《接触力学》课件 王战江讲授.pdf

序 言 本课件是 2013 2015 年我在重庆大学给机械学院研究生授课所用 共讲授过三次 并 经过几次修改 当时课程名称是 界面力学 但实际上是讲授 接触力学 的基本知 识 国内开设 接触力学 的课程的高校不多 作者在这方面进行了十余年的研究 希望 能将所学以点到面讲授给广大接触力学的初学者 避免少走弯路 本课件按内容分十个章 节 初学者可根据自己需求选择学习 但前四章节建议仔细阅读 第一章 预备知识 第二章 弹性半平面问题 第三章 弹性半空间问题 第四章 Hertz接触理论 第五章 滑动接触和微滑接触 第六章 滚动接触 第七章 粘着接触 第八章 弹塑性接触 第九章 粘弹性接触 第十章 涂层接触 关于作者的最新研究可关注 google scholar网址 0 yxs yxs yxq yxq yxpyxq yxpyxqss yx 21 21 x u x u cxVs x u x u cyVs yy yy xx xx 求解方程 6 滚动接触 6王战江王战江授课授课 6 滚动接触 皮带轮问题皮带轮问题 伸张应变 V dx dt是皮带轮未被伸长的速度 系统打滑比 dx长度皮带伸长dl RMtTT 2 21 0 0 2 1 2 1 2 1 RTM RTM T T e T dxdl 1 VT dt dx dt dl v 1 1 RMTT V vv V R 21 2121 2016 12 20 2 7王战江王战江授课授课 6 滚动接触 两个同质材料的牵引滚动两个同质材料的牵引滚动 切向力叠加切向力叠加 dcxacdxp a c xq axaaxpxq x x 1 1 2 122 0 2 122 0 dcxadxp cEa c x u axaxp aEx u x x 1 2 1 2 0 2 0 2 由它们产生的应变由它们产生的应变 dcxatconsdp aEx u x u xx tan 1 2 0 2 粘着区为常值粘着区为常值 Leading edge Trailing edge 1 二维问题 二维问题 8王战江王战江授课授课 6 滚动接触 两个同质材料的牵引滚动两个同质材料的牵引滚动 由于两接触表面的切向力大小相等方向相反由于两接触表面的切向力大小相等方向相反 在粘着区在粘着区 dcxa 蠕变率 2 1 0 2 11 1 4 PQ R a aEdp x x 2 1 111PQ a c a d x 1 二维问题 二维问题 9王战江王战江授课授课 6 滚动接触 两个同质材料的牵引滚动两个同质材料的牵引滚动 从静止状态到滚动的动态过程从静止状态到滚动的动态过程 1 二维问题 二维问题 10王战江王战江授课授课 6 滚动接触 两个同质材料的牵引滚动两个同质材料的牵引滚动 2 三维问题 三维问题 3 1 2 11 16 34 3 P Q Ga P x x 椭圆接触理论的蠕变率 窄带理论 将接触区沿着滚动方向划分成窄带 每个 窄带带看成一个二维问题 11王战江王战江授课授课 6 滚动接触 数值解可以实现数值解可以实现 2 三维问题 三维问题 Kalker s solutionWang s solution Wang Z J Jin X Q Keer L M and Wang Q 2012 A Numerical Approach For Analyzing Three Dimensional Steady State Rolling Contact Including Creep Using a Fast Semi Analytical Method Tribology Transactions 55 4 pp 446 457 参见文献参见文献 12王战江王战江授课授课 References 1 Wang Z J Jin X Q Keer L M and Wang Q 2012 A Numerical Approach For Analyzing Three Dimensional Steady State Rolling Contact Including Creep Using a Fast Semi Analytical Method Tribology Transactions 55 4 pp 446 457 2 Kalker J J 1990 Three Dimensional Elastic Bodies in Rolling Contact Springer 3 Johnson K L 1985 Contact Mechanics Cambridge University Press Cambridge UK 2016 12 20 1 1王战江王战江授课授课 接触力学 西南交通大学 王战江 2王战江王战江授课授课 7 粘着接触 JKR 接触 考虑接触界面的表面能 接触界面存在粘着力接触 考虑接触界面的表面能 接触界面存在粘着力 K L Johnson and K Kendall and A D Roberts Surface energy and the contact of elastic solids Proc R Soc London A 324 1971 301 313 3王战江王战江授课授课 典型的粘着接触理论典型的粘着接触理论 Bradley model of rigid contact Johnson Kendall Roberts JKR model Derjaguin Muller Toporov DMT model Maugis Dugdale model 7 粘着接触 4王战江王战江授课授课 7 粘着接触 3 0 9 00 3 8 z z z z z zF Assume that the surface force between two atomic planes at a distance z from each other can be derived from the Lennard Jones potential F is the force is the total surface energy of both surfaces per unit area z0is the equilibrium separation of the two atomic planes Bradley model applied the to find the force of adhesion between two rigid spheres The maximum adhesive force the pull off force Pc RFF ca 2 21 2 0 8 0 111 4 1 3 8 RRRz z z zR zFa Bradley model 5王战江王战江授课授课 7 粘着接触 Hertz接触公式接触公式 Johnson Kendall Roberts JKR model 2 0 21 21 2 21 23 2 2 2 2 1 2 1 1 0 21 