﻿ 一类多孔固体的等效偶应力动力学梁模型
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 力学学报  2016, Vol. 48 Issue (1): 111-126  DOI: 10.6052/0459-1879-15-210 0

### 引用本文 [复制中英文]

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Su Wenzheng, Liu Shutiany. EFFECTIVE COUPLE-STRESS DYNAMIC BEAM MODEL OF A TYPICAL CELLULAR SOLID[J]. Chinese Journal of Ship Research, 2016, 48(1): 111-126. DOI: 10.6052/0459-1879-15-210.
[复制英文]

### 文章历史

2015-06-09收稿
2015-09-01录用
2015-09-09网络版发表

1. 大连交通大学土木与安全工程学院, 大连 116028;
2. 大连理工大学工业装备结构分析国家重点实验室/工程力学系, 大连 116023

 图 1 多孔固体类梁结构变形示意图 Fig.1 Sketch of the deformation of a cellular solid beam-like structure

 图 2 一类多孔固体类梁结构及单胞构型 Fig.2 A typical cellular solid beam-like structure and its unit cell
1 多孔固体的等效偶应力连续介质模型 1.1 偶应力理论简介

 \left. \begin{align} & {{\varepsilon }_{x}}=\partial u/\partial x,\quad {{\varepsilon }_{y}}=\partial v/\partial y,\quad {{\gamma }_{xy}}=\partial u/\partial y+\partial v/\partial x \\ & {{\kappa }_{xz}}=\partial \phi /\partial x,\quad {{\kappa }_{yz}}=\partial \phi /\partial y \\ \end{align} \right\} (1)

 \left. \begin{align} & \frac{\partial {{\sigma }_{x}}}{\partial x}+\frac{\partial {{\tau }_{yx}}}{\partial y}=\rho \frac{{{\partial }^{2}}u}{\partial {{t}^{2}}} \\ & \frac{\partial {{\tau }_{xy}}}{\partial x}\ +\frac{\partial {{\sigma }_{y}}}{\partial y}=\rho \frac{{{\partial }^{2}}v}{\partial {{t}^{2}}} \\ & \frac{\partial {{m}_{xz}}}{\partial x}+\frac{\partial {{m}_{yz}}}{\partial y}+{{\tau }_{xy}}-{{\tau }_{yx}}=\Theta \frac{{{\partial }^{2}}\phi }{\partial {{t}^{2}}} \\ \end{align} \right\} (2)
 图 3 平面问题的应力和偶应力分量 Fig.3 Components of stress and couple-stress in a planar problem

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\sigma _x}\\ {\sigma _y}\\ {\tau _S} \end{array} \end{array}} \right\} = \left[ \begin{array}{l} {C_{11}}\;\;{C_{12}}\;\;\;0\\ {C_{12}}\;\;{C_{22}}\;\;\;0\\ 0\;\;\;\;\;\;0\;\;\;{C_{66}} \end{array} \right]\left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\varepsilon _x}\\ {\varepsilon _y}\\ {\gamma _{xy}} \end{array} \end{array}} \right\}\\ \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {m_{xz}}\\ {m_{yz}} \end{array} \end{array}} \right\} = \left[ \begin{array}{l} {D_{11}}\;\;\;0\\ 0\;\;\;\;\;{D_{22}} \end{array} \right]\left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\kappa _{xz}}\\ {\kappa _{yz}} \end{array} \end{array}} \right\} \end{array} \end{array}} \right\}$ (3)

 $\sigma =C\varepsilon ,m=D\kappa$ (4)

 \left. \begin{align} & {{l}_{G1}}=\sqrt{{{B}_{11}}/G},\ {{l}_{E1}}=\sqrt{2\left( 1+{{\mu }_{21}} \right){{B}_{11}}/{{E}_{11}}} \\ & {{l}_{G2}}=\sqrt{{{B}_{22}}/G},\ {{l}_{E2}}=\sqrt{2\left( 1+{{\mu }_{12}} \right){{B}_{22}}/{{E}_{22}}} \\ \end{align} \right\} (5)

