MATS1

This entry is used if a MAT1 entry is specified with the same MID in a nonlinear subcase.

Format A (HR = 1 to 5)

Format B.1 (HR = 6: Kinematic Hardening (NLKIN), TYPKIN=PARAM)

Format B.2 (HR = 6: Kinematic Hardening (NLKIN), TYPKIN=HALFCYCL)

Format B.3 (HR = 6: Isotropic Hardening (NLISO), TYPISO=PARAM)

Format B.4 (HR = 6: Isotropic Hardening (NLISO), TYPISO=TABLE)

Example A (HR = 1, 2, 3)

Example (HR = 6): Kinematic Hardening

Example (HR = 6): Combined Hardening

TYPKIN=PARAM, NLKIN=2, Temperature-dependent

Example (HR = 6): Kinematic Hardening

Definitions

Comments

1. String based labels allow for easier visual identification, including when being referenced by other entries. (example, the MID field of properties). For more details, refer to String Label Based Input File.
2. For nonlinear elastic material, the stress-strain data given in the TABLES1 or TABLEG entry will be used to determine the stress for a given value of strain. The values H, YF, HR, and LIMIT1 will not be used in this case. Nonlinear elastic material is only available in EXPDYN subcases.
3. For elastoplastic materials, the elastic stress-strain matrix is computed from a MAT1 entry, and the isotropic plasticity theory is used to perform the plastic analysis. In this case, either the table identification TID or the work hardening slope $H$ may be specified, but not both. If the TID is omitted, the work hardening slope $H$ must be specified, unless the material is perfectly plastic. The plasticity modulus ( $H$ ) is related to the tangential modulus ( $E_T$ ) by:
   (1)

   $$H=\frac{E_T}{1-\frac{E_T}{E}}$$
   Where, $E$ is the elastic modulus and $E_T=dY/d\epsilon$ is the slope of the uniaxial stress-strain curve in the plastic region.

   
   
   Figure 1.

4. If TID is given, TABLES1 or TABLEG entries (Xi,Yi) of stress-strain data ( $\text{ε}$ _x,Y_x) must conform to the following rules:

   If TYPE=PLASTIC, the curve must be defined in the first quadrant. The data points must be in ascending order. If the table is defined in terms of total strain (TYPSTRN=0), the first point must be at the origin (X1=0, Y1=0) and the second point (X2, Y2) must be at the initial yield point (Y₁) specified on the MATS1 entry. The slope of the line joining the origin to the yield stress must be equal to the value of E. If the table is defined in terms of plastic strain (TYPSTRN=1), the first point (X1, Y1), corresponding to yield point (Y₁), must be at X1=0. TID may reference a TABLEST entry. In this case, the above rules apply to all TABLES1 tables pointed to by TABLEST.

   If TYPE=NLELAST, the full stress-strain curve may be defined in the first and third quadrants to accommodate different uniaxial compression data. If the curve is defined only in the first quadrant, then the curve must start at the origin (X1=0.0, Y1=0.0).

   For analysis where small deformations are assumed, there should be little or no difference between the true stress-strain curve and the engineering stress-strain curve, so either of them may be used in the TABLES1 definition. For analyses where small deformations are not assumed, the true stress-strain curve should be used.

   If the deformations go past the values defined in the table, the curve is extrapolated linearly.

5. Kinematic hardening and Mixed hardening are supported only for solids.
6. The conversion of the relation stress versus total strain (TYPSTRN=0) into stress versus plastic strain (TYPSTRN=1) is illustrated below. This is clearly different than simply shifting the entire table along the epsilon-axis.

   
   
   Figure 2.

7. The temperature-dependence of the MATS1 material is defined by referencing a TABLEST entry via the TID field.
8. Large strain elasto-plasticity can be activated using MATS1 (TYPE=PLASTIC) in conjunction with PARAM, LGDISP,1.
9. MATS1 is not supported in conjunction with second order shell elements (CTRIA6 and CQUAD8).
10. MATS1 is supported for CROD, CONROD, CBAR and CBEAM elements in the axial translational direction only. The behaviors in other directions remain elastic.

    The torsional deformation of CROD/CONROD elements or the shear, bending and torsional deformations of CBAR/CBEAM elements remain elastic.

