γ辐射下聚合物应力松弛的分数Zener模型研究
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摘要
高聚物材料在生产和生活的各个领域的应用非常广泛,了解高聚物的粘弹性是合理使用这种材料的前提。在过去的几十年中,关于高聚物粘弹性的研究取得了巨大的进展,但它依然是人们致力研究的一个重大课题。研究高聚物的粘弹性对于高聚物力学性能的测试和表征,高聚物材料的合理使用和制品的设计、高聚物的改性都具有重要的意义。
从宏观上唯象地描述粘弹行为的经典模型是利用弹簧和粘壶进行串并联组合形成不同的结构,并建立相应结构的动力学方程,这些本构方程具有经典整数阶微分或积分的形式。将分数阶微积分引入本构方程使粘弹性理论有了突破性地发展,由此建立的本构关系具有更广的普适性,能用较简单的模型和较少的参数对复杂粘弹行为给出很好的描述。最基本的粘弹性现象有应力松弛、蠕变和动态粘弹性三个方面。本论文主要利用分数Zener模型对γ辐射下高聚物的应力松弛过程进行了研究。
由于分数Zener模型的应力松弛模量中含有H-Fox函数H_(12)~(11) (z) ,所以本文中分析了H-Fox函数的收敛性和数值计算问题。当|z|较小时,函数H_(12)~(11) (z)的收敛是较快的,但当|z|较大时,该级数收敛很慢,计算十分困难。本文讨论了z较大时级数收敛慢的原因,研究了H-Fox函数的数值计算问题,给出了收敛条件。
对广义H-Fox函数提出了利用遗传算法结合共轭梯度法编写的参数优化程序来确定参数的方法。首先利用最小二乘法给出参数的近似值,从而构成了遗传算法的初始解群,然后利用遗传算法的变异和杂交算子对各参数在其允许的范围内进行全局搜索,其次以对实验数据的拟合误差最小为判断依据,利用共轭梯度法结合遗传算法对给出的参数进行局域优化,最后可以得到最优参数。
利用分数Zener模型研究了γ辐射下未取向和取向超高分子量聚乙烯(UHMWPE)的应力松弛过程。首先根据分数Zener模型应力松弛模量的渐近解分析了此分数模型的应力松弛特征,从而为参数的拟合提供了理论依据。然后利用参数优化程序得到的最优结构参数可以算出相应的应力松弛模量,做出拟合曲线。最后根据参数的值以及应力松弛曲线对UHMWPE在γ射线辐射后的参数与结构的变化进行了分析。
分析结果表明,对于未取向的样品,应力松弛过程在起始时段表现出较大的结构流动性,而长时间行为表现出结构的稳定性。对于取向的样品,γ辐射后结构流动性增大,各时段的应力松弛指数增大,松弛时间减少。表明分数Zener模型可以对γ辐射下UHMWPE的应力松弛特征及结构变化给出很好的描述。
Polymer materials are very popular in every field of production and subsistence,the viscoelasticity of polymer is the foundation to use these materials reasonably.During the past years, the studies about the viscoelasticity of polymer have madegreat progress, yet it remains the focus for many scholars. To study the viscoelasticityof polymer is of great significance to test and describe the mechanical property,reasonable use, design of polymer material and change the properties of polymer.
From a macroscopical point of view, phenomenological classical viscoelasticmodels are based on springs and dashpots, different structure is formed byseries-parallel connection combination of springs and dashpots. Different structureshave corresponding dynamics constitutive equations, which have classical form ofinteger order calculus. Fractional calculus was incorporated into the standardconstitutive equations in the field of viscoelasticity, which made great progress in theviscoelastic theory. These models involve a fairly small number of single elementsand parameters, but can describe complex viscoelastic behavior very well. Theessential viscoelasticity phenomenon includes stress relaxation, creep and dynamicviscoelasticity. In this paper the fractional Zener model is used to investigate the stressrelaxation ofγ-irradiation Ultra High Molecular Weight Polyethylene.
