Classical phenomenological viscoplasticity models for
small strains are usually categorized into two types: • the Perzyna formulation • the Duvaut–Lions formulation
Perzyna formulation In the Perzyna formulation the plastic strain rate is assumed to be given by a
constitutive relation of the form \dot{\varepsilon}_{\mathrm{vp}} = \cfrac{\left\langle f(\boldsymbol{\sigma}, \boldsymbol{q})\right\rangle}{\tau} \cfrac{\partial f}{\partial \boldsymbol{\sigma}} = \begin{cases} \cfrac{f(\boldsymbol{\sigma}, \boldsymbol{q})}{\tau} \cfrac{\partial f}{\partial \boldsymbol{\sigma}}& \text{if}~f(\boldsymbol{\sigma}, \boldsymbol{q}) > 0 \\ 0 & \text{otherwise} \\ \end{cases} where f(.,.) is a
yield function, \boldsymbol{\sigma} is the
Cauchy stress, \boldsymbol{q} is a set of internal variables (such as the
plastic strain \boldsymbol{\varepsilon}_{\mathrm{vp}}), \tau is a relaxation time. The notation \langle \dots \rangle denotes the
Macaulay brackets. The flow rule used in various versions of the
Chaboche model is a special case of Perzyna's flow rule and has the form \dot{\varepsilon}_{\mathrm{vp}} = \left\langle \frac{f}{f_0} \right\rangle^n \sgn (\boldsymbol{\sigma} - \boldsymbol{\chi}) where f_0 is the quasistatic value of f and \boldsymbol{\chi} is a
backstress. Several models for the backstress also go by the name
Chaboche model.
Duvaut–Lions formulation The Duvaut–Lions formulation is equivalent to the Perzyna formulation and may be expressed as \dot{\varepsilon}_{\mathrm{vp}} = \begin{cases} \mathsf{C}^{-1}:\cfrac{\boldsymbol{\sigma} - \mathcal{P}\boldsymbol{\sigma}}{\tau} & \text{if}~f(\boldsymbol{\sigma}, \boldsymbol{q}) > 0 \\ 0 & \text{otherwise} \end{cases} where \mathsf{C} is the elastic stiffness tensor, \mathcal{P}\boldsymbol{\sigma} is the closest point projection of the stress state on to the boundary of the region that bounds all possible elastic stress states. The quantity \mathcal{P}\boldsymbol{\sigma} is typically found from the rate-independent solution to a plasticity problem.
Flow stress models The quantity f(\boldsymbol{\sigma}, \boldsymbol{q}) represents the evolution of the
yield surface. The yield function f is often expressed as an equation consisting of some invariant of stress and a model for the yield stress (or plastic
flow stress). An example is
von Mises or J_2 plasticity. In those situations the plastic strain rate is calculated in the same manner as in rate-independent plasticity. In other situations, the yield stress model provides a direct means of computing the plastic strain rate. Numerous empirical and semi-empirical flow stress models are used the computational plasticity. The following temperature and strain-rate dependent models provide a sampling of the models in current use: • the Johnson–Cook model • the Steinberg–Cochran–Guinan–Lund model. • the Zerilli–Armstrong model. • the Mechanical threshold stress model. • the Preston–Tonks–Wallace model. The Johnson–Cook (JC) model is purely empirical and is the most widely used of the five. However, this model exhibits an unrealistically small strain-rate dependence at high temperatures. The Steinberg–Cochran–Guinan–Lund (SCGL) model is semi-empirical. The model is purely empirical and strain-rate independent at high strain-rates. A dislocation-based extension based on is used at low strain-rates. The SCGL model is used extensively by the shock physics community. The Zerilli–Armstrong (ZA) model is a simple physically based model that has been used extensively. A more complex model that is based on ideas from dislocation dynamics is the Mechanical Threshold Stress (MTS) model. This model has been used to model the plastic deformation of copper,
tantalum, alloys of steel, and aluminum alloys. However, the MTS model is limited to strain-rates less than around 107/s. The Preston–Tonks–Wallace (PTW) model is also physically based and has a form similar to the MTS model. However, the PTW model has components that can model plastic deformation in the overdriven shock regime (strain-rates greater that 107/s). Hence this model is valid for the largest range of strain-rates among the five flow stress models.
