# /NONLOCAL/MAT

Block Format Keyword Non-local regularization for elasto-plastic failure criteria (as in, dependent to plastic strain) and shell 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)

${R}_{len}$ Non-local internal length.

(Real)

$\left[\text{m}\right]$
${L}_{e}{}^{MAX}$ Mesh convergence element length target.

(Real)

$\left[\text{m}\right]$

1. The non-local 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 imply a mesh convergence for mesh sizes ${L}_{e}$ less than or equal to the maximum value you set, ${L}_{e}\le {L}_{e}^{MAX}$ . This maximum mesh size ${L}_{e}{}^{MAX}$ is then the highest mesh size used for which results are mesh convergent.

The non-local formulation is compatible with elasto-plastic 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 "non-local" plastic strain calculated on the entire mesh. The non-local 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:

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 non-local "internal length" corresponding to a radius of influence in the non-local variable computation. This defines the size of the non-local regularization band ${L}_{r}=f\left({R}_{len}\right)$ (Figure 1).

The failure criterion damage variable is then computed using the non-local plastic strain.

$D=\sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_{p}^{nl}}{{\epsilon }_{f}}$

Where, ${\epsilon }_{f}$ is the plastic strain at failure depending on the failure criterion formulation.

2. To set the non-local 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 needed. 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 non-local 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 converged and mesh-independent results for a mesh size of 5mm, ${L}_{e}^{MAX}=5$ mm. In this case, the results will be converged, mesh-size and mesh orientation independent for ${L}_{e}\le 5$ mm.

3. When the non-local 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 z-direction, and the thickness variation is locally computed at Gauss points. By introducing the non-local plastic strain in the "in-thickness" strain increment, the compatibility is restored (Figure 3).
Where, $\text{Δ}{\lambda }_{nl}=f\left({\epsilon }_{p}^{nl}\right)$ is the non-local plastic multiplier.
 Figure 2. Transverse strain incompatibility (local) Figure 3. Transverse strain compatibility (non-local)
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 $-2}{3}\le \eta \le 2}{3}$ .
4. This option is compatible with:
Note: The method is not yet compatible with quadratic elements /TETRA10 and /BRIC20.
5. 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) /MAT/LAW121 (PLAS_RATE)
6. List of elasto-plastic failure model and coupled damage model compatible with non-local regularization:
 MMC damage model in /MAT/LAW72 Damage model in /MAT/LAW76 /FAIL/BIQUAD /FAIL/COCKROFT /FAIL/EMC /FAIL/HC_DSSE (for shells) /FAIL/INIEVO /FAIL/JOHNSON /FAIL/ORTHBIQUAD /FAIL/RTCL /FAIL/SPALLING /FAIL/SYAZWAN /FAIL/TAB1 /FAIL/TAB2 /FAIL/USERi /FAIL/WIERZBICKI /FAIL/WILKINS
7. List of material laws with non-local regularized temperature computation:
• /MAT/LAW104 (JOHNS_VOCE_DRUCKER)
• /MAT/LAW109
8. Two additional specific outputs, non-local plastic strain (NL_EPSP) and non-local plastic strain rate (NL_EPSD) are available in ANIM and H3D files. These are also available in time histories with NL_PLAS and NL_EPSD for shells and NL_PLAS and NL_PLSR for solids, respectively. For more information, refer to Output Database.
1 Valentin Davaze, Sylvia Feld-Payet, Nicolas Vallino, Bertrand Langrand, Jacques Besson,A non-local approach for Reissner–Mindlin shell elements in dynamic simulations: Application with a Gurson model, Computer Methods in Applied Mechanics and Engineering 415 (2023), 116142, ISSN 0045-7825.
2 Valentin Davaze, Nicolas Vallino, Bertrand Langrand, Jacques Besson, Sylvia Feld-Payet,A non-local damage approach compatible with dynamic explicit simulations and parallel computing, International Journal of Solids and Structures 228 (2021), 110999, ISSN 0020-7683.
3 Valentin Davaze, Numerical modelling of crack initiation and propagation in ductile metallic sheets for crash simulations. Mechanics of materials. University Paris sciences et lettres, 2019. English.