﻿ 摩擦接触问题的比例边界等几何B可微方程组方法
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 力学学报  2016, Vol. 48 Issue (3): 615-623  DOI: 10.6052/0459-1879-15-329 0

### 引用本文 [复制中英文]

[复制中文]
Xue Binghan, Lin Gao, Hu Zhiqiang, Pang Lin. ANALYSIS OF FRICTIONAL CONTACT PROBLEMS BY SBIGA-BDE METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 615-623. DOI: 10.6052/0459-1879-15-329.
[复制英文]

### 文章历史

2015-09-01 收稿
2015-11-03 录用
2015-11-05 网络版发表.

0 引言

1 接触问题的基本方程

 $\mathbf{L}\text{d}{{\sigma }^{\alpha }}+\text{d}{{\bar{b}}^{\alpha }}=\mathbf{0}$ (1)

 $\text{d}{{\varepsilon }^{\alpha }}=\widehat{\mathbf{L}}\text{d}{{u}^{\alpha }}$ (2)

 $\text{d}{{\sigma }^{\alpha }}=D\text{d}{{\varepsilon }^{\alpha }}$ (3)

 $\text{d}{{u}^{\alpha }}=\text{d}{{\bar{u}}^{\alpha }}$ (4)
 ${{({{n}^{\alpha }})}^{\text{T}}}\text{d}{{\sigma }^{\alpha }}=\text{d}{{\bar{q}}^{\alpha }}$ (5)

 $\left. \begin{matrix} \text{d}u_{n}^{1}-\text{d}u_{n}^{2}+{{\delta }_{n}}\ge 0,\ \ {{p}_{n}}\le 0 \\ {{p}_{n}}\left( \text{d}u_{n}^{1}-\text{d}u_{n}^{2}+{{\delta }_{n}} \right)=0 \\ \left| {{p}_{\tau }} \right|<-\mu {{p}_{n}}\Rightarrow \left| \text{d}u_{\tau }^{1}-\text{d}u_{\tau }^{2}+{{\delta }_{\tau }} \right|=0 \\ \left| {{p}_{\tau }} \right|=-\mu {{p}_{n}}\Rightarrow \left| \text{d}u_{\tau }^{1}-\text{d}u_{\tau }^{2}+{{\delta }_{\tau }} \right|\ge 0 \\ \end{matrix} \right\}$ (6)

2 B样条与非均匀有理B样条基函数 2.1 B样条

B样条基函数通过一组节点向量进行定义
 $s=\left[ {{s}_{0}},{{s}_{1}},\cdots ,{{s}_{n+p}} \right]$ (7)

 $\left. \begin{array}{*{35}{l}} {{N}_{i,0}}=\left\{ \begin{array}{*{35}{l}} 1, & s\in [{{s}_{i}},{{s}_{i+1}}] \\ 0, & 否则 \\ \end{array} \right.\text{ } \\ {{N}_{i,p}}=\frac{s-{{s}_{i}}}{{{s}_{i+p}}-{{s}_{i}}}{{N}_{i,p-1}}(s)+ \\ ~~~~~~~~~~\frac{{{s}_{i+p+1}}-s}{{{s}_{i+p+1}}-{{s}_{i+1}}}{{N}_{i+1,p-1}}(s),\ p\ge 1 \\ \end{array} \right\}$ (8)

B样条基函数的基本性质：非负性，单位分解性，高阶连续性，紧支性，线性无关性等.

2.2 非均匀有理B样条基函数

 $R_{i}^{p}\left( s \right)=\frac{{{\omega }_{i}}{{N}_{i,p}}\left( s \right)}{\sum\limits_{j=0}^{n}{{{\omega }_{j}}{{N}_{j,p}}\left( s \right)}}$ (9)

 $C\left( s \right)=\sum\limits_{i=0}^{n}{R_{i}^{p}\left( s \right)}{{P}_{i}}$ (10)

 图1 二次非均匀有理B样条曲线 Fig.1 Quadratic NURBS curve
 图2 二次非均匀有理B样条基函数 Fig.2 Quadratic NURBS basis function
3 弹性静力摩擦接触问题的比例边界等几何分析方程推导 3.1 比例边界坐标变换

 图3 比例边界等几何分析分析计算示意图 Fig.3 Sketch for scaled boundary isogeometric analysis

 $\left. \begin{array}{*{35}{l}} \hat{x}=\xi (x(\eta )-{{{\hat{x}}}_{0}}))+{{{\hat{x}}}_{0}}=\xi (R(\eta )x-{{{\hat{x}}}_{0}})+{{{\hat{x}}}_{0}} \\ \hat{y}=\xi (y(\eta )-{{{\hat{y}}}_{0}}))+{{{\hat{y}}}_{0}}=\xi (R(\eta )y-{{{\hat{y}}}_{0}})+{{{\hat{y}}}_{0}} \\ \end{array} \right\}$ (11)

