CONTACT ANALYSIS OF POROUS HALF-SPACE UNDER HOLLOW COLUMN LOADING
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摘要: 多孔结构具有诸多优良特性被广泛应用于航空航天、生物医疗以及各种工程装备中, 而空心柱具有独特的结构常被用于承担结构载荷. 因此, 研究空心柱与多孔半空间的接触问题显得尤为重要. 文章运用Hankel变换将轴对称接触问题转化为积分方程的求解问题, 推导出表面接触应力和位移的精确表达式. 发展了高斯-切比雪夫结合乘积型Bessel函数的无穷积分的数值求解方法, 并退化对比验证了方法的正确性, 结果表明该方法在奇异值处的计算精度和效果更佳. 数值分析了表面接触应力随泊松比、孔隙率、力载荷、内径、壁厚(空心圆柱)和半径(碗形抛物柱)的变化情况, 并给出了位移随孔隙率、内径、壁厚、半径及深度的变化情况. 研究结果表明基体的接触区域中心产生明显的叠加变形, 并且接触区域外侧的奇异性要高于内侧, 所以外侧更有可能是裂纹的起始位置. 此外, 在较小的内径下, 当空心柱壁厚与实心柱半径一致时, 空心柱比实心柱所导致的基体接触区域外边缘的应力奇异性小1倍以上, 并且随着孔隙率的增大该比值降低; 当空心柱外径与实心柱半径一致时, 空心柱比实心柱所导致的基体接触区域外边缘的应力奇异性大1倍以上, 并且在大孔隙下该比值更大. 研究结果对多孔材料的设计和应用具有重要的指导意义.
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关键词:
- 空心柱 /
- 多孔材料轴对称接触 /
- micro-dilatation 理论 /
- 积分方程方法
Abstract: Porous structures, owing to their excellent characteristics, are widely used in aerospace, biomedical, and various engineering applications. Hollow columns, due to their unique structure, are extensively employed to bear structural loads. Therefore, the study of the axisymmetric contact problem between hollow columns and porous half space is particularly important. In this paper, the Hankel transform is used to transform the contact problem into the problem of solving an integral equation, and exact expressions for surface contact stress and displacement are derived. The Gauss-Chebyshev method coupled with the infinite integration of product-type Bessel functions for solving integral equations is developed, and degradation comparisons are carried out to verify the correctness of the method. The results indicate that the method has better computational accuracy and performance at singular points. The numerical results show the variations of surface contact stress with respect to Poisson's ratio, porosity, mechanical loading, inner diameter, wall thickness (hollow cylinder), and radius (bowl shaped parabolic column). The study additionally provides insights into the displacement variation concerning porosity, inner diameter, wall thickness, radius, and depth. The results further indicate that the center of the contact region of the substrate has obvious superimposed deformation, and the singularity of the outer side of the contact region is higher than that of the inner side. Therefore, it is more likely that the outer side is the initial position for cracks. Furthermore, for a smaller inner diameter, when the wall thickness of the hollow column matches the radius of the solid column, the stress singularity at the outer edge of the contact area of the substrate caused by the hollow column is more than 1 times smaller compared to that caused by the solid column, and this ratio decreases as the porosity increases. On the other hand, when the outer diameter of the hollow column matches the radius of the solid column, the stress singularity at the outer edge of the contact area of the substrate caused by the hollow column is more than 1 times larger compared to that caused by the solid column, and this ratio increases with an increase in the porosity. The research results have important guiding significance in the design and application of porous materials. -
引 言
多孔或颗粒状结构的材料在刚度、强度、轻质、韧性、冲击吸能、隔热防热、减振降噪、可设计性以及生物相容性等方面展现优良的特性, 被广泛应用于建筑工业、航空航天、生物医疗及各种军、民用工程装备中[1-3], 也被国际上认为是最有应用前景的新型结构之一, 引起广大学者的浓厚兴趣[4-7].
随着智能材料的快速发展, 压头与智能材料的接触研究已有重要进展[8-12], 多孔材料中的接触问题也同样得到了研究者非常广泛的关注. Biot等[13]用于模拟饱和流体多孔固体的固结理论和无流体多孔弹性micro-dilatation理论[14-15]已被广泛用于多孔材料的断裂和接触分析. 在无摩擦micro-dilatation理论线性多孔弹性模型的背景下, 前人研究了均质各向同性多孔材料的诸多轴对称接触问题[16-20]. Ieşan等[16]利用Hankel积分变换研究了多孔半空间的狄利克雷和诺伊曼型的轴对称接触问题, 讨论了径向位移和体积分数函数的行为. Chebakov等[17-18]讨论了刚性压头(底部呈平面或抛物面)与多孔弹性半空间以及刚性压头与多孔弹性层相互作用的轴对称无摩擦接触问题, 利用积分方程方法, 研究了压头的作用力与其位移之间的关系, 并对不同孔隙率的影响进行了比较分析. Kolosova等[19]研究了刚性压头(底部呈平面或抛物面)与孔弹性层相互作用的轴对称无摩擦接触问题, 得到接触应力和接触面积的解析表达式, 给出了压头上作用力与其位移之间的关系. 在此工作基础上, Chebakov等[20]进一步推广了Kolosova等[19]的结果, 研究了压头与固定在多孔弹性基上的弹性层的轴对称接触问题, 分析了弹性层与多孔弹性半空间界面上的应力响应.
空心柱作为一种常见的支撑形式, 具有自重轻、基础载荷小及节约材料等特点, 可以提供稳定的支撑和较好的抗压、抗弯与抗扭能力, 所以被广泛用于承担一定的结构载荷, 如桥梁、建筑物的支撑, 大型高铁站和航站楼等底部大空间结构等. 王学峰等[21]采用多尺度方法研究了空心柱-箱型转换层-剪力墙结构的抗震性能, 分析了空心柱的受力情况. 相泽辉等[22]采用将纤维、轻骨料混凝土以及空心柱相结合的思路, 研究了纤维增韧轻骨料混凝土空心柱的抗震性能试验. 刘小立[23]开展了钢纤维高强轻骨料混凝土角钢组合空心柱的抗震性能研究, 表明适当的轴压比和空心率会显著提高抗震性能和承载性能. Ahmad等[24]研究了空心玻璃纤维增强聚合物混凝土柱的轴向性能, 得知空心玻璃纤维增强聚合物混凝土柱比实心土柱具有更大的轴向载荷和变形能力. Ahmed等[25]提出了一种统一的数值模型来预测不同形状(包括矩形、圆形和圆端形)的钢管混凝土柱的性能, 将不同形式的截面转换为等效圆柱, 分析了钢管混凝土柱不同半径比和长宽比对其性能影响. 此外, 空心柱压头也可以用于对材料进行压缩或弯曲测试, 如空心柱力学性能研究[26-31]、空心柱轴向压缩性能研究[32-33]、空心圆柱的瞬态响应[34-35]与静力学响应[36]等.
