Viscoplastic Constitutive Equations

Abstract

Plastic deformations of solids depend on the rate of loading or deformation, exhibiting the time-dependence or rate-dependence in general, which is called the viscoplastic deformation. Constitutive equations for the viscoplastic deformation is described in this chapter.

Plastic deformations of solids depend on the rate of loading or deformation, exhibiting the time-dependence or rate-dependence in general, which is called the viscoplastic deformation. Constitutive equations for the viscoplastic deformation is described in this chapter. The physical background of rate-dependent plastic deformation, i.e. viscoplastic deformation and the history of the viscoplastic constitutive equation are reviewed first. Then, the pertinent formulation of the viscoplastic constitutive equation is formulated based on the subloading surface model, which is capable of describing the smooth elastic-viscoplastic transition and the rate-dependence for the general rate ranging from the quasi-static to the impact loadings.

13.1 Rate-Dependent Deformation of Solids

The elastic and the plastic deformations of solids are induced by the deformation of solid particles themselves (crystals in metals, soil particles in soils, etc.) and the mutual slips between them, respectively. Therefore,

• (1) Elastic tangent modulus is high.

• (2) High stress has to apply to solids in order that plastic deformation is induced, which overcomes the friction between solid particles. The stress inducing the plastic deformation is macroscopically the so-called yield stress.

• (3) Tangent modulus in the elastoplastic deformation process lowers from that in the elastic deformation process.

The rate-dependent elastic deformation induced in the stress lower than the yield stress is called the viscoelastic deformation. On the other hand, the rate-dependent plastic deformation induced over the yield stress is called the viscoplastic deformation. Then, they are classified as

which is illustrated in Fig. 13.1.

Fig. 13.1

Rate-dependent deformation of solids

13.2 History of Viscoplastic Constitutive Equations

The most pertinent viscoplastic model would be the overstress model. The development of this model is reviewed in this section, while the overview of the history is portrayed in Fig. 13.2.

Fig. 13.2

History of viscoplastic model

The elastic constitutive equation extended so as to describe the rate-dependence is called the viscoelastic constitutive equation and one of the typical models is the Maxwell model, in which the spring and the dashpot are connected in series. Therefore, the strain rate $\mathop \varepsilon \limits^{ \bullet }$ is additively decomposed into the elastic strain rate ${\mathop \varepsilon \limits^{ \bullet }} {^{e}} = E^{ - 1} \mathop \sigma \limits^{ \bullet }$ and the viscous strain rate ${\mathop \varepsilon \limits^{ \bullet }} {^{v}} = \mu^{ - 1} \sigma$, where $\sigma$ designates the stress, $E$ is the elastic modulus and $\mu$ is the viscous coefficient, leading to

$$ \mathop \varepsilon \limits^{ \bullet } = {\mathop \varepsilon \limits^{ \bullet }} {^{e} }+ {\mathop \varepsilon \limits^{ \bullet }}{^{v}} = E^{ - 1} \mathop \sigma \limits^{ \bullet } + \mu^{ - 1} \sigma $$

(13.1)

This model is concerned with the rate-dependent deformation at the low stress level below the yield stress.

On the other hand, the elastoplastic constitutive equation can be schematically expressed by the Prandtl model in which the dashpot is replaced with the slider in the Maxwell model, whereas the slider begins to move in the state that the stress $\sigma$ reaches the yield stress $\sigma_{y}$, by which the plastic strain rate is induced (see Fig. 13.2). Then, the strain rate is additively composed of the elastic and the plastic strain rates, i.e.

$$ \mathop \varepsilon \limits^{ \bullet } = \left\{ {\begin{array}{l} {{\mathop \varepsilon \limits^{ \bullet }}{^{e}} = E^{ - 1} \mathop \sigma \limits^{ \bullet } \quad{\text{for}}\quad\sigma < \sigma_{y} } \hfill \\ {{\mathop \varepsilon \limits^{ \bullet }} {^{e} }+ {\mathop \varepsilon \limits^{ \bullet }}{^{p} } = E^{ - 1} \mathop \sigma \limits^{ \bullet } + M^{{p{ - 1}}} \mathop \sigma \limits^{ \bullet } \quad{\text{for}}\quad \sigma = \sigma_{y} } \hfill \\ \end{array} } \right. $$

(13.2)

where $M^{p}$ is the plastic modulus.