21 21 3 0 16 9 11 4 3 P RR RR kk E k E k P RR RR kka 6王战江王战江授课授课 7 粘着接触 引入表面能引入表面能 surface energy Us 单位面积的表面能单位面积的表面能 x 物体运动的位移 约等于 物体运动的位移 约等于 Johnson Kendall Roberts JKR model 2 aUS dxdUF SS 2121 2 RRRRax 2121 RRRRFS 2016 12 20 2 7王战江王战江授课授课 7 粘着接触 两球体接触 令 则仅考虑Hertz接触时 3 13 23 22 RKPRa KRPa 0 3 0 KRPa 1 3 1 总能量总能量 total energy UT 弹性能 弹性能 elastic energy UE 机械能 机械能 mechanical energy UM 表面能 表面能 surface energy Us UT UE UM US Johnson Kendall Roberts JKR model 3 4 21 2121 kkK RRRRR 8王战江王战江授课授课 7 粘着接触 弹性能弹性能 elastic energy UE UE U1 U2 分两步 1 弹性加载到P1 对于接触半径a1 2 固定接触半径a1 载荷减小到P0 3 13 2 3 5 1 3 13 2 3 2 0 3 2 2 2 2 0 22 00 1 5 2 3 2 3 2 3 2 1 1 11 RK P dP RK P dP P RK RK P P dP RK P PPdU P P P 2 2 2 2 3 2 3 RK PdP d RK P Johnson Kendall Roberts JKR model 9王战江王战江授课授课 7 粘着接触 卸载时载荷 位移关系满足 3 1 1 2 0 2 1 3 13 2 1 3 13 2 3 2 2 3 1 3 2 3 2 1 0 1 0 P PP RK dP Ka P dP RK P U P P P P 3 1 1 Ka P R 3 1 15 1 1 3 1 5 2 3 1 1 2 0 3 5 1 3 13 2 3 1 1 3 13 2 2 0 2 1 3 13 2 3 5 1 PPP RK PRK PP RK P UE Johnson Kendall Roberts JKR model 10王战江王战江授课授课 7 粘着接触 机械能机械能 mechanical energy UM 表面能表面能 surface energy Us 总能量总能量 total energy UT Johnson Kendall Roberts JKR model 3 1 10 3 2 1 3 13 2 0 01 3 1 1 3 13 2 3 2 1 0 1 01 10 20 3 2 3 1 3 2 3 2 PPP RK P K PP RP K RK P P Ka PP P PUM 3 2 1 2 1 KRP aUS 3 2 3 2 1 3 2 3 1 1 2 0 3 2 10 3 13 2 3 1 1 2 0 3 5 1 3 13 2 3 2 3 1 3 1 15 11 K PR PP PP RK PPP RK UUUU SMET 11王战江王战江授课授课 7 粘着接触 最小能量满足最小能量满足 取正号 Hertz载荷真实载荷 Johnson Kendall Roberts JKR model 2 001 3 63RRPRPP 0 1 dadUT 0 1 dPdUT 1 2 010 2 0 2 1 3 1 3 2 3 4 1 3 1 1 3 2 3 2 3 4 1 2 0 3 1 10 3 4 1 2 0 3 2 1 3 1 3 2 1 622 9 3 2 9 2 9 2 9 1 9 11 RPPPPPP RK P P K R PPPPPPP RK dP dUT 2 0 2 001 2 001 2 1 3 3 0 3 2 PRPRPP PRPPP 12王战江王战江授课授课 7 粘着接触 真实载荷 1 当 当P 0时时 2 两个球体恰好分离时的载荷 特殊状态 两个球体恰好分离时的载荷 特殊状态 2 1 0 0 2 1 2 2 0 2 1 2 2 0 2 2 1 1 a E p R aE p a r p a r prp 压力分布压力分布 Johnson Kendall Roberts JKR model 23 3 63RRPRP K R a KRRa 6 3 RP 2 3 2016 12 20 3 13王战江王战江授课授课 Derjaguin Muller Toporov DMT model 7 粘着接触 粘着发生在接触区外部粘着发生在接触区外部 RP RP E R a 2 2 4 3 3 Tabor coefficient z0is the equilibrium separation between the two surfaces in contact The JKR theory applies to large compliant spheres for which is large The DMT theory applies for small stiff spheres with small values of Assume that the contact profile remains the same as in Hertzian contact but with additional attractive interactions outside the area of contact 3 1 3 0 2 2 0 zE R z dc 14王战江王战江授课授课 7 粘着接触 Maugis Dugdale model The work of adhesion energy per unit contact area where 1 2are the adhesive energies of the two surfaces and 12is an interaction term 1 2 12 where 0is the maximum force predicted by the Lennard Jones potential and h0is the maximum separation obtained by matching the areas under the Dugdale and Lennard Jones curves 0h0 Attractive force is constant for z0 z z0 h0 The ratio m is defined as m c a 15王战江王战江授课授课 7 粘着接触 Maugis Dugdale model The surface traction distribution is divided into two parts 1 