金属热黏塑性本构关系的研究进展

王强; 王建军; 张晓琼; 张天辉; 王怀坤; 吴桂英

doi:10.11883/bzycj-2021-0443

金属热黏塑性本构关系的研究进展

doi: 10.11883/bzycj-2021-0443

太原理工大学机械与运载工程学院，山西太原 030024

基金项目: 国家自然科学基金（11902272, 12172245）

详细信息

作者简介:
王　强（1995－　），男，博士研究生，wangqiang0004@link.tyut.edu.cn

通讯作者:
王建军（1987－　），男，博士，副研究员，wangjianjun@tyut.edu.cn

中图分类号: O347.3
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出版历程
- 收稿日期: 2021-10-28
- 修回日期: 2022-03-22
- 网络出版日期: 2022-03-29
- 刊出日期: 2022-09-29

Advances in the research of metallic thermo-viscoplastic constitutive relationships

College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China

摘要

摘要: 金属材料的塑性流动行为依赖于温度和应变率，温度和应变率敏感性是金属材料塑性流动的最重要的本质特性之一，建立合适的热黏塑性本构关系来准确描述金属塑性流动行为的温度和应变率依赖性，是金属材料能被广泛应用的必要前提。为此，对金属热黏塑性本构关系的最新研究进展进行了综述，介绍了常见的几种金属热黏塑性本构关系并进行了详细讨论，给出了各本构关系的优势与不足，最后系统介绍了包含金属塑性流动行为中出现的第三型应变时效、或K-W锁位错结构引起的流动应力随温度变化出现的反常应力峰以及拉压不对称等行为的金属热黏塑性本构关系的研究进展。
- 热黏塑性 /
- 本构关系 /
- 应变率效应 /
- 温度效应
Abstract: The studies of the plastic flow behaviour of metallic materials show that the plastic deformation process of metallic materials is dependent on temperature and strain rate, so the temperature and strain rate sensitivities are the most important essential properties of plastic deformation of metallic materials. It is therefore necessary to establish appropriate thermo-viscoplastic constitutive relations to accurately describe the temperature and strain rate dependences of the plastic flow behaviour of metals over a wide range of temperatures and strain rates. Advantages and disadvantages of these constitutive relationships are first reviewed in the present paper. With the increasing applications of metallic materials and the emergence of new materials, the 3rd type strain aging, K-W lock induced anomalous stress peak, and tensile-compression asymmetry are often observed in the plastic flow behaviour of metals. Due to the occurrence of those phenomena, the traditional metal thermo-viscoplastic constitutive relations may no longer be applicable. In view of the significant roles played by the 3rd type strain aging, K-W lock dislocation structure-induced anomalous stress peaks, and tensile-compression asymmetry in the plastic flow behaviour of metals, especially in high temperature loading, it is necessary to take those particular phenomena into account in the framework of the thermo-viscoplastic constitutive relationship of metals. Thus, a large variety of constitutive relation, which considers the interaction of strain, temperature and strain rate, has been established to predict the deformation behaviors of metals. In this context, this paper presents a systematic review of the thermo-viscoplastic constitutive relationships of metals, which includes the anomalous stress peaks in the flow stresses with temperature due to the 3rd type strain aging or K-W-locked dislocation structures, and the tensile-compression asymmetry. In addition, the forms of these thermo-viscoplastic constitutive relationship considering the 3rd type strain aging, K-W lock dislocation structure-induced anomalous stress peaks and tensile-compression asymmetry in the flow stress of metals, are discussed and analysed.
- thermal-viscosity plasticity /
- constitutive relationship /
- strain-rate effect /
- temperature effect

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图 1 不同应变率下93W-Ni-Fe的流动应力随温度的变化曲线^[4]

Figure 1. Flow stress-temperature curves of 93W-Ni-Fe at different strain rates^[4]

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图 2 不同应变率下，流动应力随温度变化的模型预测结果与试验结果比较^[23]

Figure 2. Comparison of model predictions and experimental results of flow stress variation with temperature at different strain rates^[23]

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图 3 试验应力-应变曲线与修正的J-C本构关系预测结果的比较^[27]

Figure 3. Comparison betwwen experimental stress-strain curves with MJC model predictions^[27]

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图 4 不同应变率下，Ti-6Al-4V(296 K)的准静态和动态加载试验结果与K-H-L和J-C本构关系预测结果的比较^[13]

