1. Introduction#

1.1. An elasto-damageable law of behavior#

The behavior model MAZARS] is a simple model, known to be robust, based on the mechanics of damage, which makes it possible to describe the decrease in the stiffness of the material under the effect of the creation of micro-cracks in concrete. It is based on a single scalar internal variable \(D\) describing damage isotropically, but still distinguishing between tensile damage and compression damage. The version implemented in Aster corresponds to the 2012 reformulation. The major modification compared to the original model] is the introduction of a new internal variable, noted \(Y\), corresponding to the maximum reached during loading by the equivalent deformation defined in the 1980s. As a result, damage is no longer the internal variable in the revisited model. In addition, its law of evolution is simplified in order to eliminate the concepts of damage, traction and compression.

Unlike model ENDO_ISOT_BETON, this model does not make it possible to translate the phenomenon of crack closure (stiffness restoration). Moreover, the Mazars model does not take into account the possible plastic deformations or viscous effects that may be observed during the deformations of a concrete.

Version \(\mathrm{1D}\) of the Mazars model is described in the document [R5.03.09] « Nonlinear Behavioral Relationships ». In this specific case, the model is able to account for the phenomenon of reclosing cracks. Version \(\mathrm{1D}\) of the model can only be used with multi-fiber beams.

1.2. Regularization limits and methods#

Like all softening laws, the Mazars model poses difficulties related to the phenomenon of localization of deformations.

Physically, the heterogeneity of the microstructure of concrete induces interactions at a distance between the cracks formed]. Thus, the deformations are located in a thin band, called the localization band, to form the macro-cracks. The state of the stresses at a material point can no longer be described only by the characteristics of the point but must also take into consideration its environment. In the case of this (local) model, no indication is included concerning the scale of the crack. Consequently, no information is given on the width of the location band, which then becomes zero. This results in mechanical behavior with failure without energy dissipation, which is physically unacceptable.

Mathematically, the location makes the problem to be solved poorly posed because softening causes a loss of ellipticity in the differential equations that describe the deformation process]. Digital solutions do not converge into physically acceptable solutions despite mesh refinements.

1.3. Coupling with thermal#

For some studies, it may be interesting to be able to take into account the modification of material parameters under the effect of temperature. This is possible in*Aster* (MAZARS_FO combined or not with ELAS_FO). The hypotheses made for coupling with thermal energy are as follows:

  • thermal expansion is assumed to be linear, i.e.:

\[\]

: label: EQ-None

{varepsilon} ^ {text {th}} =alpha (T- {T} _ {text {ref}}) {I} _ {d}

with \(\alpha\) = constant or function of temperature,

  • we do not take into account thermo-mechanical interactions, that is to say that we do not model the effect of the state of mechanical stress on the thermal deformation of concrete,

  • concerning the evolution of material parameters with temperature, it is considered that they depend not on the current temperature but on the maximum temperature \(\mathit{Tmax}\) seen by the material during its history, (effect cited in the literature),

  • only elastic (mechanical) deformation causes damage.

Note:

Due to computer constraints, the initial value of \(\mathrm{Tmax}\) is initialized to 0. As a result, you cannot use the material parameters defined for negative temperatures (if necessary, you can however get around this problem by entering all the temperatures in Kelvin instead of \(°C\) ) .

1.4. Mazars law in the presence of a drying or hydration field#

The use of ELAS_FO and/or MAZARS_FO under the operator DEFI_MATERIAU makes it possible to make the material parameters depend on drying or hydration.

Moreover, the deformations related to the withdrawal of endogenous \({\varepsilon }_{\text{re}}\) and to the withdrawal of desiccation \({\varepsilon }_{\text{rd}}\) are taken into account in the model, in the following (linear) form (cf. [R7.01.12]):

\[\]

: label: EQ-None

{varepsilon} _ {text {re}} =-betaxi {I} _ {d}

\[\]

: label: EQ-None

{varepsilon} _ {text {rd}}mathrm {}}mathrm {=}mathrm {-}}mathrm {-}}mathrm {-} C) {mathrm {-} C) {mathrm {-} C) {mathrm {-} C) {mathrm {-} C) {mathrm {-} C) {mathrm {-} C) {mathrm {I}}}

where \(\xi\) is hydration, \(C\) the water concentration (drying field in Code_Aster terminology), \({C}_{\mathit{ref}}\) the initial water concentration (or reference drying). Finally \(\beta\) is the endogenous shrinkage coefficient and \(\kappa\) is the desiccation shrinkage coefficient to be entered in DEFI_MATERIAU, keyword factor ELAS_FO, operands B_ ENDO and K_ DESSIC. As we said in the previous paragraph, the choice that was made in implementing the MAZARS model is that only elastic deformation induces damage. Consequently, if we model a concrete test specimen that dries or that hydrates freely and uniformly, we will indeed obtain a non-zero deformation field and a stress field that is perfectly zero.

First, we present the writing of the model and then some data on the identification of the parameters. Finally, we outline the principles of digital integration in Code_Aster.