从耗散率函数看黏性与塑性间的内在联系
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摘要
利用热力学的基本定律和内应变率空间中的最大Homilton耗散率原理,建立了黏性和塑性相统一的广义黏弹性本构理论.利用凸分析数学理论,阐明了内应变率空间中的最大Homilton耗散率原理等价于内应力空间中的最大Lagrange耗散率原理.当Homilton耗散率函数为内应变速率的一次欧拉齐次函数时,材料的黏性特性退化为塑性特性,最大Lagrange耗散率原理退化为Hill最大塑性功原理,因此,塑性是材料广义黏性的一种特例.选取不同的自由能函数和耗散率函数,利用最大Homilton耗散率原理和最大Lagrange耗散率原理,建立了准Maxwell模型、准Kelvin模型和准临界状态黏弹性模型,证明了广义Von Mises屈服准则、广义Von Mises线性运动硬化屈服准则和魏汝龙临界状态屈服准则是准Maxwell模型、准Kelvin模型和准临界状态黏弹性模型当耗散率函数为一次欧拉齐次函数时的特例.通过数值算例和图可以直观地验证上述结论的可靠性.
A generalized viscoelastic constitutive theory in which visco-elasticity and plasto-elasticity were unified was established based on the elemental discipline of thermodynamics and the principle of maximum Homilton's dissipation rate.Convex analysis demonstrated that the principle of maximum Homilton's dissipation rate in the internal strain rate space is equivalent to the principle of maximum Lagrange's dissipation rate in the internal stress space.When the function of Homilton's dissipation rate is an Euler's homogenous function of degree one in internal strain rate,viscoelasticity is degenerated into plasto-elasticity and the principle of maximum Lagrange's dissipation rate is degenerated into the principle of Hill's maximum plastic work.Therefore,plastic behavior is a specific case of generalized viscosity.Quasi-Maxell's models and quasi-Kelvin's models and quasi-critical state visco-elastic models are derived respectively from different free energy functions and different dissipation rate functions by the principle of maximum Homilton's dissipation rate and the principle of maximum Lagrange's dissipation rate.It is proved that as their dissipation rate function are Euler's homogeneous functions of degree one,the generalized Von Mises yield criterion and the generalized Von Mises linear kinematic hardening criterion and Wei Ru-long's critical state yielding criterion are specific cases of quasi-Maxwell's models and quasi-Kelvin's models and quasi-critical state visco-elastic models respectively.The value results and charts visually verified the reliability of above-mentioned conclusions.
A generalized viscoelastic constitutive theory in which visco-elasticity and plasto-elasticity were unified was established based on the elemental discipline of thermodynamics and the principle of maximum Homilton's dissipation rate.Convex analysis demonstrated that the principle of maximum Homilton's dissipation rate in the internal strain rate space is equivalent to the principle of maximum Lagrange's dissipation rate in the internal stress space.When the function of Homilton's dissipation rate is an Euler's homogenous function of degree one in internal strain rate,viscoelasticity is degenerated into plasto-elasticity and the principle of maximum Lagrange's dissipation rate is degenerated into the principle of Hill's maximum plastic work.Therefore,plastic behavior is a specific case of generalized viscosity.Quasi-Maxell's models and quasi-Kelvin's models and quasi-critical state visco-elastic models are derived respectively from different free energy functions and different dissipation rate functions by the principle of maximum Homilton's dissipation rate and the principle of maximum Lagrange's dissipation rate.It is proved that as their dissipation rate function are Euler's homogeneous functions of degree one,the generalized Von Mises yield criterion and the generalized Von Mises linear kinematic hardening criterion and Wei Ru-long's critical state yielding criterion are specific cases of quasi-Maxwell's models and quasi-Kelvin's models and quasi-critical state visco-elastic models respectively.The value results and charts visually verified the reliability of above-mentioned conclusions.
引文
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[2]赵彭年.松散介质力学[M].北京:地震出版社,1995:32-35.
[3]龚晓南.土塑性力学[M].杭州:浙江大学出版社,2001:223-231.
[4]ZIEGLER H.An introduction to thermomechanics[M].North-Holland:Amsterdam,1983:253-256.
[5]HOULSBY G T.A study of plasticity theories and theirapplication to soils[D].Cambridge:University of Cam-bridge,1981.
[6]COLLINS I F,HOULSBY G T.Application of thermo-mechanical principles to the modeling of geotechnicalmaterials[J].Proceedings of the Royal Society A,1997,45(3):1975-2001.
[7]LUBLINER J.A simple theory of plasticity[J].Inter-na-tioal Journal of Solids and Structure,1974,10:313-319.
[8]LUBLINER J,AURICCHIO F.Generalized plasticityand shape-memory alloy[J].Internatioal Journal of Sol-ids and Structure,1996,33:991-1003.
[9]ERLICHER S,POINT N.Endochronic theory,non-lin-gear kinematic hardening rule and generalized plasticity:a new interpretation based on generalized normalityassumption[J].Journal of Solids and Structure,2006,43:4175-4200.
[10]POINT N,ERLICHER S.Application of the orthogo-nality principle to the endochronic and Mroz models ofplasticity[J].Materials Science and Engineering,A,2008,483-484(15):47-50.
[11]HOULSBY G T,PUZRIN A M.Rate-dependent plas-ticity models derived from potential punctions[J].Jour-nal of Rheology,2002,46(1):113-126.
[12]周筑宝.最小耗能原理及其应用[M].北京,科学出版社,2001:18-52.
[13]CHEN Zu-yu.A generalized solution for tetrahedral rockwedge stability analysis[J].International Journal of Rock-Mechanics&Mining Sciences,2004,41:613-628.
[14]王秋生.基于超塑性力学的软黏土本构理论研究[D].杭州:浙江大学,2007.WANG Qiu-sheng.Research on constitutives theory ofSoft clay base on hyper-plascity[D].Hangzhou:Zhe-jiang Unversity,2007.
[15]史树中.凸分析[M].上海:上海科学技术出版社,1990:33-65.
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