Figure 4. Quasi-static and dynamic loading experimental results of Ti-6Al-4V (at a temperature of 296 K) for different strain rates with correlations using K-H-L and J-C models^[13]

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图 5 不同晶粒尺寸纳米晶铝在不同应变率下的流动应力-应变试验结果与K-H-L本构关系预测结果^[38]

Figure 5. Observed and calculated responses for nanocrystalline aluminium at different strain rates by using KHL model for various grain sizes^[38]

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图 6 不同温度和应变率下铜的真实应力-真实应变曲线与预测结果的对比^[41]

Figure 6. Comparison of true stress - true strain curves and model predictions for copper at different temperatures and strain rates^[41]

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图 7 30CrMnSiA热软化情况的理论和试验值的比较^[44]

Figure 7. Comparison between theoretical and experimental values on thermal softening of 30CrMnSiA^[44]

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图 8 Sn60Pb40合金的计算结果与试验结果的比较^[45]

Figure 8. Comparison of calculated results and experimental results for Sn60Pb40 alloy^[45]

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图 9 不同应变率和温度下，退火OFHC铜的N-N-L本构关系预测结果与试验结果的比较^[57]

Figure 9. Comparison of model prediction predictions with experimental results for annealed OFHC copperat different strain rates and temperatures using N-N-L constitutive model^[57]

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图 10 不同应变率下N-N-L模型预测与试验结果的比较^[58]

Figure 10. Comparison between N-N-L model predictions with experimental results at different strain rate^[58]

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图 11 用M-R-K本构关系描述的流动应力随塑性应变的变化^[65]

Figure 11. Change of the flow stress with plastic strain described using the M-R-K model^[65]

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图 12 DH36钢流动应力随温度变化曲线上出现的反常应力峰^[71]

Figure 12. Anomalous stress peaks in the flow stress curves of DH36 steel with temperature^[71]

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图 13 镍基高温合金的拉伸强度随温度的变化^[97-98]

Figure 13. Tensile strength of nickel base superalloy as a function of temperature^[97-98]

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图 14 高温合金K403在不同应变率下的流动应力随温度的变化

Figure 14. Flow stress of superalloy K403 as function of temperature at different strain rates

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图 15 考虑拉压不对称行为的金属热黏塑性本构关系预测结果与试验结果的对比

Figure 15. Comparison between the predicted and experimental results of thermo-viscoplastic constitutive relationships of metal considering the asymmetrical behavior of tension and compression

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表 1 唯象型本构关系的模型对比

Table 1. Comparison of phenomenological constitutive relations

本构关系	年份	模型	应变率范围/s⁻¹	主要特征
J-C	1983	Johnson-Cook^[20]	>10⁴	（1）形式简单，材料常数容易获得（2）考虑了应变率效应（3）考虑了温度效应（4）不能准确描述流动应力随对数应变率呈非线性关系的金属材料的塑性流动行为的应变率敏感性（5）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	1991	Holmquist^[24]	10⁻³～500	（1）应变率项修正为幂函数（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象（3）对高应变率范围内的应变率敏感性的加强描述依旧有限
	1998	Rule-Jones^[25]	>10³	（1）高速率情况下，可以更好地描述应变率敏感性的作用（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	1999	Kang-Cho^[26]	10⁻³～5000	（1）应变率项中引入了对数应变率的二次型项（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象（3）对高应变率范围内的应变率敏感性的加强描述依旧有限
	2009	Vural-Cairo^[27]	10⁻²～10⁴	（1）在应变和应变率项中加入了温度效应（2）可以描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	2010	Lin-Xia^[28]	10⁻²～10	（1）考虑温度和应变率耦合效应（2）应变项为抛物线形式
K-H	1992	Khan-Huang^[16]	10⁻⁵～10⁴	（1）考虑了应变率效应（2）假定依赖于J₂不变量（3）没有考虑温度效应
	2009	Yu-Guo^[35]	10⁻⁴～1.6×10³	没有考虑温度效应
	1999	Khan-Liang^[36]	10⁻⁶～10⁴	（1）考虑了应变率对应变硬化的影响。（2）应变率项为幂函数形式
	2000	Khan-Zhang^[37]	10⁻⁴～10³	（1）考虑了晶粒尺寸对金属流动应力的影响（2）考虑了应变率对应变硬化的影响（3）应变率项为幂函数形式
	2004	Farrokh-Khan^[38]	10⁻⁴～10³	（1）可以描述晶粒细化引起的不同多晶金属的塑性流动行为（2）考虑了应变率对应变硬化的影响（3）应变率项为幂函数形式
其他	1976	Voce-Kocks^[39-40]	10	考虑温度和应变率对饱和应力${\sigma _{\rm{s}}}$的影响
其他	2005	Molinari-Ravichandran^[41]	10⁻³～8.5×10⁴	考虑了微观结构的演化