注意到空心柱与多孔结构相结合的接触研究工作相对较少. 而空心柱与多孔材料的接触问题的理论研究能够为空心柱相关结构提供可参考的理论支撑. 所以, 本文利用积分方程方法研究空心圆柱和碗形抛物柱与多孔弹性半空间的接触问题, 主要讨论了材料参数、压痕深度、半径、空心柱的壁厚与内径等对接触变形的影响.
1. 模型及问题的基本解
假设多孔材料是各向同性且不受体力影响, 外加压载的空心圆柱和碗形抛物柱分别放置在多孔半空间上. 图1(a)是空心圆柱与多孔半空间接触的三维模型图, 图1(b)是其轴对称接触的模型图. 图2(a)是碗形抛物柱与多孔半空间接触的三维模型图, 图右侧的环是柱底部的放大图, 图2(b)是其轴对称接触的模型图, 图2(c)是图2(b)中取a为常数时的对应图, 此接触问题由如下方程控制[15,37-38]
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图 1 空心圆柱与多孔半空间接触模型图Figure 1. Diagrammatic sketch of contact model between hollow cylinder and porous half space![]()
图 2 碗形抛物柱与多孔半空间接触模型图Figure 2. Diagrammatic sketch of contact model between bowl shaped parabolic column and porous half space$$ \left.\begin{split} & {(\lambda + \mu )\frac{{\partial \vartheta }}{{\partial r}} + \mu \left(\Delta {u_r} - \frac{{{u_r}}}{{{r^2}}}\right) + \beta \frac{{\partial \phi }}{{\partial r}} = 0} \\ & {(\lambda + \mu )\frac{{\partial \vartheta }}{{\partial z}} + \mu \Delta {u_z} + \beta \frac{{\partial \phi }}{{\partial z}} = 0} \\ & {\alpha \Delta \phi - \xi \phi - \beta \vartheta = 0} \end{split}\right\} $$ (1) 其中
$$ \varDelta = \frac{{{\partial ^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{{{\partial ^2}}}{{\partial {z^2}}},\quad \vartheta = \frac{{{u_r}}}{r} + \frac{{\partial {u_r}}}{{\partial r}} + \frac{{\partial {u_z}}}{{\partial z}} $$ 式中, $ \phi $是体积分数改变量, $ \lambda 和\mu $为Lame常数, $ \alpha $为孔隙扩散参数, $ \xi $为孔隙刚度模量, $ \beta $为耦合模量, $ {u_r} $和$ {u_z} $分别是沿r ($r=\sqrt{{{x}^{2}}+{{y}^{2}}} $)轴和z轴的位移.
当$ \beta = 0 $时问题退化为弹性基体的变形问题. 应力张量的分量如下
$$\left. \begin{split} & {\sigma _{rr}}(r,z) = \lambda \vartheta + 2\mu \frac{{\partial {u_r}}}{{\partial r}} + \beta \phi \\ & {\sigma _{zz}}(r,z) = \lambda \vartheta + 2\mu \frac{{\partial {u_z}}}{{\partial z}} + \beta \phi \\ & {\sigma _{rz}}(r,z) = \mu \left(\frac{{\partial {u_z}}}{{\partial r}} + \frac{{\partial {u_r}}}{{\partial z}}\right)\end{split}\right\} $$ (2) 这里$ {\sigma _{rr}}(r,z),{\sigma _{zz}}(r,z)和{\sigma _{rz}}(r,z) $分别为径向应力、法向应力和剪切应力. 当$ z \to \infty $时, 应力和位移均为0. 边界条件如下
$$ \left.\begin{split} & {\sigma _{rz}}(r,0) = 0,\quad \frac{{\partial \phi }}{{\partial z}} = 0,\;{u_z}(r,0) = \delta (r)\;(a \leqslant r \leqslant b) \\ & {\sigma _{zz}}(r,0) = \left\{ \begin{aligned} & { - q(r),}\quad{a \leqslant r \leqslant b} \\ & {0,}\quad{0 \leqslant r \lt a\;\;{\mathrm{or}}\;\;r \gt b} \end{aligned} \right. \end{split}\right\} $$ (3) 对于施加在刚性压头上的z方向的力载荷P, 未知接触应力$ q(r) $满足$ 2\text{π} \displaystyle\int_a^b {rq(r)} {\mathrm{d}}r = P $
空心柱平压的情形下, 表面沉降为常数, 即 $ \delta (r) = {\delta _0} = {\mathrm{const.}} $; 碗形抛物柱抛物压的情形下, 表面沉降是关于接触半径的函数[39-40], 即 $ \delta (r) = {\delta _0} - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {(2 R)}}} \right. } {(2 R)}} $, 其中R是抛物线顶点处的曲率半径.
引入Hankel变换
$$\left. \begin{split} & {u_r}(r,z) = \int_0^\infty {\zeta U(\zeta ,z){{\mathrm{J}}_1}(\zeta r)} {\mathrm{d}}\zeta \\ & {u_z}(r,z) = \int_0^\infty {\zeta W(\zeta ,z){{\mathrm{J}}_0}(\zeta r)} {\mathrm{d}}\zeta \\ & \phi (r,z) = \int_0^\infty {\zeta \varPhi (\zeta ,z){{\mathrm{J}}_0}(\zeta r)} {\mathrm{d}}\zeta \end{split}\right\} $$ (4) 其中$ {{\mathrm{J}}_i}(\zeta r)\;(i = 0,1) $分别是0阶和1阶Bessel函数, $ U,W和\varPhi $分别为$ {u_r},{u_z}和\phi $的变换函数.
