功能梯度材料轴对称接触力学及微动分析
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摘要
功能梯度材料作为一种新型的复合材料,由于其优越的特性和广阔的应用前景而受到广泛关注。近年来一些研究者的实验研究结果表明,功能梯度材料用作涂层可有效地抵制接触变形和接触破坏,因此研究功能梯度材料的接触力学具有重要的意义。目前关于功能梯度涂层接触力学的理论研究更多地局限于二维问题,本文主要研究功能梯度材料涂层的轴对称接触及微动问题。由于功能梯度材料参数可以在空间某个方向上按任意连续函数变化,这给实际的接触力学分析带来数学上的困难。针对这一点本文建立两种数学模型来模拟梯度材料:第一种假设功能梯度材料的弹性模量按照指数函数形式变化;第二种采用线性分层模型来模拟弹性模量按任意函数形式变化但泊松比为1/3的功能梯度材料,即将功能梯度材料分成若干个子层,在每个子层中,假设材料弹性模量沿厚度方向按线性函数变化,在各子层界面处连续并等于真实值。利用这两个模型,本文研究了若干典型的轴对称接触问题,主要工作包括:
1)分别利用指数函数模型和线性分层模型得到涂层半空间在任意轴对称荷载作用下的基本解。
2)利用上述基本解,求解刚性柱形、球形和锥形压头作用下的功能梯度材料涂层半空间的无摩擦轴对称接触问题。
3)利用上述基本解,求解功能梯度材料涂层半空间的Reissner-Sagoci问题。
4)利用上述基本解,求解刚性球形压头作用下的功能梯度材料涂层半空间的有限摩擦接触问题。
5)在以上工作的基础上,研究功能梯度材料涂层弹性体之间的扭转微动接触问题。
本文应用Hankel积分变换技术以及传递矩阵方法,将轴对称接触及微动问题转化为奇异积分方程(组),然后采用数值方法进行求解。首先对两种模型得到的结果进行比较,然后利用线性分层模型求解弹性模量按照其他函数形式变化的情况。研究结果表明:
1)两种模型的计算结果吻合很好。与传统层合板模型相比,线性分层模型不但保留了允许弹性参数按任意函数形式变化的特性,同时消除了材料界面处弹性参数不连续的弊端。在求解轴对称接触问题时,线性分层模型具有更高的计算效率。
2)通过改变功能梯度材料涂层的剪切模量变化形式,可以调节涂层表面的接触应力分布,从而有效地抑制接触变形和接触破坏。另外材料剪切模量的梯度变化对力与压痕的关系产生明显影响,这表明通过压痕实验可以得到材料的弹性参数梯度。涂层剪切模量的梯度变化可以改变Reissner-Sagoci扭转时的接触力分布、表面位移和扭矩的大小,对于梯度材料的无损实验具有指导意义。
3)通过调整功能梯度材料涂层剪切模量的变化形式,可以改变微动接触时的接触力分布,影响黏着/滑移区和剪切力峰值出现的位置。特别地,减小涂层表面的剪切模量或减小梯度变化指数可以减小有限摩擦接触时的径向接触力和扭转微动接触时的周向剪切力的峰值。因此通过调整剪切模量的变化规律可以抑制微动接触破坏。
4)泊松比的变化对轴对称无摩擦接触和扭转微动接触时的接触力分布影响不是很大,但对有限摩擦接触时的径向剪切力分布影响较大。
5)发生有限摩擦接触时,摩擦系数对法向接触力影响非常小,但是对径向剪切力分布影响较大。
本文工作不仅完善了功能梯度材料接触及微动问题的理论研究,而且对功能梯度材料的优化设计与工程应用具有重要意义。
Functionally graded material(FGM)--a new kind of nonhomogeneous composites materials has predominant properties and can be used in many fields,so it has been the interest of scientists.In recent years,the experimental results which have been obtained by many reseachers indicate that FGM used as coatings can improve the resistance to contact deformation and damage,thus it is very important to study the contact problems of FGM coatings.The theoretical results which have been obtained localized on the two-dimension contact mechanics of FGM coatings.In this thesis the axisymmetric contact and fretting problems of FGM coatings are considered.The properties of non-nomogeneous may vary aribitarily along a certain spatial direction,which makes the solution of the contact problem very difficult in mathematics.In the present work, we adopt two methods to model the FGM:First,assume the elastic modulus of the FGM varies as the exponent functuion.Second,simulate the FGM with arbitrarily varying material modulus by the linear multi-layered model limited to Possion's ratio of 1/3.That is,the FGM is divided into a series of sub-layers,with linear variation of the shear modulus in each sub-layer.The shear modulus is taken to be continuous on the sub-interfaces and equal to their real values.With these two models,several axisymmetric contact problems are solved:
1 ) The fundamental solutions of a functionally graded coated half-space under arbitrary axisymmetric loads are obtained by both exponential functiuon model and the linear multi-layered model.