 $\hat{J}(\xi ,\eta )=\left[ \begin{matrix} {{{\hat{x}}}_{,\xi }} & {{{\hat{y}}}_{,\xi }} \\ {{{\hat{x}}}_{,\eta }} & {{{\hat{y}}}_{,\eta }} \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & {} \\ {} & \xi \\ \end{matrix} \right]{{J}_{\xi }}(\eta )$ (12)

 $J_{\xi }^{-1}(\eta )=\left[ \begin{matrix} {{j}_{11}} & {{j}_{12}} \\ {{j}_{21}} & {{j}_{22}} \\ \end{matrix} \right]$ (13)

 $\text{d}\Omega =\left| {\hat{J}} \right|\text{d}\xi \text{d}\eta =\xi \left| J \right|\text{d}\xi \text{d}\eta$ (14)

 $\mathbf{L}={{b}_{1}}\frac{\partial }{\partial \xi }+\frac{1}{\xi }{{b}_{2}}\frac{\partial }{\partial \eta }$ (15)

 ${{b}_{1}}=\left[ \begin{matrix} {{j}_{11}} & {} \\ {} & {{j}_{21}} \\ {{j}_{21}} & {{j}_{11}} \\ \end{matrix} \right],\quad {{b}_{2}}=\left[ \begin{matrix} {{j}_{12}} & {} \\ {} & {{j}_{22}} \\ {{j}_{22}} & {{j}_{12}} \\ \end{matrix} \right]$ (16)

 $\text{d}u(\xi ,\eta )=R\text{d}u(\xi )$ (17)

 $\text{d}\varepsilon =\mathbf{L}\text{d}u(\xi )={{B}_{1}}\text{d}u{{(\xi )}_{,\xi }}+\frac{1}{\xi }{{B}_{2}}\text{d}u(\xi )$ (18)

 $\left. \begin{array}{*{35}{l}} {{B}_{1}}={{b}_{1}}R \\ {{B}_{2}}={{b}_{2}}{{R}_{,\eta }} \\ \end{array} \right\}$ (19)

3.2 域边界等几何离散

 \begin{align} & \begin{matrix} \int_{{{\Omega }^{1}}+{{\Omega }^{2}}}{\delta \text{d}{{\varepsilon }^{\text{T}}}\text{d}\sigma }\text{d}\Omega -\int_{{{\Omega }^{1}}+{{\Omega }^{2}}}{\delta \text{d}{{u}^{\text{T}}}\text{d}\bar{b}}\text{d}\Omega - \\ ~\int_{\Gamma _{\text{bq}}^{1}+\Gamma _{\text{bq}}^{2}}{\delta \text{d}{{u}^{\text{T}}}\text{d}{{{\bar{q}}}_{\text{b}}}}\text{d}\Gamma -\int_{\Gamma _{\text{sq}}^{1}+\Gamma _{\text{sq}}^{2}}{\delta \text{d}{{{\hat{u}}}^{\text{T}}}\text{d}{{{\bar{q}}}_{\text{s}}}}\text{d}\xi - \\ \end{matrix} \\ & \int_{{{\Gamma }_{\text{c}}}}{{{(\delta \text{d}u_{\text{c}}^{1}-\delta \text{d}u_{\text{c}}^{2})}^{\text{T}}}}\text{d}{{p}_{\text{c}}}\text{d}\Gamma =0\text{ } \\ \end{align} (20)
 $\text{d}{{u}_{i}}\in {{D}_{\text{u}}}=\left\{ \text{d}{{u}_{i}}:\text{d}{{u}_{i}}=\text{d}{{{\bar{u}}}_{i}}\quad {{x}_{i}}\in {{\Gamma }_{\text{u}}} \right\}~$ (21)

 \begin{align} & \begin{array}{*{35}{l}} {{\left. \delta \text{d}{{u}^{\text{T}}}({{E}_{0}}\text{d}{{u}_{,\xi }}+{{({{E}_{1}})}^{\text{T}}}\text{d}u-\text{d}P-\text{d}R) \right|}_{\xi =1}}- \\ \int_{{{\xi }_{0}}}^{{{\xi }_{1}}}{\{\delta }\text{d}u{{(\xi )}^{\text{T}}}\frac{1}{\xi }[{{E}_{0}}{{\xi }^{2}}\text{d}u{{(\xi )}_{,\xi \xi }}+ \\ ({{E}_{0}}+{{({{E}_{1}})}^{\text{T}}}-{{E}_{1}})\xi \text{d}u{{(\xi )}_{,\xi }}- \\ \end{array} \\ & {{E}_{2}}\text{d}u(\xi )+\text{d}F(\xi )]\}\text{d}\xi =0~~ \\ \end{align} (22)