将式(4)代入控制方程式(1)中得到变换域下的控制方程为
$$ \left. \begin{split} & - {\zeta ^2}U(\zeta ,z) - (1 - {c^2})\zeta \frac{{\partial W(\zeta ,z)}}{{\partial z}} + \\ &\qquad {c^2}\frac{{{\partial ^2}U(\zeta ,z)}}{{\partial {z^2}}} - H\zeta \varPhi (\zeta ,z) = 0 \\ & \frac{{{\partial ^2}W(\zeta ,z)}}{{\partial {z^2}}} + (1 - {c^2})\zeta \frac{{\partial U(\zeta ,z)}}{{\partial z}} - \\ &\qquad {c^2}{\zeta ^2}W(\zeta ,z) + H\frac{{\partial \varPhi (\zeta ,z)}}{{\partial z}} = 0 \\ & l_1^2\left( {\frac{{{\partial ^2}\varPhi (\zeta ,z)}}{{\partial {z^2}}} - {\zeta ^2}\varPhi (\zeta ,z)} \right) - \frac{{l_1^2}}{{l_2^2}}\varPhi (\zeta ,z) - \\ &\qquad \zeta U(\zeta ,z) - \frac{{\partial W(\zeta ,z)}}{{\partial z}} = 0\end{split}\right\} $$ (5) 其中
$$ l_1^2 = \frac{\alpha }{\beta },\quad l_2^2 = \frac{\alpha }{\xi },\quad {c^2} = \frac{{1 - 2\nu }}{{2(1 - \nu )}},\quad H = \frac{\beta }{{\lambda + 2\mu }} $$ (6) 由式(2) ~ 式(4)得变换域下的边界条件为
$$\left. \begin{split} & \frac{{\partial \varPhi (\zeta ,0)}}{{\partial z}} = 0,\;\; - \zeta W(\zeta ,0) + \frac{{\partial U(\zeta ,0)}}{{\partial z}} = 0 \\ & - \frac{{{c^2}}}{\mu }Q(\zeta ) = (1 - 2{c^2})\zeta U(\zeta ,0) + \frac{{\partial W(\zeta ,0)}}{{\partial z}} + H\varPhi (\zeta ,0)\end{split}\right\} $$ (7) 其中
$$\left.\begin{split} & Q(\zeta ) = \int_0^\infty {q(r){{\mathrm{J}}_0}(\zeta r)r} {\mathrm{d}}r \\ & q(r) = \int_0^\infty {Q(\zeta ){{\mathrm{J}}_0}(\zeta r)\zeta } {\mathrm{d}}\zeta \end{split} \right\}$$ (8) 根据常微分方程(ODE)的一般理论, 方程组式(5)的形式解如下
$$ \left( {U,\;W,\;\varPhi } \right) = \left( {A,\;B,\;C} \right){{\mathrm{e}}^{\lambda z}} $$ (9) 对式(9)求1阶和2阶导数得
$$ \begin{split} & \left( {{\mathrm{D}}U,\;{\mathrm{D}}W,\;{\mathrm{D}}\varPhi ,\;{{\mathrm{D}}^2}U,\;{{\mathrm{D}}^2}W,\;{{\mathrm{D}}^2}\varPhi } \right) = \\ &\qquad \left( {\lambda A,\;\lambda B,\;\lambda C,\;{\lambda ^2}A,\;{\lambda ^2}B,\;{\lambda ^2}C} \right){{\mathrm{e}}^{\lambda z}}\end{split} $$ (10) 其中D代表微分算子$ {{\mathrm{d}} \mathord{\left/ {\vphantom {{\mathrm{d}} {{\mathrm{d}}z}}} \right. } {{\mathrm{d}}z}} $. 将式(9)和式(10)代入式(5)得
$$ \left.\begin{split} & ({c^2}{\lambda ^2} - {\zeta ^2})A - (1 - {c^2})\zeta \lambda B - H\zeta C = 0 \\ & (1 - {c^2})\zeta \lambda A + ({\lambda ^2} - {c^2}{\zeta ^2})B + H\lambda C = 0 \\ & \zeta A + \lambda B + ({{l_1^2} \mathord{\left/ {\vphantom {{l_1^2} {l_2^2}}} \right. } {l_2^2}} + l_1^2{\zeta ^2} - l_1^2{\lambda ^2})C = 0\end{split}\right\} $$ (11) 上式是关于$ A,B和C $的齐次代数方程组, 其系数行列式为$- {{H{c^2}} \mathord{\left/ {\vphantom {{H{c^2}} {(Nl_2^4)}}} \right. } {(Nl_2^4)}}{({\tilde \lambda ^2} - {\tilde \zeta ^2})^2}[{\tilde \lambda ^2} - ({\tilde \zeta ^2} - N + 1)] $, 其中无量纲参数孔隙率$ N = \left( {{{l_2^2} \mathord{\left/ {\vphantom {{l_2^2} {l_1^2}}} \right. } {l_1^2}}} \right)H $, 且$ 0 \leqslant N \lt 1 - {c^2} $, $ \tilde \lambda = {l_2}\lambda ,\;\tilde \zeta = {l_2}\zeta $. 我们要寻找该方程组的非平凡解, 得其特征值为
$$ {\lambda }_{1,2} = \pm \zeta \;(\text{double root}),\quad {\lambda }_{3} = \pm \frac{\sqrt{{l}_{2}^{2}{\zeta }^{2} + 1-N}}{{l}_{2}} $$ 考虑到半空间中应力在无穷远处为0, 则ODE系统式(5)的一般解为
$$ \left.\begin{split} & U(\zeta ,z) = ({d_1} + {d_2}z){{\mathrm{e}}^{ - \zeta z}} + {d_3}{{\mathrm{e}}^{ - mz}} \\ & W(\zeta ,z) = ({s_1} + {s_2}z){{\mathrm{e}}^{ - \zeta z}} + {s_3}{{\mathrm{e}}^{ - mz}} \\ & \varPhi (\zeta ,z) = ({t_1} + {t_2}z){{\mathrm{e}}^{ - \zeta z}} + {t_3}{{\mathrm{e}}^{ - mz}}\end{split}\right\} $$ (12) 其中$ m = {{\sqrt {l_2^2{\zeta ^2} + 1 - N} } \mathord{\left/ {\vphantom {{\sqrt {l_2^2{\zeta ^2} + 1 - N} } {{l_2}}}} \right. } {{l_2}}} $. 将式(12)代入式(5)可得
