Combined Hardening of von Mises Plasticity

Combining hardening can be used for analysis with cyclic loading, in order to capture shakedown, ratcheting effect, and so on.

It consists of two nonlinear hardening rules, the nonlinear kinematic (NLKIN) and nonlinear isotropic (NLISO) hardening methods.

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

Combined hardening can be activated by setting HR=6 on the MATS1 Bulk Data.

Isotropic Hardening (NLISO): Nonlinear Yield Function

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

f (S, α, {\bar{ε}}_{p}) = \sqrt{\frac{3}{2} (S - α) : (S - α)} - σ_{y} ({\bar{ε}}_{p})

Where,

$S$: Deviatoric stress tensor.
$α$: Back stress tensor.
$σ_{y}$: Yield stress as a function of equivalent plastic strain ${\bar{ε}}_{p}$ .

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

{\dot{ε}}_{p} = \dot{λ} \frac{\partial f}{\partial σ}

Where, $λ$ is the rate of plastic multiplier, which is also the rate of equivalent plastic strain.

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

N = \frac{\partial f}{\partial σ} = \sqrt{\frac{3}{2}} \frac{(S - α)}{| | S - α | |} = \sqrt{\frac{3}{2}} \frac{η}{| | η | |}

Where, $η$ is the relative stress tensor, which is the difference between deviatoric stress and back stress.

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

σ_{y} ({\bar{ε}}_{p}) = σ_{y 0} + Q (1 - e^{- b {\bar{ε}}_{p}})

Where, $Q$ and $b$ are two parameters which can be directly input via the Q and B fields on the MATS1 data (HR=6, TYPISO=PARAM) or these parameters are computed by parameter fitting algorithms, based on the stress-strain curve from experiment. The isotropic part of the yield stress and the equivalent plastic stress are provided via the SIG and EPS fields on the MATS1 data (HR=6, TYPISO=TABLE).

Nonlinear isotropic hardening is based on the von Mises plasticity criteria and; therefore, is associated with the flow rule.

Kinematic Hardening (NLKIN): Evolution Law of Back Stress

Compared with traditional linear hardening (HR=1 or 2), or the mixed hardening (HR=3), the main difference of NLKIN via HR=6 is the extended evolution law of back stress, which consists of a set of evolution equations for each back stress component:

\dot{α} = \sum_{k = 1}^{m} {\dot{α}}_{k}

and

{\dot{α}}_{k} = \frac{2}{3} C_{k} {\dot{ε}}_{p} - γ_{k} α_{k} {\dot{\bar{ε}}}_{p}

Where,

$k$: Denotes the component number of the back stresses.
$m$: Total number of back stress components.
$C_{k}$ and $γ_{k}$: Corresponding parameter pair of component $k$ .; Two parameters which can be directly input via the Ci and Gi fields on the MATS1 data (HR=6, TYPKIN=PARAM) or these parameters are computed by parameter fitting algorithms, based on the stress-strain curve from experiment, which are defined via the SIG and EPS fields on the MATS1 data (HR=6, TYPKIN=HALFCYCL).

As the evolution of back stress ${\dot{α}}_{k}$ depends both on the flow direction that is parallel to ${\dot{ε}}_{p}$ and the back stress component $α_{k}$ itself; thus, the evolution law of back stress is non-associated. This leads to the unsymmetric elasto-plastic consistent tangent modulus. The unsymmetric solver is always turned on as long as combined hardening is active.

Time Integration Scheme

The plasticity problem is usually solved using the return mapping method, which means it is first assumed to be an elastic trial stage, and then the stress is pulled back onto the yield surface if plastic flow occurs. Backward-Euler algorithm is used during the return mapping process.

Parameter Fitting

If the stress-strain curve is provided (TYPKIN=HALFCYCL or TYPISO=TABLE), the data points provided are used to compute the optimal parameters for combined hardening. For parameter fitting, the Levenberg-Marquardt method is used, which is an extension of Newton method. The fitted parameters are printed in the .out file.

For NLISO with TYPISO=TABLE, the provided data in the continuation line is the yield stress versus equivalent plastic strain. This curve is usually generated based on the cyclic experiment with constant strain. The same curve is used for isotropic hardening (HR=1) for example, with TABLES1/TABLEG or TABLEST input.

For NLKIN with TYPKIN=HALFCYCL, the provided data is the yield stress versus equivalent plastic strain, where the yield stress is the total yield stress that is measured directly from experiment, and the equivalent plastic strain is modified by subtracting the elastic strain. 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 larger than those for NLISO with TYPISO=TABLE.

Figure 1. Parameter fitting of NLKIN or NLISO (demonstration)

Temperature-dependent combined hardening

If temperature-dependent combined hardening is active, then all the parameters are temperature dependent, for example,

$\begin{array}{l} C_{k} (T) \\ γ_{k} (T) \\ σ_{y 0} (T) \\ Q (T) \\ b (T) \end{array}$

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.

Using isotropic hardening NLISO as an example, if the parameters are provided at two temperatures $T_{1}$ and $T_{2}$ , then there are two yield stress functions for interpolation:

σ_{y} ({\bar{ε}}_{p}, T_{1}) = σ_{y 0} + Q (T_{1}) \cdot (1 - e^{- b (T_{1}) \cdot {\bar{ε}}_{p}})

σ_{y} ({\bar{ε}}_{p}, T_{2}) = σ_{y 0} + Q (T_{2}) \cdot (1 - e^{- b (T_{2}) \cdot {\bar{ε}}_{p}})

The interpolated yield stress at the current temperature $T_{c u r r}$ in [ $T_{1}$ , $T_{2}$ ] is then,

σ_{y} ({\bar{ε}}_{p}, T_{c u r r}) = f_{1} \cdot σ_{y} ({\bar{ε}}_{p}, T_{1}) + f_{2} \cdot σ_{y} ({\bar{ε}}_{p}, T_{2})

with two factors for different temperatures $f_{1} = \frac{T_{2} - T_{c u r r}}{T_{2} - T_{1}}$ , $f_{2} = \frac{T_{c u r r} - T_{1}}{T_{2} - T_{1}}$ .

As an example, Figure 2 illustrates the principle of interpolation.

It is worthy to mention that the parameters (Q and B) are not interpolated directly. Instead, the yield functions at two temperatures are interpolated. This is the similar case for NLKIN, where the evolution laws at two temperatures are interpolated.

Figure 2. Principle of interpolation for temperature dependent NLISO

If the temperature, $T$ is beyond the available temperature range [ $T_{1}$ , $T_{2}$ ], then closest temperature will be selected. In other words, no extrapolation of temperature is performed. Furthermore, it is suggested to provide the material parameter of NLKIN and NLISO at the same temperature.

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⁸ Broggiato, Giovanni B., Francesca Campana, and Luca Cortese. "The Chaboche nonlinear kinematic hardening model: calibration methodology and validation." Meccanica 43.2 (2008): 115-124.

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