The stress relaxation modulus of fractional Zener model contain H-Fox function,therefore, this paper analyze the convergence and the numerical calculation of H-Foxfunction H_(12)~(11) (z) .For small |z| ,the H function is convergent, but for large |z |,thespeed of convergence is slow, thus the calculation is difficult. This paper discuss thecauses of slow convergence, investigate the different formula for different value of|z| ,then get the convergent condition.
This paper put forward the Genetic Algorithm combined with ConjugateGradient method, which were used to optimize the model parameters for the H-Foxfunction. Firstly, Least Squares method was used to find the approximation of theparameters, which constructs the initial solution group. Secondly, crossover andmutation were used to search every scope of parameters within their fields in overall.Thirdly, according to the fitting error of experimental data, the optimizingparameters were presented after Conjugate Gradient method optimizing local parameters that were given by the method of Genetic Algorithm.
The paper study the stress relaxation process of oriented and unorientedγ-irradiated UHMWPE by using the fractional Zener model. According to theasymptotic solution，the paper analyze the stress relaxation characteristics of thefractional Zener model, which provides optimizing of the model parameters with atheoretical basis. By using the best parameters, stress relaxation modulus was given,and the stress relaxation process ofγ-irradiated UHMWPE were fitted. Through theparameters and stress relaxation curve, the relation between the parameters and thestructure of UHMWPE were got.
To the unoriented UHMWPE, the results of analysis show that the process ofstress relaxation performed strong structural mobility at the very beginning, while itperformed structural stability in the long term. To the oriented UHMWPE, flow rateincreased remarkably, stress relaxation parameters increased with a reducingrelaxation time. This study indicates that the fractional Zener model can describe therelaxation process and structure change of UHMWPE very well.
从宏观上唯象地描述粘弹行为的经典模型是利用弹簧和粘壶进行串并联组合形成不同的结构,并建立相应结构的动力学方程,这些本构方程具有经典整数阶微分或积分的形式。将分数阶微积分引入本构方程使粘弹性理论有了突破性地发展,由此建立的本构关系具有更广的普适性,能用较简单的模型和较少的参数对复杂粘弹行为给出很好的描述。最基本的粘弹性现象有应力松弛、蠕变和动态粘弹性三个方面。本论文主要利用分数Zener模型对γ辐射下高聚物的应力松弛过程进行了研究。
由于分数Zener模型的应力松弛模量中含有H-Fox函数H_(12)~(11) (z) ,所以本文中分析了H-Fox函数的收敛性和数值计算问题。当|z|较小时,函数H_(12)~(11) (z)的收敛是较快的,但当|z|较大时,该级数收敛很慢,计算十分困难。本文讨论了z较大时级数收敛慢的原因,研究了H-Fox函数的数值计算问题,给出了收敛条件。
对广义H-Fox函数提出了利用遗传算法结合共轭梯度法编写的参数优化程序来确定参数的方法。首先利用最小二乘法给出参数的近似值,从而构成了遗传算法的初始解群,然后利用遗传算法的变异和杂交算子对各参数在其允许的范围内进行全局搜索,其次以对实验数据的拟合误差最小为判断依据,利用共轭梯度法结合遗传算法对给出的参数进行局域优化,最后可以得到最优参数。
利用分数Zener模型研究了γ辐射下未取向和取向超高分子量聚乙烯(UHMWPE)的应力松弛过程。首先根据分数Zener模型应力松弛模量的渐近解分析了此分数模型的应力松弛特征,从而为参数的拟合提供了理论依据。然后利用参数优化程序得到的最优结构参数可以算出相应的应力松弛模量,做出拟合曲线。最后根据参数的值以及应力松弛曲线对UHMWPE在γ射线辐射后的参数与结构的变化进行了分析。
分析结果表明,对于未取向的样品,应力松弛过程在起始时段表现出较大的结构流动性,而长时间行为表现出结构的稳定性。对于取向的样品,γ辐射后结构流动性增大,各时段的应力松弛指数增大,松弛时间减少。表明分数Zener模型可以对γ辐射下UHMWPE的应力松弛特征及结构变化给出很好的描述。
Polymer materials are very popular in every field of production and subsistence,the viscoelasticity of polymer is the foundation to use these materials reasonably.During the past years, the studies about the viscoelasticity of polymer have madegreat progress, yet it remains the focus for many scholars. To study the viscoelasticityof polymer is of great significance to test and describe the mechanical property,reasonable use, design of polymer material and change the properties of polymer.