2) With the above fundamental solutions,the frictionless axisymmetric contact problem of the FGM coating indented by a rigid cylinder and spheical and conical punch are solved.
3) With the above fundamental solutions,the Reissner-Sagoci problem of the FGM coated half-space is solved.
4) With the above fundamental solutions,the finite friction contact problem of the FGM coated half-space which indented by a rigid spherical punch is solved.
5) Based on the above analysis,fretting contact of two FGM coated elastic bodies under torsion is discussed.
The Hankel integral transform technique and the transfer matrix method are applied to reduce the axisymmetric contact and fretting problems to a set of Cauchy singular integral equations that can be numerically caculated.The results obtained by these two methods are compared.The problems with other functional forms of the elatic modulus are caulated by using linear multi-layered model.The results show:
1 ) The numerical results with the two models are in pretty agreement.Compared with the piecewise multi-layered(PWML) model,the linear multi-layered model can simulate FGM coatings with arbitrarily varying elatic modulus and involves no discontinuties of the materials properties at the sub-layers.The linear multi-layered model is more efficient in solving the axisymmetric contact problem of the FGM than thePWML model.
2) The varying form of the elastic modulus can alter the contact tractions at the surface of FGMs.So control of the gradients of the elastic modulus offers the opportunities for the design of surfaces with resisitance to contact deformation and damage.The results also show that the relation between the load vs the indenter displacement is greatly effected by the gradient of the coating,so the results provide a way to measure the gradient of the coating using the indentation-testing method. Furthermore,the contact stress,surface displacement and torque can be changed by adjusting the gradient of the coating for Reissner-Sagoci problem.This is of interest to materials non-desreuctive testing.
3) The varying forms of the elastic modulus can change the distribution of contact tractions and alter the stick/slip region and the position where the maximum contact shear tractions appearing in fretting problem.In particular,reducing the shear modulus or the gradient of the coating can lead to decreasing of the maximum shear tractions in finit frctional contact as well as the maximum circumferical shear traction in the fretting problem under torsion.Therefore,we can enhance the contact damage resistance under fretting conditions by adjusting the gradient of the coating.
4) The varying of Possion's ratio has no significiant effect on the contact tractions in axisymmetric frictionless contact and the fretting problem under torsion,but it has influence on the shear tractons in the contact problem with finite friction.
5) In the contact problem with finite frction,the friction coefficient has slight effect on the normal tractions but a significant effect on the distribution of shear tractions.
The present investigation will be expected to perfect the theory and analytical methods for the axisymmetric contact and fretting problem of FGM coatings,and to provid a guidance for design considertations and applications of FGMs.
1)分别利用指数函数模型和线性分层模型得到涂层半空间在任意轴对称荷载作用下的基本解。
2)利用上述基本解,求解刚性柱形、球形和锥形压头作用下的功能梯度材料涂层半空间的无摩擦轴对称接触问题。
3)利用上述基本解,求解功能梯度材料涂层半空间的Reissner-Sagoci问题。
4)利用上述基本解,求解刚性球形压头作用下的功能梯度材料涂层半空间的有限摩擦接触问题。
5)在以上工作的基础上,研究功能梯度材料涂层弹性体之间的扭转微动接触问题。
本文应用Hankel积分变换技术以及传递矩阵方法,将轴对称接触及微动问题转化为奇异积分方程(组),然后采用数值方法进行求解。首先对两种模型得到的结果进行比较,然后利用线性分层模型求解弹性模量按照其他函数形式变化的情况。研究结果表明:
1)两种模型的计算结果吻合很好。与传统层合板模型相比,线性分层模型不但保留了允许弹性参数按任意函数形式变化的特性,同时消除了材料界面处弹性参数不连续的弊端。在求解轴对称接触问题时,线性分层模型具有更高的计算效率。
2)通过改变功能梯度材料涂层的剪切模量变化形式,可以调节涂层表面的接触应力分布,从而有效地抑制接触变形和接触破坏。另外材料剪切模量的梯度变化对力与压痕的关系产生明显影响,这表明通过压痕实验可以得到材料的弹性参数梯度。涂层剪切模量的梯度变化可以改变Reissner-Sagoci扭转时的接触力分布、表面位移和扭矩的大小,对于梯度材料的无损实验具有指导意义。
3)通过调整功能梯度材料涂层剪切模量的变化形式,可以改变微动接触时的接触力分布,影响黏着/滑移区和剪切力峰值出现的位置。特别地,减小涂层表面的剪切模量或减小梯度变化指数可以减小有限摩擦接触时的径向接触力和扭转微动接触时的周向剪切力的峰值。因此通过调整剪切模量的变化规律可以抑制微动接触破坏。
4)泊松比的变化对轴对称无摩擦接触和扭转微动接触时的接触力分布影响不是很大,但对有限摩擦接触时的径向剪切力分布影响较大。
5)发生有限摩擦接触时,摩擦系数对法向接触力影响非常小,但是对径向剪切力分布影响较大。
本文工作不仅完善了功能梯度材料接触及微动问题的理论研究,而且对功能梯度材料的优化设计与工程应用具有重要意义。