Hertz contact pressure 2 The Dugdale adhesive stress Hertz contact is assumed in the region 0 r 0 and K 0 are given constants K is often called the plastic modulus The variable 0 T R is a nonnegative function of the amount of plastic flow slip called an internal hardening variable If K 0 and K is plastic modulus In many metals subjected to cyclic loading it is experimentally observed that the center of the yield surface experiences a motion in the direction of the plastic flow The evolution of the back stress q is defined by Ziegler s rule as H is called the kinematic hardening modulus 14王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Tangent elastoplastic modulus Elastoplastic tangent modulus Consistency condition 15王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 One Dimensional Rate Independent Plasticity Combined Kinematic and Isotropic Hardening Kinematic Hardening Plasticity 16王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 17王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 18王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 常微分方程数值解法 2016 12 20 4 19王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 隐式Euler 弹性试探解 20王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 1 2 21王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 2 并且 和 22王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm Return Mapping Algorithm for 1 D Rate Independent Plasticity Isotropic Hardening 23王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm Return Mapping Algorithm for One Dimensional Rate Independent Plasticity Combined Isotropic Kinematic Hardening 24王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 Return Mapping Algorithm 三维问题三维问题 K T条件 关联塑性 Associative plasticity 2016 12 20 5 25王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 J2 Plasticity Nonlinear Isotropic Kinematic Hardening q Equivalent plastic strain The center of the von Mises yield surface K Isotropic hardening modulus H Kinematic hardening modulus 26王战江王战江授课授课 8 弹塑性接触 塑性力学理论塑性力学理论 J2 Plasticity Nonlinear Isotropic Kinematic Hardening 径向返回算法径向返回算法 Radial Return Mapping 27王战江王战江授课授课 References 1 Simo J C Hughes T J R 1997 Computational Inelasticity Springer Verlag New York 2 Belytschko T Liu W K and Moran B 2000 Nonlinear Finite Elements for Continua and Structures Wiley New York 2016 12 20 1 1王战江王战江授课授课 接触力学 西南交通大学 王战江 2王战江王战江授课授课 接触模型介绍 Greenwood和Williamson 1966 GW模型 Persson B N J 2001 分形模型 有限元法 FEM Wriggers 1995 Liu等 2001 Pei等 2005 半解析法 SAM 确定性模型 统计模型 利用解析方法得到影响系数 使用快速傅里 叶变换 FFT 计算叠加效应 得到数值解 Kalker等 1972 Nogi等 1997 Polonsky等 1999 Hu等 1999 统计模型不能得到局部接触信息 粗糙峰之间的相互作用没有考虑 FEM求解粗糙表面接触需要划分大量网格 效率低 SAM仅需要在关心的区域划分网格 计算速度快 解析模型 解析模型 弹性点 线接触的解析解 弹性点 线接触的解析解 Hertz理论理论 Elasto plastic model 3王战江王战江授课授课 半解析法 SAM 利用解析的方法得到影响系数 使用FFT计算 叠加效应 得到数值解 本章研究目标 基于SAM并使用FFT对接触问题进行研究 弹性接触中的应力 位移关系弹性接触中的应力 位移关系 弹塑性接触中的残余应力和表面残余位移弹塑性接触中的残余应力和表面残余位移 层状材料弹性接触中的应力 位移关系层状材料弹性接触中的应力 位移关系 本研究通过半解析法得到以下关系 Elasto plastic model 4王战江王战江授课授课 Kuhn Tucker