# MATS1

Bulk Data Entry Specifies strain-, rate-, and temperature-dependent material properties for use in applications involving nonlinear materials.

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

## Format A (HR = 1 to 3)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 MID TID TYPE H YF HR LIMIT1
TYPSTRN TYPSTRT
Optional continuation lines for Johnson-Cook hardening (for Explicit Dynamic analysis only)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
JHCOOK A B N C RSTRT
Optional continuation lines for Crushable Foam model (for Explicit Dynamic analysis only)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
CFOAM TSC
Optional continuation lines for Cowper-Symonds model (for Explicit Dynamic analysis only)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
CSYMONDS D p

## Example A (HR = 1, 2, 3)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 17 28 PLASTIC 0.0 1 1 2.0E04

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

Format B illustrates the syntax of the MATS1 entry for HR=6. Both NLKIN and NLISO continuation lines can be defined on the same MATS1 entry.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 MID HR
+
NLKIN TYPKIN NKIN
SIGY0 C1 G1 C2 G2 etc. TEMP
etc. etc. etc.

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

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 MID HR
+
NLKIN TYPKIN NKIN
SIG EPS TEMP
etc. etc. etc.

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

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 MID HR
+
NLISO TYPISO NISO
SIGY0 Q B TEMP
etc. etc. etc. etc.

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

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 MID HR
+
NLISO TYPISO NISO
SIG EPS TEMP
etc. etc. etc.

## Example (HR = 6): Kinematic Hardening

TYPKIN=PARAM, NLKIN=10, Temperature-independent
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 17 6
+
+ NLKIN PARAM 10
120.0 1000.0 10.0 2000.0 20.0 3000.0 30.0 4000.0
40.0 5000.0 50.0 6000.0 60.0 7000.0 70.0 8000.0
80.0 9000.0 90.0 10000.0 100.0

## Example (HR = 6): Combined Hardening

TYPKIN=PARAM, NLKIN=2, Temperature-dependent

TYPISO=PARAM, NLISO=1, Temperature-dependent
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 17 6
+
+ NLKIN PARAM 2
120.0 1000.0 10.0 2000.0 20.0 23.0
100.0 800.0 8.0 1800.0 18.0 27.0
80.0 600.0 6.0 1600.0 16.0 35.0
NLISO PARAM 1
100.0 70 5.0 23.0
120.0 80 6.0 27.0
80.0 60 4.0 35.0

## Example (HR = 6): Kinematic Hardening

TYPKIN=HALFCYCL, NLKIN=2, Temperature-dependent
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATS1 17 6
+
+ NLKIN HALFCYCL 2
120.0 0.0 23.0
125.0 0.01 23.0
130.0 0.03 23.0
80.0 0.0 35.0
86.0 0.01 35.0
92.0 0.03 35.0

## Definitions

Field Contents SI Unit Example
MID Identification of a MAT1 entry.
Integer
Specifies an identification number for this material.
<String>
Specifies a user-defined string label for this material entry. 1

No default (Integer > 0 or <String>)

TID Identification of a TABLES1, TABLEST, TABLEG or TABLEMD entry. If H is given, this field must be blank. 4
Integer
Specifies an identification number for this material.
This is supported for all the table entries listed above.
<String>
Specifies a user-defined string label for this material entry. 1
This is supported only for the TABLES1 entry.
blank

(Integer ≥ 0, <String> (only for TABLES1), or blank)

TYPE Material nonlinearity type.
PLASTIC (Default)
Elastoplastic material.
In an LGDISP analysis, this activates large strain elastoplasticity.
NLELAST
Nonlinear elastic material.
blank

H Work hardening slope (slope of stress versus plastic strain) in units of stress. For elastic-perfectly plastic cases, H = 0.0. For more than a single slope in the plastic range, the stress-strain data must be supplied on a TABLES1 or TABLEG entry referenced by TID, and this field must be blank. 3

(Real)

YF Yield function criterion.
1
von Mises.
Supported only with Implicit analysis (nonlinear static and nonlinear transient).
2
Maximum principal stress.
Supported only with Explicit Dynamic analysis. 2

(Integer)

HR Hardening rule.
1 (Default)
Isotropic hardening.
2
Kinematic hardening.
3
Mixed hardening with 30% contribution of kinematic hardening and 70% contribution of isotropic hardening.
6
Combined hardening. 13
Adjustable mixed hardening is selected by choosing a (Real) value for HR:
0 < HR < 1
The contribution of the kinematic hardening is HR whereas the contribution of the isotropic hardening is 1 - HR. 7

(1, 2, 3, 6, or 0.0 < Real < 1.0) (Integer)

LIMIT1 Initial yield point.