Furthermore, the model which describes the rate-dependent plastic strain rate ${\mathop \varepsilon \limits^{ \bullet }}{^{vp}}$ induced for the state of stress over the yield stress, called the viscoplastic strain rate, was introduced by Bingham (1922), combining the above-mentioned Maxwell model and Prandtl model so as to connect the dashpot and the slider in parallel as shown in Fig. 13.2, where $\bar{\mu }$ is the viscoplastic coefficient and $n$ is the material constant, while n is chosen to be 4–8 in practice. Then, the strain rate is given by

$$ \mathop \varepsilon \limits^{ \bullet } = {\mathop \varepsilon \limits^{ \bullet } }{^{e}} + {\mathop \varepsilon \limits^{ \bullet }}{^{vp} }= E^{ - 1} \mathop \sigma \limits^{ \bullet } + \bar{\mu }^{ - 1} \left\langle { \sigma - \sigma_{y}} \right\rangle^{n} $$

(13.3)

The Bingham model for the elasto-viscoplastic deformation is the origin of the overstress model based on the concept that the viscoplastic strain rate is induced by the overstress, i.e. stress over the yield stress.

The above-mentioned Bingham model for the one-dimensional deformation was extended by Hohenemser and Prager (1932) and Prager (1961) to describe the three-dimensional deformation of metals, adopting the Mises yield condition for the slider as shown in Fig. 13.2. In this model, the strain rate ${\mathbf{d}}$ is additively decomposed into the elastic strain rate ${\mathbf{d}}^{e}$ and the viscoplastic strain rate ${\mathbf{d}}^{vp}$, i.e.

$$ {\mathbf{d}} = {\mathbf{d}}^{e} + {\mathbf{d}}^{vp} $$

(13.4)

with

$$ {\mathbf{d}}^{vp} = \frac{1}{{\bar{\mu }}}\left\langle \frac{{\sigma^{eq} }}{{F(\varepsilon^{eqvp} )}} - 1\right\rangle^{n} \frac{{{\varvec{\upsigma}}^{\prime } }}{{||{\varvec{\upsigma}}^{\prime } ||}} $$

(13.5)

where $\varepsilon^{eqvp} \equiv \sqrt {2/3} \int {||{\mathbf{d}}^{vp\prime } ||dt}$ is the equivalent viscoplastic strain given by replacing the plastic strain rate ${\mathbf{d}}^{p}$ to the viscoplastic strain rate ${\mathbf{d}}^{vp}$ in the plastic equivalent strain $\varepsilon^{eqp} \equiv \sqrt {2/3} \int {||{\mathbf{d}}^{p\prime } ||dt}$ in Eq. (6.56). The viscoplastic coefficient $\bar{\mu }$ depends on stress, internal variables and temperature in general.

Furthermore, the viscoplastic strain rate in the Prager’s overstress model was extended by Perzyna (1963, 1966) for materials having the general yield condition unlimited to the Mises yield condition as

$$ {\mathbf{d}}^{vp} = \frac{1}{{\bar{\mu }}}\left\langle \frac{{f({\varvec{\upsigma}})}}{F(H)} - 1\right\rangle^{n} {\mathbf{n}},\quad {\mathbf{n}} \equiv \frac{{\partial f({\varvec{\upsigma}})}}{{\partial {\varvec{\upsigma}}}}/ \left\| {\frac{{\partial f({\varvec{\upsigma}})}}{{\partial {\varvec{\upsigma}}}}} \right\| $$

(13.6)

Then, substituting Eqs. (6.29) and (13.6) into Eq. (13.4), we have

$$ {\mathbf{d}} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\left\langle \frac{{f({\varvec{\upsigma}})}}{F(H)} - 1\right\rangle^{n} {\mathbf{n}} $$

(13.7)

and thus

$$ \mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - \frac{1}{{\bar{\mu }}}\left\langle \frac{{f({\varvec{\upsigma}})}}{F(H)} - 1\right\rangle^{n} {\mathbf{E:n}} $$

(13.8)

where the hardening variable $H$ evolves by

$$ \mathop H\limits^{ \bullet } = \mathop H\limits^{ \bullet } ({\varvec{\upsigma}},H; \, {\mathbf{d}}^{vp} ) $$

(13.9)

by replacing the plastic strain rate ${\mathbf{d}}^{p}$ to the viscoplastic strain rate ${\mathbf{d}}^{vp}$ in the evolution rule of the isotropic hardening variable in Eq. (6.37) for the plastic constitutive equation. Therefore, $\mathop H\limits^{ \bullet }$ is the homogeneous function of ${\mathbf{d}}^{vp}$ in degree-one. In what follows, the isotropic yield condition $f({\varvec{\upsigma}}) = F(H)$ in Eq. (6.30) is used below for the sake of simplicity in explanation up to Sect. 13.4.

