基于分数阶导数的静态黏弹性本构模型与应用
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摘要
高聚物材料应用极为广泛,黏弹性是其最重要的力学性能之一。黏弹性按载荷变化规律可以分为静态黏弹性和动态黏弹性。本文基于分数阶微积分理论对高聚物的静态黏弹性进行研究,主要工作及结论如下:
1、详细探讨分数阶微积分的定义,并推导Riemann-Liouville型分数阶微积分的表达形式,介绍分数阶微积分的Laplace变换、Laplace逆变换及Fourier变换、Fourier逆变换以及Mittag-Leffler函数的使用条件。构建分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型和自相似分数阶导数线性流变固体模型,并推导其存储模量、损耗模量、存储柔量、损耗柔量、损耗因子、蠕变柔量、松弛模量等。
2、分析分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型蠕变柔量表达式中各参数的物理意义。用分数阶导数Maxwell模型拟合试验数据,结果表明:弹性模量E与松弛时间τ随着温度升高而减小,但分数阶次α却变大;在一定时间内E ,τ随着对数老化时间增加而线性增大,但α基本保持不变;用低温拟合参数计算的长期蠕变柔量与通过时温等效原理平移得到的主曲线几乎完全吻合,说明该模型能很好地预测高聚物长期蠕变行为。
3、分析分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型松弛模量表达式中各参数的物理意义。由Mittag-Leffler函数变换性质可得分数阶导数线性流变固体模型的两种松弛模量表达形式,研究发现其中一种只需要增加取项即可精确表征应力松弛行为,并对该形式在不同项数时的误差进行了分析。分数阶导数线性流变固体模型与标准线性流变固体模型比较,结果说明分数阶导数模型能够更好地表征高聚物的黏弹性力学行为。
本文研究受到国家自然科学基金面上项目(10772156)、教育部科学技术重点项目(209085)、湖南省教育厅重点项目(08A069)和新世纪优秀人才支持计划项目(NCET-08-0685)的资助。
Polymer materials are widely used; viscoelasticity is one of the most important mechanical properties. According to time dependence of the applied load ,viscoelasticity can be classified into two types: static viscoelasticity and dynamic viscoelasticity. Based on the fractional calculus, the static viscoelasticity of polymer is investigated in this paper. The contents of the work are outlined in the following:
1. First, the definition of fractional calculus is discussed in detail and the expression of Riemann-Liouville is derived. And then Laplace transform, inverse Laplace transform,Fourier transform, inverse Fourier transform of the fractional calculus and the applicability of Mittag-Leffler are introduced. The fractional derivative Maxwell model, fractional derivative Kelvin model, and fractional derivative linear solid model, self-similar fractional derivative linear solid model are developed, and the expression of storage modulus, loss modulus, storage compliance, loss compliance, loss factor, creep compliance, relaxation modulus, etc are derived.
2. The physical meanings of the parameters in the expressions of creep compliances of different models are analyzed. Employing the fractional derivative Maxwell model to study the experimental data, the results reveal that the elastic modulus E and relaxation timeτdecrease with the increase in temperature, nevertheless, the fractional orderαincreases with the increasing temperature. Under a certain range of time, E andτlinearly decrease with the increases in logarithmic ageing time, andαis almost the constant. The long-term creep compliances, expressed by master curve, was determined by time-temperature superposition and modeled with the fractional derivative Maxwell model. The model prediction coincides with the master curve very well; therefore, the fractional derivative Maxwell model can be used to predict the long-term creep behavior of polymers accurately.
3. The physical meanings of the parameters in the expressions of relaxation modulus of different models are analyzed. According to the transformation properties of the Mittag-Leffler function, two different kinds of expression of relaxation modulus of the fractional order derivative linear solid model are obtained. The investigation reveals increase number of the term of one of the expressions can characterize the stress relaxation behavior well, and the error of the expression with varied number of term are analyzed. Comparing the fractional order derivative linear solid model with the classical linear solid model, the results show that the fractional order derivative model can characterize viscoelastic behavior of polymer better.
This work was supported by National Natural Science Foundation of China (10772156) , Program for New Century Excellent Talents in University (No.NCET-08-0685), Key Project of Chinese Ministry of Education (No.209085) and Scientific Research Fund of Hunan provincial Education Department (No.08A069).