 $\left. \begin{array}{*{35}{l}} {{E}_{0}}=\int_{{{\eta }_{i}}}^{{{\eta }_{i+1}}}{{{B}^{1}}{{(\eta )}^{\text{T}}}D{{B}^{1}}(\eta )\left| J \right|\text{d}\eta } \\ {{E}_{1}}=\int_{{{\eta }_{i}}}^{{{\eta }_{i+1}}}{{{B}^{2}}{{(\eta )}^{\text{T}}}D{{B}^{1}}(\eta )\left| J \right|\text{d}\eta } \\ {{E}_{2}}=\int_{{{\eta }_{i}}}^{{{\eta }_{i+1}}}{{{B}^{2}}{{(\eta )}^{\text{T}}}D{{B}^{2}}(\eta )\left| J \right|\text{d}\eta } \\ \text{d}P=\int_{\Gamma _{\text{bq}}^{\text{1e}}+\Gamma _{\text{bq}}^{\text{2e}}}{{{R}^{\text{T}}}\text{d}{{{\bar{q}}}_{\text{b}}}(\eta )}\text{d}\Gamma \\ \text{d}R=\int_{\Gamma _{\text{c}}^{\text{e}}}{{{R}^{\text{T}}}\text{d}{{p}_{\text{c}}}(\eta )\text{d}\Gamma } \\ \text{d}F(\xi )={{\xi }^{2}}\int_{\Gamma _{\text{b}}^{\text{e}}}{{{R}^{\text{T}}}\text{d}\bar{b}(\xi ,\eta )\left| J \right|\text{d}\eta }+\xi \text{d}{{{\bar{q}}}_{\text{s}}}(\xi ) \\ \end{array} \right\}$ (23)

 \begin{align} & {{E}_{0}}{{\xi }^{2}}\text{d}u{{(\xi )}_{,\xi \xi }}+({{E}_{0}}+{{({{E}_{1}})}^{\text{T}}}-{{E}_{1}})\xi \text{d}u{{(\xi )}_{,\xi }}- \\ & {{E}_{2}}\text{d}u(\xi )+\text{d}F(\xi )=0 \\ \end{align} (24)
 ${{E}_{0}}\text{d}{{u}_{,\xi }}+{{({{E}_{1}})}^{\text{T}}}\text{d}u-\text{d}P-\text{d}R=0~$ (25)

4 弹性静力摩擦接触问题的求解 4.1 比例边界等几何分析方程的求解

 $\text{d}f(\xi )={{E}_{0}}\xi \text{d}{{u}_{,\xi }}+{{({{E}_{1}})}^{\text{T}}}\text{d}u$ (26)

 $\xi X{{(\xi )}_{,\xi }}=-ZX(\xi )~$ (27)
 $Z=\left[ \begin{matrix} {{({{E}_{0}})}^{-1}}{{({{E}_{1}})}^{\text{T}}} & -{{({{E}_{0}})}^{-1}} \\ {{E}_{1}}{{({{E}_{0}})}^{-1}}{{({{E}_{1}})}^{\text{T}}}-{{E}_{2}} & -{{E}_{1}}{{({{E}_{0}})}^{-1}} \\ \end{matrix} \right]$ (28)
 $X(\xi )=\left\{ \begin{matrix} \text{d}u(\xi ) \\ \text{d}f(\xi ) \\ \end{matrix} \right\}~$ (29)

 $Z\Phi =\Phi \text{ }\Lambda =\left[ \begin{matrix} \text{ }{{\Phi }_{11}} & \text{ }{{\Phi }_{12}} \\ \text{ }{{\Phi }_{21}} & \text{ }{{\Phi }_{22}} \\ \end{matrix} \right]\left[ \begin{matrix} {{\lambda }_{n}} & {} \\ {} & {{\lambda }_{p}} \\ \end{matrix} \right]$ (30)

 $X(\xi )=\left[ \begin{matrix} \text{ }{{\Phi }_{11}} & \text{ }{{\Phi }_{12}} \\ \text{ }{{\Phi }_{21}} & \text{ }{{\Phi }_{22}} \\ \end{matrix} \right]\left[ \begin{matrix} {{\xi }^{-{{\lambda }_{n}}}} & {} \\ {} & {{\xi }^{-{{\lambda }_{p}}}} \\ \end{matrix} \right]\left\{ \begin{matrix} {{c}_{1}} \\ {{c}_{2}} \\ \end{matrix} \right\}$ (31)