Johnson–Cook flow stress model The Johnson–Cook (JC) model T_0 is a reference temperature, and T_m is a reference
melt temperature. For conditions where T^* , we assume that m = 1.
Steinberg–Cochran–Guinan–Lund flow stress model The Steinberg–Cochran–Guinan–Lund (SCGL) model is a semi-empirical model that was developed by Steinberg et al. is based on simplified dislocation mechanics. The general form of the equation for the flow stress is \text{(3)} \qquad \sigma_y(\varepsilon_{\text{p}},\dot{\varepsilon}_\text{p},T) = \sigma_a + B\exp(-\beta T) + B_0\sqrt{\varepsilon_{\text{p}}}\exp(-\alpha T) ~. In this model, \sigma_a is the athermal component of the flow stress given by \sigma_a := \sigma_g + \frac{k_h}{\sqrt{\ell}} + K\varepsilon_{\text{p}}^n, where \sigma_g is the contribution due to solutes and initial dislocation density, k_h is the microstructural stress intensity, \ell is the average grain diameter, K is zero for fcc materials, B, B_0 are material constants. In the thermally activated terms, the functional forms of the exponents \alpha and \beta are \alpha = \alpha_0 - \alpha_1 \ln(\dot{\varepsilon}_{\!\text{p}}); \quad \beta = \beta_0 - \beta_1 \ln(\dot{\varepsilon}_{\!\text{p}}); where \alpha_0, \alpha_1, \beta_0, \beta_1 are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The Zerilli–Armstrong model has been modified by for better performance at high temperatures.
Mechanical threshold stress flow stress model The Mechanical Threshold Stress (MTS) model) has the form \text{(4)} \qquad \sigma_y(\varepsilon_{\text{p}},\dot{\varepsilon},T) = \sigma_a + (S_i \sigma_i + S_e \sigma_e)\frac{\mu(p,T)}{\mu_0} where \sigma_a is the athermal component of mechanical threshold stress, \sigma_i is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, \sigma_e is the component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (S_i, S_e) are temperature and strain-rate dependent scaling factors, and \mu_0 is the shear modulus at 0 K and
ambient pressure. The scaling factors take the
Arrhenius form \begin{align} S_i & = \left[1 - \left(\frac{k_\text{B} T}{g_{0i}b^3\mu(p,T)} \ln\frac{\dot{\varepsilon}_{\!0}}{\dot{\varepsilon}}\right)^{1/q_i} \right]^{1/p_i} \\ S_e & = \left[1 - \left(\frac{k_\text{B} T}{g_{0e}b^3\mu(p,T)} \ln\frac{\dot{\varepsilon}_{\!0}}{\dot{\varepsilon}}\right)^{1/q_e} \right]^{1/p_e} \end{align} where k_b is the Boltzmann constant, b is the magnitude of the Burgers' vector, (g_{0i}, g_{0e}) are normalized activation energies, (\dot{\varepsilon}, \dot{\varepsilon}_0) are the strain-rate and reference strain-rate, and (q_i, p_i, q_e, p_e) are constants. The strain hardening component of the mechanical threshold stress (\sigma_e) is given by an empirical modified
Voce law \text{(5)} \qquad \frac{d\sigma_e}{d\varepsilon_{\text{p}}} = \theta(\sigma_e) where \begin{align} \theta(\sigma_e) & = \theta_0 [ 1 - F(\sigma_e)] + \theta_{IV} F(\sigma_e) \\ \theta_0 & = a_0 + a_1 \ln \dot{\varepsilon}_{\!\text{p}} + a_2 \sqrt{\dot{\varepsilon}_{\!\text{p}}} - a_3 T \\ F(\sigma_e) & = \cfrac{\tanh\left(\alpha \frac{\sigma_e}{\sigma_{es}}\right)} {\tanh(\alpha)}\\ \ln\cfrac{\sigma_{es}}{\sigma_{0es}} & = \frac{k_\text{B} T}{g_{0es} b^3 \mu(p,T)} \ln \cfrac{\dot{\varepsilon}_{\!\text{p}}}{\dot{\varepsilon}_{\!\text{p}}} \end{align} and \theta_0 is the hardening due to dislocation accumulation, \theta_{IV} is the contribution due to stage-IV hardening, (a_0, a_1, a_2, a_3, \alpha) are constants, \sigma_{es} is the stress at zero strain hardening rate, \sigma_{0es} is the saturation threshold stress for deformation at 0 K, g_{0es} is a constant, and \dot{\varepsilon}_\text{p} is the maximum strain-rate. Note that the maximum strain-rate is usually limited to about 10^7/s.
