Viscoplasticity

Figure 2. Stress–strain response of a viscoplastic material at different strain rates. The dotted lines show the response if the strain-rate is held constant. The blue line shows the response when the strain rate is changed suddenly.

One consequence of yielding is that as plastic deformation proceeds, an increase in stress is required to produce additional strain. This phenomenon is known as Strain/Work hardening.[18] For a viscoplastic material the hardening curves are not significantly different from those of rate-independent plastic material. Nevertheless, three essential differences can be observed.

1. At the same strain, the higher the rate of strain the higher the stress
2. A change in the rate of strain during the test results in an immediate change in the stress–strain curve.
3. The concept of a plastic yield limit is no longer strictly applicable.

The hypothesis of partitioning the strains by decoupling the elastic and plastic parts is still applicable where the strains are small,[3] i.e.,

${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }+{\boldsymbol {\varepsilon }}_{\mathrm {vp} }}$

where ${\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {e} }}$ is the elastic strain and ${\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}$ is the viscoplastic strain. To obtain the stress–strain behavior shown in blue in the figure, the material is initially loaded at a strain rate of 0.1/s. The strain rate is then instantaneously raised to 100/s and held constant at that value for some time. At the end of that time period the strain rate is dropped instantaneously back to 0.1/s and the cycle is continued for increasing values of strain. There is clearly a lag between the strain-rate change and the stress response. This lag is modeled quite accurately by overstress models (such as the Perzyna model) but not by models of rate-independent plasticity that have a rate-dependent yield stress.

Figure 3a. Creep test

Figure 3b. Strain as a function of time in a creep test.

Creep is the tendency of a solid material to slowly move or deform permanently under constant stresses. Creep tests measure the strain response due to a constant stress as shown in Figure 3. The classical creep curve represents the evolution of strain as a function of time in a material subjected to uniaxial stress at a constant temperature. The creep test, for instance, is performed by applying a constant force/stress and analyzing the strain response of the system. In general, as shown in Figure 3b this curve usually shows three phases or periods of behavior[9]

1. A primary creep stage, also known as transient creep, is the starting stage during which hardening of the material leads to a decrease in the rate of flow which is initially very high. ${\displaystyle (0\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{1})}$.
2. The secondary creep stage, also known as the steady state, is where the strain rate is constant. ${\displaystyle ({\boldsymbol {\varepsilon }}_{1}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{2})}$.
3. A tertiary creep phase in which there is an increase in the strain rate up to the fracture strain. ${\displaystyle ({\boldsymbol {\varepsilon }}_{2}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{R})}$.

Figure 4. a) Applied strain in a relaxation test and b) induced stress as functions of time over a short period for a viscoplastic material.

As shown in Figure 4, the relaxation test[19] is defined as the stress response due to a constant strain for a period of time. In viscoplastic materials, relaxation tests demonstrate the stress relaxation in uniaxial loading at a constant strain. In fact, these tests characterize the viscosity and can be used to determine the relation which exists between the stress and the rate of viscoplastic strain. The decomposition of strain rate is

${\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}={\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}+{\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}~.}$

The elastic part of the strain rate is given by

${\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}={\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}$

For the flat region of the strain-time curve, the total strain rate is zero. Hence we have,

${\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}=-{\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}$

Therefore, the relaxation curve can be used to determine rate of viscoplastic strain and hence the viscosity of the dashpot in a one-dimensional viscoplastic material model. The residual value that is reached when the stress has plateaued at the end of a relaxation test corresponds to the upper limit of elasticity. For some materials such as rock salt such an upper limit of elasticity occurs at a very small value of stress and relaxation tests can be continued for more than a year without any observable plateau in the stress.

