1. Field of application

1.1. Purposes

The purpose of the law of behavior ENDO_LOCA_EXP, available in command STAT_NON_LINE, is to describe the diffuse, almost fragile damage to concrete under mechanical tensile stresses. The damage is described isotropically there. 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.

In the presence of localized damage, the non-local model ENDO_FISS_EXP, whose parameter definition is close to that of ENDO_LOCA_EXP, will be preferred.

Interested readers can find more detailed information on the formulation of the law and its validation in reference [Lorentz, 2020]. Given the proximity to the laws of behavior ENDO_ISOT_BETON and ENDO_FISS_EXP, he can also refer to the references [Badel et al., 2007], [Lorentz, 2017], [R7.01.04] and [R5.03.28].

1.2. Material parameters

The law of behavior is based on six internal parameters (including elasticity constants), as will be seen in the presentation of the equations in the next chapter. However, in a diffuse cracking regime where an average inter-crack distance emerges, it is possible to reformulate the 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 a parameter to describe the form of the softening response. Finally, the average inter-crack distance will be fixed based on the characteristics of the reinforcement, for example. The DEFI_MATER_GC command takes care of the transformation of these engineering parameters into internal characteristics of the law and can therefore replace the DEFI_MATERIAU command, but in this case it requires also entering the mean inter-crack distance.

1.3. Internal variables

Behavioral law ENDO_LOCA_EXP gives access to five internal variables. The degree of damage \(b\), between 0 for a healthy material and 1 for a totally damaged material, is a theory the only true internal variable of the model. For reasons of numerical efficiency, the state of the current time step is also maintained, with the values 0=elastic, 1=damaging, 2=saturated. Finally, three variables are also grouped together to facilitate post-processing.

The internal variables are grouped in the following table:

ENDO	V1	Damage \(b\)
INDIENDO	V2	State of the current time step (0, 1, or 2)
ENDORIGI	V3	Stiffness loss \(1-B(b)\)
ENERELAS	V4	Elastic volume energy
ENERDISS	V5	Volume energy consumed