The LIMIT1 field can be blank, if the initial yield point value is defined via a referenced TABLES1, TABLEG, TABLEST or TABLEMD entries on the TID field. OptiStruct will error out if LIMIT1 is blank and TID does not reference a TABLES1, TABLEG, TABLEST or TABLEMD entry.

(Real > 0 or blank)

TYPSTRN Specifies the type of strain used on the x-axis of the table pointed to by TID. The strain type is selected by one of the following values. 4, 6, 8, 11
0 (Default)
Total strain is used on the x-axis.
1
Plastic strain or volumetric strain is used on the x-axis. 11

(Integer)

TYPSTRT Specifies the type of strain rate used on the x-axis of the table pointed to by TID of TABLEMD. The strain rate type is selected by one of the following values. ,
0 (Default)
Total strain rate is used on the x-axis.
1
Plastic strain rate or volumetric strain rate is used on the x-axis.

(Integer)

CFOAM Flag that identifies that the Crushable Foam model parameters are to follow. 11
TSC Tensile stress cutoff. A nonzero, positive value is recommended for realistic behavior.

Default = 0.0 (Real ≥ 0.0)

JHCOOK Flag that identifies that the Johnson-Cook hardening method parameters are to follow. 10
A Material yield stress.

No default (Real)

B Coefficient to the plastic strain.

Default = 0.0 (Real)

N Exponent to the plastic strain.

Default = 1.0 (Real)

C Coefficient to the strain rate.

Default = 0.0 (Real)

RSTRT Reference strain rate.

Default = 1.0 (Real)

CSYMONDS Flag that identifies that the Cowper-Symonds method parameters are to follow. 12
D Cowper-Symonds strain rate parameter.

No default (Real)

p Cowper-Symonds strain rate parameter.

No default (Real)

NLKIN Continuation line flag indicating that data input for kinematic hardening is to follow. 7, 14
TYPKIN Kinematic hardening data input type.
HALFCYCL (Default)
Table input providing stress-strain curve. Total stress from experiment is provided as a column via the SIG fields, while the Equivalent Plastic Strain column is provided via the EPS fields. For NLKIN, the equivalent plastic strain is usually sourced directly from the first cycle of the experiment.
If temperature-dependent data is to be provided, the final column is TEMP, which is the temperature. This column should be provided in ascending order.
PARAM
Parameter input which provides the parameters directly. The parameters are SIGY0, Ci, and Gi for kinematic hardening. These parameters can be temperature dependent via the TEMP column, which should be specified in an ascending order.

NKIN Number of back stresses for kinematic hardening definition via NLKIN.

Default = 1 (Integer)

NLISO Continuation line flag indicating that data input for isotropic hardening is to follow. 14
TYPISO Isotropic hardening data input type.
TABLE (Default)
Table input providing isotropic part of yield stress versus the equivalent plastic strain (similar to the curve used for isotropic hardening, HR=1). Isotropic part of the yield stress from experiment is provided as a column via the SIG fields, while the Equivalent Plastic Strain column is provided via the EPS fields. For NLISO, the equivalent plastic strain is usually sourced from cyclic loading experiments.
If temperature-dependent data is to be provided, the final column is TEMP, which is the temperature. This column should be provided in ascending order.
PARAM
Parameter input which provides the parameters directly. The parameters are SIGY0, Q, and B for isotropic hardening. These parameters can be temperature dependent via the TEMP column, which should be specified in an ascending order.

NISO Number of parameters for isotropic hardening definition via NLISO.

Default = 1 (Integer)

SIGY0 Initial yield stress via the PARAM option for NLKIN or NLISO.

No default (Real > 0.0)

Ci Parameter(s) Ci of back stress components for NLKIN (PARAM). Up to 10 parameters (C1 to C10) can be specified.

No default (Real > 0.0)

Gi Parameter(s) Gi of back stress components for NLKIN (PARAM). Up to 10 parameters (G1 to G10) can be specified.

No default (Real > 0.0)

Q Parameter Q for NLISO (PARAM).

No default (Real > 0.0)

B Parameter B for NLISO (PARAM).

No default (Real > 0.0)

SIG Stress input for data curve input for NLKIN (HALFCYCL) or NLISO (TABLE).

No default (Real > 0.0)

EPS Equivalent plastic strain input for data curve input for NLKIN (HALFCYCL) or NLISO (TABLE).

No default (Real > 0.0)

TEMP Temperature for temperature-dependent data specification for NLKIN or NLISO.

No default (Real)

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. Information for TYPE=NLELAST:
Analysis type
Supported for Explicit Dynamic analysis.
Tables
Supported with TABLES1 and TABLEG.
H, YF, HR, and LIMIT1 will not be used.
3. For elastoplastic materials, the elastic stress-strain matrix is computed from the MAT1 entry, and the isotropic plasticity theory is used to perform the plastic analysis.

