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Energy integral in fracture mechanics (J-integral) and Gauss-Bonnet theorem. (English) Zbl 1138.74004
Summary: 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 homotopy operator corresponds to J-integral for dislocation-disclination fields and gives the force on dislocation-disclination fields a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of J-integral. This geometric interpretation gives that the J-integral is an alternative expression of a well-known theorem in differential geometry, i.e., 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 Einstein 3-form (geometric objects of material space).

74A45 Theories of fracture and damage
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