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表 2 唯象型本构关系的方程形式

Table 2. Equations of phenomenological constitutive relationships relations

模型	方程形式
Johnson-Cook^[20]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left( {1 + C\ln {{\dot \varepsilon }^ * }} \right)\left( {1 - {T^{ * m}}} \right)$
Holmquist^[24]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left( {{{\dot \varepsilon }^{ * C}}} \right)\left( {1 - {T^{ * m}}} \right)$
Rule-Jones^[25]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left[ {1 + C\ln {{\dot \varepsilon }^ * } + {C_4}\left( {\dfrac{1}{{{C_5} - \ln {{\dot \varepsilon }^ * }}} - \dfrac{1}{{{C_5}}}} \right)} \right]\left( {1 - {T^{ * m}}} \right)$
Kang-Cho^[26]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left[ {1 + {C_1}\ln {{\dot \varepsilon }^ * } + {C_2}{{\left( {\ln {{\dot \varepsilon }^ * }} \right)}^2}} \right]\left( {1 - {T^{ * m}}} \right)$
Vural-Cairo^[27]	$\sigma = \left\{ {A + {B_0}\left[ {1 - {{\left( {\dfrac{{T - {T_0}}}{{{T_{\text{m}}} - {T_0}}}} \right)}^p}} \right]{\varepsilon ^n}} \right\}\left[ {1 + \left( {{c_1}T_{\text{r}}^{ * p} + {c_2}H} \right)\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)} \right]\left[ {1 - {{\left( {\dfrac{{T - {T_0}}}{{{T_{\text{r}}} - {T_0}}}} \right)}^p}} \right]$
Lin-Xia^[28]	$\sigma = \left( {A + {B_1}\varepsilon + {B_2}{\varepsilon ^2}} \right)\left( {1 + C\ln {{\dot \varepsilon }^ * }} \right)\exp \left[ {\left( {{\lambda _1} + {\lambda _2}\ln {{\dot \varepsilon }^ * }} \right)\left( {T - {T_{\text{r}}}} \right)} \right]$
Khan-Huang^[16]	${J_2} = {f_1}\left( \varepsilon \right){f_2}\left( {D_2^{\text{p}}} \right)$
Yu-Guo^[35]	$\sigma = f\left( {\varepsilon ,\dot \varepsilon } \right) = {\sigma _0}{\hat f_2}\left( {\dot \varepsilon } \right) + {E_\infty }\varepsilon - a{{\text{e}}^{ - \alpha \varepsilon }}$
Khan-Liang^[36]	$\sigma = \left[ {A + B{ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right)}^{ {n_1} } }{\varepsilon ^{ {n_0} } } } \right]\left( {1 - {T^{*m} } } \right){\dot \varepsilon ^C}$
Khan-Zhang^[37]	$\sigma = \left[ { \left({a{}_1 + \dfrac{ { {k_1} } }{ {\sqrt d } } }\right) + B{ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right)}^{ {n_1} } }{\varepsilon ^{ {n_0} } } } \right]\left( {1 - {T^{ * m} } } \right){\dot \varepsilon ^C}$
Farrokh-Khan^[38]	$\sigma {\text{ = } }\left\{ {\left( { {a_1} + \dfrac{ { {k_1} } }{ {\sqrt d } } } \right) + B{ {\left( {\dfrac{d}{ { {d_0} } } } \right)}^{ {n_2} } }{ {\left[ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right) {\dfrac{ { {T_{\text{m} } } } }{T} } } \right]}^{ {n_1} } }{ {\left( \varepsilon \right)}^{ {n_0} } } } \right\}{\left( {\dfrac{ { {T_{\text{m} } } - T} }{ { {T_{\text{m} } } - {T_{\text{r} } } } } } \right)^m}{\left( {\dfrac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }^ * } } } } \right)^C}$
Voce-Kocks^[39-40]	$\sigma = {\sigma _{\text{s}}} + \left[ {\left( {{\sigma _0} - {\sigma _{\text{s}}}} \right)\exp \left( { - \dfrac{\varepsilon }{{{\varepsilon _{\text{r}}}}}} \right)} \right]$
Molinari-Ravichandran^[41]	$\dfrac{ { {\delta _{\text{r} } } } }{ { {\delta _{ {\text{r0} } } } } } = {\left[ {1 - { {\left( { {k_{\text{r} } } {\dfrac{T}{ { {T_{ {\text{r0} } } } } } } \lg {\dfrac{ { { {\dot \varepsilon }_{ {\text{r0} } } } } }{ {\dot \varepsilon } } } } \right)}^{ {p_{\text{r} } } } } } \right]^{ {q_{\text{r} } } } }\text{，}\dfrac{ { {\delta _{\text{s} } } } }{ { {\delta _{ {\text{s0} } } } } } = \dfrac{1}{ { { {\left[ {1 - { {\left( { {k_{\text{s} } } {\dfrac{T}{ { {T_{ {\text{s0} } } } } } } \lg {\dfrac{ { { {\dot \varepsilon }_{ {\text{r0} } } } } }{ {\dot \varepsilon } } } } \right)}^{ {p_{\text{s} } } } } } \right] }^{ {q_{\text{s} } } } } } }$

