/FAIL/LEMAITRE

Block Format Keyword A stress triaxiality-based failure model dedicated to ductile failure. It is based on plastic strain evolution, and the computed damage variable generates a softening of the stress tensor.

This failure model can be used for shell and solid elements and is compatible with all elasto-plastic materials.

Format


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/FAIL/LEMAITRE/`mat_ID`/`unit_ID`
$ε_{D}$		$S_{D}$		$D_{C}$			`FAILIP`	`P_thick`_fail

Optional line:


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fail_ID`

Definition


Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	(Optional) Unit identifier. (Integer, maximum 10 digits)
$ε_{D}$	Plastic strain value triggering the damage evolution. Default = 1.0E30 (Real)
$S_{D}$	Strain energy release rate. Default = 1.0E30 (Real)	$[Pa]$
$D_{C}$	Critical damage value to trigger element deletion. Default = 1.0 (Real)
`FAILIP`	Number of failed integration points prior to solid element deletion. Default = 1
`P_thick`_fail	Fraction of failed thickness for shell element deletion. -1.0 ≤ `P_thick`_fail ≤ 0 Fraction of failed integration point through thickness. 0.0 ≤ `P_thick`_fail ≤ 1.0 Fraction of failed thickness. Default = 1.0 (Real)
`fail_ID`	(Optional) Failure criteria identifier. (Integer, maximum 10 digits)

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
                  Mg                  mm                   s
/FAIL/LEMAITRE/25
#              EPS_D                 S_D                  DC              FAILIP         Pthick_fail       
                 .27                0.05                  1.                   1                 0.5
#  fail_ID
      1234
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The LEMAITRE failure criterion is based on the damage coupled model proposed by Lemaitre & Chaboche and later Lemaitre & Desmorat. It considers the continuum damage mechanics framework introduced by Kachanov. To describe this framework let us consider a Representative Volume Elementary (RVE) of a section $S$ to which a normal load $F$ is applied.
Figure 1.

The stress in the undamaged volume (a) is then:
$σ = \frac{F}{S}$
If the same elementary volume is damaged (b), the remaining effective surface $\tilde{S}$ decreases and the damaged surface $S_{d a m}$ occupied by cavities increases. The damage variable is then defined as the ratio between the damaged and the initially sound surface.
$D = \frac{S_{d a m}}{S}$
This variable evolves between 0 (no damage) and 1 for a complete RVE failure. In this case, the cavities are occupying the entire initial section of the elementary volume. The stress in the damaged volume can be expressed by:
$\tilde{σ} = \frac{F}{\tilde{S}} = \frac{F}{S (1 - D)} = \frac{σ}{(1 - D)}$
One can then differentiate the effective stress of the elasto-plastic section denoted $\tilde{σ}$ and the equivalent damaged stress of the entire finite element section denoted $σ = \tilde{σ} (1 - D) = (1 - D) E ε$ . The stress in the element then decreases as the damage variable increases, until complete failure, in which the cavities occupy the entire element section ( $D = 1$ ).
The proposed LEMAITRE damage model considers an evolution of the damage variable governed by:
$Δ D = \{\begin{matrix} \frac{Y}{S_{D} (1 - D)} Δ ε_{p} & if ε_{p} > ε_{D} and σ_{1} > 0 \\ 0 & otherwise \end{matrix}$
Where, $Y$ is a variable based on thermodynamic theory so-called elastic energy release rate, computed with:
$Y = \frac{1}{2} \tilde{σ} : C_{e}^{- 1} : \tilde{σ} = \frac{1}{2} \frac{σ}{(1 - D)} : C_{e}^{- 1} : \frac{σ}{(1 - D)}$
Using some assumptions, this variable can be rewritten as:
$Y = \frac{{\tilde{σ}}_{V M} R_{v}}{2 E}$
Where,

${\tilde{σ}}_{V M}$

Effective von Mises stress

$E$

Material Young's modulus

$R_{v}$

Stress triaxiality dependent coefficient that is obtained with:
$R_{v} = \frac{2}{3} (1 + ν) + 3 (1 - 2 ν) η^{2}$

Where,

$ν$

Poisson ratio

$η$

Stress triaxiality

According to this equation, you can see that the damage rate is higher at large stress triaxiality, implying a less ductile behavior with the increasing triaxiality.
It is worth noting that the damage variable only evolves under tensile loading condition ( $σ_{1} > 0$ ), as ductile fracture under compression is less likely.
This failure model has three parameters to be set up:

$ε_{D}$

Plastic strain value threshold to trigger the damage variable evolution

$S_{D}$

Fracture elastic energy release rate. If an accurate identification of this parameter is not performed, it is recommended to use $\frac{R_{e}}{200}$ with $R_{e}$ the initial yield stress.

$D_{C}$

The critical damage value at which element / integration point is deleted, and stress tensor is set to zero (even if the evolution of the damage variable has not been completed).
This damage model is compatible with /NONLOCAL/MAT to obtain regularized and mesh independent results.
The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL. There is no default value. If the line is blank, no value is output for failure model variables in the /INIBRI/FAIL (written in the .sta file with /STATE/BRICK/FAIL option). It can also be used for some specific failure output /H3D/ELEM/FAILURE.

¹ Lemaitre, J., & Chaboche, J.-L. (1992). Mechanics of Solid Materials. Cambridge University Press

² Lemaitre, J., & Desmorat, R. (2005). Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer

³ Kachanov, L. M. (1958). Time of the rupture process under creep conditions. Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, 8, 26–31