Preston–Tonks–Wallace flow stress model The Preston–Tonks–Wallace (PTW) model attempts to provide a model for the flow stress for extreme strain-rates (up to 1011/s) and temperatures up to melt. A linear Voce hardening law is used in the model. The PTW flow stress is given by \text{(6)} \qquad \sigma_y(\varepsilon_{\text{p}},\dot{\varepsilon}_\text{p},T) = \begin{cases} 2\left[\tau_s + \alpha\ln\left[1 - \varphi \exp\left(-\beta-\cfrac{\theta\varepsilon_{\text{p}}}{\alpha\varphi}\right)\right]\right] \mu(p,T) & \text{thermal regime} \\ 2\tau_s\mu(p,T) & \text{shock regime} \end{cases} with \alpha := \frac{s_0 - \tau_y}{d}; \quad \beta := \frac{\tau_s - \tau_y}{\alpha}; \quad \varphi := \exp(\beta) - 1 where \tau_s is a normalized work-hardening saturation stress, s_0 is the value of \tau_s at 0K, \tau_y is a normalized yield stress, \theta is the hardening constant in the Voce hardening law, and d is a dimensionless material parameter that modifies the Voce hardening law. The saturation stress and the yield stress are given by \begin{align} \tau_s & = \max\left\{s_0 - (s_0 - s_{\infty}) \text{erf}\left[\kappa \hat{T}\ln\cfrac{\gamma\dot{\xi}}{\dot{\varepsilon}_\text{p}}\right], \, s_0\left(\cfrac{\dot{\varepsilon}_\text{p}}{\gamma\dot{\xi}}\right)^{s_1}\right\} \\ \tau_y & = \max\left\{y_0 - (y_0 - y_{\infty}) \text{erf}\left[\kappa \hat{T}\ln\cfrac{\gamma\dot{\xi}}{\dot{\varepsilon}_\text{p}}\right], \, \min\left\{ y_1\left(\cfrac{\dot{\varepsilon}_\text{p}}{\gamma\dot{\xi}}\right)^{y_2}, \, s_0\left(\cfrac{\dot{\varepsilon}_\text{p}}{\gamma\dot{\xi}}\right)^{s_1}\right\}\right\} \end{align} where s_{\infty} is the value of \tau_s close to the melt temperature, (y_0, y_{\infty}) are the values of \tau_y at 0 K and close to melt, respectively, (\kappa, \gamma) are material constants, \hat{T} = T/T_m, (s_1, y_1, y_2) are material parameters for the high strain-rate regime, and \dot{\xi} = \frac{1}{2}\left(\cfrac{4\pi\rho}{3M}\right)^{1/3} \left(\cfrac{\mu(p,T)}{\rho}\right)^{1/2} where \rho is the density, and M is the
atomic mass. == See also ==