$$ \left.\begin{split} & {t_2} = 0,\quad {s_2} = {d_2} \\ & {s_3} = \frac{m}{\zeta }{d_3},\quad {t_3} = \frac{{1 - N}}{{Nl_1^2\zeta }}{d_3} \\ & {t_1} = \frac{{ - 2l_2^2{c^2}}}{{(N - 1 + {c^2})l_1^2}}{d_2}\\ &{s_1} = {d_1} + \frac{1}{\zeta }\frac{{N - 1 - {c^2}}}{{N - 1 + {c^2}}}{d_2}\end{split}\right\} $$ (13) 因此只需要确定系数$ {d_1},{d_2}和{d_3} $的值就可以得到ODE系统式(5)的一般解. 由式(7)、式(12)和式(13)知
$$\left.\begin{split} & {d_1} = \frac{{m{c^2}(2l_2^2N{\zeta ^2} - 1 + N)}}{\zeta }Q(\zeta )F(\zeta ) \\ & {d_2} = - m(1 - N)(N - 1 + {c^2})Q(\zeta )F(\zeta ) \\ & {d_3} = - 2l_2^2{c^2}N{\zeta ^2}Q(\zeta )F(\zeta )\end{split}\right\} $$ (14) 其中
$$ F(\zeta ) = \frac{1}{{2\mu [2l_2^2{c^2}N{\zeta ^2}(m - \zeta ) - m(1 - N)(N - 1 + {c^2})]}} $$ 2. 积分方程推导及其数值方案
当$ z = 0 $时, 由式(12) ~ 式(14)得
$$ W(\zeta ,0) = \frac{{Q(\zeta ){l_2}}}{{2\mu (1 - {c^2})}}L(\zeta {l_2}) $$ 其中
$$ \begin{split} &L(\zeta {l_2}) = \tilde L(\tilde \zeta ) = \\ &\quad \frac{{(1 - {c^2}){{(N - 1)}^2}}}{{ 2{c^2}N{{\tilde \zeta }^3}\left( {1 - {{\tilde \zeta } / {\sqrt {{{\tilde \zeta }^2} + 1 - N} }}} \right) - \tilde \zeta (1 - N)(N - 1 + {c^2})}}\end{split} $$ (15) 由式(4)和式(12) ~ 式(15)得
$$ {u_z}(r,0) = \int_0^\infty {\zeta \frac{{Q(\zeta ){l_2}}}{{2\mu (1 - {c^2})}}L(\zeta {l_2}){J_0}(\zeta r)} {\mathrm{d}}\zeta $$ 令$ \tilde \zeta = \zeta {l_2},\;\tilde \rho = {\rho \mathord{\left/ {\vphantom {\rho {{l_2}}}} \right. } {{l_2}}},\;\tilde r = {r \mathord{\left/ {\vphantom {r {{l_2}}}} \right. } {{l_2}}} $, 则有
$$\begin{split} & {u_z}(\tilde r{l_2},0) = \\ &\quad {l_2}\frac{{1 - \nu }}{\mu }\int_{{a \mathord{\left/ {\vphantom {a {{l_2}}}} \right. } {{l_2}}}}^{{b \mathord{\left/ {\vphantom {b {{l_2}}}} \right. } {{l_2}}}} {\left[ {q({l_2}\tilde \rho )\tilde \rho \int_0^\infty {\tilde \zeta \tilde L(\tilde \zeta ){{\mathrm{J}}_0}(\tilde \zeta \tilde \rho ){{\mathrm{J}}_0}(\tilde \zeta \tilde r)} {\mathrm{d}}\tilde \zeta } \right]} {\mathrm{d}}\tilde \rho \end{split} $$ 采用$ {\tilde u_z}(\tilde r) = {u_z}(\tilde r{l_2},0)l_2^{ - 1} $表示无量纲法向位移, 则有
$$ {\tilde u_z}(\tilde r) = \int_{\tilde a}^{\tilde b} {\tilde q(\tilde \rho )\tilde \rho } k(\tilde \rho ,\tilde r){\mathrm{d}}\tilde \rho = \tilde \delta (\tilde r) $$ (16) 其中
$$ k(\tilde \rho ,\tilde r) = \int_0^\infty {\tilde \zeta \tilde L(\tilde \zeta ){{\mathrm{J}}_0}(\tilde \zeta \tilde \rho ){{\mathrm{J}}_0}(\tilde \zeta \tilde r)} {\mathrm{d}}\tilde \zeta $$ (17) $$ \tilde{\delta }(\tilde{r}) = \delta (\tilde{r}){l}_{2}^{-1} = \left\{\begin{aligned} &{\tilde{\delta }}_{0},\quad \text{flat punch}\\ &{\tilde{\delta }}_{0}-{\tilde{r}}^{2}/(2\tilde{R}),\quad \text{paraboloid punch}\end{aligned}\right. $$ (18) 式中
$$ \begin{split} & \tilde q(\tilde \rho ) = \frac{{1 - \nu }}{\mu }q({l_2}\tilde \rho ),\quad \;\tilde a = {a / {{l_2}}},\quad \tilde b = {b \mathord{\left/ {\vphantom {b {{l_2}}}} \right. } {{l_2}}} \\ & {{\tilde \delta }_0} = {{{\delta _0}} \mathord{\left/ {\vphantom {{{\delta _0}} {{l_2}}}} \right. } {{l_2}}},\quad \tilde R = {R \mathord{\left/ {\vphantom {R {{l_2}}}} \right. } {{l_2}}},\quad \tilde r = {r \mathord{\left/ {\vphantom {r {{l_2}}}} \right. } {{l_2}}}\end{split} $$ 方程式(16)对$ \tilde r $求导得
$$ {\displaystyle {\int }_{\tilde{a}}^{\tilde{b}}\tilde{q}(\tilde{\rho })\tilde{\rho }{k}_{D}(\tilde{\rho },\tilde{r})}{\mathrm{d}}\tilde{\rho } = \left\{\begin{aligned} &0,\quad \text{flat punch}\\ &\tilde{r}/\tilde{R},\quad \text{paraboloid punch}\end{aligned}\right. $$ (19) 其中
$$ {k_D}(\tilde \rho ,\tilde r) = \int_0^\infty {{{\tilde \zeta }^2}\tilde L(\tilde \zeta ){{\mathrm{J}}_1}(\tilde \zeta \tilde r){{\mathrm{J}}_0}(\tilde \zeta \tilde \rho )} {\mathrm{d}}\tilde \zeta $$ (20) 以上求解涉及到的乘积型Bessel函数的无穷积分的数值计算采用我们发展的文献[41]的方法, 涉及到的定积分数值求解采用高斯-切比雪夫算法. 对于方程(19), 我们分别给出了平压和抛物压的高斯-切比雪夫求积方案.
