﻿ 正交各向异性板带有一般孔形时应力分析
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (4): 491-497.  DOI: 10.11908/j.issn.0253-374x.2018.04.011 0

### 引用本文

LÜ Aizhong, ZHANG Xiaoli, WANG Shaojie. Stress Analysis for an Orthotropic Plate with a General Shaped Hole[J]. Journal of Tongji University (Natural Science), 2018, 46(4): 491-497. DOI: 10.11908/j.issn.0253-374x.2018.04.011

### 文章历史

Stress Analysis for an Orthotropic Plate with a General Shaped Hole
LÜ Aizhong , ZHANG Xiaoli , WANG Shaojie
School of Renewable Energy, North China Electric Power University, Beijing 102206, China
Abstract: The complex variables function method was used for analyzing the stresses in an infinite orthotropic plate with a hole under uniform in-plane loadings at infinity. When the hole was hexagonal or irregular shaped, the stress distributions along/near the hole boundary were obtained for different fiber angles and loading directions. The research shows that the maximum tangential stress occurs on the hole boundary and exactly in the sharp corner point, when the fiber orientation angles are 0° and -90.0° and the uniaxial loadings are perpendicular to the pointing direction of the sharp corner. However, the position of the maximum tangential stress will get farther from the sharp corner point with the rotating of the fiber orientation angles, and the magnitude of the maximum tangential stress decreases correspondingly. Therefore, the stress concentration of the orthotropic plate can be decreased by adjusting the fiber orientation angles. In addition, the tangential stresses at the intersections of the axis of the uniaxial loadings σ and the hole boundary are always exactly the same value -σ for the orthotropic plate.
Key words: orthotropic plate    analytical solution    stress analysis    stress concentration

1 基本方程和应力解析解的求解

 $z = \omega \left( \zeta \right) = R\left( {\zeta + \sum\limits_{k = 0}^\infty {{C_k}{\zeta ^{ - k}}} } \right)$ (1)

 $\left\{ \begin{array}{l} {z_1} = x + {\mu _1}y = {\gamma _1}z + {\delta _1}\bar z\\ {z_2} = x + {\mu _2}y = {\gamma _2}z + {\delta _2}\bar z \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {z_1} = {\omega _1}\left( {{\zeta _1}} \right) = {\gamma _1}R\left( {{\zeta _1} + } \right.\\ \;\;\;\left. {\sum\limits_{k = 0}^n {{C_k}\zeta _1^{ - k}} } \right) + {\delta _1}\bar R\left( {\frac{1}{{{\zeta _1}}} + \sum\limits_{k = 0}^n {{{\bar C}_k}\zeta _1^k} } \right)\\ {z_2} = {\omega _2}\left( {{\zeta _2}} \right) = {\gamma _2}R\left( {{\zeta _2} + } \right.\\ \;\;\;\left. {\sum\limits_{k = 0}^n {{C_k}\zeta _2^{ - k}} } \right) + {\delta _2}\bar R\left( {\frac{1}{{{\zeta _2}}} + \sum\limits_{k = 0}^n {{{\bar C}_k}\zeta _2^k} } \right) \end{array} \right.$ (3)
 $\left\{ \begin{array}{l} {\gamma _1}R\left( {{\zeta _1} + \sum\limits_{k = 0}^n {{C_k}\zeta _1^{ - k}} } \right) + {\delta _1}\bar R\left( {\frac{1}{{{\zeta _1}}} + \sum\limits_{k = 0}^n {{{\bar C}_k}\zeta _1^k} } \right) = \\ \;\;\;\;{\gamma _1}R\left( {\zeta + \sum\limits_{k = 0}^n {{C_k}{\zeta ^{ - k}}} } \right) + {\delta _1}\bar R\left( {\bar \zeta + \sum\limits_{k = 0}^n {{{\bar C}_k}{{\bar \zeta }^{ - k}}} } \right)\\ {\gamma _2}R\left( {{\zeta _2} + \sum\limits_{k = 0}^n {{C_k}\zeta _2^{ - k}} } \right) + {\delta _2}\bar R\left( {\frac{1}{{{\zeta _2}}} + \sum\limits_{k = 0}^n {{{\bar C}_k}\zeta _2^k} } \right) = \\ \;\;\;\;{\gamma _2}R\left( {\zeta + \sum\limits_{k = 0}^n {{C_k}{\zeta ^{ - k}}} } \right) + {\delta _2}\bar R\left( {\bar \zeta + \sum\limits_{k = 0}^n {{{\bar C}_k}{{\bar \zeta }^{ - k}}} } \right) \end{array} \right.$ (4)