complementarity conditions Contact regions p 0 h 0 Non contact regions p 0 h 0 h0 initial body separation between two surfaces ue the surface displacement due to elastic effect up the permanent deformation due to plastic effect Plastic strain The plastic strains can be treated as eigenstrains p h Residual stress Contact model Surface gap h h0 ue up Elasto plastic model 5王战江王战江授课授课 Plastic strain p h Residual stress rp xdC ii r u k p D p Induced by elastic deformation K the pressure displacement influence coefficients Induced by plastic strain D the plastic displacement influence coefficients The plastic strains can be treated as eigenstrains 2 2 3 13 0ijijij I ijji xxe pAIFFT ii p Elastic stress Residual stress Stress Surface displacement 3D FFT for the four terms Elasto plastic model 6王战江王战江授课授课 1 Conjugate Gradient Method CGM is applied to minimize the quadratic function with inequality constrain 2 Fast Fourier Transform FFT is used to calculate ue upand sub surface stresses and residual stress h h0 k p D p hi j 0 pi j 0 hi j pi j 0 Pi j W Plastic strain p h Residual stress Elasto plastic model 2016 12 20 2 7王战江王战江授课授课 共轭梯度法求互补问题 FFT算法加快弹性变形 ue 塑性变形 up 和应力的计算速度 H H0 C P D p Hi j 0 pi j 0 Hi j pi j 0 残余变形 弹性变形 1 共轭梯度法求解互补问题得到压力分布共轭梯度法求解互补问题得到压力分布 4 径向返回算法求解塑性应变径向返回算法求解塑性应变 3 基于半解析法求解残余应力基于半解析法求解残余应力 5 根据体内塑性应变计算表面残余位移 根据体内塑性应变计算表面残余位移 6 将残余变形叠加到表面形貌将残余变形叠加到表面形貌 求解流程 求解流程 p r 2 基于半解析法求解压力引起的应力基于半解析法求解压力引起的应力 求解线性互补问题 残余应力求解方案残余应力求解方案 互补算法推广到弹塑性接触的关键 表面残余变形的计算 基于SAM实现了残余应力和残余变形 的求解 Elasto plastic model 8王战江王战江授课授课 ikkjjii FFu 1 2 2 x 1 ikkjjkkijikkjimmkkji FFFF x jikkji ikkjjkkijikkjimmkkji ee FFFF 2 1 x 2 Stresses outside 3 Stresses inside The elastic field due to eigenstrains can be expressed in terms of Galerkin vectors F Yu and Sanday 1991 1 Displacements Yu H Y and Sanday S C 1991 Proc R Soc London Ser A 434 1892 pp 521 530 Displacement and stress induced by eigenstrains Cuboidal inclusion in half space 9王战江王战江授课授课 A cuboidal inclusion in an elastic half space 1 Y P Chiu s solution JAM 1978 Complicated formulation with recursive relations involving Legendre polynomials Did not explicitly demonstrate the convolution correlation properties in the depth direction 2 S Liu RRRRRR R 21 D II R 1 Cuboidal inclusion in half space 1H User defined variables Half space model 11王战江王战江授课授课 ijijij I ijij xx 2 2 3 13 0 When solving multiple cubic eigenstrain problem one can conduct combined 3D DC FFT for the first term 3D DCR FFT for the second term take x3out and do 3D DCR FFT for the third term and take x32out and do 3D DCR FFT for the forth term where i and j represent the six components 11 22 33 12 13 23 and Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity In the half space the Galerkin vectors include two parts one is induced by the infinite space and the other by the mirror domain Finally the displacements and stresses can be expressed as Liu et all 2012 Induced by the infinite spaceInduced by the mirror domain