令$ \tilde \rho = \dfrac{{\tilde b - \tilde a}}{2}{\rho ^*} + \dfrac{{\tilde b + \tilde a}}{2} $, $ \tilde r = \dfrac{{\tilde b - \tilde a}}{2}{r^*} + \dfrac{{\tilde b + \tilde a}}{2} $, 可得
$$ \tilde{q}(\tilde{\rho }) = \left\{\begin{aligned} &\frac{{q}^{*}({\rho }^{*})}{\sqrt{1-{({\rho }^{*})}^{2}}},\quad \text{flat punch}\\ &{q}^{*}({\rho }^{*})\sqrt{1-{({\rho }^{*})}^{2}},\quad \text{paraboloid punch}\end{aligned}\right. $$ 由式(19)知
$$ \left.\begin{split} &{\displaystyle {\int }_{-1}^{1}\frac{\varOmega \left({\rho }^{*},{r}^{*}\right)}{\sqrt{1-{({\rho }^{*})}^{2}}}}{\mathrm{d}}{\rho }^{*} = 0,\quad \text{flat punch},\\ &{\displaystyle {\int }_{-1}^{1}\sqrt{1-{({\rho }^{*})}^{2}}\varOmega \left({\rho }^{*},{r}^{*}\right)}{\mathrm{d}}{\rho }^{*} = \frac{2\tilde{r}}{\tilde{b}-\tilde{a}}\frac{1}{\tilde{R}},\;\; \text{paraboloid punch}\end{split}\right\} $$ 其中
$$ \varOmega \left( {{\rho ^*},{r^*}} \right) = {q^*}({\rho ^*})\tilde \rho {k_D}\left( {\tilde \rho ,\tilde r} \right) $$ 且
$$ \left.\begin{split} &{\displaystyle {\int }_{-1}^{1}\frac{\tilde{\rho }{q}^{*}({\rho }^{*})}{\sqrt{1-{({\rho }^{*})}^{2}}}}{\mathrm{d}}{\rho }^{*} = \frac{\tilde{P}}{\text{π} (\tilde{b}-\tilde{a})},\quad \text{flat punch}\\ &{\displaystyle {\int }_{-1}^{1}\tilde{\rho }{q}^{*}({\rho }^{*})\sqrt{1-{({\rho }^{*})}^{2}}}{\mathrm{d}}{\rho }^{*} = \frac{\tilde{P}}{\text{π} (\tilde{b}-\tilde{a})},\;\; \text{paraboloid punch}\end{split}\right\} $$ (21) 式中$ \tilde P = \dfrac{{1 - \nu }}{\mu }\dfrac{P}{{l_2^2}} $. 积分方程式(19)和式(21)被转换成关于$ q_j^* = {q^*}\left( {\rho _j^*} \right) $的n阶代数方程式(22)和式(23)
$$ \left.\begin{split} & {\sum\limits_{j = 1}^n {{{\tilde \rho }_j}{q^*}_j{k_D}\left( {{{\tilde \rho }_j},{{\tilde r}_i}} \right)} = 0} \\ & {\sum\limits_{j = 1}^n {{{\tilde \rho }_j}q_j^*} = \frac{n}{\text{π} }\frac{{\tilde P}}{{\text{π} (\tilde b - \tilde a)}}} \end{split}\right\}$$ (22) 及
$$ \left. \begin{split} & {\sum\limits_{j = 1}^n {{{\sin }^2}\left( {\frac{{\text{π} j}}{{n + 1}}} \right){{\tilde \rho }_j}{q^*_j}{k_D}\left( {{{\tilde \rho }_j},{{\tilde r}_i}} \right)} = \frac{{n + 1}}{\text{π} }{{\frac{{2{{\tilde r}_i}}}{{\tilde b - \tilde a}}} \frac{1}{\tilde R}}} \\ & {\sum\limits_{j = 1}^n {{{\sin }^2}\left( {\frac{{\text{π} j}}{{n + 1}}} \right){{\tilde \rho }_j}q_j^*} = \frac{{n + 1}}{\text{π} }\frac{{\tilde P}}{{\text{π} (\tilde b - \tilde a)}}} \end{split} \right\} $$ (23) 其中
$$ {\tilde \rho _j} = \frac{{\tilde b - \tilde a}}{2}\rho _j^* + \frac{{\tilde b + \tilde a}}{2},\quad {\tilde r_i} = \frac{{\tilde b - \tilde a}}{2}r_i^* + \frac{{\tilde b + \tilde a}}{2} $$ $$ {\rho }_{j}^{*} = \left\{\begin{aligned} &\mathrm{cos}\frac{\text{π} (2j-1)}{2n},\quad j = 1,2,\cdots ,n, \text{flat punch}\\ &\mathrm{cos}\frac{j\text{π} }{n + 1},\quad j = 1,2,\cdots ,n, \text{paraboloid punch}\end{aligned}\right. $$ $$ {r}_{i}^{*} = \left\{\begin{aligned} &\mathrm{cos}\frac{i\text{π} }{n + 1},\;\; i = 1,2,\cdots ,n-1,2,\text{flat punch}\\ &\mathrm{cos}\frac{\text{π} (2i-1)}{2(n + 1)},\;\; i = 1,2,\cdots ,n + 1, \text{paraboloid punch}\end{aligned}\right.$$ 若取$ N = 0 $, 问题简化为压头和线弹性半空间的接触问题, 其接触应力$ p(\tilde r) $的精确解如下[17]