 $\begin{array}{*{20}{c}} {{a_{22}}\frac{{{\partial ^4}F}}{{\partial {x^4}}} - 2{a_{26}}\frac{{{\partial ^4}F}}{{\partial {x^3}\partial y}} + \left( {2{a_{12}} + {a_{36}}} \right)\frac{{{\partial ^4}F}}{{\partial {x^2}\partial {y^2}}} - }\\ {2{a_{16}}\frac{{{\partial ^4}F}}{{\partial x\partial {y^3}}} + {a_{11}}\frac{{{\partial ^4}F}}{{\partial {y^4}}} = 0} \end{array}$ (5)
 图 1 受面内均布荷载作用且带有任意形状孔的正交各向异性板 Fig.1 An orthotropic plate with an arbitrary shaped hole under in-plane loadings

 ${a_{11}}{\mu ^4} - 2{a_{16}}{\mu ^3} + \left( {2{a_{12}} + {a_{66}}} \right){\mu ^2} - 2{a_{26}}\mu + {a_{22}} = 0$ (6)

 $F = {F_1}\left( {{z_1}} \right) + \overline {{F_1}\left( {{z_1}} \right)} + {F_2}\left( {{z_2}} \right) + \overline {{F_2}\left( {{z_2}} \right)}$ (7)

Φ1(z1)=dF1(z1)/dz1Φ2(z2)=dF2(z2)/dz2，这样，求解各向异性板平面应力问题就转化为寻找满足相应的边界条件的应力解析函数Φ1(z1)和Φ2(z2)的问题.

Φ1(z1)、Φ2(z2)表示的应力边界条件为

 $\left\{ \begin{array}{l} {\mathit{\Phi }_1}\left( {{z_1}} \right) + \overline {{\mathit{\Phi }_1}\left( {{z_1}} \right)} + {\mathit{\Phi }_2}\left( {{z_2}} \right) + \overline {{\mathit{\Phi }_2}\left( {{z_2}} \right)} = {f_1}\\ {\mu _1}{\mathit{\Phi }_1}\left( {{z_1}} \right) + \overline {{\mu _1}} \overline {{\mathit{\Phi }_1}\left( {{z_1}} \right)} + {\mu _2}{\mathit{\Phi }_2}\left( {{z_2}} \right) + \\ \;\;\;\;\;\;\;\overline {{\mu _2}} \overline {{\mathit{\Phi }_2}\left( {{z_2}} \right)} = {f_2} \end{array} \right.$ (8)

 $\left\{ \begin{array}{l} {\mathit{\Phi }_1}\left( {{z_1}} \right) = {B^ * }{z_1} + \mathit{\Phi }_1^0\left( {{z_1}} \right)\\ {\mathit{\Phi }_2}\left( {{z_2}} \right) = \left( {{{B'}^ * } + {\rm{i}}{{C'}^ * }} \right){z_2} + \mathit{\Phi }_2^0\left( {{z_2}} \right) \end{array} \right.$ (9)

 $\left\{ \begin{array}{l} \mathit{\Phi }_1^0\left( {{z_1}} \right) = \mathit{\Phi }_1^0\left[ {{\omega _1}\left( {{\zeta _1}} \right)} \right] = \mathit{\Phi }_{1 * }^0\left( {{\zeta _1}} \right) = \sum\limits_{k = 0}^\infty {{a_k}\zeta _1^{ - k}} \\ \mathit{\Phi }_2^0\left( {{z_2}} \right) = \mathit{\Phi }_2^0\left[ {{\omega _2}\left( {{\zeta _2}} \right)} \right] = \mathit{\Phi }_{2 * }^0\left( {{\zeta _2}} \right) = \sum\limits_{k = 0}^\infty {{b_k}\zeta _2^{ - k}} \end{array} \right.$ (10)

 $\left\{ \begin{array}{l} {\mathop{\rm Re}\nolimits} \left[ {\mathit{\Phi }_{1 * }^0\left( \sigma \right) + \mathit{\Phi }_{2 * }^0\left( \sigma \right)} \right] = \\ \;\;\;\; - {\mathop{\rm Re}\nolimits} \left[ {{B^ * }{\omega _1}\left( \sigma \right) + \left( {{{B'}^ * } + {\rm{i}}{{C'}^ * }} \right){\omega _2}\left( \sigma \right)} \right]\\ {\mathop{\rm Re}\nolimits} \left[ {{\mu _1}\mathit{\Phi }_{1 * }^0\left( \sigma \right) + {\mu _2}\mathit{\Phi }_{2 * }^0\left( \sigma \right)} \right] = \\ \;\;\;\; - {\mathop{\rm Re}\nolimits} \left[ {{\mu _1}{B^ * }{\omega _1}\left( \sigma \right) + {\mu _2}\left( {{{B'}^ * } + {\rm{i}}{{C'}^ * }} \right){\omega _2}\left( \sigma \right)} \right] \end{array} \right.$ (11)

