In mechanics, when studying the evolution of a system from initial time t1 to later time t2, the procedure followed is as follows:
a) Have a precise knowledge of the system's configuration at time t1; this configuration is defined by Cartesian or generalized coordinates.
b) Consider the system at an arbitrary time t between t1 and t2; use a schematic representation (conform to observation) of the system where the coordinates take arbitrary values.
c) Apply the laws of mechanics at t, which yields equations to solve; for example, Newton's second law leads to an inventory of the forces applied to the system at t.
d) Solve the equations to determine the system's evolution.
Let us now consider, as example, a large solid subjected to external stress in mode I+III (tension in the x2 -direction and shear in the x1x3-plane along x3-direction), in which a crack propagates over large distances. Experience shows that the general shape of the crack is non-planar, oscillatory about the mean plane x1x3 (perpendicular to x2); a model in which the crack front runs indefinitely along the x3-direction with an oscillatory form (the fracture propagation direction being x1) is an acceptable representation for large cracks. Under such conditions, at an arbitrary time t, the crack configuration is non-planar; it is in such configuration that the available laws of mechanics must apply. Crack configurations at t where the crack-front is straight (as frequently observed in articles) are irrelevant in this example of loaded crack. For the treatment, it is comfortable to view the crack-front as a dislocation and represent the crack by a continuous distribution of infinitesimal dislocations (Friedel; Bilby, Cottrell and Swinden; Chang and Ohr; Anongba and Vitek; Anongba…). The crack extension force G, per unit length of the crack-front (i.e. energy release rate), appears to be useful appropriate physical quantity…
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