1. Field of application#

1.1. Purposes#

The purpose of the ENDO_FISS_EXP law of behavior, available in command STAT_NON_LINE, is to describe the almost fragile damage to concrete under mechanical tensile stresses. The damage is described isotropically, which positions the model to describe cracks individually and not homogenously. In terms of phenomenology, the model aims to reflect in particular:

  • the contrast between the limits of elasticity in tension and in compression, thus avoiding premature damage in compression under prestress, for example;

  • the specific shape of the damage surface under multiaxial traction, close to a Rankine criterion;

  • the impact on the stiffness of crack closure in compression directions, which also reinforces the robustness of simulations in the presence of through cracks.

Moreover, law ENDO_FISS_EXP falls within the framework of non-local models with an internal variable gradient (modeling *_ GRAD_VARI). The particular shape of the law ensures a fundamental property: when the characteristic length of the model tends to zero (in practice, when it becomes small compared to the characteristic size of the structure studied), the results tend towards those of a cohesive model, well suited to predict propagation kinematics and crack opening. Therefore, except when looking for information on a sub-centimeter scale (which is not recommended), the characteristic length does not appear as a parameter to be adjusted but can be fixed directly from the dimensions of the structure studied.

Interested readers can find detailed information on the formulation of the law and its physical validation in reference [Lorentz, 2016].

1.2. Material parameters#

The law of behavior is based on eight internal parameters (including elasticity constants), as will be seen in the presentation of the equations in the next chapter. However, thanks to consistency with a cohesive law, we can reformulate these internal parameters in terms of quantities that are more accessible to the engineer. In addition to the Young’s modulus and the Poisson’s ratio, we will provide the tensile and compressive strengths (the latter being less significant for the targeted problems), the fracturing energy and two parameters (one of which is optional) to describe the form of the softening response. Finally, the characteristic length will be fixed on the basis of the dimensions of the work or piece studied. The DEFI_MATER_GC command takes care of the transformation of these engineering parameters into the internal characteristics of the law and replaces the DEFI_MATERIAU command.

1.3. Internal variables#

The degree of damage a, between \(0\) for a healthy material and \(1\) for a completely damaged material, is in principle the only internal variable of interest for the user. It is stored in first place (V1) in the array of internal variables stored by the code. As a post-treatment, the component V3 corresponds to the impact of this level of damage on the tensile stiffness, i.e. \(A(a)\). Finally, for information, questions of performance and numerical robustness also lead to archiving in field VARI_ELGA the state of the current time step in the component V2 (0=elastic, 1=damaging, 2=saturated) as well as the level of deformation reached at the end of the time step in the components V4 to V9 (where the shear components are affected by a factor \(\sqrt{(2)}\)).