13.3 On the Creep Model

Based on a concept different from the overstress model, the creep model, which also aims at describing the viscoplastic deformation, has been studied widely. The typical one is the Norton law (Norton 1929) in which the creep strain rate is given as follows:

$$ {\mathbf{d}}^{c} = d_{0}^{c} ||{\varvec{\upsigma}}||^{m} {\mathbf{n}} $$

(13.10)

where $d_{0}^{c}$ and $m\; ( \gg 1)$ are the material constants. Further, it is modified for the crystal plasticity by Nakada and Keh (1966), Hutchinson (1976), Pan and Rice (1983), Peirce et al. (1983), etc. as follows:

$$ {\mathbf{d}}^{c} = ||{\mathbf{d}}^{c} ||{\mathbf{n}} = d_{0}^{c} \left( {\frac{{f({\varvec{\upsigma}})}}{F(H)}} \right)^{m} {\mathbf{n}} $$

(13.11)

where the rate of the isotropic hardening variable is given by

$$ \mathop H\limits^{ \bullet } = \mathop H\limits^{ \bullet } ({\varvec{\upsigma}},H; {\mathbf{d}}^{c} ) $$

(13.12)

instead of Eq. (13.9).

The strain rate is given by

$$ {\mathbf{d}} = {\mathbf{d}}^{e} + {\mathbf{d}}^{c} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + d_{0}^{c} ||{\varvec{\upsigma}}||^{m} {\mathbf{n}} $$

(13.13)

for Eq. (13.10) and

$$ {\mathbf{d}} = {\mathbf{d}}^{e} + {\mathbf{d}}^{c} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + d_{0}^{c} \left(\frac{{f({\varvec{\upsigma}})}}{F(H)}\right)^{m} {\mathbf{n}} $$

(13.14)

for Eq. (13.11).

The creep model described above has different structures from the overstress model because Eqs. (13.10) and (13.11) possesses no threshold value for the generation of the creep strain rate. Especially, its deformation behavior is not reduced to that of the elastoplastic constitutive equation at quasi-static deformation. Therefore, this model cannot describe appropriately the deformation behavior at a low rate. In fact, the creep strain rate does not diminish even if the stress decreases into the inside of yield surface, exhibiting the overrunning stress-strain curve and the creep deformation proceeds unlimitedly under a constant stress state in the creep model as shown in Fig. 13.3. As a concrete example, it is quite unnatural that a bar subjected to any low tension continues to be elongated endlessly as a time elapse. In addition, it is incapable of describing appropriately the impact loading behavior as it describes an elastic deformation behavior with an infinite strength. On the other hand, the viscoplastic strain rate diminishes immediately after the stress decreases into the inside of yield surface in the overstress model as is observed in real materials. Consequently, the creep model is not physically accepted but the overstress model is appropriate for the description of the rate-dependent plastic behavior.

Fig. 13.3

Comparison of overstress model and creep model: the latter exhibits unrealistic behavior predicting always creep strain rate

Furthermore, various constitutive models including the time itself elapsed after a loading mode (loading/unloading) changed have been proposed to date. Here, note that the judgment whether or not the loading mode changes contains an ambiguity depending on the subjectivity of observers, especially in a fluctuating state of deformation rate. Therefore, these models are impertinent, lacking an objectivity.

13.4 Mechanical Response of Past Overstress Model

The development of rate-dependent elastoplastic constitutive equation is reviewed above and it is described that the overstress model would have a pertinent basic structure. Here, let the mechanical responses at the infinitesimal and the infinite rates of deformation be examined in order to clarify the basic property of this model.

The past overstress model advocated by Bingham (1922) and extended by Prager (1961) and Perzyna (1963, 1966) describes the elastoplastic deformation in the quasi-static loading since the viscous resistance of the dashpot diminishes but the elastic deformation in the impact loading since the viscous resistance of the dashpot becomes infinite. Therefore, it cannot be applied to the deformation behavior in a high rate as an impact loading. In what follows, we will show this fact on the past overstress model.

Equations (13.7) and (13.8) are expressed in the following equations for the incremental forms.

$$ d{\varvec{\varepsilon}}= {\mathbf{E}}^{ - 1} {\mathbf{:}}d{\varvec{\upsigma}} + \frac{1}{{\bar{\mu }}}\left\langle {\frac{{f({\varvec{\upsigma}})}}{F(H)} - 1} \right\rangle^{n} {\mathbf{n}}\,dt $$

(13.15)

and thus

$$ d{\varvec{\upsigma}} = {\mathbf{E}}\,{\mathbf{:}\,}d{\varvec{\varepsilon}} - \frac{1}{{\bar{\mu }}}\left\langle {\frac{{f({\varvec{\upsigma}})}}{F(H)} - 1} \right\rangle^{n} {\mathbf{E:n}}\, dt $$

(13.16)

designating $d{\varvec{\varepsilon}}\equiv {\mathbf{d}}\,dt$.

