Introduction ============ Characteristics of concrete damage -------------------------------------------- Concrete is a complex material, composed of aggregates and a paste ensuring cohesion between these aggregates, and within which microcracks formed during the various manufacturing stages pre-exist. Concrete is generally considered to be an initially isotropic material, as microcracks do not have a preferred orientation. This isotropy is maintained if the load applied remains within the elastic range. From a certain level of loading, the microcracks will develop in particular directions, which induces the appearance of anisotropy in the non-linear phase. The cracks develop preferentially orthogonally to the directions of greatest traction or of smallest compression. The damage process therefore results in a loss of rigidity caused by the decohesion of the material. Stiffness can be regained when cracks are closed (unilateral effect). In addition, there is another strong asymmetry in the behavior between traction and compression: the stresses supported in compression are 10 times (or even more) greater than the stresses supported in tension. Finally, let us mention other phenomena such as the formation of irreversible deformations (caused for example by the blocking of the crack lips by friction or the presence of degraded material between these lips) or phenomena of energy dissipation by friction of the crack lips. Objectives of the ENDO_ORTH_BETON law ----------------------------------- Despite the existence of various anisotropic damage models for concrete, isotropic models are still exclusively used in design offices to report on the behavior of concrete structures. For anisotropic models, this is due to the complexity of their numerical implementation, to the difficulty of identifying their sometimes numerous parameters, to the mismatch of the objectives of the model and of the industrial study, to the difficulty in coupling the model with other physical phenomena (creep, plasticity), and in most cases to the significant calculation times required by anisotropic models. Moreover, the use of anisotropic models is not necessary in cases where isotropic models describe the same behavior of the structure. However, there are cases where anisotropic models can be interesting, as long as they predict behavior different from isotropic models. The objective is to have a simple anisotropic model (low number of parameters), numerically robust, and respecting the rules of thermodynamics (positive dissipation). It is obvious that a certain number of phenomena observed experimentally cannot be taken into account. A specification was therefore defined prior to the development of the model in order to define its framework. Two categories of requirements can be distinguished, the objective being to obtain a reasonable result regardless of the load. One concerns the physical consistency of the prediction of the model (1) to (4), and the other concerns the numerical robustness (5) and 6). The framework of our model is composed of the following points: 1. Taking into account anisotropy thanks to the introduction of a symmetric tensor of order 2 representing the effects of damage. It is therefore more accurate to speak of an orthotropic model insofar as the use of such a tensor only makes it possible to define three specific directions of damage. A higher-order tensor (4 or even 8) is necessary to account for complete anisotropy. 2. Cancellation of the compulsion to ruin. This leads us to define free energy, a function of deformations, rather than a free enthalpy, a function of stresses, because it seems easier to obtain zero stress from finite deformations rather than the other way around. 3. Increasing and bounded damage eigenvalues. This point reflects the irreversible nature of the damage process (growth of eigenvalues) whose ruin constitutes the limit (bounded eigenvalues). It also makes it possible to reach the ruin in several directions. 4. Taking into account the unilateral behavior of concrete: closure of cracks under compression, asymmetry of damage thresholds between traction and compression. 5. Continuity of the stress-deformation response, particularly in the open-closed passage of cracks. In addition to the fact that a discontinuity would be physically doubtful, it would lead to problems of convergence of the numerical algorithm. 6. Respect for the framework of generalized standard materials. This makes it possible to ensure the thermodynamic coherence of the model (positive dissipation) and this provides pleasant mathematical properties for numerical resolution (existence and uniqueness of the solution of the problem of calculating the stresses and of the final damage at a fixed deformation increment, called the "local projection problem" by analogy with plasticity). **Note:** .. csv-table:: "The framework for generalized standard materials (CSMG) as we understand it here is not strictly the one defined by Halphen and Nguyen [:ref:`bib11 `]. Indeed, the strict CSMG ensures the existence and the uniqueness of the solution of the global problem if the energy is convex in relation to all the variables simultaneously. This cannot be verified in the case of softening laws of behavior. The CSMG "degradation" that we define only ensures the existence and the uniqueness of the solution of the local projection problem (calculation of the evolution of damage with fixed deformation). To respect CSMG, we must check the convexity of the free energy, on the one hand in relation to the deformation, and on the other hand in relation to the internal variables: * *Convexity in relation to deformation is necessary to ensure the stability of the elastic problem.* * **Convexity in relation to all internal variables concurrently***is required to have the right mathematical properties for the local projection problem. In the case where several internal variables are used, the convexities separated with respect to each of these variables are not sufficient. * * *The global convexity in relation to the deformation and to the internal variables simultaneously is not required since the deformation increment is fixed for the local projection to calculate the evolution of the internal variables. Moreover, it seems impossible to obtain this global convexity in the case of softening laws of behavior.* *The framework that has been defined omits a number of physical phenomena associated with concrete damage:* * *appearance of irreversible deformities* * *volume expansion of the material under compression* * *hysteretic behavior for charge-discharge cycles, caused by friction between the crack lips.*"