$$ p(\tilde{r}) = \left\{\begin{split} &\frac{\tilde{P}}{2\text{π} \tilde{b}\sqrt{{\tilde{b}}^{2}-{\tilde{r}}^{2}}},\quad \left|\tilde{r}\right| \lt \tilde{b}, \text{flat punch}\\ &\frac{3\tilde{P}\sqrt{{\tilde{b}}^{2}-{\tilde{r}}^{2}}}{2\text{π} \tilde{b}\tilde{\delta }\tilde{R}},\quad \left|\tilde{r}\right| \lt \tilde{b}, \text{paraboloid punch}\end{split}\right. $$ 令$ \tilde \zeta = \zeta {l_2},\;\tilde \rho = {\rho \mathord{\left/ {\vphantom {\rho {{l_2}}}} \right. } {{l_2}}},\;\tilde r = {r \mathord{\left/ {\vphantom {r {{l_2}}}} \right. } {{l_2}}},\;\tilde z = {z \mathord{\left/ {\vphantom {z {{l_2}}}} \right. } {{l_2}}} $, 由式(4)和式(12) ~ 式(14)得法向位移为
$$\begin{split} &{u_z}(\tilde r,\tilde z) = {l_2}\frac{{1 - \nu }}{\mu }\int_{\tilde a}^{\tilde b} q(\tilde \rho {l_2})\tilde \rho\cdot \\ &\qquad \int_0^\infty {\left[ G(\tilde \zeta )\sqrt {{{\tilde \zeta }^2} + 1 - N} \cdot X(\tilde z,\tilde \zeta ){J_0}(\tilde \zeta \tilde r){J_0}(\tilde \zeta \tilde \rho ) \right]} {\mathrm{d}}\tilde \zeta {\mathrm{d}}\tilde \rho \end{split}$$ 其中
$$ \begin{split} &G(\tilde \zeta ) ={{(1 - {c^2})}}/ \left[ 2{{\tilde \zeta }^2}{c^2}N(\sqrt {{{\tilde \zeta }^2} + 1 - N} -\right.\\ &\left. \tilde \zeta ) - \sqrt {{{\tilde \zeta }^2} + 1 - N} (1 - N)(N - 1 + {c^2}) \right]\end{split} $$ $$ \begin{split} & X(\tilde z,\tilde \zeta ) = [{c^2}(2N{{\tilde \zeta }^2} - 1 + N) - \left( {1 - \frac{{2{c^2}}}{{N - 1 + {c^2}}} + \tilde z\tilde \zeta } \right) \cdot \\ &\qquad (1 - N)(N - 1 + {c^2})]{{\mathrm{e}}^{ - \tilde \zeta \tilde z}} - 2{c^2}N{{\tilde \zeta }^2}{{\mathrm{e}}^{ - \tilde z\sqrt {{{\tilde \zeta }^2} + 1 - N} }}\end{split} $$ 采用$ {\tilde u_z}(\tilde r,\tilde z) = {u_z}(\tilde r,\tilde z)l_2^{ - 1} $表示无量纲法向位移, 则
$$ {\tilde u_z}(\tilde r,\tilde z) = \int_{\tilde a}^{\tilde b} {\tilde q(\tilde \rho )\tilde \rho \int_0^\infty {\varPsi (\tilde \zeta ,\tilde z){{\mathrm{J}}_0}(\tilde \zeta \tilde r)} {{\mathrm{J}}_0}(\tilde \zeta \tilde \rho )} {\mathrm{d}}\tilde \zeta {\mathrm{d}}\tilde \rho $$ 其中, $ \varPsi (\tilde \zeta ,\tilde z) = G(\tilde \zeta )\sqrt {{{\tilde \zeta }^2} + 1 - N} X(\tilde z,\tilde \zeta ) $. 由方程式(22)和式(23)可求得$ \tilde q(\tilde \rho ) $, 进而得无量纲法向位移.
3. 数值结果与讨论
未知接触应力$ \tilde q(\tilde r) $, 内径$ 2\tilde a $, 壁厚$ \tilde b - \tilde a $, 孔隙率N, 力载荷$ \tilde P $, 曲率半径$ \tilde R $, 泊松比v均为无量纲化参数, 材料参数对应力分布和基体变形的影响讨论如下.
3.1 退化结果验证
3.1.1 空心圆柱
利用高斯-切比雪夫算法(GCA)结合乘积型Bessel函数的无穷积分的数值方法求解了积分方程, 为体现我们发展算法的优越性, 我们也用直接配点法(DCM)[17]结合乘积型Bessel函数的无穷积分的数值结果与本文的求解方法对比.
图3分别给出了已知表面沉降$ \tilde \delta {\text{ = }}1 $和已知力载荷$ \tilde P = 1 $的退化结果与纯弹性解析解的对比情况, 可以看到退化情况的数值解与解析解吻合很好, 验证了本文方法的有效性. 图3(b)表明, 在相同的n值下, 高斯-切比雪夫算法结合乘积型Bessel函数的无穷积分的数值方法在奇异值处的计算效果更佳. 经验证本文算法收敛且稳定, 但当n取值在30左右时, 本研究误差相对最小, 故下文所有数值计算均取$ n = 30 $.
3.1.2 碗形抛物柱
当$ N = 0,\;a = 0 $时, 本文结果退化成抛物型单压头纯弹性情形.
图4分别给出了当$ \tilde b = 1.395\;5 $, $ \tilde b = $1.41[42], $ \tilde b = $1.43, 1.44[17]时, 本文结果与文献[17, 42]给出的纯弹性解析解的对比情况, 可以看到退化情况的数值解与弹性解析解吻合较好.
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3.2 未知接触应力变化
根据所提算法绘制了空心圆柱和碗形抛物柱对基体表面接触应力影响的平面图和三维空间图, 进而给出了现象描述及解释.