It is important to note that relaxation tests are extremely difficult to perform because maintaining the condition ${\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}=0}$ in a test requires considerable delicacy.[20]

Figure 5. Norton-Hoff model for perfectly viscoplastic solid

In a perfectly viscoplastic solid, also called the Norton-Hoff model of viscoplasticity, the stress (as for viscous fluids) is a function of the rate of permanent strain. The effect of elasticity is neglected in the model, i.e., ${\displaystyle {\boldsymbol {\varepsilon }}_{e}=0}$ and hence there is no initial yield stress, i.e., ${\displaystyle \sigma _{y}=0}$. The viscous dashpot has a response given by

${\displaystyle {\boldsymbol {\sigma }}=\eta ~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }\implies {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\eta }}}$

where ${\displaystyle \eta }$ is the viscosity of the dashpot. In the Norton-Hoff model the viscosity ${\displaystyle \eta }$ is a nonlinear function of the applied stress and is given by

${\displaystyle \eta =\lambda \left[{\cfrac {\lambda }{||{\boldsymbol {\sigma }}||}}\right]^{N-1}}$

where ${\displaystyle N}$ is a fitting parameter, λ is the kinematic viscosity of the material and ${\displaystyle ||{\boldsymbol {\sigma }}||={\sqrt {{\boldsymbol {\sigma }}:{\boldsymbol {\sigma }}}}={\sqrt {\sigma _{ij}\sigma _{ij}}}}$. Then the viscoplastic strain rate is given by the relation

${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {||{\boldsymbol {\sigma }}||}{\lambda }}\right]^{N-1}}$

In one-dimensional form, the Norton-Hoff model can be expressed as

${\displaystyle \sigma =\lambda ~\left({\dot {\varepsilon }}_{\mathrm {vp} }\right)^{1/N}}$

When ${\displaystyle N=1.0}$ the solid is viscoelastic.

If we assume that plastic flow is isochoric (volume preserving), then the above relation can be expressed in the more familiar form[21]

${\displaystyle {\boldsymbol {s}}=2K~\left({\sqrt {3}}{\dot {\varepsilon }}_{\mathrm {eq} }\right)^{m-1}~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }}$

where ${\displaystyle {\boldsymbol {s}}}$ is the deviatoric stress tensor, ${\displaystyle {\dot {\varepsilon }}_{\mathrm {eq} }}$ is the von Mises equivalent strain rate, and ${\displaystyle K,m}$ are material parameters. The equivalent strain rate is defined as

${\displaystyle {\dot {\bar {\epsilon }}}={\sqrt {{\frac {2}{3}}{\dot {\bar {\bar {\epsilon }}}}:{\dot {\bar {\bar {\epsilon }}}}}}}$

These models can be applied in metals and alloys at temperatures higher than two thirds[21] of their absolute melting point (in kelvins) and polymers/asphalt at elevated temperature. The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 6.

Figure 6: The response of perfectly viscoplastic solid to hardening, creep and relaxation tests.

Figure 7. The elastic perfectly viscoplastic material.

Two types of elementary approaches can be used to build up an elastic-perfectly viscoplastic mode. In the first situation, the sliding friction element and the dashpot are arranged in parallel and then connected in series to the elastic spring as shown in Figure 7. This model is called the Bingham–Maxwell model (by analogy with the Maxwell model and the Bingham model) or the Bingham–Norton model.[22] In the second situation, all three elements are arranged in parallel. Such a model is called a Bingham–Kelvin model by analogy with the Kelvin model.

For elastic-perfectly viscoplastic materials, the elastic strain is no longer considered negligible but the rate of plastic strain is only a function of the initial yield stress and there is no influence of hardening. The sliding element represents a constant yielding stress when the elastic limit is exceeded irrespective of the strain. The model can be expressed as

${\displaystyle {\begin{aligned}&{\boldsymbol {\sigma }}={\mathsf {E}}~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\eta }}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}\end{aligned}}}$

where ${\displaystyle \eta }$ is the viscosity of the dashpot element. If the dashpot element has a response that is of the Norton form

${\displaystyle {\cfrac {\boldsymbol {\sigma }}{\eta }}={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}}$

we get the Bingham–Norton model

${\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}$

Other expressions for the strain rate can also be observed in the literature[22] with the general form

${\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y})~{\boldsymbol {\sigma }}\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}$

The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 8.

