7. Keyword factor ENDO_FISS_EXP

Behavioral model ENDO_FISS_EXP is a non-local damagable elastic behavior model (available in GRAD_VARI modeling). It makes it possible to describe the softening behavior of concrete. In particular, it distinguishes between tensile and compressive behavior, partially restores compressive stiffness and describes shear or bi-traction states well. When the characteristic length tends to zero, it approaches a cohesive model, which explains why parameters characteristic of a cohesive law are entered more than of a volume damage model.

7.1. Operands

E	Young’s modulus (Pa)
NU	Poisson’s ratio
FT	Single pull damage threshold (Pa)
FC	Compression damage threshold (Pa)
GF	Cracking energy (N/m)
P	Main work hardening parameter of the asymptotic cohesive law
G_ INIT	Partial cracking energy resulting from the initial slope
Q	Secondary work hardening parameter of the asymptotic cohesive law
Q_ REL	Secondary work hardening parameter (between 0 and 1)
LARG_BANDE	Location band width (m)
REST_RIGI_FC	Stiffness restoration coefficient (0 = without restoration)
COEF_RIGI_MINI	Switching threshold to the fixed secant tangent matrix

Table 7.1-a: Material parameters of law ENDO_FISS_EXP

7.2. How it works

Some characteristics do not require additional explanations: Young’s modulus, Poisson’s ratio, tensile and compression damage thresholds, and cracking energy. These are common characteristics for modeling concrete damage.

With regard to the parameters P and Q that characterize the standardized response of the asymptotic cohesive model (the constraint is normalized by FT, the aperture is normalized by GF/FT), several ways are available to inform them. It is possible to directly enter P (greater than 1) or the effect of P on the initial slope of the cohesive response via the corresponding crack energy data G_ INIT if the cohesive softening model was linear with the initial slope as a slope. This way of characterizing the initial slope is used by some authors, in particular in the context of a cohesive bilinear law (we then speak of Gf, with a lowercase letter, as opposed to GF, with a capital letter, which designates the entirety of the cracking energy). For Q, you can directly enter its value (zero by default) but it is required to remain between 0 and a maximum value that depends on P. To simplify the Q data, we offer the possibility of entering Q in relation to this maximum value using the keyword Q_ REL (with a value between 0 and 1, therefore).

As far as stiffness restoration is concerned, model ENDO_FISS_EXP introduces a regulation parameter \(\mathrm{\gamma }\) which smoothes the stiffness jump between traction and compression. Rather than directly entering the value of \(\mathrm{\gamma }\), which is not necessarily very meaningful, we prefer to indicate via REST_RIGI_FC the level of stiffness restored with respect to the initial stiffness for a deformation (in compression) equal to FC/E. This value is therefore between 0 (no restoration of rigidity) and 1 (strictly less than 1, the total restoration of stiffness not being possible with the chosen regularization function). A value of 0.9 is proposed by default.

Finally, the width of the localization band, which is supposed to reflect a cohesive crack, is introduced. This is twice the parameter D (which measures the half-bandwidth in 1D, as described in the theoretical reference of the model).

As for the COEF_RIGI_MINI parameter, it’s the one introduced in DEFI_MATERIAU [U4.43.01]. When the residual stiffness normalized by E is less than COEF_RIGI_MINI, the tangent matrix is replaced with the secant matrix corresponding to this stiffness, which limits the problems associated with possible zones that are completely destroyed (i.e. without residual stiffness). This parameter has no impact on the physics of the model; it only affects the convergence properties of the Newton algorithm. By default, this function is not activated (COEF_RIGI_MINI = 0), which seems to be very suitable in most cases.

In general, reference is made to the model reference documentation [R5.03.28] for more detailed explanations of the meaning of the various parameters of the model.