Equation (13.16) reduces to the following relation for the infinitesimal rate of deformation (quasi-static deformation) fulfilling $dt \to \infty {:}$ $d{\varvec{\upsigma}}/dt \to {\mathbf{O}}$ and ${\mathbf{d}}= d{\varvec{\varepsilon}}/dt\to {\mathbf{O}}$.

$$ {\mathbf{O}} \cong {\mathbf{O}} - \frac{1}{{\bar{\mu }}}\left\langle \frac{{f({\varvec{\upsigma}})}}{F(H)} - 1\right\rangle^{n} {\mathbf{E:n}} $$

(13.17)

leading to

$$ \frac{{f({\varvec{\upsigma}})}}{F(H)} - 1 \to 0 $$

(13.18)

Then, the stress changes fulfilling the yield condition $f({\varvec{\upsigma}}) = F(H)$, i.e. obeying the plastic constitutive relation, while the elastic deformation is given by the change of stress, so that Eqs. (13.7) and (13.8) exhibit the response of the elastoplastic constitutive relation in the quasi-static deformation as shown in Fig. 13.4. Then, the overstress model is the extension of the elastoplastic constitutive equation for the non-zero rate of deformation. In fact, the elastoplastic constitutive relation is reproduced for the quasi-static deformation. Here, it is reproduced easily for the non-viscous material, i.e. the material with a small viscoplastic coefficient $\bar{\mu } \cong 0$. It should be noted that the viscoplastic strain rate depends on the rate of stress but it depends always on the state of stress in the creep model as described in Sect. 13.3.

Fig. 13.4

Past overstress model which is inapplicable to prediction of deformation behavior at high rate

On the other hand, in the infinite rate of deformation fulfilling $dt \cong 0{:}$ $d{\varvec{\upsigma}}/dt \to \infty$ and ${\mathbf{d}}= d{\varvec{\varepsilon}}/dt \to \infty$, Eq. (13.8) is reduced to

$$ d{\varvec{\upsigma}} = {\mathbf{E}}\,{\mathbf{:}\,}d{\varvec{\varepsilon}} - {\mathbf{O}},\quad {\text{i}} . {\text{e}} .\; \mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - {\mathbf{O}} $$

(13.19)

approaching the elastic response as shown in Fig. 13.4. Therefore, it predicts the unrealistic response that the material can bear an infinite load. Equation (13.19) is also known directly from Eq. (13.8) by setting $\bar{\mu } \to \infty$ resulting in the material with an infinite viscoplastic coefficient. The past overstress model is inapplicable to the prediction of deformation at high rate in general. In order to modify this defect, Lemaitre and Chaoboche (1990) and Chaboche (2008) proposed to add the creep strain rate in addition to the elastic and the viscoplastic strain rate irrationally, although the viscoplastic strain rate describes the time-dependent plastic strain rate. The rigorous modification of the overstress model so as to describe the elasto-viscoplastic behavior in the general rate ranging from the quasi-static to the impact loading will be given in the next section.

Eventually, the existing formulation of overstress model in Eq. (13.7), i.e. (13.8) is inapplicable to the prediction of deformation at a high rate. The material constant $n$ included as the power form in Eq. (13.7), i.e. (13.8) is usually selected to be larger than five, but the fitting to the test data for impact load is impossible even if $n$ is selected as one hundred which, needless to say, results in the inappropriate prediction of deformation in a slow loading process. In addition, the inclusion of a high power in the equation induces difficulty in numerical calculations.

The so-called flow stress model was proposed by Johnson and Cook (1983) in which the yield stress, i.e. flow stress depends not only on the viscoplastic strain rate but also on the viscoplastic strain rate. It is the empirical model without a generality, which is concerned with the fast loading behavior but inapplicable to the slow loading behavior. However, unfortunately it is used widely for the deformation analyses in the fast loading behavior by adopting the commercial FEM software, e.g. Abaqus and LD-DYNA. Hereinafter, it is desirable to adopt the subloading-overstress model for the analyses of the general loading behavior ranging from the quasi-static to the impact loadings, which will be described in the next section.

