/NONLOCAL/MAT
Block Format Keyword Nonlocal regularization for elastoplastic failure criteria (as in, dependent to plastic strain) and shells thickness variation.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/NONLOCAL/MAT/mat_ID/unit_ID  
${R}_{len}$  ${L}_{e}{}^{MAX}$ 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  Unit identifier. (Integer, maximum 10 digits) 

${R}_{len}$  Nonlocal internal
length. (Real) 
$\left[\text{m}\right]$ 
${L}_{e}{}^{MAX}$  Mesh convergence element length
target. (Real) 
$\left[\text{m}\right]$ 
Comments
 The nonlocal regularization is used to get mesh independent results
(size, orientation) in case of instabilities such as failure and/or
thickness variation (for shells). The mesh independent results implies a
mesh convergence for mesh sizes
${L}_{e}$
less than or equal to the maximum value set
by you,
${L}_{e}\le {L}_{e}^{MAX}$
. This maximum mesh size
${L}_{e}{}^{MAX}$
is then the highest mesh size used for which
results are mesh convergent. The nonlocal formulation is compatible with elastoplastic material laws only. When activated, the computation of the attached failure criteria based on plastic strain and/or the shell thickness variation depends on a regularized nodal "nonlocal" plastic strain calculated on the entire mesh. The nonlocal plastic strain at nodes denoted ${\epsilon}_{p}^{nl}$ is computed accounting for its own gradient and its local counterpart ${\epsilon}_{p}$ is computed at the Gauss points following the set of equations:
(1) $$\begin{array}{c}{R}_{len}^{2}\text{\Delta}{\epsilon}_{p}^{nl}\gamma {\dot{\epsilon}}_{p}^{nl}+\left({\epsilon}_{p}{\epsilon}_{p}^{nl}\right)=\zeta {\ddot{\epsilon}}_{p}^{nl}\\ \overrightarrow{\nabla}{\epsilon}_{p}^{nl}\cdot \overrightarrow{n}=0\end{array}\begin{array}{c}on\\ on\end{array}\begin{array}{c}\Omega \\ \Gamma \end{array}$$The parameters $\gamma $ and $\zeta $ are automatically set. You have to set the parameter ${R}_{len}$ (or ${L}_{e}{}^{MAX}$ , Comment 2) which defines a nonlocal "internal length" corresponding to a radius of influence in the nonlocal variable computation. This defines the size of the nonlocal regularization band ${L}_{r}=f\left({R}_{len}\right)$ (Figure 1).The failure criterion damage variable is then computed using the nonlocal plastic strain.(2) $$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}^{nl}}{{\epsilon}_{f}}}$$Where, ${\epsilon}_{f}$ is the plastic strain at failure depending on the failure criterion formulation.
 To set the nonlocal
length parameter
${R}_{len}$
, you can select:
 Directly input the value of ${R}_{len}$ in the input card, if a direct control on this parameter is wanted. In this case, the parameter ${L}_{e}{}^{MAX}$ must be ignored.
 Input the maximum mesh size
${L}_{e}{}^{MAX}$
for which results are mesh
convergent. The nonlocal regularization will then be effective for
all mesh sizes
${L}_{e}$
such as
${L}_{e}\le {L}_{e}^{MAX}$
. In this case, an automatic set of
${R}_{len}$
is realized according to the value
of
${L}_{e}{}^{MAX}$
, and the input value of
${R}_{len}$
is ignored.
For instance, if you want to get converged and meshindependent results for a mesh size of 5mm, ${L}_{e}^{MAX}=5$ mm. In this case, the results will be converged, meshsize and mesh orientation independent for ${L}_{e}\le 5$ mm.
 When the nonlocal regularization is used for shell elements, an
additional regularization is made on the thickness variation computation
avoiding an additional localization issue. In the common local case (Figure 2), the compatibility of thickness
between shell elements is not ensured, due to the lack of kinematic
equations in the zdirection, and the thickness variation is locally
computed at Gauss points. By introducing the nonlocal plastic strain in the
"inthickness" strain increment, the compatibility is restored (Figure 3).
(3) $$\text{\Delta}{\epsilon}_{zz}=\frac{\nu}{1\nu}\left(\text{\Delta}{\epsilon}_{xx}\text{\Delta}{\lambda}_{nl}{n}_{xx}+\text{\Delta}{\epsilon}_{yy}\text{\Delta}{\lambda}_{nl}{n}_{yy}\right)+\text{\Delta}{\lambda}_{nl}{n}_{zz}$$Where, $\text{\Delta}{\lambda}_{nl}=f\left({\epsilon}_{p}^{nl}\right)$ is the nonlocal plastic multiplier.Note: This last point implies that the identified parameters can be used on solid and shells, as results will be identical within the same range of stress triaxiality $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{1ex}{$3$}\right.\le \eta \le \raisebox{1ex}{$2$}\!\left/ \!\raisebox{1ex}{$3$}\right.$ .  List of compatible material laws for shells thickness variation
regularization:
 /MAT/LAW2 (PLAS_JOHNS)
 /MAT/LAW22 (DAMA)
 /MAT/LAW27 (PLAS_BRIT)
 /MAT/LAW32 (HILL)
 /MAT/LAW36 (PLAS_TAB)
 /MAT/LAW43 (HILL_TAB)
 /MAT/LAW44 (COWPER)
 /MAT/LAW48 (ZHAO)
 /MAT/LAW57 (BARLAT3)
 /MAT/LAW60 (PLAS_T3)
 /MAT/LAW63 (HANSEL)
 /MAT/LAW64 (UGINE_ALZ)
 /MAT/LAW72 (HILL_MMC)
 /MAT/LAW76 (SAMP)
 /MAT/LAW78
 /MAT/LAW87 (BARLAT2000)
 /MAT/LAW93 (ORTH_HILL) (CONVERSE)
 /MAT/LAW104 (JOHNS_VOCE_DRUCKER)
 /MAT/LAW109
 /MAT/LAW110 (VEGTER)
 List of elastoplastic failure model and coupled damage model
compatible with nonlocal regularization:
 MMC damage model in /MAT/LAW72
 Damage model in /MAT/LAW76
 /FAIL/BIQUAD
 /FAIL/COCKROFT
 /FAIL/EMC
 /FAIL/HC_DSSE (for shells)
 /FAIL/JOHNSON
 /FAIL/ORTHBIQUAD
 /FAIL/RTCL
 /FAIL/SPALLING
 /FAIL/TAB1
 /FAIL/USERi
 /FAIL/WIERZBICKI
 /FAIL/WILKINS
 List of material laws with nonlocal regularized temperature computation:
 /MAT/LAW104 (JOHNS_VOCE_DRUCKER)
 /MAT/LAW109