1、详细探讨分数阶微积分的定义,并推导Riemann-Liouville型分数阶微积分的表达形式,介绍分数阶微积分的Laplace变换、Laplace逆变换及Fourier变换、Fourier逆变换以及Mittag-Leffler函数的使用条件。构建分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型和自相似分数阶导数线性流变固体模型,并推导其存储模量、损耗模量、存储柔量、损耗柔量、损耗因子、蠕变柔量、松弛模量等。
2、分析分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型蠕变柔量表达式中各参数的物理意义。用分数阶导数Maxwell模型拟合试验数据,结果表明:弹性模量E与松弛时间τ随着温度升高而减小,但分数阶次α却变大;在一定时间内E ,τ随着对数老化时间增加而线性增大,但α基本保持不变;用低温拟合参数计算的长期蠕变柔量与通过时温等效原理平移得到的主曲线几乎完全吻合,说明该模型能很好地预测高聚物长期蠕变行为。
3、分析分数阶导数Maxwell模型、分数阶导数Kelvin模型、分数阶导数线性流变固体模型松弛模量表达式中各参数的物理意义。由Mittag-Leffler函数变换性质可得分数阶导数线性流变固体模型的两种松弛模量表达形式,研究发现其中一种只需要增加取项即可精确表征应力松弛行为,并对该形式在不同项数时的误差进行了分析。分数阶导数线性流变固体模型与标准线性流变固体模型比较,结果说明分数阶导数模型能够更好地表征高聚物的黏弹性力学行为。
本文研究受到国家自然科学基金面上项目(10772156)、教育部科学技术重点项目(209085)、湖南省教育厅重点项目(08A069)和新世纪优秀人才支持计划项目(NCET-08-0685)的资助。
Polymer materials are widely used; viscoelasticity is one of the most important mechanical properties. According to time dependence of the applied load ,viscoelasticity can be classified into two types: static viscoelasticity and dynamic viscoelasticity. Based on the fractional calculus, the static viscoelasticity of polymer is investigated in this paper. The contents of the work are outlined in the following:
1. First, the definition of fractional calculus is discussed in detail and the expression of Riemann-Liouville is derived. And then Laplace transform, inverse Laplace transform,Fourier transform, inverse Fourier transform of the fractional calculus and the applicability of Mittag-Leffler are introduced. The fractional derivative Maxwell model, fractional derivative Kelvin model, and fractional derivative linear solid model, self-similar fractional derivative linear solid model are developed, and the expression of storage modulus, loss modulus, storage compliance, loss compliance, loss factor, creep compliance, relaxation modulus, etc are derived.
2. The physical meanings of the parameters in the expressions of creep compliances of different models are analyzed. Employing the fractional derivative Maxwell model to study the experimental data, the results reveal that the elastic modulus E and relaxation timeτdecrease with the increase in temperature, nevertheless, the fractional orderαincreases with the increasing temperature. Under a certain range of time, E andτlinearly decrease with the increases in logarithmic ageing time, andαis almost the constant. The long-term creep compliances, expressed by master curve, was determined by time-temperature superposition and modeled with the fractional derivative Maxwell model. The model prediction coincides with the master curve very well; therefore, the fractional derivative Maxwell model can be used to predict the long-term creep behavior of polymers accurately.
3. The physical meanings of the parameters in the expressions of relaxation modulus of different models are analyzed. According to the transformation properties of the Mittag-Leffler function, two different kinds of expression of relaxation modulus of the fractional order derivative linear solid model are obtained. The investigation reveals increase number of the term of one of the expressions can characterize the stress relaxation behavior well, and the error of the expression with varied number of term are analyzed. Comparing the fractional order derivative linear solid model with the classical linear solid model, the results show that the fractional order derivative model can characterize viscoelastic behavior of polymer better.
This work was supported by National Natural Science Foundation of China (10772156) , Program for New Century Excellent Talents in University (No.NCET-08-0685), Key Project of Chinese Ministry of Education (No.209085) and Scientific Research Fund of Hunan provincial Education Department (No.08A069).
引文
[1]柯杨船等.高分子物理教程[M].北京:化学工业出版社,2006.
[2]马德柱等.高聚物的结构与性能(第二版)[M].北京:科学出版社,2006.
[3]杨挺青,罗文波,徐平,危银涛,刚芹果.黏弹性理论与应用[M].北京:科学出版社,2004.
[4]过梅丽.高聚物与复合材料的动态力学热分析[M].北京:化学工业出版社,2002.
[5]何平笙.高聚物的力学性能[M].合肥:中国科学技术大学出版社,2008.
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[7] Findley J D W N, Lai J S, Onaran K. Creep and relaxation of nonlinear viscoelastic materials[M]. North-Holland Pub.Co.,1976.
[8] Christensen R M. Theory of viscoelasticity [M]. New York:Academic Press,1982.(中译本:科学出版社,1992)
[9]杨挺青.黏弹性力学[M].武汉:华中理工大学出版社,1990.
[10]穆霞英.蠕变力学[M].西安:西安交通大学出版社,1990.
[11]罗文波,杨挺青.固态高聚物的应力松弛行为[J].高分子材料科学与工程,2002,18 (2):97-99.