13.5 Extension to General Rate of Deformation: Subloading Overstress Model

Equation (13.7) can be rewritten concisely in terms of the normal-yield ratio $R$ incorporated in the subloading surface model as follows:

$$ {\mathbf{d}}= {\mathbf{d}}^{e} + {\mathbf{d}}^{vp} = {\mathbf{E}}^{ - 1} \,{\mathbf{:}\,}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\langle R - 1\rangle^{n} {\mathbf{n}} $$

(13.20)

where $R = f ({\varvec{\upsigma}})F(H)$ takes value larger than unity when the stress goes over the yield surface so that $R \ge 1$ in general, and thus let it be renamed as the dynamic-loading ratio. The surface which passes through the current stress and is similar to the yield surface, called the dynamic-loading surface, is described by

$$ f({\varvec{\upsigma}}) = RF(H) $$

(13.21)

Equation (13.21) is formally identical to Eq. (7.6) for the subloading surface. Further, Eq. (13.21) is extended by incorporating the kinematic hardening as follows:

$$ f({\hat{\boldsymbol{\upsigma }}}) = RF(H) $$

(13.22)

where

$$ \mathop H\limits^{ \bullet } = \mathop H\limits^{ \bullet } ({\varvec{\upsigma}}, H;{\mathbf{d}}^{vp} ), \quad \mathop {\varvec{\upalpha}}\limits^{ \circ } = \mathop {\varvec{\upalpha}}\limits^{ \circ } ({\varvec{\upsigma}},{\varvec{\upalpha}}, {F};{\mathbf{d}}^{vp} ) $$

(13.23)

by replacing the plastic strain rate ${\mathbf{d}}^{p}$ to the viscoplastic strain rate ${\mathbf{d}}^{vp}$ in the plastic evolution equations.

Equation (13.20) can be extended so as to be applicable to the description of deformation in the general rate ranging from the quasi-static to the impact loading by incorporating the limit for the dynamic-loading ratio as follows (Hashiguchi 2007c):

$$ {\mathbf{d}}= {\mathbf{E}}^{ - 1} {\mathbf{:}\,}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\langle R - 1\rangle^{n} }}{{R_{m} - R}}{\mathbf{n}} $$

(13.24)

where $R_{m} \; ( \gg 1)$ is the material constant, called the limit dynamic-loading ratio. By virtue of this modification, the stress cannot increase over the limit dynamic-loading surface described by $f({\varvec{\upsigma}}) = R_{m} F(H)$ to which the stress reaches at an infinite rate of deformation, i.e. impact loading. The response of Eq. (13.24) is illustrated in Fig. 13.5.

Fig. 13.5

Stress–strain curve predicted by the modified overstress model

The power equation has been used for the viscoplastic strain rate after Prager (1961) and Perzyna (1963, 1966). However, the numerical problem is caused by the calculation of high power of infinitesimal number. It can be remedied by using the exponential function as follows:

$$ {\mathbf{d}}= {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\langle \,\exp [ n(R - 1)] - 1\rangle }}{{R_{m} - R}}{\mathbf{n}} $$

(13.25)

The necessity of incorporation of the plastic strain rate in addition to the elastic and the viscoplastic strain rates has been insisted in order to relax the elastic response in Eq. (13.19) such that the inelastic deformation is induced even in the infinite strain rate (cf. Lemaitre and Chaboche 1990; Hashiguchi et al. 2005a; Chaboche 2008). However, it would lead to the physical contradiction that the inelastic strain rate is described redundantly by both terms of the viscoplastic and the plastic strain rates.

Further, let the above-mentioned constitutive equation be extended such that the deformation behavior in the subloading surface model is induced in the quasi-static deformation, exhibiting the smooth elastic-viscoplastic transition. Then, assume that the subloading surface develops by the evolution rule of the normal-yield ratio in Eq. (7.9) in the viscoplastic state, provided that the plastic strain rate is replaced to the viscoplastic strain rate, i.e.,

$$ \hbox{\fbox{$ {\mathop R\limits^{\bullet}}_{s} = \left\{ {\begin{array}{*{20}l} {U(R_{s} )\left\| {{\mathbf{d}}^{vp} } \right\|} \hfill & {{\text{for}}\;{\mathbf{d}}^{vp} \ne {\mathbf{O}}} \hfill \\ {\mathop { \, R}\limits^{ \bullet } \,(R_{s} = R<1)} \hfill & {{\text{for}}\;{\mathbf{d}}^{vp} = {\mathbf{O}}} \hfill \\ \end{array} } \right. $}} $$

(13.26)

where the normal-yield ratio R in Eq. (7.6) is renamed as the subloading-yield ratio and denoted by the symbol $R_{s} \;(0 \le R_{s} \le 1)$. Let the function $U(R_{s} )$ be given by Eq. (7.18) with the replacement of R to $R_{s}$, i.e.

