## NONLINEAR HARDENING MODELLING IN OPTISTRUCT

NONLINEAR HARDENING MODELLING IN OPTISTRUCT

Strain hardening is the process that makes a material harder and stronger as a result of plastic deformation. In general, plastic deformation moves existing microstructural dislocations in the material and generates additional ones which result in a reduction of ductility and strengthening.

Strain hardening can be modeled in FE analysis through various approaches such as Isotropic hardening, Linear Kinematic hardening, Nonlinear Kinematic Hardening, Nonlinear Kinematic Hardening and Combined Hardening. Nonlinear hardening models can be used in a cyclic loading analysis, in order to capture Bauschinger effect, shakedown, ratcheting, and so on.

In Optistruct Nonlinear Kinematic, Nonlinear Isotropic and Combined Hardening models are available by defining NLKIN, NLISO and NLKIN+NLISO fields in MATS1 material card, with Hardening Rule HR=6, respectively. These material models are supported only for Implicit Analysis (Nonlinear Static and Transient for SMDISP/LGDISP).

NLKIN and NLISO can be set up in two different ways, via a direct parameter input (PARAM) or via tabular stress - plastic strain values input (TABLE). Both of them support either parameter or stress-strain curve input and they can be combined flexibly, for example NLKIN with parameter input (PARAM) and NLISO with stress-strain curve input (TABLE).

Generally, the kinematic part is described by the evolution law of back stress, and the isotropic part is closely related to the von Mises criteria.

Nonlinear Kinematic Hardening model:

Evolution law of back stress:

Kinematic hardening will increase yield in one load direction but will reduce it in the other load direction. In kinematic hardening, the yield surface translates from its original position during plastic deformation without changing shape.

The extended evolution law of back stress, which consists of a set of evolution equations for each back stress component:

And

Where C = kinematic hardening modulus, ɣ = rate at which hardening modulus decreases with plastic straining.

As the evolution of back stress, depends both on the flow direction that is parallel to ,  and the back stress component , itself. Thus, the evolution law of back stress is non-associated. This leads to the unsymmetric elasto-plastic consistent tangent modulus.

Optistruct input:

1) With parameter input (NLKIN, PARAM):

Parameter input which provides the parameters directly. The parameters are SIGY0, Ci, and Gi for kinematic hardening. Where NKIN=Number of back stresses, SIGY0=Initial yield stress, Ci=kinematic hardening modulus, Gi = rate at which hardening modulus decreases with plastic straining, i=1,2.3...

Optistruct deck:
NLKIN, PARAM, NKIN
+SIGY0, Ci, Gi…

The Ck(T), γk(T), σy0(T) parameters, can be temperature dependent via the TEMP column, which should be specified in an ascending order.

Optistruct deck:
NLKIN, PARAM, NKIN
+SIGY0, Ci, Gi, TEMP…

2) With table input (NLKIN, HALFCYCL):

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, stresses and their equivalent plastic strains are sourced directly from the first cycle (half cycle) of the experiment.

Optistruct deck:
NLKIN, HALFCYCL, NKIN
+SIG, EPS
+…

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.

Optistruct deck:
NLKIN, HALFCYCL, NKIN
+SIG, EPS, TEMP                                                                                                                                                                                                                                  +…

Nonlinear Isotropic Hardening model:

Von Mises yield:

Isotropic hardening will increase yield strength of your material both for tension and compression. With isotropic hardening, the yield surface increases in size as a result of plastic hardening but remains the same shape and position.

The yield function of von Mises plasticity can be expressed in a general form as:

The flow rule is defined as change of plastic strain, expressed in rate form as:

The flow direction, N, can be introduced which is the derivative of the yield function with respect to the stress tensor,

For the nonlinear isotropic hardening, the yield stress is assumed to be a power law function of equivalent plastic strain:

Where Q = maximum change in the size of the yield surface, b = rate at which the size of the yield surface change with plastic strain.

Optistruct input:

1) With parameter input (NLISO, PARAM):

Parameter input which provides the parameters directly. The parameters are SIGY0, Q, and B for isotropic hardening.

Optistruct deck:
NLISO, PARAM,
+SIGY0, Q, B                                                                                                                                                                                                                                        +…

The Q(T), b(T) parameters, can be temperature dependent via the TEMP column, which should be specified in an ascending order.

Optistruct deck:
NLISO, PARAM,
+SIGY0, Q, B, TEMP                                                                                                                                                                                                                              +…

Where SIGY0=Initial yield stress, Q = maximum change in the size of the yield surface, B = rate at which the size of the yield surface change with plastic strain.

The parameter B determines the yield stress increase rate, depending on the number of cycles up to the saturation. A small B value denotes that the material stabilizes relatively slow, while a high value responds to a rapid stabilization.

2) With table input (NLISO TABLE):

Table input providing isotropic part of yield stress versus the equivalent plastic strain. 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, yield stresses and their equivalent plastic strains are sourced from test's stabilized cyclic data.

Optistruct deck:
NLISO, TABLE
+SIG, EPS                                                                                                                                                                                                                                              +…

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.

Optistruct deck:
NLISO, TABLE
+SIG, EPS, TEMP                                                                                                                                                                                                                                 +…

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.

REMARKS

1) If both NLKIN and NLISO use PARAM format, then the initial yield stress SIGY0 should be the same for the same temperature.

2)For computing the parameters Ci and Gi, the isotropic part will be first subtracted from the curve, and the difference is the hardening part due to kinematic hardening. The subtracted data will be used for parameter fitting. Thus, the provided yield stress values for NLKIN with TYPKIN=HALFCYCL, should always be larger than those for NLISO with TYPISO=TABLE.

3)If temperature-dependent combined hardening is active, then all the parameters are temperature dependent, for example: Ck(T), γk(T), σy0(T), Q(T), b(T). From experiment is only possible to test a limited number of temperatures. Interpolation is used to solving for plasticity at a temperature in between the test temperatures.

EXAMPLES:

Example with Table Input:

(Find them attached to the article: "SPCECIMEN_combined_table.fem", "Table_combined_material.fem")

In this example we compare the numerical results of a specimen subjected to a stress control cyclic loading experiment with experimental data. Optistruct given the appropriate experimental values for kinematic and isotropic hardening will predict the Chaboche hardening parameters implementing a parameter fitting algorithm.

Experimental stress-strain curve

Optistruct deck MAT1, MATS1

! Both NLKIN and NLISO have to be set up manually in the .fem file under the MAT1, MATS1 card manually with a BULK_UNSUPPORTED_CARDS. These options will be available in a future release of Hyperworks.

In order to define the kinematic hardening, from the stress-strain curve we extract the first half cycle (stress- equivalent plastic strain).

In order to define the isotropic hardening, from the cyclic stress-strain curve we extract the yield stress vs equivalent plastic strain.

In the .out file we can review the Chaboche hardening parameters computed from the parameter fitting algorithm.

The results obtained in OptiStruct, using a combined hardening material model, are similar to the experimental result.

Experimental test stress-strain curve VS Numerical stress-strain curve

Example with PARAM Input:

(Find them attached to the article: "SPCECIMEN_combined_param.fem", "Param_combined_material.fem")

If we have the Chaboche hardening parameters already available, we can define the combined hardening with the parameter input.

If we take the above (computed from the solver) Chaboche hardening parameters and include them with PARAM like:

The results obtained with PARAM are identical with the results obtained with TABLE.

Numerical stress-strain curve obtained with TABLE VS Numerical stress-strain curve obtained with PARAM