 $\left. \begin{matrix} \text{d}u(\xi )={{\Phi }_{11}}{{\xi }^{-{{\lambda }_{n}}}}{{c}_{1}}+{{\Phi }_{12}}{{\xi }^{-{{\lambda }_{p}}}}{{c}_{2}} \\ \text{d}f(\xi )={{\Phi }_{21}}{{\xi }^{-{{\lambda }_{n}}}}{{c}_{1}}+{{\Phi }_{22}}{{\xi }^{-{{\lambda }_{p}}}}{{c}_{2}} \\ \end{matrix} \right\}$ (32)

 $\text{d}f(\xi =1)={{E}_{0}}\text{d}{{u}_{,\xi }}+{{({{E}_{1}})}^{\text{T}}}\text{d}u=\text{d}P+\text{d}R$ (33)

 $K\text{d}u-\text{d}P-\text{d}R=0$ (34)
 $K={{\Phi }_{21}}{{({{\Phi }_{11}})}^{-1}}$ (35)

4.2 摩擦接触条件的B可微方程组

 $H_{2}^{i}=\min \left\{ r\Delta \text{d}u_{n}^{i},p_{n}^{i} \right\}=0$ (36)
 $H_{3}^{i}=p_{\tau }^{i}-\lambda p_{\tau }^{i}\left( r \right)=0~$ (37)

 $\left. \begin{array}{*{35}{l}} p_{\tau }^{i}\left( r \right)=p_{\tau }^{i}-r\Delta \text{d}u_{\tau }^{i} \\ \lambda =\min \left\{ \mu \max \left\{ p_{n}^{i},0 \right\}/\left| p_{\tau }^{i} \right|,1 \right\} \\ \end{array} \right\}$ (38)

 $H\left( x \right)={{\left\{ {{H}_{1}},{{H}_{2}},{{H}_{3}} \right\}}^{\text{T}}}={{\left\{ 0,0,0 \right\}}^{\text{T}}}$ (39)

4.3 真实接触区域的识别

 图4 真实接触区域的识别 Fig.4 The identification of the real contact surfaces

5 数值算例 5.1 精确的几何描述

 图5 圆弧的离散模型 Fig.5 The discrete model of the arc
 图6 几何形状描述的准确性对比 Fig.6 Comparison of the exactness of geometric representation
5.2 赫兹接触问题

 图7 圆柱体与刚性基础接触 Fig.7 Contact of the cylinder and rigid base

 $a=2\sqrt{\frac{PR\left( 1-{{\nu }^{2}} \right)}{\pi LE}}$ (40)
 $F=\frac{P}{2}-\int_{r}^{a}{\frac{2P}{\pi {{a}^{2}}L}}\sqrt{{{a}^{2}}-{{x}^{2}}}\text{d}x,\quad r\in \left[ 0,a \right]$ (41)

 图8 赫兹问题接触力合力对比 Fig.8 Comparison of total contact force for Hertz problem
5.3 悬臂梁接触问题

 图9 二维悬臂梁接触 Fig.9 Contact of two cantilever beams

 图10 $X$向位移分布图 Fig.10 Distribution of $X$-displacemen
 图11 $Y$向位移分布图 Fig.11 Distribution of $Y$-displacement
6 结论

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ANALYSIS OF FRICTIONAL CONTACT PROBLEMS BY SBIGA-BDE METHOD
Xue Binghan, Lin Gao, Hu Zhiqiang, Pang Lin
Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China
Abstract: Frictional contact analysis is one of the most challenging problems in computational mechanics. The functional system of the contact problem is not only nonlinear, but also non-smooth, so in general the convergence and accuracy of contact algorithms are di cult to be guaranteed. For 2D elastic frictional contact problem, the scaled boundary isogeometric analysis combined with B di erential equation method (SBIGA-BDE method) is developed. Based on the scaled boundary isogeometric transformation, the contact equilibrium equation is derived by using virtual principle. The contact conditions are formulated as B di erential equation and satisfied rigorously. The convergence of the algorithm to solve the B di erential equation is guaranteed by the theory of mathematical programming. In the proposed method, only the outer boundary including the contact boundary need to be discretized isogeometrically, which reduces the spatial dimension by one and the boundary are represented accurately. The real contact length can be detected by the knot insertion algorithm. In addition, as the interpolatory functions used in geometry modeling and numerical analyzing are the same, the time costs in mesh generation is saved. The numerical examples, including Hertz contact problem and cantilever beams frictional contact problem, are presented and compared with analytic solution and ANSYS results. It validates the e ectiveness and accuracy of the proposed method in solving 2D elastic frictional contact problem.
Key words: elastic frictional contact problem    scaled boundary isogeometric analysis    scaled boundary finite element method    B dierential equation    numerical modeling