 \left. \begin{array}{*{35}{l}} \begin{align} & {{E}_{11}}={{C}_{11}}\left( 1-{{\mu }_{12}}{{\mu }_{21}} \right) \\ & {{E}_{22}}={{C}_{22}}\left( 1-{{\mu }_{12}}{{\mu }_{21}} \right),G={{C}_{66}} \\ & {{\mu }_{12}}={{C}_{12}}/{{C}_{11}},{{\mu }_{21}}={{C}_{21}}/{{C}_{22}} \\ & {{B}_{11}}={{D}_{11}}/4,{{B}_{22}}={{D}_{22}}/4 \\ \end{align} \\ \end{array} \right\} (6)
1.2 等效连续介质模型

2 偶应力介质的铁木辛柯梁动力学模型

2.1 梁的横向振动微分方程

 $\frac{\partial {{v}^{*}}}{\partial {{x}^{*}}}={{\alpha }^{*}}+{{\beta }^{*}}\text{ }$ (7)
 图 4 悬臂梁的横向挠度与横截面转角 Fig.4 Transverse deflection and angle of rotation of the cross-section of a cantilever beam

 \left. \begin{array}{*{35}{l}} \begin{align} & {{u}^{*}}\left( {{x}^{*}},{{y}^{*}} \right)=-{{y}^{*}}{{\alpha }^{*}}\left( {{x}^{*}} \right) \\ & {{v}^{*}}\left( {{x}^{*}},{{y}^{*}} \right)={{v}^{*}}\left( {{x}^{*}} \right) \\ & {{\phi }^{*}}\left( {{x}^{*}},{{y}^{*}} \right)={{\alpha }^{*}}\left( {{x}^{*}} \right) \\ \end{align} \\ \end{array} \right\} (8)

 \left. \begin{array}{*{35}{l}} \begin{align} & \varepsilon _{x}^{*}=-{{y}^{*}}\frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}},\varepsilon _{y}^{*}=0,\gamma _{xy}^{*}=\frac{\partial {{v}^{*}}}{\partial {{x}^{*}}}-{{\alpha }^{*}} \\ & \kappa _{xz}^{*}=\frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}},\kappa _{yz}^{*}=0 \\ \end{align} \\ \end{array} \right\} (9)

 \begin{align} & {{V}^{*}}=\frac{1}{2}\int{_{{{V}^{*}}}}{{\sigma }^{*\text{T}}}{{\varepsilon }^{*}}d{{V}^{*}}+\frac{1}{2}\int{_{{{V}^{*}}}}{{m}^{*\text{T}}}{{\kappa }^{*}}d{{V}^{*}}= \\ & \qquad \frac{1}{2}\int{_{{{V}^{*}}}}\left( \sigma _{x}^{*}\varepsilon _{x}^{*}+\tau _{S}^{*}\gamma _{xy}^{*} \right)\text{d}{{V}^{*}}+\frac{1}{2}\int{_{{{V}^{*}}}}m_{xz}^{*}\kappa _{xz}^{*}d{{V}^{*}}= \\ & \qquad \frac{1}{2}\int{_{{{V}^{*}}}}\left( E_{11}^{*}\varepsilon _{x}^{*2}+{{G}^{*}}\gamma _{xy}^{*2} \right)d{{V}^{*}}+\frac{1}{2}\int{_{{{V}^{*}}}}D_{11}^{*}\kappa _{xz}^{*2}\text{d}{{V}^{*}}= \\ & \qquad \frac{1}{2}\int{_{{{V}^{*}}}}\left[E_{11}^{*}{{\left( -{{y}^{*}}\frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}} \right)}^{2}}+{{G}^{*}}{{\left( \frac{\partial {{v}^{*}}}{\partial {{x}^{*}}}-{{\alpha }^{*}} \right)}^{2}} \right]\text{d}{{V}^{*}}+ \\ & \qquad \frac{1}{2}\int{_{{{V}^{*}}}}D_{11}^{*}{{\left( \frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}} \right)}^{2}}d{{V}^{*}}= \\ & \qquad \frac{1}{2}\int{_{{{L}^{*}}}}\left[E_{11}^{*}{{I}^{*}}{{\left( \frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}} \right)}^{2}}+{{G}^{*}}{{A}^{*}}{{\left( \frac{\partial {{v}^{*}}}{\partial {{x}^{*}}}-{{\alpha }^{*}} \right)}^{2}} \right]d{{x}^{*}}+ \\ & \qquad \frac{1}{2}\int{_{{{L}^{*}}}}D_{11}^{*}{{A}^{*}}{{\left( \frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}} \right)}^{2}}d{{x}^{*}} \\ \end{align} (10)

