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Energy integral in fracture mechanics (J‐integral) and Gauss‐Bonnet Theorem
Author(s) -
Yamasaki K.,
Nagahama H.
Publication year - 2008
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200700140
Subject(s) - path integral formulation , disclination , integral geometry , line integral , kelvin–stokes theorem , classical mechanics , volume integral , mathematics , topological defect , physics , mathematical analysis , integral equation , quantum mechanics , picard–lindelöf theorem , liquid crystal , danskin's theorem , fixed point theorem , optics , quantum
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).