11. If Johnson-Cook is selected (HR=4), use the following formulations:(2)
    $$\sigma =\left(a+b{\epsilon }_p^n\right)\left(1+c\ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right)$$
    (3)
    $$\overline{\sigma }=\left(A+B{\left({\overline{\epsilon }}^{pl}\right)}^n\right)\left(1+C\ln \left(\frac{{\dot{\overline{\epsilon }}}^{pl}}{{\dot{\epsilon }}_0}\right)\right)$$
    Johnson-Cook strain rate dependence assumes that,(4)

    $$\overline{\sigma }={\sigma }^0\left({\overline{\epsilon }}^{pl},\theta \right)R\left({\dot{\overline{\epsilon }}}^{pl}\right)$$
    and (5)

    $${\dot{\overline{\epsilon }}}^{pl}={\dot{\epsilon }}_0\exp \left(\frac{1}{C}\left(R-1\right)\right)$$

    for $\overline{\sigma }\ge {\sigma }^0$

    Where,

    $\overline{\sigma }$

    Yield stress for non-zero strain rate.

    ${\dot{\overline{\epsilon }}}^{pl}$

    Equivalent plastic strain rate.

    ${\dot{\epsilon }}_0$ and $C$

    Are material parameters measured at or below the transition temperature ${\theta }_{transition}$ .

    ${\sigma }^0\left({\overline{\epsilon }}^{pl},\theta \right)$

    Static yield stress.

    $R\left({\dot{\overline{\epsilon }}}^{pl}\right)$

    Ratio of the yield stress at nonzero strain rate to the static yield stress (so that $R\left({\dot{\epsilon }}_0\right)=1.0$ ).

    Johnson-Cook hardening is supported only with Explicit Dynamic Analysis.

12. When Crushable Foam is selected (HR=5), only maximum principal stress as yield criterion (YF=2) is used, and a table TID must be given.

    The default for TSC (tensile stress cutoff) is zero, unless the user specifies a value. TSC is defined as a positive stress value which indicates the yield stress of crushable foam under tensile loading.

    The yield stress of crushable foam under compression loading can be given by a rate independent table (TABLES1).

    Where,

    x

    Values in the table are the volumetric strain (all positive values indicate the volume is compressed). The volumetric strain is defined as $\gamma =1-\frac{V}{V_0}$ .

    y

    Values in the table are the compression yield stress (all positive values).

    Note: Since crushable foam is based on volumetric strain-based definition, TYPSTRN = 0 (default) is invalid and TYPSTRN = 1 must be specified.

    First entry in the TABLES1 entry will be x=0, y=y_0, (the initial compressive yield stress). All xi should be positive and in increasing order.

    For crushable foam, in place of the equivalent plastic strain in H3D file, the integrated volumetric strain (natural logarithm of the relative volume $-In\left(\frac{V}{V_0}\right)$ is output.

    Crushable Foam hardening is supported only with Explicit Dynamic Analysis.

13. When TID refers to TABLEMD.
    For rate-independent problems:

    HR

    X1

    1, 2 or 3

    Plastic strain/Total strain

    5

    Volumetric strain

    For rate-dependent problems:

    HR

    X1

    1, 2 or 3

    Total strain rate

    5

    Volumetric strain rate

14. Combined hardening can be used by specifying hardening, HR=6. Combined hardening is currently supported only for solid elements and for both SMDISP/LGDISP NLSTAT and NLTRAN analysis.
15. For combined hardening, at least one type of nonlinear hardening should be defined. Both NLKIN and NLSIO support either parameter or stress-strain curve input. They can be combined flexibly, for example NLKIN with parameter input (PARAM) and NLISO with stress-strain curve input (TABLE).
16. For NLKIN hardening, the number of back stresses can be specified from 1 up to 10.
17. After the continuation line with NLKIN or NLISO, you can input the data block with an unlimited number of continuation lines.
18. Using the HALFCYCL format for NLKIN, you can provide a stress-strain curve for curve fitting. The stress (column SIG) is the total stress from experiment. The strain (column EPS) should be the equivalent plastic strain. In the temperature dependent case, multiple curves can be provided one after each other. The last column temperature (TEMP) should be in ascending order.
19. Using the PARAM input format for NLKIN, you can provide the parameters directly. The parameters can be temperature dependent, with TEMP specified in ascending order. If the number of back stresses is equal or greater than 4, then multiple continuation lines will be used to define the complete set of parameters (C1-C10, G1-G10), check the example above.
20. For parameter fitting using options TABLE or HALFCYCL, the Levenberg-Marquardt method is used to determine the optimal parameters for both NLKIN and NLISO. The parameters are printed in the .out file for each temperature.
21. If both NLKIN and NLISO use PARAM format, then the initial yield stress SIGY0 should be the same for the same temperature.
22. The units of the parameter C1-C10 should be the same as the Young’s modulus on the MAT1 entry.
23. For more information regarding Combined Hardening (HR=6), refer to Combined Hardening of von Mises Plasticity in the User Guide.