[12] Ferry J D, Grandine L D, Fitzgerald E R. The relaxation distribution function of polyisobutylene in the transition from rubber‐like to glass‐like behavior [J]. Journal of Applied Physics,1953,24(7):911-916.
[13] Plazek D J, Chay I C, Ngai K L. et. al. Viscoelastic properties of polymers. 4. Thermorheological complexity of the softening dispersion in polyisobutylene[J]. Macromolecules,1995,28(19):6432-6436.
[14]罗文波,唐欣,谭江华,赵荣国.流变材料长期力学性能加速表征的若干进展[J].材料导报,2007,21(7):8-10.
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[18] Gemant A. A method of analyzing experimental results obtained from elasto-viscous bodies [J]. Physics,1936,7:311-317.
[19] Koeller RC. Applications of fractional calculus to the theory of viscoelasticity[J]. Appl Mech,1984,51:299-307.
[20] Friedrich C. Relaxation and retardation functions of the Maxwell model with fractional derivatives[J]. Rheol Acta, 1991,30:151-158.
[21] Nonnenmacher T F,Gl?ckle W G . A fractional model for mechanical stress relaxation[J]. Philosophical Magazine Letters,1991,64(2):89–93.
[22] Schiessel H, Blumen A. Hierarchical analogues to fractional relaxation equations[J]. J. Phys. A: Math. Gen., 1993, 26:5057-5069.
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[24] Heymans N, Bauwens JC. Fractal rheological models and fractional differential equations for viscoelastic behavior[J]. Rheologica Acta,1994,33(3),210-219.
[25] Schiessel H, Metzler R, Blumen A and Nonnenmacher TF.Generalized visco-elastic models: their fractional equations with solutions[J]. J. Phys. A: Math. Gen.,1995,28(23),6567-6584.
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[30] Atanackovic T M. A modified Zener model of a viscoelastic body[J]. Continuum Mech. Thermodyn, 2002,14: 137–148.
[31]朱正佑,李根国,程昌钧.具有分数导数本构关系的粘弹性Timoshenko梁的静动力学行为分析[J].应用数学和力学,2002,23(1):1-10.
[32] Pritz T. Five-parameter fractional derivative model for polymeric damping materials[J]. Journal of Sound and Vibration, 2003,265:935–952.
[33] Tan Wenchang, Pan Wenxiao, Xu Mingyu. A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates[J]. International Journal of Non-Linear Mechanics, 2003, 38: 645-650.
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[35] Xu MY, Tan WC. Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions[J]. Science in China Series G, 2003, 46(2) :145-157.
[36] Haneczok G, Weller M. A fractional model of viscoelastic relaxation[J]. Materials Science and Engineering A ,2004,370 :209–212.
[37] Heymans N. Fractional Calculus Description of Non-Linear Viscoelastic Behaviour of Polymers[J]. Nonlinear Dynamics, 2004, 38: 221–231.
[38] Heymans N, Kitagawa M, Modelling‘‘unusual’’behaviour after strain reversal with hierarchical fractional models[J]. Rheol Acta ,2004,43: 383–389.
[39]刘甲国,徐明瑜.人颅骨粘弹性的分数阶模型研究[J].中国生物医学工程学报,2005,24(1):12-16.
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[42]殷德顺,任俊娟,和成亮,陈文.一种新的岩土流变模型元件[J].岩石力学与工程学报, 2007,26(9):1899-1903.
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[2]马德柱等.高聚物的结构与性能(第二版)[M].北京:科学出版社,2006.
[3]杨挺青,罗文波,徐平,危银涛,刚芹果.黏弹性理论与应用[M].北京:科学出版社,2004.
[4]过梅丽.高聚物与复合材料的动态力学热分析[M].北京:化学工业出版社,2002.
[5]何平笙.高聚物的力学性能[M].合肥:中国科学技术大学出版社,2008.
[6] Lockett F J. Nonlinear viscoelastic solids[M]. London:Academic Press,1972.
[7] Findley J D W N, Lai J S, Onaran K. Creep and relaxation of nonlinear viscoelastic materials[M]. North-Holland Pub.Co.,1976.
[8] Christensen R M. Theory of viscoelasticity [M]. New York:Academic Press,1982.(中译本:科学出版社,1992)
[9]杨挺青.黏弹性力学[M].武汉:华中理工大学出版社,1990.
[10]穆霞英.蠕变力学[M].西安:西安交通大学出版社,1990.
[11]罗文波,杨挺青.固态高聚物的应力松弛行为[J].高分子材料科学与工程,2002,18 (2):97-99.
[12] Ferry J D, Grandine L D, Fitzgerald E R. The relaxation distribution function of polyisobutylene in the transition from rubber‐like to glass‐like behavior [J]. Journal of Applied Physics,1953,24(7):911-916.