 \begin{align} & {{V}^{*}}=\frac{1}{2}\int_{0}^{L*}{\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right)}{{\left( \frac{\partial {{\alpha }^{*}}}{\partial {{x}^{*}}} \right)}^{2}}d{{x}^{*}}+ \\ & \qquad \frac{1}{2}\int_{0}^{L*}{{k}'{{G}^{*}}{{A}^{*}}}{{\left( \frac{\partial {{v}^{*}}}{\partial {{x}^{*}}}-{{\alpha }^{*}} \right)}^{2}}\text{d}{{x}^{*}} \\ \end{align} (11)

 ${k}'=\frac{10\left( 1+\mu \right)}{12+11\mu }\text{ }$ (12)

 \left. \begin{array}{*{35}{l}} \begin{align} & x={{x}^{*}}/{{L}^{*}} \\ & v={{v}^{*}}/{{L}^{*}} \\ & A={{A}^{*}}/{{L}^{*2}} \\ & G={{G}^{*}}{{L}^{*4}}/\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right) \\ & V={{V}^{*}}{{L}^{*}}/\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right) \\ \end{align} \\ \end{array} \right\} (13)

 $V=\frac{1}{2}\int_{0}^{1}{{{\left( \frac{\partial \alpha }{\partial x} \right)}^{2}}}\text{ d}x+\frac{1}{2}\int_{0}^{1}{{k}'GA}{{\left( \frac{\partial v}{\partial x}-\alpha \right)}^{2}}\text{ d}x\text{ }$ (14)

 \begin{align} & {{T}^{*}}=\frac{1}{2}\int_{0}^{L*}{{{\rho }^{*}}{{A}^{*}}}{{\left( \frac{\partial {{v}^{*}}}{\partial {{t}^{*}}} \right)}^{2}}d{{x}^{*}}+ \\ & \qquad \frac{1}{2}\int_{0}^{L*}{\left( {{\rho }^{*}}{{I}^{*}}+{{\Theta }^{*}}{{A}^{*}} \right)}{{\left( \frac{\partial {{\alpha }^{*}}}{\partial {{t}^{*}}} \right)}^{2}}d{{x}^{*}} \\ \end{align} (15)

 \left. \begin{array}{*{35}{l}} \begin{align} & t={{t}^{*}}\omega _{1}^{*} \\ & \rho ={{\rho }^{*}}{{L}^{*6}}\omega _{1}^{*2}/\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right) \\ & \Theta ={{\Theta }^{*}}{{L}^{*4}}\omega _{1}^{*2}/\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right) \\ & T={{T}^{*}}{{L}^{*}}/\left( E_{11}^{*}{{I}^{*}}+D_{11}^{*}{{A}^{*}} \right) \\ \end{align} \\ \end{array} \right\} (16)

 \begin{align} & T=\frac{1}{2}\int_{0}^{1}{\rho A}{{\left( \frac{\partial v}{\partial t} \right)}^{2}}dx+ \\ & \qquad \frac{1}{2}\int_{0}^{1}{\left( \rho I+\Theta A \right)}{{\left( \frac{\partial \alpha }{\partial t} \right)}^{2}}dx \\ \end{align} (17)

 $\delta \text{ }\int_{{{t}_{1}}}^{{{t}_{2}}}{\left( T-V \right)}\text{ d}t=0$ (18)