Either the table identification TID or the work hardening slope H may be specified, but not both.

4. If TID is given, TABLES1, TABLEG , TABLEST or TABLEMD entries (Xi,Yi) of stress-strain data ( $\text{ε}$ x,Yx) must conform to the following rules:
Entity TYPE = PLASTIC TYPE = NLELAST
Quadrant Plastic stress-strain curve must be defined in the first quadrant only. Full stress-strain curve may be defined in the first and third quadrants to accommodate different uniaxial compression data.
First point If TYPSTRN = 0,
• First point must be at the origin (X1 = 0, Y1 = 0).
• Second point (X2, Y2) must be the initial yield stress (Y2=LIMIT1) at initial yield strain (X2=LIMIT1/E).

The slope of the line joining the origin to the initial yield stress must be equal to the value of E.

If TYPSTRN = 1,
• First point (X1, Y1) is corresponding to initial yield stress (Y1=LIMIT1), with zero equivalent plastic strain (X1=0).
If the curve is defined only in the first quadrant, then the curve must start at the origin (X1 = 0.0, Y1 = 0.0).
Additional details The data points must be in ascending order.
If TYPSTRN = 1,
• TID may reference a TABLEST entry.

In this case, the above rules apply to all TABLES1 tables pointed to by TABLEST.

• In case of small deformations, the true and the engineering stress-strain curves are almost the same, so either of them can be used in the table definition.
• In case of large deformations, the true stress-strain curve should be used.
• If the deformations exceed the values defined in the table, linear extrapolated is done.
5. When TID refers to TABLEMD, the following applies:
Xi_j Implicit analysis (NLSTAT/NLTRAN for SMDISP/LGDISP) Explicit analysis
Xi_1 Equivalent plastic strain. This can be used for both rate-dependent and independent problems. Represents plastic or total strain in case of rate-independent problems.
Xi_2 Represents plastic strain rate in case of rate-dependent problems

If only one value of X2 is specified, it is still rate independent.

Represents plastic or total strain rate in case of rate-dependent problems.
Xi_3 Temperature. N/A
6. Information about rate dependent plasticity:
Entity Implicit analysis (NLSTAT/NLTRAN for SMDISP/LGDISP) Explicit analysis
Activation Rate dependent plasticity can be activated by specifying TABLEMD ID in the TID field. This uses a piecewise linear function. Rate dependent plasticity can be activated by several ways –
• Specifying TABLEMD ID in TID field. This uses a piecewise linear function.
• Johnson Cook model (only when strain rate is greater than RSTRT).
• Crushable Foam model.
• Cowper Symonds model.
Elements Solid elements only. Shell and solid elements.
Temperature dependent rate dependent plasticity Supported. The temperature-dependence is defined by referencing a TABLEST entry via the TID field. Not supported.
TABLEMD definition A maximum of 4 fields can be used in TABLEMD, which represent –
• Yield stress
• Equivalent plastic strain
• Plastic strain rate
• Temperature

Currently, only experimental data can be used. A stress-strain curve at zero plastic strain rate must be provided.

A maximum of 3 fields can be used in TABLEMD, which represent –
• Yield stress
• Equivalent plastic strain
• Plastic/total/volumetric strain rate
Supported strain rates Only plastic strain rate is supported. Hence, total strain rate input (TYPSTRT=0) is ignored. Plastic/total/volumetric strain rate is supported.
Additional remarks Total strain input (TYPSTRN=0) is not supported and ignored.

Hence, the second column in TABLEMD must be the equivalent plastic strain.

Mixed hardening (HR=1, 2, 3 or real value) can be combined with rate dependent plasticity (TID=TABLEMD).