From a macroscopical point of view, phenomenological classical viscoelasticmodels are based on springs and dashpots, different structure is formed byseries-parallel connection combination of springs and dashpots. Different structureshave corresponding dynamics constitutive equations, which have classical form ofinteger order calculus. Fractional calculus was incorporated into the standardconstitutive equations in the field of viscoelasticity, which made great progress in theviscoelastic theory. These models involve a fairly small number of single elementsand parameters, but can describe complex viscoelastic behavior very well. Theessential viscoelasticity phenomenon includes stress relaxation, creep and dynamicviscoelasticity. In this paper the fractional Zener model is used to investigate the stressrelaxation ofγ-irradiation Ultra High Molecular Weight Polyethylene.
The stress relaxation modulus of fractional Zener model contain H-Fox function,therefore, this paper analyze the convergence and the numerical calculation of H-Foxfunction H_(12)~(11) (z) .For small |z| ,the H function is convergent, but for large |z |,thespeed of convergence is slow, thus the calculation is difficult. This paper discuss thecauses of slow convergence, investigate the different formula for different value of|z| ,then get the convergent condition.
This paper put forward the Genetic Algorithm combined with ConjugateGradient method, which were used to optimize the model parameters for the H-Foxfunction. Firstly, Least Squares method was used to find the approximation of theparameters, which constructs the initial solution group. Secondly, crossover andmutation were used to search every scope of parameters within their fields in overall.Thirdly, according to the fitting error of experimental data, the optimizingparameters were presented after Conjugate Gradient method optimizing local parameters that were given by the method of Genetic Algorithm.
The paper study the stress relaxation process of oriented and unorientedγ-irradiated UHMWPE by using the fractional Zener model. According to theasymptotic solution，the paper analyze the stress relaxation characteristics of thefractional Zener model, which provides optimizing of the model parameters with atheoretical basis. By using the best parameters, stress relaxation modulus was given,and the stress relaxation process ofγ-irradiated UHMWPE were fitted. Through theparameters and stress relaxation curve, the relation between the parameters and thestructure of UHMWPE were got.
To the unoriented UHMWPE, the results of analysis show that the process ofstress relaxation performed strong structural mobility at the very beginning, while itperformed structural stability in the long term. To the oriented UHMWPE, flow rateincreased remarkably, stress relaxation parameters increased with a reducingrelaxation time. This study indicates that the fractional Zener model can describe therelaxation process and structure change of UHMWPE very well.
引文
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[3]郝如江.硕士毕业论文.郑州大学,2002.
[4]何曼君,陈维孝,董西侠.高分子物理.上海：复旦大学出版社,1990.
[5]符若文,李谷,冯开才.高分子物理.北京：化学工业出版社.
[6]金日光,华幼卿.高分子物理.北京：化学工业出版社,2000.
[7] Flory P J.链状分子的统计力学.四川：四川科学技术出版社,1990.
[8]钱人元.中国科学B,1983,1080.
[9]钱保功,许观藩,余赋生.高聚物的转变与松弛.北京：科学出版社,1986.
[10]刘林超.硕士毕业论文.暨南大学,2005.
[11] Ward I M.固体高聚物的力学性能.北京：科学出版社,1980.
[12] Aklonis J J, MacKnight W J.聚合物粘弹性引论.北京：科学出版社,1986.
[13]何平笙.高聚物的力学性能.北京：中国科学技术大学出版社,1997.
[14]翟茂林,伊敏,哈鸿飞.高分子材料辐射加工技术及进展.北京：化学工业出版社，2004.
[15]哈鸿飞,吴季兰.高分子辐射化学—原理与应用.北京：北京大学出版社,2002.
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[20] F.Zoepfl,V.Markovic,J.Silerman.J.Polym.Sci.PolymChem.1984,22:2017.
[21] A.Shinde,R.Salovey.J.Polym.Sci.Polym.Chem.1985,23:1681.
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[28]王国英,姜炳政.应用化学,1996,13(4):68.
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