Figure 8. The response of elastic perfectly viscoplastic solid to hardening, creep and relaxation tests.

An elastic-viscoplastic material with strain hardening is described by equations similar to those for an elastic-viscoplastic material with perfect plasticity. However, in this case the stress depends both on the plastic strain rate and on the plastic strain itself. For an elastoviscoplastic material the stress, after exceeding the yield stress, continues to increase beyond the initial yielding point. This implies that the yield stress in the sliding element increases with strain and the model may be expressed in generic terms as

${\displaystyle {\begin{aligned}&{\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }={\mathsf {E}}^{-1}~{\boldsymbol {\sigma }}=~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y},{\boldsymbol {\varepsilon }}_{\mathrm {vp} })~{\boldsymbol {\sigma }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||\geq \sigma _{y}\end{aligned}}}$.

This model is adopted when metals and alloys are at medium and higher temperatures and wood under high loads. The responses for strain hardening, creep, and relaxation tests of such a material are shown in Figure 9.

Figure 9. The response of elastoviscoplastic hardening solid to hardening, creep and relaxation tests.

In the Perzyna formulation the plastic strain rate is assumed to be given by a constitutive relation of the form

${\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\cfrac {\left\langle f({\boldsymbol {\sigma }},{\boldsymbol {q}})\right\rangle }{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}={\begin{cases}{\cfrac {f({\boldsymbol {\sigma }},{\boldsymbol {q}})}{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}&{\rm {if}}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0\\0&{\rm {otherwise}}\\\end{cases}}}$

where ${\displaystyle f(.,.)}$ is a yield function, ${\displaystyle {\boldsymbol {\sigma }}}$ is the Cauchy stress, ${\displaystyle {\boldsymbol {q}}}$ is a set of internal variables (such as the plastic strain ${\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}$), ${\displaystyle \tau }$ is a relaxation time. The notation ${\displaystyle \langle \dots \rangle }$ denotes the Macaulay brackets. The flow rule used in various versions of the Chaboche model is a special case of Perzyna's flow rule[23] and has the form

${\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }=\left\langle {\frac {f}{f_{0}}}\right\rangle ^{n}({\boldsymbol {\sigma }}-{\boldsymbol {\chi }})}$

where ${\displaystyle f_{0}}$ is the quasistatic value of ${\displaystyle f}$ and ${\displaystyle {\boldsymbol {\chi }}}$ is a backstress. Several models for the backstress also go by the name Chaboche model.

The Duvaut–Lions formulation is equivalent to the Perzyna formulation and may be expressed as

${\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\begin{cases}{\mathsf {C}}^{-1}:{\cfrac {{\boldsymbol {\sigma }}-{\mathcal {P}}{\boldsymbol {\sigma }}}{\tau }}&{\rm {{if}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0}}\\0&{\rm {otherwise}}\end{cases}}}$

where ${\displaystyle {\mathsf {C}}}$ is the elastic stiffness tensor, ${\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}$ 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 ${\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}$ is typically found from the rate-independent solution to a plasticity problem.

The quantity ${\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {q}})}$ represents the evolution of the yield surface. The yield function ${\displaystyle f}$ 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 ${\displaystyle J_{2}}$ 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:

1. the Johnson–Cook model
2. the Steinberg–Cochran–Guinan–Lund model.
3. the Zerilli–Armstrong model.
4. the Mechanical threshold stress model.
5. the Preston–Tonks–Wallace model.

The Johnson–Cook (JC) model [24] 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 [25][26] is semi-empirical. The model is purely empirical and strain-rate independent at high strain-rates. A dislocation-based extension based on [27] is used at low strain-rates. The SCGL model is used extensively by the shock physics community. The Zerilli–Armstrong (ZA) model [28] 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.[29] This model has been used to model the plastic deformation of copper, tantalum,[30] alloys of steel,[31][32] and aluminum alloys.[33] However, the MTS model is limited to strain-rates less than around 10⁷/s. The Preston–Tonks–Wallace (PTW) model [34] 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.

