The paper presents t he Strong Discontinuity A pproach for the a n alysis and s i m ulation of strong discontinuities in solids using continuum p lasticity m o d els. Kinematics of weak and strong discontinuities are discussed, and a regularized kinematic state of discontinuity i s p r o posedasamean to model the f o r m ation of a strong discontinuity a s t he c o llapsedstate o f a w eak discontinuity (w i t h a c haracteristic bandwidth), induced by a bifurcation of the stress-strain eld, which p r o pagates in the solid domain. The analysis of the conditions to induce the bifurcation provides a critical value for the bandwidth a t t he onset of the w eak discontinuity a n d t he direction of propagation. Then a variable bandwidth m o d el is proposedto c haracterize the transition between the w eak and strong d i s c o n tinuity r e g i m es. Several aspects related to t he c o n tinuum a n d, their associated,discrete c o n s t itutive equations, the expendedpower in the formation of the discontinuity a n d relevant computational details related to the nite element simulations are also discussed. Finally, some representative n umerical simulations are shown to illustrate t he p r o posedapproach.

SUMMARYIn this paper a rate-dependent isotropic damage model developed for the numerical analysis of concrete dams subjected to seismic excitation is presented. The model is shown to incorporate two features essential for seismic analysis: stiffness degradation and stiffness recovery upon load reversals and strain-rate sensitivity. The issue of mesh objectivity is addressed using the concept of the 'characteristic length' of the fracture zone, to show that both the softening modulus and the fluidity parameter must depend on it to provide consistent results as the computational mesh is refined. Some aspects of the numerical implementation of the model are also treated, to show that the model can be easily incorporated in any standard non-linear finite element code. The application of the proposed model to the seismic analysis of a large gravity concrete dam shows that the structural response may vary significantly in terms of the development of damage. The inclusion of rate sensitivity is able to reproduce the experimental observation that the tensile peak strength of concrete can be increased up to 50 percent for the range of strain rates that appear in a structural safety analysis of a dam subjected to severe seismic actions.

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The paper presents a new methodology to model material failure, in twodimensional reinforced concrete members, using the Continuum Strong Discontinuity Approach (CSDA). The Hill's mixture theory is used as the methodological approach to model reinforced concrete as a composite material, constituted by a plain concrete matrix reinforced with two embedded orthogonal long fiber bundles (rebars). Matrix failure is modeled on the basis of a continuum damage model, equipped with strain softening, whereas the rebars effects are modeled by means of phenomenological constitutive models devised to reproduce the axial non-linear behavior, as well as the bond-slip and dowel effects. The proposed methodology extends the fundamental ingredients of the standard Strong Discontinuity Approach, and the embedded discontinuity finite element formulations, in homogeneous materials, to matrix/fiber composite materials, as reinforced concrete. The specific aspects of the material failure modeling for those composites are also addressed. A number of available experimental tests are reproduced in order to illustrate the feasibility of the proposed methodology.