Keywords: damage mechanics, interface elements, cohesive crack model, brittle matrices, fracture in concrete.

In the work is discussed a new class of interface elements especially devised for the analysis of the deformation process of concrete-like materials, characterised by the occurrence of fractures that dissipate energy in a domain whose dimension is smaller than the dimension of the structure (actually, its dimension is fractal). This phenomenon is associated with an unstable behaviour at the material level and with the characteristic scale effect, that is, the local stress-strain response changes with the dimension of the volume used for measuring it. The main properties of the interface model proposed are that it derives from thermodynamic principles, and that it can is compatible with a continuum damage mechanics constitutive model recently proposed by the author for the numerical analysis of damaging materials [1]. These properties are especially convenient for using the interface model in connection with enhanced elements with embedded discontinuities, that have been recently proposed by many authors.

The interface model is proposed in a plane stress state, and accounts for mode I and mode II fracture. The constitutive model is a generalisation of the Cohesive Crack Model and is consistently developed starting from an unilateral interface. Internal variables are used, in addition to relative displacements, for defining the state of the interface. Each state variable is decomposed in a reversible and an irreversible component. The evolution of the reversible components derive from dual non differentiable internal energy potentials, that turn out to reduce to the complementary unilateral laws of the interface with the introduction of a dependency on the damage internal variable. The unilateral behaviour of the interface is exactly enforced using an Augmented Lagrangian regularisation of the unilateral (contact) potential. The evolution law for the irreversible part of the internal variable and of the relative displacements is derived from the normality rule to a generalised "plastic" surface of the interface, expressed in terms of normal and tangential tractions, of hardening variables and of the dual damage variable. If only the latter is included in the potential, only damage of the interface develops, and a cohesive stress-CMOD relation is obtained, without residual displacements after unloading. However, additional dissipation phenomena like fibre yielding and slippage can be introduced, by means of additional (convex) dissipation potentials, so that irreversible relative displacements can be obtained.

A number of interesting consequences derive form the general law of the model, that is, the limit stress states in the plane evolve as damage progresses, and in the same time, as can be observed from fig.1, the slope of the limit curve tends to zero. Since an associative rule is used, this means that the dilatancy initially introduced in the model tends quickly to zero, as observed in experimental results (for instance, those performed by Hassanzadeh), and this is obtained authomatically from the energy potentials, with no need for introducing any "ad hoc" hypotheses.

1

Contrafatto, L., Cuomo, M., A new thermodynamically consistent continuum model for hardening plasticity coupled with damage, submitted to Int. J. of Solids and Structures. doi:10.1016/S0020-7683(02)00470-5

2

Braides A., Approximation of Free-Discontinuity Problems, Springer Verlag, Berlin, 1998

3

Camacho, G.T., Ortiz, M., Computational modelling of impact damage in brittle materials, Int. J. Solids Struct., 33(20-22), 2899-2938 (1996) doi:10.1016/0020-7683(95)00255-3

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