Functionally graded material(FGM)--a new kind of nonhomogeneous composites materials has predominant properties and can be used in many fields,so it has been the interest of scientists.In recent years,the experimental results which have been obtained by many reseachers indicate that FGM used as coatings can improve the resistance to contact deformation and damage,thus it is very important to study the contact problems of FGM coatings.The theoretical results which have been obtained localized on the two-dimension contact mechanics of FGM coatings.In this thesis the axisymmetric contact and fretting problems of FGM coatings are considered.The properties of non-nomogeneous may vary aribitarily along a certain spatial direction,which makes the solution of the contact problem very difficult in mathematics.In the present work, we adopt two methods to model the FGM:First,assume the elastic modulus of the FGM varies as the exponent functuion.Second,simulate the FGM with arbitrarily varying material modulus by the linear multi-layered model limited to Possion's ratio of 1/3.That is,the FGM is divided into a series of sub-layers,with linear variation of the shear modulus in each sub-layer.The shear modulus is taken to be continuous on the sub-interfaces and equal to their real values.With these two models,several axisymmetric contact problems are solved:
1 ) The fundamental solutions of a functionally graded coated half-space under arbitrary axisymmetric loads are obtained by both exponential functiuon model and the linear multi-layered model.
2) With the above fundamental solutions,the frictionless axisymmetric contact problem of the FGM coating indented by a rigid cylinder and spheical and conical punch are solved.
3) With the above fundamental solutions,the Reissner-Sagoci problem of the FGM coated half-space is solved.
4) With the above fundamental solutions,the finite friction contact problem of the FGM coated half-space which indented by a rigid spherical punch is solved.
5) Based on the above analysis,fretting contact of two FGM coated elastic bodies under torsion is discussed.
The Hankel integral transform technique and the transfer matrix method are applied to reduce the axisymmetric contact and fretting problems to a set of Cauchy singular integral equations that can be numerically caculated.The results obtained by these two methods are compared.The problems with other functional forms of the elatic modulus are caulated by using linear multi-layered model.The results show:
1 ) The numerical results with the two models are in pretty agreement.Compared with the piecewise multi-layered(PWML) model,the linear multi-layered model can simulate FGM coatings with arbitrarily varying elatic modulus and involves no discontinuties of the materials properties at the sub-layers.The linear multi-layered model is more efficient in solving the axisymmetric contact problem of the FGM than thePWML model.
2) The varying form of the elastic modulus can alter the contact tractions at the surface of FGMs.So control of the gradients of the elastic modulus offers the opportunities for the design of surfaces with resisitance to contact deformation and damage.The results also show that the relation between the load vs the indenter displacement is greatly effected by the gradient of the coating,so the results provide a way to measure the gradient of the coating using the indentation-testing method. Furthermore,the contact stress,surface displacement and torque can be changed by adjusting the gradient of the coating for Reissner-Sagoci problem.This is of interest to materials non-desreuctive testing.
3) The varying forms of the elastic modulus can change the distribution of contact tractions and alter the stick/slip region and the position where the maximum contact shear tractions appearing in fretting problem.In particular,reducing the shear modulus or the gradient of the coating can lead to decreasing of the maximum shear tractions in finit frctional contact as well as the maximum circumferical shear traction in the fretting problem under torsion.Therefore,we can enhance the contact damage resistance under fretting conditions by adjusting the gradient of the coating.
4) The varying of Possion's ratio has no significiant effect on the contact tractions in axisymmetric frictionless contact and the fretting problem under torsion,but it has influence on the shear tractons in the contact problem with finite friction.
5) In the contact problem with finite frction,the friction coefficient has slight effect on the normal tractions but a significant effect on the distribution of shear tractions.
The present investigation will be expected to perfect the theory and analytical methods for the axisymmetric contact and fretting problem of FGM coatings,and to provid a guidance for design considertations and applications of FGMs.
引文
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