3.2.1 空心圆柱
空心圆柱情形下, 图5给出了当$ N = 0.1, $ $ \tilde P = 1, \tilde a = 0.3和\tilde b = 1 $时未知接触应力随泊松比$ \nu $的二维及三维变化情况. 图6给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde a = 0.3和\tilde b = 1 $时未知接触应力随孔隙率N的二维及三维变化情况. 图7给出了当$ N = 0.1,\;\nu = 0.3,\;\tilde a = 0.3和\tilde b = 1 $时未知接触应力随力载荷$ \tilde P $的二维及三维变化情况. 图8给出了当$ N = 0.1,\;\nu = 0.3,\;\tilde P = 1和\tilde b = \tilde a + 1 $时未知接触应力随内径$ 2\tilde a $的二维及三维变化情况. 图9给出了当$ N = 0.1,\;\nu = 0.3,\;\tilde P = 1和\tilde a = 0.3 $时未知接触应力随壁厚$ \tilde b - \tilde a $的二维及三维变化情况. 由图5 ~ 图9可知, 接触边缘明显存在应力奇异性, 由于圆柱内壁之间相互的影响, 削弱了内侧的奇异性, 所以接触区域外侧的奇异性要高于内侧. 虽然未知接触应力随多孔材料的泊松比和孔隙率的变化不显著, 但是外侧奇异性随孔隙率的增大而增大, 内侧奇异性随孔隙率的增大而减小. 与弹性材料相比, 接触外侧奇异性更大, 但是内侧相对更小, 说明压头对大孔隙率的应力奇异性的抑制效果更好. 也显示力载荷与接触应力呈正相关, 但表面接触应力对空心柱内径和壁厚的变化比较敏感, 内径越小应力奇异性越大, 壁越厚应力越小. 我们给出了参数对应力的三维影响情况.
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图 5 参数$ \nu $对未知接触应力的影响Figure 5. Effect of parameter $ \nu $ on unknown contact stresses![]()
图 7 参数$ \tilde P $对未知接触应力的影响Figure 7. Effect of parameter $ \tilde P $ on unknown contact stresses![]()
图 8 参数$ 2\tilde a $对未知接触应力的影响Figure 8. Effect of parameter $ 2\tilde a $ on unknown contact stresses![]()
图 9 参数$ \tilde b - \tilde a $对未知接触应力的影响Figure 9. Effect of parameter $ \tilde b - \tilde a $ on unknown contact stresses表1给出了当孔隙率变化时圆柱的实心与空心分别引起的接触外边缘应力奇异性的对比情况, 可知在较小的内径下, 当空心柱壁厚与实心柱半径一致时, 空心柱所导致的基体接触区域外边缘的应力奇异性比实心柱要小1倍以上, 并且随着孔隙率的增大该比值在降低, 但随着内径的增大该比值在增大, 在壁厚增大到一定程度时, 比值趋于稳定. 当空心柱外径与实心柱半径一致时, 空心柱所导致的基体接触区域外边缘的应力奇异性比实心柱要大1倍以上, 并且随着孔隙率的增大该比值增大, 但随着内径的增大该比值降低, 在壁厚增大到一定程度时, 比值趋于稳定.
表 1 孔隙率变化时, 圆柱的实心与空心所引起的接触外边缘应力奇异性的对比Table 1. Comparison of stress singularity at the contact outer edge caused by solid and hollow cylinders with changes of porosityCylinder ($ \nu = {\text{0}}{\text{.3,}}\;\tilde P{\text{ = 1,}}\;N = [0,0.05,0.2,0.6] $) The wall thickness of the hollow column is consistent with the radius of the solid column 
solid ($ \tilde a = 0 $) hollow ($ \tilde a = 0.3 $) solid ($ \tilde a = 0 $) hollow ($ \tilde a = 2 $) (a) stress ratio at the outer edge The outer diameter of the hollow column is consistent with the radius of the solid column 
solid ($ \tilde a = 0 $) hollow ($ \tilde a = 0.3 $) solid ($ \tilde a = 0 $) hollow ($ \tilde a = 2 $) (b) stress ratio at the outer edge 3.2.2 碗形抛物柱
碗形抛物柱情形下, 图10给出了当$ N = 0.5,\;\tilde P = 2.5, $ $ \tilde R = 2和\tilde a = 0.5 $时未知接触应力随泊松比$ \nu $的二维及三维变化情况. 图11给出了当$ \nu = 0.3,\;\tilde P = 1, \tilde a = 0.5和\tilde R = 2 $时未知接触应力随孔隙率N的二维及三维变化情况. 图12给出了当$ N = 0.5,\;\nu = 0.3,\;\tilde a = 0.5 和\tilde R = 0.5 $时未知接触应力随力载荷$ \tilde P $的二维及三维变化情况. 图13给出了当$ \nu = 0.3,\;\tilde P = 0.6,\;\tilde a = 0.5和 N = 0.5 $时未知接触应力随半径$ \tilde R $的二维及三维变化情况. 图14给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde R = 0.5和N = 0.5 $时未知接触应力随内径$ 2\tilde a $的二维及三维变化情况. 由图10 ~ 图14可知, 泊松比越大, 接触应力越大, 但在接触外边缘的变化情况刚好相反; 孔隙率越小, 接触应力越大; 力载荷越小, 接触范围越小, 这是因为压力小会导致压入深度浅. 半径越大, 压入深度越小, 导致接触范围越大. 内径越大, 接触区域内侧的奇异性越大.
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图 10 参数$ \nu $对未知接触应力的影响Figure 10. Effect of parameter $ \nu $ on unknown contact stresses![]()
图 12 参数$ \tilde P $对未知接触应力的影响Figure 12. Effect of parameter $ \tilde P $ on unknown contact stresses![]()
图 13 参数$ \tilde R $对未知接触应力的影响Figure 13. Effect of parameter $ \tilde R $ on unknown contact stresses![]()
图 14 参数$ 2\tilde a $对未知接触应力的影响Figure 14. Effect of parameter $ 2\tilde a $ on unknown contact stresses3.3 法向位移
根据上述给出的表面未知接触应力及位移表达式绘制了空心圆柱和碗形抛物柱影响下的基体位移随参数的变化情况.
3.3.1 空心圆柱
在$ \tilde z = 1 $平面上空心圆柱情形下, 图15(a)给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde a = 0.3,\;\tilde b = 1和\tilde z = 1 $时法向位移随孔隙率N的变化情况. 图15(b)给出了当$ \nu = 0.3, \tilde P = 1,\;\tilde a = 0.3,\;\tilde b = 1和N = 0.1 $时法向位移随深度$ \tilde z $的变化情况. 图16给出了当$ N = 0.1,\;\nu = 0.3,\;\tilde P = 1,\;\tilde b = \tilde a + 1和 \tilde z = 1 $时法向位移随内径$ 2\tilde a $的变化情况. 图17给出了当$ N = 0.1,\;\nu = 0.3,\;\tilde P = 1,\;\tilde a = 0.3和\tilde z = 1 $时法向位移随壁厚$ \tilde b - \tilde a $的变化情况. 由图15 ~ 图17可知, 孔隙率越大, 法向位移越大. 深度越小, 法向位移越大, 内径越大, 法向位移越大, 壁厚越大, 法向位移越小. 位移在中心产生明显的叠加凸起现象.