The Johnson–Cook (JC) model [24] is purely empirical and gives the following relation for the flow stress (${\displaystyle \sigma _{y}}$)

${\displaystyle {\text{(1)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[A+B(\varepsilon _{\rm {p}})^{n}\right]\left[1+C\ln({\dot {\varepsilon _{\rm {p}}}}^{*})\right]\left[1-(T^{*})^{m}\right]}$

where ${\displaystyle \varepsilon _{\rm {p}}}$ is the equivalent plastic strain, ${\displaystyle {\dot {\varepsilon _{\rm {p}}}}}$ is the plastic strain-rate, and ${\displaystyle A,B,C,n,m}$ are material constants.

The normalized strain-rate and temperature in equation (1) are defined as

${\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}:={\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p0}}}}}\qquad {\text{and}}\qquad T^{*}:={\cfrac {(T-T_{0})}{(T_{m}-T_{0})}}}$

where ${\displaystyle {\dot {\varepsilon _{\rm {p0}}}}}$ is the effective plastic strain-rate of the quasi-static test used to determine the yield and hardening parameters A,B and n. This is not as it is often thought just a parameter to make ${\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}}$ non-dimensional.[35] ${\displaystyle T_{0}}$ is a reference temperature, and ${\displaystyle T_{m}}$ is a reference melt temperature. For conditions where ${\displaystyle T^{*}<0}$, we assume that ${\displaystyle m=1}$.

The Steinberg–Cochran–Guinan–Lund (SCGL) model is a semi-empirical model that was developed by Steinberg et al.[25] for high strain-rate situations and extended to low strain-rates and bcc materials by Steinberg and Lund.[26] The flow stress in this model is given by

${\displaystyle {\text{(2)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[\sigma _{a}f(\varepsilon _{\rm {p}})+\sigma _{t}({\dot {\varepsilon _{\rm {p}}}},T)\right]{\frac {\mu (p,T)}{\mu _{0}}};\quad \sigma _{a}f\leq \sigma _{\text{max}}~~{\text{and}}~~\sigma _{t}\leq \sigma _{p}}$

where ${\displaystyle \sigma _{a}}$ is the athermal component of the flow stress, ${\displaystyle f(\varepsilon _{\rm {p}})}$ is a function that represents strain hardening, ${\displaystyle \sigma _{t}}$ is the thermally activated component of the flow stress, ${\displaystyle \mu (p,T)}$ is the pressure- and temperature-dependent shear modulus, and ${\displaystyle \mu _{0}}$ is the shear modulus at standard temperature and pressure. The saturation value of the athermal stress is ${\displaystyle \sigma _{\text{max}}}$. The saturation of the thermally activated stress is the Peierls stress (${\displaystyle \sigma _{p}}$). The shear modulus for this model is usually computed with the Steinberg–Cochran–Guinan shear modulus model.

The strain hardening function (${\displaystyle f}$) has the form

${\displaystyle f(\varepsilon _{\rm {p}})=[1+\beta (\varepsilon _{\rm {p}}+\varepsilon _{\rm {p}}i)]^{n}}$

where ${\displaystyle \beta ,n}$ are work hardening parameters, and ${\displaystyle \varepsilon _{\rm {p}}i}$ is the initial equivalent plastic strain.

The thermal component (${\displaystyle \sigma _{t}}$) is computed using a bisection algorithm from the following equation.[26][27]

${\displaystyle {\dot {\varepsilon _{\rm {p}}}}=\left[{\frac {1}{C_{1}}}\exp \left[{\frac {2U_{k}}{k_{b}~T}}\left(1-{\frac {\sigma _{t}}{\sigma _{p}}}\right)^{2}\right]+{\frac {C_{2}}{\sigma _{t}}}\right]^{-1};\quad \sigma _{t}\leq \sigma _{p}}$

where ${\displaystyle 2U_{k}}$ is the energy to form a kink-pair in a dislocation segment of length ${\displaystyle L_{d}}$, ${\displaystyle k_{b}}$ is the Boltzmann constant, ${\displaystyle \sigma _{p}}$ is the Peierls stress. The constants ${\displaystyle C_{1},C_{2}}$ are given by the relations