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表 3 物理概念本构关系的模型对比

Table 3. Comparison of physically based constitutive relations

本构关系	年份	模型	应变率范围/s⁻¹	本构关系的主要特征
B-P	1975	Bodner-Partom^[18]	10⁻³～1	（1）将材料的总变形率分为弹性和塑性两部分（2）通过塑性功项合并应变硬化效应（3）没有考虑温度效应
Z-A	1987	Zerilli-Armstrong^[17]	4×10³	（1）考虑了晶粒尺寸的影响（2）基于热激活理论
	2009	Zhang-Wen^[48]	10⁻⁵～10⁻²	考虑了温度、应变率和变形过程对Z-A模型中参数的影响
	2009	Samantaray-Mandal^[49]	10⁻³～1	考虑了温度与应变、温度与应变率的耦合效应对流动应力的影响
	2005	Abed-Voyiadjis^[50]	10⁻⁴～10⁴	（1）可用于预测等温和绝热塑性变形的应力应变曲线（2）将模型参数准确的与微观结构物理参数联系起来
M-T-S	1988	Follansbee^[10,51]	10⁻⁴～10⁴	（1）认为应变率敏感性的上升应归因于结构演化的速率敏感性（2）考虑了阈值应力
N-N-L	1998	Nemat-Nasser-Li^[57]	10⁻³～10⁴	考虑了位错密度随应变和温度的变化
其他	2001	Rusinek-Klepaczko^[61,62]	10⁻⁴～10³	考虑了应变率历史效应对金属材料塑性流动行为的影响
	2009	Rusinek-Rodrguez-Martnez^[63]	10⁻⁴～10⁴	（1）添加一个第三项来扩展该本构关系的应用范围（2）考虑负应变率敏感性和粘性阻力
	2010	Sung^[66]	10⁻³～10	通过Hollomon和Voce应变硬化方程的线性组合来揭示应变硬化率的温度敏感
	2010	Gao-Zhang^[67]	10⁻³～10⁴	考虑FCC金属变形过程中微观结构的演变建立阈值应力与应变、温度和应变率的关系