[13] Plazek D J, Chay I C, Ngai K L. et. al. Viscoelastic properties of polymers. 4. Thermorheological complexity of the softening dispersion in polyisobutylene[J]. Macromolecules,1995,28(19):6432-6436.
[14]罗文波,唐欣,谭江华,赵荣国.流变材料长期力学性能加速表征的若干进展[J].材料导报,2007,21(7):8-10.
[15]张淳源.粘弹性断裂力学[M].武昌:华中理工大学出版社,1994.
[16] Podlubny I. Fractional Differential Equations[M]. New York:Academic Press,1999.
[17] Mandelbrot B B. The fractal geometry of nature[M]. San Francisco:Freeman, 1982.
[18] Gemant A. A method of analyzing experimental results obtained from elasto-viscous bodies [J]. Physics,1936,7:311-317.
[19] Koeller RC. Applications of fractional calculus to the theory of viscoelasticity[J]. Appl Mech,1984,51:299-307.
[20] Friedrich C. Relaxation and retardation functions of the Maxwell model with fractional derivatives[J]. Rheol Acta, 1991,30:151-158.
[21] Nonnenmacher T F,Gl?ckle W G . A fractional model for mechanical stress relaxation[J]. Philosophical Magazine Letters,1991,64(2):89–93.
[22] Schiessel H, Blumen A. Hierarchical analogues to fractional relaxation equations[J]. J. Phys. A: Math. Gen., 1993, 26:5057-5069.
[23] Gl(o|¨)ckle W G , Nonnenmacher T F. Fractional relaxation and the time-temperature superposition principle[J]. Rheol Acta, 1994,33:337-343.
[24] Heymans N, Bauwens JC. Fractal rheological models and fractional differential equations for viscoelastic behavior[J]. Rheologica Acta,1994,33(3),210-219.
[25] Schiessel H, Metzler R, Blumen A and Nonnenmacher TF.Generalized visco-elastic models: their fractional equations with solutions[J]. J. Phys. A: Math. Gen.,1995,28(23),6567-6584.
[26] Nonnenmacher T F, Metzler R.On the Riemann-Liouville Fractional Calculus and Some Recent Applications[J ] . Fractals,1995 ,3 (3) :557-566.
[27] Welch S W J , Rorrer R A L , Duren R G.Application of time-based f ractional calculus method to viscoelastic creep and stress relaxation of materials[J]. Mechanics of Time-Dependent Materials,1999,3 : 279-303.
[28]张为民.一种采用分数阶导数的新流变模型理论[J].湘潭大学自然科学学报,2001,23(1):30-36.
[29] Hernández-Jiménez A, J. Hernández-Santiago, A. Macias-Garíca,J. Sánchez-González,Relaxation modulus in PMMA and PTFE fitting by fractional Maxwell model[J]. Polymer Testing, 2002, 21:325–331.
[30] Atanackovic T M. A modified Zener model of a viscoelastic body[J]. Continuum Mech. Thermodyn, 2002,14: 137–148.
[31]朱正佑,李根国,程昌钧.具有分数导数本构关系的粘弹性Timoshenko梁的静动力学行为分析[J].应用数学和力学,2002,23(1):1-10.
[32] Pritz T. Five-parameter fractional derivative model for polymeric damping materials[J]. Journal of Sound and Vibration, 2003,265:935–952.
[33] Tan Wenchang, Pan Wenxiao, Xu Mingyu. A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates[J]. International Journal of Non-Linear Mechanics, 2003, 38: 645-650.
[34] Ralf Metzler, Theo F. Nonnenmacher. Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials[J]. International Journal of Plasticity, 2003,19 :941–959.
[35] Xu MY, Tan WC. Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions[J]. Science in China Series G, 2003, 46(2) :145-157.
[36] Haneczok G, Weller M. A fractional model of viscoelastic relaxation[J]. Materials Science and Engineering A ,2004,370 :209–212.
[37] Heymans N. Fractional Calculus Description of Non-Linear Viscoelastic Behaviour of Polymers[J]. Nonlinear Dynamics, 2004, 38: 221–231.
[38] Heymans N, Kitagawa M, Modelling‘‘unusual’’behaviour after strain reversal with hierarchical fractional models[J]. Rheol Acta ,2004,43: 383–389.
[39]刘甲国,徐明瑜.人颅骨粘弹性的分数阶模型研究[J].中国生物医学工程学报,2005,24(1):12-16.
[40] Jia Guo Liu , Ming Yu Xu. Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions[J]. Mech Time-Depend Mater ,2006, 10:263-279.
[41]孙海忠,张卫.一种分析软土黏弹性的分数导数开尔文模型[J].岩土力学, 2007,28(9): 1983-1986.
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