${\displaystyle C_{1}:={\frac {\rho _{d}L_{d}ab^{2}\nu }{2w^{2}}};\quad C_{2}:={\frac {D}{\rho _{d}b^{2}}}}$

where ${\displaystyle \rho _{d}}$ is the dislocation density, ${\displaystyle L_{d}}$ is the length of a dislocation segment, ${\displaystyle a}$ is the distance between Peierls valleys, ${\displaystyle b}$ is the magnitude of the Burgers vector, ${\displaystyle \nu }$ is the Debye frequency, ${\displaystyle w}$ is the width of a kink loop, and ${\displaystyle D}$ is the drag coefficient.

The Zerilli–Armstrong (ZA) model [28][36][37] is based on simplified dislocation mechanics. The general form of the equation for the flow stress is

${\displaystyle {\text{(3)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\sigma _{a}+B\exp(-\beta T)+B_{0}{\sqrt {\varepsilon _{\rm {p}}}}\exp(-\alpha T)~.}$

In this model, ${\displaystyle \sigma _{a}}$ is the athermal component of the flow stress given by

${\displaystyle \sigma _{a}:=\sigma _{g}+{\frac {k_{h}}{\sqrt {l}}}+K\varepsilon _{\rm {p}}^{n},}$

where ${\displaystyle \sigma _{g}}$ is the contribution due to solutes and initial dislocation density, ${\displaystyle k_{h}}$ is the microstructural stress intensity, ${\displaystyle l}$ is the average grain diameter, ${\displaystyle K}$ is zero for fcc materials, ${\displaystyle B,B_{0}}$ are material constants.

In the thermally activated terms, the functional forms of the exponents ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are

${\displaystyle \alpha =\alpha _{0}-\alpha _{1}\ln({\dot {\varepsilon _{\rm {p}}}});\quad \beta =\beta _{0}-\beta _{1}\ln({\dot {\varepsilon _{\rm {p}}}});}$

where ${\displaystyle \alpha _{0},\alpha _{1},\beta _{0},\beta _{1}}$ are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The Zerilli–Armstrong model has been modified by [38] for better performance at high temperatures.

The Mechanical Threshold Stress (MTS) model [29][39][40]) has the form

${\displaystyle {\text{(4)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon }},T)=\sigma _{a}+(S_{i}\sigma _{i}+S_{e}\sigma _{e}){\frac {\mu (p,T)}{\mu _{0}}}}$

where ${\displaystyle \sigma _{a}}$ is the athermal component of mechanical threshold stress, ${\displaystyle \sigma _{i}}$ is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, ${\displaystyle \sigma _{e}}$ is the component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (${\displaystyle S_{i},S_{e}}$) are temperature and strain-rate dependent scaling factors, and ${\displaystyle \mu _{0}}$ is the shear modulus at 0 K and ambient pressure.

The scaling factors take the Arrhenius form

${\displaystyle {\begin{aligned}S_{i}&=\left[1-\left({\frac {k_{b}~T}{g_{0i}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{i}}\right]^{1/p_{i}}\\S_{e}&=\left[1-\left({\frac {k_{b}~T}{g_{0e}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{e}}\right]^{1/p_{e}}\end{aligned}}}$

where ${\displaystyle k_{b}}$ is the Boltzmann constant, ${\displaystyle b}$ is the magnitude of the Burgers' vector, (${\displaystyle g_{0i},g_{0e}}$) are normalized activation energies, (${\displaystyle {\dot {\varepsilon }},{\dot {\varepsilon _{\rm {0}}}}}$) are the strain-rate and reference strain-rate, and (${\displaystyle q_{i},p_{i},q_{e},p_{e}}$) are constants.