$$ \hbox{\fbox{${U(R_{s} ) = u \cot \left(\frac{\pi }{2}\frac{{\left\langle R_{s} - R_{e} \right\rangle }}{{1 - R_{e} }}\right)}$}} $$

(13.27)

$R_{s}$ can be calculated analytically through the integration of Eq. (13.26)₁ with Eq. (13.27) in the viscoplastic deformation process ${\mathbf{d}}^{vp} \ne {\mathbf{O}}$ similarly to Eq. (7.19) as

$$ R_{s} = \frac{2}{\pi }(1 - R_{e} )\cos^{{ - {1}}} \left[ {{\cos}\left( {\frac{\pi }{2}\frac{{R_{{s {0}}} - R_{e} }}{{1 - R_{e} }}} \right) \exp \left( { - \frac{\pi }{2}u\frac{{\varepsilon^{vp} - \varepsilon_{0}^{vp} }}{{1 - R_{e} }}} \right)} \right] + R_{e} $$

(13.28)

under the initial condition $\varepsilon^{vp} = \varepsilon_{0}^{vp} {:}\;R_{s} = R_{s0}$, defining $\varepsilon^{vp} \equiv \int {||{\mathbf{d}}^{vp} ||dt.}$ Then, introducing the subloading-yield ratio, Eqs. (13.24) and (13.25) can be extended to describe the smooth elastic-plastic transition as follows:

$$ {\mathbf{d}} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\langle R{ - }R_{s} \rangle^{n} }}{{R_{m} - R}}{\mathbf{n}},\,\mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - \frac{1}{{\bar{\mu }}}\frac{{\left\langle {R - R_{s} } \right\rangle^{n} }}{{R_{m} - R}}{\mathbf{E:n}} $$

(13.29)

$$ \hbox{\fbox{${{\mathbf{d}} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\langle \,\exp [n(R{ - }R_{s} )] - 1\rangle }}{{R_{m} - R}}{\mathbf{n}}, \quad \mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - \frac{1}{{\bar{\mu }}}\frac{{\langle \,\exp [n(R{ - }R_{s} )] - 1\rangle }}{{R_{m} - R}}{\mathbf{E:n}}}$}} $$

(13.30)

The response of the subloading-overstress model is illustrated in Fig. 13.6.

Fig. 13.6

Stress–strain curve predicted by the subloading-overstress model

Incorporating the tangential inelastic strain rate formulated in Sect. 9.9 into Eqs. (13.29) and (13.30), the strain rate and the stress rate for the extended subloading surface model are given as follows:

$$ {\mathbf{d}}= {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\langle R - R_{s} \rangle^{n} }}{{R_{m} - R}}{\bar{\mathbf{n}}} + \frac{T(R)}{{2{G}}}{\mathop {\varvec{\upsigma}}\limits^{{\overline{ \circ }}}}{_{t}}^{\prime } $$

(13.31)

$$ \mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - \frac{1}{{\bar{\mu }}}\frac{{\left\langle {R - R_{s} } \right\rangle^{n} }}{{R_{m} - R}}{\mathbf{E}}\,{\mathbf{:}\,}\bar{\mathbf{n}} - 2{G}\frac{T(R)}{1 + T(R)}\bar{\mathbf{d}}_{t}^{\prime } $$

(13.32)

or

$$ {\mathbf{d}} = {\mathbf{E}}^{ - 1} {\mathbf{:}}\mathop {\varvec{\upsigma}}\limits^{ \circ } + \frac{1}{{\bar{\mu }}}\frac{{\left\langle \,{\exp [n(R - R_{s} )] - 1} \right\rangle }}{{R_{m} - R}}{\bar{\mathbf{n}}} + \frac{T(R)}{{2{G}}}{\mathop {\varvec{\upsigma}}\limits^{{\overline{ \circ }}}} {_{t}}^{\prime } $$

(13.33)

$$ \mathop {\varvec{\upsigma}}\limits^{ \circ } = {\mathbf{E:d}} - \frac{1}{{\bar{\mu }}}\frac{{\left\langle \,{\exp [n(R - R_{s} )] - 1} \right\rangle }}{{R_{m} - R}}{\mathbf{E}}{\mathbf{:}}\bar{\mathbf{n}} - 2{G}\frac{T(R)}{1 + T(R)}\bar{\mathbf{d}}_{t}^{\prime } $$

(13.34)

where all the evolution rules of internal variables H, α, c (in ${\bar{\mathbf{n}}}$) and $R_{s}$ are given by those in the elastoplastic constitutive equation (Chapter 9) with the replacements of the plastic strain rate ${\mathbf{d}}^{p}$ to the viscoplastic strain rate ${\mathbf{d}}^{vp}$. The subloading ratio R is calculated in general from Eq. (9.45) which is explicitly described by Eq. (10.32) for the Mises metals.