 \left. \begin{array}{*{35}{l}} \begin{align} & \rho A\frac{{{\partial }^{2}}v}{\partial {{t}^{2}}}-{k}'GA\left( \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}-\frac{\partial \alpha }{\partial x} \right)=0 \\ & \left( \rho I+\Theta A \right)\frac{{{\partial }^{2}}\alpha }{\partial {{t}^{2}}}-\frac{{{\partial }^{2}}\alpha }{\partial {{t}^{2}}}-{k}'GA\left( \frac{\partial v}{\partial x}-\alpha \right)=0 \\ \end{align} \\ \end{array} \right\} (19)
2.2 梁的自振频率及振型

 ${{\omega }_{\text{c}}}=\sqrt{{k}'GA/\left( \rho I+\Theta A \right)}$ (20)

 \left. \begin{align} & W\left( x \right)={{C}_{1}}\sin ax+{{C}_{2}}\cos ax+ \\ & \qquad {{C}_{3}}\sinh bx+{{C}_{4}}\cosh bx \\ & \Psi \left( x \right)={{D}_{1}}\sin ax+{{D}_{2}}\cos ax+ \\ & \qquad {{D}_{3}}\sinh bx+{{D}_{4}}\cosh bx \\ \end{align} \right\} (21)

 \left. \begin{align} & W\left( x \right)={{{\tilde{C}}}_{1}}\sin ax+{{{\tilde{C}}}_{2}}\cos ax+ \\ & {{{\tilde{C}}}_{3}}\sin \tilde{b}x+{{{\tilde{C}}}_{4}}\cos \tilde{b}x \\ & \Psi \left( x \right)={{{\tilde{D}}}_{1}}\sin ax+{{{\tilde{D}}}_{2}}\cos ax+ \\ & \qquad {{{\tilde{D}}}_{3}}\sin \tilde{b}x+{{{\tilde{D}}}_{4}}\cos \tilde{b}x \\ \end{align} \right\} (22)

 \left. \begin{array}{*{35}{l}} \begin{align} & {{\left. W\left( x \right) \right|}_{x=0}}=0 \\ & {{\left. \Psi \left( x \right) \right|}_{x=0}}=0 \\ \end{align} \\ \end{array} \right\} (23)

 \left. \begin{array}{*{35}{l}} \begin{align} & {{\left. \frac{\text{d}\Psi \left( x \right)}{dx} \right|}_{x=1}}=0 \\ & {{\left. \left[\frac{dW\left( x \right)}{dx}-\Psi \left( x \right) \right] \right|}_{x=1}}=0 \\ \end{align} \\ \end{array} \right\} (24)

3 模型准确性验证

 ${MAC}_{ij} = \dfrac{\left| {\left( {{ X}_i^{\rm e} } \right)^{\rm T}{ X}_j^{\rm d} } \right|^2}{\left[{\left( {{ X}_i^{\rm e} } \right)^ {\rm T}{ X}_i^{\rm e} } \right]\left[{\left( {{ X}_j^{\rm d} } \right)^ {\rm T}{ X}_j^{\rm d} } \right]}\,,\ \ \ i,j = 1,2,\cdots,r$ (25)

4 算例及分析

4.1 算例结果

 图 5 不同的单胞孔径相对大小以及不同的单胞绝对尺寸下，多孔固体短梁结构($L^{\ast } =50$，$H^{\ast } =10$)的等效偶应力 介质铁木辛柯梁的计算频率. 各图中两条虚线分别对应0.95及1.05(下同) Fig.5 Frequencies of cellular solid short ($L^{\ast } =50$, $H^{\ast } =10$) beam-like structures with different relative sizes of void diameter and different absolute sizes of unit cells from the effective couple-stress continuum Timoshenko beam model. The two dotted lines denote the values of 0.95 and 1.05 (similarly hereinafter)

 图 6 不同的单胞孔径相对大小以及不同的单胞绝对尺寸下，多孔固体中梁结构($L^{\ast } =100$，$H^{\ast } =10$)的等效偶应力介质 铁木辛柯梁的计算频率 Fig.6 Frequencies of cellular solid middle long ($L^{\ast } =100$, $H^{\ast } =10$) beam-like structures with different relative sizes of void diameter and different absolute sizes of unit cells from the effective couple-stress continuum Timoshenko beam model
 图 7 不同的单胞孔径相对大小以及不同的单胞绝对尺寸下，多孔固体长梁结构($L^{\ast} =200$，$H^{\ast } =10$)的等效偶应力介质铁木辛柯梁的计算频率 Fig.7 Frequencies of cellular solid long ($L^{\ast } =200$, $H^{\ast } =10$) beam-like structures with different relative sizes of void diameter and different absolute sizes of unit cells from the effective couple-stress continuum Timoshenko beam model
4.2 分析与讨论