The strain hardening component of the mechanical threshold stress (${\displaystyle \sigma _{e}}$) is given by an empirical modified Voce law

${\displaystyle {\text{(5)}}\qquad {\frac {d\sigma _{e}}{d\varepsilon _{\rm {p}}}}=\theta (\sigma _{e})}$

where

${\displaystyle {\begin{aligned}\theta (\sigma _{e})&=\theta _{0}[1-F(\sigma _{e})]+\theta _{IV}F(\sigma _{e})\\\theta _{0}&=a_{0}+a_{1}\ln {\dot {\varepsilon _{\rm {p}}}}+a_{2}{\sqrt {\dot {\varepsilon _{\rm {p}}}}}-a_{3}T\\F(\sigma _{e})&={\cfrac {\tanh \left(\alpha {\cfrac {\sigma _{e}}{\sigma _{es}}}\right)}{\tanh(\alpha )}}\\\ln({\cfrac {\sigma _{es}}{\sigma _{0es}}})&=\left({\frac {kT}{g_{0es}b^{3}\mu (p,T)}}\right)\ln \left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p}}}}}\right)\end{aligned}}}$

and ${\displaystyle \theta _{0}}$ is the hardening due to dislocation accumulation, ${\displaystyle \theta _{IV}}$ is the contribution due to stage-IV hardening, (${\displaystyle a_{0},a_{1},a_{2},a_{3},\alpha }$) are constants, ${\displaystyle \sigma _{es}}$ is the stress at zero strain hardening rate, ${\displaystyle \sigma _{0es}}$ is the saturation threshold stress for deformation at 0 K, ${\displaystyle g_{0es}}$ is a constant, and ${\displaystyle {\dot {\varepsilon _{\rm {p}}}}}$ is the maximum strain-rate. Note that the maximum strain-rate is usually limited to about ${\displaystyle 10^{7}}$/s.

The Preston–Tonks–Wallace (PTW) model [34] 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

${\displaystyle {\text{(6)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)={\begin{cases}2\left[\tau _{s}+\alpha \ln \left[1-\varphi \exp \left(-\beta -{\cfrac {\theta \varepsilon _{\rm {p}}}{\alpha \varphi }}\right)\right]\right]\mu (p,T)&{\text{thermal regime}}\\2\tau _{s}\mu (p,T)&{\text{shock regime}}\end{cases}}}$

with

${\displaystyle \alpha$ :={\frac {s_{0}-\tau _{y}}{d}};\quad \beta :={\frac {\tau _{s}-\tau _{y}}{\alpha }};\quad \varphi :=\exp(\beta )-1}

where ${\displaystyle \tau _{s}}$ is a normalized work-hardening saturation stress, ${\displaystyle s_{0}}$ is the value of ${\displaystyle \tau _{s}}$ at 0K, ${\displaystyle \tau _{y}}$ is a normalized yield stress, ${\displaystyle \theta }$ is the hardening constant in the Voce hardening law, and ${\displaystyle d}$ is a dimensionless material parameter that modifies the Voce hardening law.

The saturation stress and the yield stress are given by

${\displaystyle {\begin{aligned}\tau _{s}&=\max \left\{s_{0}-(s_{0}-s_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}}}\right\}\\\tau _{y}&=\max \left\{y_{0}-(y_{0}-y_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],\min \left\{y_{1}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{y_{2}},s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}\right\}}}\right\}\end{aligned}}}$

where ${\displaystyle s_{\infty }}$ is the value of ${\displaystyle \tau _{s}}$ close to the melt temperature, (${\displaystyle y_{0},y_{\infty }}$) are the values of ${\displaystyle \tau _{y}}$ at 0 K and close to melt, respectively, ${\displaystyle (\kappa ,\gamma )}$ are material constants, ${\displaystyle {\hat {T}}=T/T_{m}}$, (${\displaystyle s_{1},y_{1},y_{2}}$) are material parameters for the high strain-rate regime, and

${\displaystyle {\dot {\xi }}={\frac {1}{2}}\left({\cfrac {4\pi \rho }{3M}}\right)^{1/3}\left({\cfrac {\mu (p,T)}{\rho }}\right)^{1/2}}$

where ${\displaystyle \rho }$ is the density, and ${\displaystyle M}$ is the atomic mass.