7. Kinematic hardening and mixed hardening are supported:
Supported Entity Implicit analysis (NLSTAT/NLTRAN for SMDISP/LGDISP) Explicit analysis
Elements Solid elements only. Shell and solid elements.
8. The conversion of the stress versus total strain (TYPSTRN=0) into stress versus plastic strain (TYPSTRN=1) is illustrated in Figure 1. This is clearly different than simply shifting the entire table along the epsilon-axis.
9. Element support restrictions for MATS1:
Element type Restriction
Second order shell elements (CTRIA6 and CQUAD8).
• MATS1 is not supported.
CROD, CONROD, CBAR and CBEAM
• MATS1 is supported in the axial translational direction only.
• The behaviors in other directions remain elastic.
• That is, the torsional deformation of CROD/CONROD elements or the shear, bending and torsional deformations of CBAR/CBEAM elements remain elastic.
10. Information on the Johnson-Cook model:
Entity Details
Analysis type Supported only for Explicit Dynamic analysis.
Strain rate dependency Activated only when strain rate is greater than reference strain rate.
Formulation:
$\sigma =\left(a+b{\epsilon }_{p}^{n}\right)\left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
$\overline{\sigma }=\left(A+B{\left({\overline{\epsilon }}^{pl}\right)}^{n}\right)\left(1+C\mathrm{ln}\left(\frac{{\stackrel{˙}{\overline{\epsilon }}}^{pl}}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
Johnson-Cook strain rate dependence assumes that
$\overline{\sigma }={\sigma }^{0}\left({\overline{\epsilon }}^{pl},\theta \right)R\left({\stackrel{˙}{\overline{\epsilon }}}^{pl}\right)$
and
${\stackrel{˙}{\overline{\epsilon }}}^{pl}={\stackrel{˙}{\epsilon }}_{0}\mathrm{exp}\left(\frac{1}{C}\left(R-1\right)\right)$
for $\overline{\sigma }\ge {\sigma }^{0}$
Where,
• $\overline{\sigma }$ is the yield stress for non-zero strain rate
• ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain rate
• ${\stackrel{˙}{\epsilon }}_{0}$ and $C$ are material parameters measured at or below the transition temperature ${\theta }_{transition}$
• ${\sigma }^{0}\left({\overline{\epsilon }}^{pl},\theta \right)$ is the static yield stress
• $R\left({\stackrel{˙}{\overline{\epsilon }}}^{pl}\right)$ is the ratio of the yield stress at nonzero strain rate to the static yield stress $R\left({\stackrel{˙}{\epsilon }}_{0}\right)=1.0$
11. Information on the Crushable Foam model:
Entity Details
Analysis type Supported only for explicit dynamic analysis.
TSC definition Defined as a positive stress value which indicates the yield stress of crushable foam under tensile loading
Table definition
The yield stress of crushable foam under compression loading can be given by a rate independent table (TABLES1), where the following rules apply.
• x values in the table are the volumetric strain (all positive values indicate that 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 will be x=0, y=y_0 (the initial compressive yield stress).
• All xi should be positive and in increasing order.
Specific output Instead of the equivalent plastic strain, the integrated volumetric strain result (natural logarithm of the relative volume $-In\left(\frac{V}{{V}_{0}}\right)$ is output.
12. Information on the Cowper-Symonds model:
Entity Details
Analysis type Supported only for Explicit Dynamic analysis.
TYPSTRT Either total strain rate or plastic strain rate can be used.
$f\left({\epsilon }^{pl}\right)$ Can be specified in linear hardening form or tabular form using TABLES1 or TABLEMD.
Formulation:

The yield stress is computed as:

${\sigma }_{y}=\beta f\left({\epsilon }^{pl}\right)$
Where,
• $f\left({\epsilon }^{pl}\right)$ is the reference rate plastic strain versus yield stress hardening function
• β is the strain rate effect term, computed as $\beta =1+{\left(\frac{\stackrel{˙}{\epsilon }}{D}\right)}^{\frac{1}{p}}$
13. Information on using combined hardening:
Entity Details
Analysis type Supported only for Implicit analysis (NLSTAT/NLTRAN for SMDISP/LGDISP)
Element type Supported only for solid elements
Rate dependent plasticity with combined hardening Not supported, thus TID field is ignored
Flexibility in combination Both NLKIN and NLISO 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).

At least one type of nonlinear hardening should be defined. For more information refer to the Combined Hardening of von Mises Plasticity.

14. Information on using NLKIN or NLISO:
Entity Details
Analysis type Supported only for Implicit analysis (NLSTAT/NLTRAN for SMDISP/LGDISP)
Temperature dependency When TYPKIN = HALFCYCL or TYPISO = TABLE, multiple curves can be provided one after each other. The last column temperature (TEMP) should be in ascending order.When TYPKIN/TYPISO = PARAM, the parameters can be temperature dependent, with TEMP specified in ascending order.
Parameter fitting (for TABLE/HALFCYCL) The Levenberg-Marquardt method is used.

The parameters are printed in the .out file for each temperature.

Equivalent plastic strain determination For NLKIN, it is usually sourced directly from the first cycle of the experiment. For NLISO, it is usually sourced from cyclic loading experiments.
Additional remarks If both NLKIN and NLISO use PARAM format, then the initial yield stress SIGY0 should be the same for the same temperature.

After the continuation line, user can input the data block with an unlimited number of continuation lines.

15. When TYPKIN=PARAM, if the number of back stresses is equal to or greater than 4, multiple continuation lines can be used to define the complete set of parameters (C1-C10, G1-G10), check the example above.