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图 15 参数N和$ \tilde z $对法向位移的影响Figure 15. Effect of parameter N and $ \tilde z $ on normal displacement![]()
图 16 法向位移随$ 2\tilde a $的变化Figure 16. Normal displacement with respect to $ 2\tilde a $![]()
图 17 法向位移随$ \tilde b - \tilde a $的变化Figure 17. Normal displacement with respect to $ \tilde b - \tilde a $3.3.2 碗形抛物柱
在$ \tilde z = 1 $平面上碗形抛物柱情形下, 图18给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde a = 0.5,\;\tilde R = 2和\tilde z = 1 $时法向位移随孔隙率N的变化情况. 图19给出了当$ \nu = 0.3,\;\tilde P = 0.6, \tilde a = 0.5,\; N = 0.5和\tilde z = 1 $时法向位移随半径$ \tilde R $的变化情况. 图20给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde R = 0.5,\;N = 0.5和 \tilde z = 1 $时法向位移随内径$ 2\tilde a $的变化情况. 图21给出了当$ \nu = 0.3,\;\tilde P = 1,\;\tilde a = 0.5,\;\tilde R = 2和N = 0.5 $时法向位移随深度$ \tilde z $的变化情况.
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图 20 法向位移随$ 2\tilde a $的变化Figure 20. Normal displacement with respect to $ 2\tilde a $由图可知, 随孔隙率的增大, 法向位移先减小后增大; 随半径和内径的增大, 法向位移先增大后减小;深度越小, 法向位移越大. 位移在中心产生明显的叠加凸起现象, 但随着深度增加, 这种现象逐渐消失.
4. 结 论
本文研究了空心柱与多孔半空间的接触问题, 利用积分方程方法给出了未知接触应力和位移的精确表达式, 发展了高斯-切比雪夫结合乘积型Bessel函数的无穷积分的数值方法求解了积分方程, 并进行退化对比验证, 得到主要结论如下.
(1)空心圆柱: 接触区域外侧的奇异性明显高于内侧, 外侧更有可能是裂纹起始位置; 空心柱对较大孔隙率的应力奇异性的抑制效果更好; 在较小的内径下, 当空心柱壁厚与实心柱半径一致时, 接触外边缘的应力奇异性比实心柱要小1倍以上, 并且在大孔隙下该比值在降低, 随着内径的增大该比值在增大; 当空心柱外径与实心柱半径一致时, 接触外边缘的应力奇异性比实心柱要大1倍以上, 并且在大孔隙下该比值在增大, 随着内径的增大该比值在降低.
(2)碗形抛物柱: 内径取常数会导致接触区域内侧产生应力奇异性, 泊松比和孔隙率的变化对接触应力的影响甚微, 但对位移的影响较大. 力载荷、半径和内径对接触应力的影响较大.
(3)在横截平面上, 参数强烈影响基体的变形能力, 产生明显的叠加变形现象.
研究结果揭示了多孔材料表面损伤的潜在机理, 为多孔材料的设计和应用提供可参考的理论依据.
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图 1 空心圆柱与多孔半空间接触模型图
Figure 1. Diagrammatic sketch of contact model between hollow cylinder and porous half space
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图 2 碗形抛物柱与多孔半空间接触模型图
Figure 2. Diagrammatic sketch of contact model between bowl shaped parabolic column and porous half space
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图 5 参数$ \nu $对未知接触应力的影响
Figure 5. Effect of parameter $ \nu $ on unknown contact stresses
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图 7 参数$ \tilde P $对未知接触应力的影响
Figure 7. Effect of parameter $ \tilde P $ on unknown contact stresses
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图 8 参数$ 2\tilde a $对未知接触应力的影响
Figure 8. Effect of parameter $ 2\tilde a $ on unknown contact stresses
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图 9 参数$ \tilde b - \tilde a $对未知接触应力的影响
Figure 9. Effect of parameter $ \tilde b - \tilde a $ on unknown contact stresses
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图 10 参数$ \nu $对未知接触应力的影响
Figure 10. Effect of parameter $ \nu $ on unknown contact stresses
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图 12 参数$ \tilde P $对未知接触应力的影响
Figure 12. Effect of parameter $ \tilde P $ on unknown contact stresses
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图 13 参数$ \tilde R $对未知接触应力的影响
Figure 13. Effect of parameter $ \tilde R $ on unknown contact stresses
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图 14 参数$ 2\tilde a $对未知接触应力的影响
Figure 14. Effect of parameter $ 2\tilde a $ on unknown contact stresses
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图 15 参数N和$ \tilde z $对法向位移的影响
Figure 15. Effect of parameter N and $ \tilde z $ on normal displacement
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图 16 法向位移随$ 2\tilde a $的变化
Figure 16. Normal displacement with respect to $ 2\tilde a $
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图 17 法向位移随$ \tilde b - \tilde a $的变化
Figure 17. Normal displacement with respect to $ \tilde b - \tilde a $
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图 20 法向位移随$ 2\tilde a $的变化
Figure 20. Normal displacement with respect to $ 2\tilde a $
表 1 孔隙率变化时, 圆柱的实心与空心所引起的接触外边缘应力奇异性的对比
Table 1 Comparison of stress singularity at the contact outer edge caused by solid and hollow cylinders with changes of porosity
Cylinder ($ \nu = {\text{0}}{\text{.3,}}\;\tilde P{\text{ = 1,}}\;N = [0,0.05,0.2,0.6] $) The wall thickness of the hollow column is consistent with the radius of the solid column solid ($ \tilde a = 0 $) hollow ($ \tilde a = 0.3 $) solid ($ \tilde a = 0 $) hollow ($ \tilde a = 2 $) (a) stress ratio at the outer edge The outer diameter of the hollow column is consistent with the radius of the solid column solid ($ \tilde a = 0 $) hollow ($ \tilde a = 0.3 $) solid ($ \tilde a = 0 $) hollow ($ \tilde a = 2 $) (b) stress ratio at the outer edge 
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