ijijij I iji xx 2 2 3 13 0 U Half space model Cuboidal inclusion in half space x de H ij Out ij 4 xUde H u ii 8 1 12王战江王战江授课授课 A cuboidal inclusion in an elastic half space Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity Cuboidal inclusion in half space xUde H u ii 8 1 2016 12 20 3 13王战江王战江授课授课 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity Cuboidal inclusion in half space jikkji Out ij In ij ee 1 2 2 xx x de H ij Out ij 4 where 14王战江王战江授课授课Surface displacements due to Eigenstrain Surface element quantities are evaluated at the center of the upper surface of the element 1 2 ss ii ue d Ux 1 111 1 221 1 1 1 3 2 121 2 1 2 0 T s D D D UR 2 112 2 222 2 2 2 3 1 122 2 2 1 0 T s D D D UR 3 11 3 22 3 3 3 2 1 12 2 2 3 T s D D D UR Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity 15王战江王战江授课授课 Analytical solution vs FEM Displacement Stress Stress Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity Displacement and stress Liu et al 2012 16王战江王战江授课授课 Comparison of computational efficiency Comparisons of the computational efficiency between the present analytical solution with Chiu s solution Liu et al 2012 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity 17王战江王战江授课授课 Mesh density test Elastic plastic model Liu et al 2012 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity 18王战江王战江授课授课 Mesh density test Residual stress Liu et al 2012 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity Test the dependence of the solution on grids size 16 16 16 32 32 32 64 64 64 128 128 128 2016 12 20 4 19王战江王战江授课授课 Residual stress Liu et al 2012 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 International journal of plasticity Test the dependence of the solution on grids size 32 32 32 mesh can roughly offer a reasonable prediction for the stress variation versus the depth Mesh density test 20王战江王战江授课授课 光滑表面接触 粗糙表面接触 接触面积 粗糙表面弹塑性接触 21王战江王战江授课授课 References 1 Wang Z J Jin X Q Zhou Q H Ai X L Keer L M and Wang Q 2013 An Efficient Numerical Method With a Parallel Computational Strategy for Solving Arbitrarily Shaped Inclusions in Elasto plastic Contact Problems ASME Journal of Tribology 135 3 p 031401 2 Liu S B Jin X Q Wang Z J Keer L M and Wang Q 2012 Analytical Solution for Elastic Fields Caused by Eigenstrains in a Half Space and Numerical Implementation Based on FFT International Journal of Plasticity 35 pp 135 154 3 Wang Z J Wang W Z Hu Y Z and Wang H 2010 A Numerical Elastic Plastic Contact Model for Rough Surfaces Tribology Transactions 53 2 pp 224 238 2016 12 20 1 1王战江王战江授课授课 接触力学 西南交通大学 王战江 2王战江王战江授课授课 9 Viscoelastic contact Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation 弹性通常是在一个有序的固体中沿 晶面键的伸缩 粘度通常是非晶质材料内部的原子 或分子扩散的结果 3王战江王战江授课授课 9 Viscoelastic contact 弹性材料 粘弹性材料 加载加载 卸载曲线卸载曲线 4王战江王战江授课授课 Creep the stress is held constant the strain increases with time Relaxation the strain is held constant the stress decreases with time 9 Viscoelastic contact 5王战江王战江授课授课 9 Viscoelastic