KINEMATIC

Description

This object defines the model of kinematic hardening (tensorial back stresses) applicable in behaviors or potential objects in the gen_evp behavior. The models generally store an additional tensorial internal variable, and therefore may support static recovery and state coupling without modification. The name of this internal variable will be determined normally by the object to which it belongs (a POTENTIAL for example). Kinematic hardening acts as a translation of the yield surface in stress space (often deviatoric).

Syntax

The syntax consists of specifying the type of kinematic object, supplying an optional name on the same line, and giving the appropriate coefficient definitions on the following lines. This is summarized below where the keyword specifying a KINEMATIC object is assumed to be *kinematic:

*kinematic type [ name ] \(~\,~\,\) coefficients

where the possible substitutions for type are:

CODE	CRITERION
linear	linear hardening
nonlinear	nonlinear Armstrong-Fredrick hardening with static recovery
nonlinear_ai	same as the nonlinear model but using more accurate asymptotic integration
ziegler	nonlinear hardening based on the Euclidean norm \(I_2\)
nonlinear_phi	nonlinear with cumulative influence
nonlinear_evrad	nonlinear with reduced ratcheting
nonlinear_with_crit	linear-nonlinear with criterion
aniso_nonlinear	nonlinear model using coefficient matrices for the coefficients

Kinematic models are all formulated in state variable form, where the stored (integrated) variable is analogous to a strain translation in tensorial strain space. The “back stress” component is then calculated using an appropriate modulus. Most of the kinematic models use the convention for calculating the back stress:

\[\ten X = \frac{2}{3}{\tt C}\ten \alpha\]

with \(\ten X\) the back stress and \(\ten \alpha\) the internal variable. Material softening through kinematic variables is not thought to be physically reasonable, and so the modulus coefficients C should always be positive.

Evolution of the internal variable is written in the form:

\[\dot{\ten \alpha}_i = \dot{\lambda}\ten m_{kin}(\ten \sigma, \ten X, p, \dots) - \dot{\ten \omega}_{kin}(\ten X)\]

where \(\ten m\) is the “hardening normal” and \(\dot{\ten \omega}_{kine}\) a static recovery function.

linear

The linear kinematic hardening has its internal variable evolve with the inelastic strain:

\[{\ten m}_{kin} = {\ten n}\]

The only coefficient for this model is the C modulus (units of stress). The slope in a uniaxial test will be equal to the C value.

nonlinear

The nonlinear model evolves much as the isotropic nonlinear model does (with a slight change in coefficient meaning). The evolution is the following:

\[\begin{split}\begin{aligned} &{\ten m_{kin}} = {\ten n} - \frac{3D}{2C}{\ten X} \\ &\ten \omega_{kin} = \frac{3}{2}\frac{\ten X}{J_2(\ten X)} \left<\frac{J_2(\ten X)}{M}\right>^m \end{aligned}\end{split}\]

The coefficients are C and D for the strain evolution, and M and m for the static recovery term. In the absence of these latter two, there will be no static recovery calculation. The saturation of this model occurs at \({\tt C}/{\tt D}\) in a uniaxial test at a rate determined by D. Increases in D yield faster saturating (more nonlinear) behavior.

Static recovery is seen to follow a Norton type formulation in the back stress tensorial space. This kinematic model supports RK, TM, or Reduced TM integrations.

nonlinear_phi

This model installs a kinematic variable with the following evolution:

\[{\ten m_{kin}} = \ten n - \phi(\lambda)\frac{3D}{2C}{\ten X}\]

where the function \(\phi\) is:

\[\phi(\lambda) = \phi_m + (1-\phi_m)\exp(-\omega \lambda)\]

where the material coefficients \(\phi_m\) and \(\omega\) are named phim and omega. Note that \(\omega\) defines the rapidity of saturation in the non-linear coefficient, and \(\phi_m\) is a factor between zero and one interpolating the D coefficient between initial and saturated values. This kinematic model works in RK, TM, or Reduced TM.

nonlinear_evrad

This model limits the ratcheting effect in biaxial conditions by comparing the flow direction with the back stress. Evolution is calculated as follows:

\[{\ten m_{kin}} = \ten n - \left(\eta\ten 1 + \dfrac{2}{3}(1-\eta)\ten n\otimes\ten n\right)\frac{3D}{2C}\ten X\]

The coefficients are defined such that uniaxial response is equivalent to the nonlinear model given equivalent coefficients. The coefficient names are C, D, and eta. This model works in RK, TM, or Reduced TM, but has no static recovery.

ziegler

This model describes a back stress with hydrostatic and deviatoric components:

\[\begin{split}\begin{aligned} &\ten m_{kin} = \frac{\ten \sigma - {\ten X}}{I_2(\ten \sigma - {\ten X})} - \frac{D}{C}{\ten X} \\ &\ten \omega_{kin} = \frac{\ten X}{I_2(\ten X)} \left<\frac{I_2(\ten X)}{M}\right>^m \end{aligned}\end{split}\]

which takes the coefficients C, D, M and m. In the absence of the latter two, there will be no static recovery. The Ziegler model may not be used with the reduced_plastic integration.

nonlinear_with_crit

\[\ten m_{kin} = \ten n - \frac{3D}{2C}\phi(\ten X)\ten X\]

\[\phi(\ten X) = \left<\frac{DJ(\ten X) - \omega}{1-\omega}\right>^{m_1} \frac{1}{DJ(\ten X)^{m_2}}\]

Example

*kinematic nonlinear_evrad
   C     40000.
   D     500.
   eta   0.4