 $\left\{ \begin{array}{l} {\sigma _x} = 2{\mathop{\rm Re}\nolimits} \left[ {\mu _1^2{{\mathit{\Phi '}}_1}\left( {{z_1}} \right) + \mu _2^2{{\mathit{\Phi '}}_2}\left( {{z_2}} \right)} \right]\\ {\sigma _y} = 2{\mathop{\rm Re}\nolimits} \left[ {{{\mathit{\Phi '}}_1}\left( {{z_1}} \right) + {{\mathit{\Phi '}}_2}\left( {{z_2}} \right)} \right]\\ {\tau _{xy}} = - 2{\mathop{\rm Re}\nolimits} \left[ {{\mu _1}{{\mathit{\Phi '}}_1}\left( {{z_1}} \right) + {\mu _2}{{\mathit{\Phi '}}_2}\left( {{z_2}} \right)} \right] \end{array} \right.$ (12)
 $\left\{ \begin{array}{l} {\sigma _\rho } + {\sigma _\theta } = {\sigma _x} + {\sigma _y}\\ {\sigma _\theta } - {\sigma _\rho } + 2{\rm{i}}{\tau _{\rho \theta }} = \left( {{\sigma _y} - {\sigma _x} + 2{\rm{i}}{\tau _{xy}}} \right)\frac{{{\zeta ^2}}}{{{\rho ^2}}}\frac{{\omega '\left( \zeta \right)}}{{\overline {\omega '\left( \zeta \right)} }} \end{array} \right.$ (13)

 $\left\{ \begin{array}{l} {\mu _1} = \frac{{{{\mu '}_1}\cos \varphi - \sin \varphi }}{{\cos \varphi + {{\mu '}_1}\sin \varphi }}\\ {\mu _2} = \frac{{{{\mu '}_2}\cos \varphi - \sin \varphi }}{{\cos \varphi + {{\mu '}_2}\sin \varphi }} \end{array} \right.$ (14)

2 算例分析和结果比较 2.1 不规则孔形的孔边切向应力分析

 图 2 带有不规则孔形的3种材料结构在σx∞=σ单独作用下的孔边切向应力分布 Fig.2 Distribution of normalized tangential stresses for three kinds of plates with irregular shaped hole under uniaxial loading σx∞=σ
 图 3 带有不规则孔形的3种材料结构在σy∞=σ单独作用下的孔边切向应力分布 Fig.3 Distribution of normalized tangential stresses for three kinds of plates with irregular shaped hole under uniaxial loading σy∞=σ

2.2 不规则孔形的孔外域切向应力分析

 图 4 沿y轴单向拉伸荷载作用下(σy∞=σ)不规则孔形在θ =0°、180.0°板内点的切向应力分布 Fig.4 Distribution of normalized tangential stresses in θ=0° and 180.0° for irregular shaped hole with uniaxial loading σy∞=σ

2.3 正六角形孔的孔边切向应力分析

 图 5 带有正六角形孔的3种材料结构在σx∞=σ单独作用下的孔边切向应力分布 Fig.5 Distribution of normalized tangential stresses for three kinds of plates with hexagon hole under uniaxial loading σx∞=σ
 图 6 带有正六角形孔的3种材料结构在σy∞=σ单独作用下的孔边切向应力分布 Fig.6 Distribution of normalized tangential stresses for three kinds of plates with hexagon hole under uniaxial loading σy∞=σ

2.4 孔边最大切向应力点位置的确定

 图 7 纤维角度按[10.0°/-80.0°]s布置的示意图 Fig.7 Sketch of the fiber orientation angles at [10.0°/-80.0°]s

3 结论

(1) 在单向均布荷载作用下，各向同性材料孔边产生最大切向应力的位置是尖点处，而正交各向异性材料则不一定.

(2) 当单向均布荷载的作用方向与孔边尖点的指向垂直时，纤维角度按[0°/-90.0°]s布置会使最大的切向应力产生在尖点处(θ=0°)；但是随着纤维角度的旋转，最大应力点的位置逐渐偏离尖点，最大切向应力的数值也逐渐减小.

(3) 各向同性材料产生最大切向应力的位置在孔边，而正交各向异性材料则不一定，当单向拉伸荷载的方向与尖点指向垂直时，最大切向应力可能是在孔边(纤维按[0°/-90.0°]s角度布置)，也可能是在孔外部的临近区域(纤维按[45.0°/-45.0°]s角度布置).

(4) 正交各向异性板切向应力的集中程度也与材料的纤维方向有很大的关系，通过调整材料纤维的方向可以获得孔边或者孔外区域较小的应力集中.

(5) 当无限平板中均布的单向拉伸荷载σ沿某坐标轴方向作用时，该坐标轴与孔边界的交点及其附近的孔边区域，切向应力均为压应力.尤其对于正交各向异性板，在该坐标轴与孔边界的交点，切向应力均为-σ.

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