (the geometric objects of the material space).