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Extra resources for Density of C inW 1,p (x ) with discontinuous exponent p (x )
Strong dissipation coupling is restricted only to a ductile-like damage in elastic-plastic materials, where both dissipation mechanisms: plasticity and damage, are governed by the single plastic multiplier λp . 25) are used alternatively, however, only for case a, when approaching critical damage D = 1 the eﬀective stress approaches zero σ = 0. In case b, for critical damage only the isotropic hardening term tends to zero, whereas the kinematic one remains ﬁnite. In case c, both the isotropic and kinematic hardening terms remain ﬁnite at the critical damage.
Limitations of the unilateral damage condition are discussed by Chaboche [34, 35], Chaboche et al. , Halm and Dragon . Some existing concepts (Krajcinovic and Fonseka , Ju ) may lead to nonsymmetric eﬀective elastic stiﬀness or compliance, or to stress–strain discontinuities under nonproportional loading. It was shown in Skrzypek and Kuna-Ciskal  that, if the unilateral damage condition inﬂuences both diagonal and oﬀ-diagonal components of the stiﬀness or compliance matrix, a stress discontinuity occurs when one of the principal strains changes sign, whereas the others remain unchanged.
Fig. 3. A plane stress concrete structure with pre-load inclined crack, subjected to compression in the direction 1: (a) the geometry and mesh, (b) maximum principal stress distribution surrounding the pre-load crack tip Contact at the pre-load crack edges was deﬁned by identifying and pairing potential contact surfaces. To deﬁne a sliding interface between two surfaces, one of the surfaces (the “slave” surface) is covered with 3-node contact elements. The second surface (the “master” surface) is deﬁned by a series of nodes ordered as the segments of the slide line.