There is the other overstress model (Duvaut and Lions 1972; Simo et al. 1988; Simo and Hughes 1998) in which the tensor connecting the current stress point and its closest point on the yield surface is used instead of the ordinary overstress which is the scalar variable described in this section. It would be beneficial for the viscoplastic constitutive models assuming the intersecting yield surfaces but the calculation becomes complicated. However, the existence of the intersecting yield surfaces leads the constitutive model to be quite complicated, and its pertinence would be doubtful.

13.6 Subloading-Viscoplastic Model Based on Multiplicative Decomposition

The deformation gradient ${\mathbf{F}}$ is multiplicatively decomposed into the elastic deformation gradient ${\mathbf{F}}^{e}$ and the plastic deformation gradient ${\mathbf{F}}^{vp}$ instead of the plastic deformation gradient ${\mathbf{F}}^{p}$ in the multiplicative elastoplasticity described in Chap. 12. Then, we first adopt the following equation instead of Eq. (12.1).

$$ {\mathbf{F}} = {\mathbf{F}}^{e} {\mathbf{F}}^{vp} $$

(13.35)

The velocity gradient $\varvec{l}$ in the current configuration is additively decomposed into the elastic and the viscoplastic parts:

$$ \varvec{l}= \varvec{l}^{e} + \varvec{l}^{vp} $$

(13.36)

where

$$ \left. {\begin{array}{*{20}l} { {\varvec{l} \equiv \mathop {\mathbf{F}}\limits^{ \bullet } {\mathbf{F}}^{ - 1} },} \hfill \\ {{\varvec{l}^{e} \equiv {{\mathop {\mathbf{F}}\limits^{ \bullet }}}^{e} {\mathbf{F}}^{e-1} , \varvec{l}^{p} \equiv {\mathbf{F}}^{e} {\mathop {\mathbf{F}}\limits^{ \bullet }}^{vp} {\mathbf{F}}^{vp-1} {\mathbf{F}}^{e-1} = {\mathbf{F}}^{e} {\overline{\mathbf{L}}}^{vp} {\mathbf{F}}^{e-1} }} \hfill \\ {{\overline{\mathbf{L}}}^{vp} \equiv {\mathop {\mathbf{F}}\limits^{ \bullet }}^{vp} {\mathbf{F}}^{vp-1} } \hfill \\ \end{array} } \right\} $$

(13.37)

Further, the velocity gradient ${\overline{\mathbf{L}}}$ in the intermediate configuration is additively decomposed into the elastic and the plastic parts as follows:

$$ {\overline{\mathbf{L}}} = {\overline{\mathbf{L}}}^{e} + {\overline{\mathbf{L}}}^{vp} $$

(13.38)

where

$$ \left. \begin{array}{l} {\overline{\mathbf{L}}} \equiv {\mathbf{F}}^{e-1} \varvec{l}{\mathbf{F}}^{e} \hfill \\ {\overline{\mathbf{L}}}^{e} \equiv {\mathbf{F}}^{e-1} \varvec{l}^{e} {\mathbf{F}}^{e} = {\mathbf{F}}^{e-1} {{\mathop {\mathbf{F}}\limits^{ \bullet }}}^{e} , {\overline{\mathbf{L}}}^{vp} \equiv {\mathbf{F}}^{e-1} \varvec{l}^{vp} {\mathbf{F}}^{e} = {\mathop {\mathbf{F}}\limits^{ \bullet }}^{vp} {\mathbf{F}}^{vp-1} \\ \end{array} \right\} $$

(13.39)

from which it follows that

$$ \left. \begin{array}{l} {\overline{\mathbf{L}}} = {\overline{\mathbf{D}}} + {\overline{\mathbf{W}}} \hfill \\ {\overline{\mathbf{L}}}^{e} = {\overline{\mathbf{D}}}^{e} + {\overline{\mathbf{W}}}^{e} , {\overline{\mathbf{L}}}^{vp} = {\overline{\mathbf{D}}}^{vp} + {\overline{\mathbf{W}}}^{vp} \hfill \\ \end{array} \right\} $$

(13.40)

$$ {\overline{\mathbf{D}}} = {\overline{\mathbf{D}}}^{e} + {\overline{\mathbf{D}}}^{vp} ,\quad {\overline{\mathbf{W}}} = {\overline{\mathbf{W}}}^{e} + {\overline{\mathbf{W}}}^{vp} $$

