Localization of Oriented Cracking into a Band
The nonlocal model based on crack interactions has been applied to the problem of localization of unidirectional cracking into an infinite planar band parallel to the cracks (Bazant and Jirasek 1994b)). The body either is infinite or is an infinite planar layer parallel to the cracks, of thickness L. This represents the most fundamental localization problem, which is one dimensional, with the coordinate x normal to the cracks as the only coordinate. Due to translational symmetry in directions x and 2 parallel to the layer, the constitutive relation given by Eq. (13.3.9) with (13.3.43) can be integrated in the direction у parallel to the layer (the original problem is considered two-dimensional, although generalization to three dimensions would be possible). For the approximate crack interaction function (Eq. (13.3.43) with (13.3.45)) with 0 = – ф — 0), which is asymptotically correct at infinity, the integral can be evaluated analytically. This yields the following one-dimensional field equations for the increments of stresses and strains
Aa — C(x)Ae(x) – AS(x) (13.3.56)
Ф(о:,0;£, 77)1/17, A(x,£)
( “ (х ■- £)/£; С — elastic material stiffness (modulus); Ф is the one-dimensional weight function for spatial averaging, corresponding to the averaging over crack surface in Kachanov’s method; and A is the one-dimensional crack influence function. Note that function A(x,£) is always positive, in contrast to the two-dimensional function Л.
The solutions of Eqs. (13.3.56)—(13.3.59) have been studied numerically, by introducing a discrete subdivision in coordinate x and reducing the equations to a matrix form. As the boundary condition, the layer of thickness L was considered fixed at both surfaces. The localization profiles of strain increment Ac, beginning from a state of uniform strain of various magnitudes, have been calculated and the evolution of the strain profile during loading has been followed. Fig. 13.3.9a shows the evolution of the strain profile across the layer, obtained for a local stress-strain relation that is linear up to the peak and then decays exponentially. Fig. 13.3.9b shows the stress-displacement diagram obtained for various ratios L/£ of the thickness of the layer to the nonlocal characteristic length. It is clear that the formulation prevents localization into a layer of zero thickness and enforces a smooth strain profile through the localization band. It is also seen that the size effect on the postpeak softening slope is obtained realistically. An interesting point is that localizations according to this formulation can happen even before attaining the maximum load. For further details, see Bazant and Jirasek (1994b).