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表 4 物理概念本构关系的方程形式

Table 4. Equations of physically based constitutive relations

模型	方程形式
Bodner-Partom^[18]	$D_2^{\text{p}} = D_0^2\exp \left[ { - \left( {\dfrac{{n + 1}}{n}} \right){{\left( {\dfrac{{{Z^2}}}{{3{J_2}}}} \right)}^n}} \right]$
Zerilli-Armstrong^[17]	对于FCC: $\sigma = {\sigma _{\text{a}}} + B{\varepsilon ^{1/2}}\exp \left( { - \alpha T} \right)$和$\alpha {\text{ = }}{\alpha _0} - {\alpha _1}\ln \dot \varepsilon $ 对于BCC：$\sigma = {\sigma _{\text{a}}} + B\exp \left( { - \beta T} \right) + {B_0}{\varepsilon ^n}$和$\beta {\text{ = }}{\beta _0} - {\beta _1}\ln \dot \varepsilon $ 对于HCP：$\sigma = {\sigma _{\text{a}}} + B\exp \left( { - \beta T} \right) + {B_0}{\varepsilon ^{^{1/2}}}\exp \left( { - \alpha T} \right)$
Zhang-Wen^[48]	对于FCC：$\sigma = {\sigma _{\text{a}}} + {C_1}{\varepsilon ^{1/2}}\exp \left\{ {\left[ { - {C''_3}T + {C'_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{r\left( \varepsilon \right)r\left( {\dot \varepsilon } \right)}}} \right)} \right]H\left( T \right)} \right\}$ 对于BCC：$\sigma = {\sigma _{\text{a}}} + {C_2}\exp \left\{ {\left[ { - {C''_3}T + {C'_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{r\left( \varepsilon \right)r\left( {\dot \varepsilon } \right)}}} \right)} \right]H\left( T \right)} \right\} + {C_5}{\varepsilon ^n}$
Samantaray-Mandal^[49]	$\sigma = \left( {{C_1} + {C_2}{\varepsilon ^n}} \right)\exp \left[ { - \left( {{C_3} + {C_4}\varepsilon } \right){T^ * } + \left( {{C_5} + {C_6}{T^ * }} \right)\ln {{\dot \varepsilon }^ * }} \right]$
Abed-Voyiadjis^[50]	对于FCC：$\sigma = {C_2}{\varepsilon ^{0.5}}\left( {1 - {X^{1/2}} - X + {X^{3/2}}} \right) + {C_6}$和$X = {C_4}T\ln \left( {1/{{\dot \varepsilon }^ * }} \right)$ 对于BCC：$\sigma = {C_1}\left( {1 - {X^{1/2}} - X + {X^{3/2}}} \right) + {C_5}{\varepsilon ^n} + {C_6}$
Follansbee^[10,51]	$\sigma = {\sigma _{\text{a}}} + \left( {\hat \sigma - {\sigma _{\text{a}}}} \right){\left[ {1 - {{\left( {\dfrac{{kT}}{{{g_0}\mu {b^3}}}\ln \dfrac{{{{\dot \varepsilon }_0}}}{{\dot \varepsilon }}} \right)}^{1/q}}} \right]^{1/p}}$
Nemat-Nasser-Li^[57]	$\sigma \left( {\dot \varepsilon ,\varepsilon ,T} \right) = {\sigma ^0}{\left\{ {1 - {{\left[ { - \dfrac{{kT}}{{{G'_0}}}\left( {\ln \dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}} + \ln \left( {1 + a\left( T \right){\varepsilon ^{1/2}}} \right)} \right)} \right]}^{\tfrac{1}{2}}}} \right\}^{\tfrac{3}{2}}}\left[ {1 + a\left( T \right){\varepsilon ^{\tfrac{1}{2}}}} \right] + \sigma _{\text{a}}^0{\varepsilon ^{{n_1}}}$
Rusinek-Klepaczko^[61,62]	$\sigma = \dfrac{{E\left( T \right)}}{{{E_0}}}\left[ {{B_0}\theta _{\text{m}}^{ - v}{{\left( {{\varepsilon _0} + {\varepsilon _{\text{p}}}} \right)}^{n\left( {1 - {D_2}{\theta _n}} \right)}} + \sigma _0^*{{\left( {1 - {D_1}{\theta _{\text{m}}}} \right)}^m}} \right]$
Rusinek-Rodrguez-Martnez^[63]	${\sigma _{{\text{ns}}}}\left( {\dot \varepsilon ,T} \right) = \sigma _0^{{\text{ns}}} \left[ {\lg \left( {\dfrac{{{{\dot \varepsilon }_{{\text{trans}}}}}}{{\dot \varepsilon }}} \right)} \right] \left[ {1 - {D_3}\left( {\dfrac{{{T_{\text{m}}}}}{T}} \right)\lg \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{\max }}}}} \right)} \right]$
Sung^[66]	$\sigma = \sigma \left( {\varepsilon ,\dot \varepsilon ,T} \right) = f\left( {\varepsilon ,T} \right)g\left( {\dot \varepsilon } \right)h\left( T \right)$
Gao-Zhang^[67]	$\sigma {\text{ = }}{\sigma _{\text{a}}} + \hat Y{\varepsilon ^n}\exp \left[ {{c_3}T\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{{\text{s0}}}}}}} \right)} \right]{\left\{ {1 - {{\left[ { - {c_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)} \right]}^{1/q}}} \right\}^{1/p}}$

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