contact Strain OA initial elastic response AB delayed elastic strain BC creep strain CD an immediate elastic response when the stress is removed DE delayed elastic response E a permanent strain 典型粘弹性应力 应变随时间变化曲线典型粘弹性应力 应变随时间变化曲线 6王战江王战江授课授课 9 Viscoelastic contact 粘弹性模型的基本元件粘弹性模型的基本元件 线性弹簧线性弹簧 Hookean elastic spring 粘滞阻尼器粘滞阻尼器 viscous damper E dt d 2016 12 20 2 7王战江王战江授课授课 1 Maxwell model 9 Viscoelastic contact 适用 热塑性聚合物 新鲜混凝土 忽略其老 化 金属在温度接近其熔点 适用 热塑性聚合物 新鲜混凝土 忽略其老 化 金属在温度接近其熔点 dt d Edt d dt d dt d SDTotal SDTotal SDTotal 1 8王战江王战江授课授课 A Maxwell material is suddenly deformed and held to a strain of 0 9 Viscoelastic contact 1 Maxwell model A Maxwell material is suddenly subjected to a stress 0 阶跃响应阶跃响应 00 t E t 0 Tt Eet Relaxation time T E 9王战江王战江授课授课 2 Kelvin Voigt model 9 Viscoelastic contact 适用 有机聚合物 橡胶 木材 当负载 不太高 适用 有机聚合物 橡胶 木材 当负载 不太高 Apply a constant stress 0 Rate of relaxation E Free the material a time t1 dt td tEt SDTotal SDTotal 1 0 t e E t 11 1 tt ettt 10王战江王战江授课授课 3 Standard linear solid model 9 Viscoelastic contact 21 1 2 2 EE E dt d E E dt d 11王战江王战江授课授课 4 Generalized Maxwell model 9 Viscoelastic contact 应用 金属和合金的在温度低于绝对熔化温度 以 应用 金属和合金的在温度低于绝对熔化温度 以K表示 的四分之一 表示 的四分之一 12王战江王战江授课授课 5 Kelvin Chain Model 9 Viscoelastic contact 2016 12 20 3 13王战江王战江授课授课 9 Viscoelastic contact t 松弛函数松弛函数 Relaxation function t 蠕变函数蠕变函数 Creep function 考虑聚合物 考虑聚合物 polymer 的泊松比一般都大于 的泊松比一般都大于0 4 应力应力 应变关系用偏张量形式表示应变关系用偏张量形式表示 对于不可压缩粘弹性材料对于不可压缩粘弹性材料 0 t te t sttdt t 0 t ts t ettdt t s 偏应力分量偏应变 e 123 1 3 123 1 3 22sGeG G是剪切模量 14王战江王战江授课授课 Prony series 9 Viscoelastic contact Shear stress relaxation modulus N i ii tGGtG 1 exp N i i GGGtG 1 0 0 exp 1 1 0i N i i tGGtG 15王战江王战江授课授课 9 Viscoelastic contact 三参数固体三参数固体 两个模量为g1和g2的弹簧与一个粘滞性为 的阻尼器联接组成 蠕变响应对阶跃变化应力s0的表达式为 T1 g2 对阶跃变化应变e0的响应为 T2 g1 g2 0212 21 1 0 exp eTtgg gg g etts 01 21 0 exp 1 11 sTt gg stte 16王战江王战江授课授课 9 Viscoelastic contact 常用粘弹性模型常用粘弹性模型 17王战江王战江授课授课 9 Viscoelastic contact 弹性解 粘弹性解 弹性解 粘弹性解 压力 载荷 接触半径 压力 载荷 接触半径 RP G Ra 2 1 8 3 2 33 2 122 2 4 raG R rp t t dtP t d d ttRta 0 3 8 3 t t drta t d d tt R trp 0 2 1 22 4 t t dta t d d tt R tP 0 3 3 8 18王战江王战江授课授课 9 Viscoelastic contact 三参数固体 Three parameter solid Maxwell Three parameter solid Viscous solid 01 21 0 exp 1 11 sTt gg stte 21 gT 0 212 21 1 0 exp eTtgg gg g etts 212 ggT 1 11 8 3 21 0 3Tt e gg RPta 2016 12 20 4 19王战江王战江授课授课 9 Viscoelastic contact Maxwell model 20王战江王战江授课授课 1 解析法 解析法 解析法发展得最早 它的优点是可得到准确的结果准确的结果 对于分 析方程及其解答的数学性质很有用 缺点是只适用于简单的边界 条件和简单的方程 难度主要来源于偏微分算子和边界的形状 特别是边界的几何形状 因此目前只求出了少数的解析解 2 数值法 数值法 数值方法是一种近似方法 在电子计算机出现之后 数值方 法发展很快 特别是产生了有限元法有限元法 它处理复杂的边界形状没 有原则上的困难 是一种普遍的方法 但有限元法并非完美无缺 比如用它计算接触问题计算量很大计算量很大 应力及靠近边界处的位移算 得不够准确 3 半解析法 半解析法 把解析法和数值法结合起来就产生半解析法 半解析法吸收 了解析法结果准确和数值方法 主要是有限元法 处理边界灵活处理边界灵活 的优点 可以在明显减少计算量减少计算量的同时提高计算精度提高计算精度 受到了人 们的重视 半解析法是综合了数值解和分析解的长处的一种方法 其方法的原理在于 首先得到一个单位激励响应解单位激励响应解 通常是以格 林函数 格 林函数的形式 最后的解由所有单位解的叠加叠加得到 并使用快速 傅里叶变换 快速 傅里叶变换完成叠加算法 9 Viscoelastic contact 21王战江王战江授课授课 粘弹性接触求解采用与弹性接触比拟 的方法 即将粘