(13.41)

where

$$ \left. {\begin{array}{*{20}l} {\overline{\mathbf{D}}}=\text{ sym}[{\overline{\mathbf{L}}}], \, {\overline{\mathbf{W}}}=\text{ ant}[{\overline{\mathbf{L}}}] \hfill \\ {\overline{\mathbf{D}}}^{e} =\text{ sym}[{\overline{\mathbf{L}}}^{e} ],{\overline{\mathbf{W}}}^{e} =\text{ ant}[{\overline{\mathbf{L}}}^{e} ] \hfill \\ {\overline{\mathbf{D}}}^{vp} =\text{ sym}[{\overline{\mathbf{L}}}^{vp} ], \, {\overline{\mathbf{W}}}^{vp} =\text{ ant}[{\overline{\mathbf{L}}}^{vp} ] \hfill \\ \end{array} } \right\} $$

(13.42)

The viscoplastic strain rate is given noting Eqs. (13.31) or (13.33) by

$$ {\overline{\mathbf{D}}}^{vp} = \frac{1}{{\bar{\mu }}}\frac{{\langle R - R_{s} \rangle^{n} }}{{R_{m} - R}}{\overline{\underline{\mathbf{N}}}} $$

(13.43)

or

$$ {\overline{\mathbf{D}}}^{vp} = \frac{1}{{\bar{\mu}}}\frac{\langle\, \exp [n(R-R_{s})] - 1\rangle}{{R_{m}} - {R}}{\overline{\underline{\mathbf{N}}}} $$

(13.44)

where ${\overline{\underline{\mathbf{N}}}}$ is given by Eq. (12.54).

The viscoplastic spin ${\overline{\mathbf{W}}}^{vp}$ is given analogously to Eq. (12.60) as follows:

$$ {\overline{\mathbf{W}}}^{vp} = \eta^{vp} ({{\overline{\mathbf{M}}\,\overline{\mathbf{D}}}}^{vp} -{\overline{\mathbf{D}}}^{vp} {\overline{\mathbf{M}}}) = \eta^{vp} \frac{1}{{\bar{\mu }}}\frac{{\langle R - R_{s} \rangle^{n} }}{{R_{m} - R}}({{\overline{\mathbf{M}}\,{\overline{\underline{\mathbf{N}}}}}} - {{\overline{\underline{\mathbf{N}}}}\,\overline{\mathbf{M}}}) $$

(13.45)

where $\eta^{vp}$ is the material parameter.

The viscoplastic velocity gradient is given by substituting Eqs. (13.43) and (13.45) into Eqs. (13.40) as follows:

$$ {\overline{\mathbf{L}}}^{vp} = \frac{1}{{\bar{\mu }}}\frac{{\langle R - R_{s} \rangle^{n} }}{{R_{m} - R}}[{\overline{\underline{\mathbf{N}}}} + \eta^{vp} ({{\overline{\mathbf{M}}\,\overline{\underline{\mathbf{N}}}}}- {{\overline{\underline{\mathbf{N}}}}\,\overline{\mathbf{M}}})] $$

(13.46)

The rate of the viscoplastic gradient is given from Eq. (13.37)$_{3}$ as follows:

$$ {\mathop {\mathbf{F}}\limits^{\bullet }}^{vp} = {\overline{\mathbf{L}}}^{vp} {\mathbf{F}}^{vp} $$

(13.47)

The time-integration for ${\mathbf{F}}^{vp}$ can be performed effectively by the tensor exponential method as described for the plastic deformation gradient ${\mathbf{F}}^{p}$ in the end of Sect. 12.8. The elastic deformation gradient ${\mathbf{F}}^{e}$ is given by substituting the time-integration ${\mathbf{F}}^{vp}$ of Eq. (13.47) into

$$ {\mathbf{F}}^{e} ={\mathbf{FF}}^{vp-1} $$

(13.48)

Then, ${\overline{\mathbf{C}}}^{e}$ is calculated by Eq. (12.3) and further the stresses ${\overline{\mathbf{S}}}$ and ${\overline{\mathbf{M}}}$ are calculated by Eqs. (12.31) and (12.32) as the hyperelastic relation as described in Sect. 12.8.

Needless to say, the internal variables $H$ ${\overline M}_{k}$, ${\overline M}_{c}$ (in ${\overline{\underline{\mathbf{N}}}}$) and ${R}_{s}$ evolve by the viscoplastic strain rate ${\overline{\mathbf{D}}}^{vp}$ as described in Sect. 13.5.