$H^{\ast }/h^{\ast }$实际上反映了宏观结构的关键尺寸与材料微结构的尺寸比，决定了多孔固体的等效连续介质模型应当基于经典连续介质理论或者广 义连续介质理论. 根据我们早期的工作，当宏观结构的关键尺寸与组成结构的材料特征长度尺寸比小于20时，结构横向振动行为存在尺寸 效应，需采用等效偶应力连续介质力学模型. 偶应力介质的特征长度可通过偶应力介质的材料常数计算获得[6, 23]，其数值与材料的单胞尺寸大致成正比. 对于本文所述的多孔固体，将特征长度折算为单胞尺寸后，大致上，$H^{\ast }/h^{\ast }$值低于5的结构应采用偶应力模型.

$d^{\ast }/h^{\ast }$体现了材料的单胞结构的几何特征，其意义体现在两个方面：(1)决定了类梁结构的局部变形强弱程度，$d^{\ast }/h^{\ast }$越大，单胞的局部刚度越小，振动中的局部变形模式越容易出现；(2)在一定程度上决定等效介质高阶性能的强弱，即，在宏微观结 构尺寸比相差不大的情况下，$d^{\ast }/h^{\ast }$越大，越容易产生等效性能的尺寸效应. 由于采用了等效偶应力连续介质模型，第2方面影响已经可以忽略不计. 前文算例表明，$d^{\ast }/h^{\ast } =0.9$ 时，单胞层面的局部变形严重影响了等效梁模型的适用性.

 图 8 不同结构尺寸及单胞尺寸下，多孔固体梁结构的等效偶应力介质欧拉-伯努利梁模型的振动分析结果 Fig.8 Results for the vibration analysis of cellular solid beam-like structures with different structural size and cell size from the effective couple-stress continuum Euler-Bernoulli beam model

5 结论

(1)若多孔固体类梁结构的高度尺寸与单胞尺寸的比值为1，或者多孔固体单胞孔径相对大小大于0.7，建议采用直接离散模型.

(2)若多孔固体类梁结构的高度尺寸与单胞尺寸的比值介于2与5之间，且多孔固体单胞孔径相对大小小于0.7，建议采用等效偶应力连续介质的铁木辛柯梁模型.

(3)若多孔固体类梁结构的高度尺寸与单胞尺寸的比值大于5，且多孔固体单胞孔径相对大小小于0.7，建议采用等效经典连续介质的铁木辛柯梁模型.

(4)长高比低于5的短梁不建议采用等效梁模型，因短梁所能够获得准确振型阶数较低.

(5)等效的欧拉-伯努利梁模型，建议只用来计算低频振动，如一阶振动，因其对大多数梁的高阶振动计算结果较差.

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EFFECTIVE COUPLE-STRESS DYNAMIC BEAM MODEL OF A TYPICAL CELLULAR SOLID
Su Wenzheng, Liu Shutian
1. School of Civil and Safety Engineering, Dalian Jiaotong University, Dalian 116028, China;
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China
Abstract: The e ective continuum beam model can be used to study the dynamic behavior of the one-dimensional cellular solid structure. However, this e ective model will bring the size e ect when the cell size is close to the height dimension of the beam-like structure. The classical continuum model, which has no length scale, cannot describe this size dependent feature. By contrast, the generalized continuum model is pertinent to the description of the size e ect. The purpose of this paper is to develop an e ective Timoshenko continuum beam model for the transverse natural vibration analysis of a periodic cellular solid beam-like structure whose unit cell has a circular void, based on the couple-stress theory. The properties of the e ective couple-stress continuum are generated under the criterion of the equivalent strain energy and the geometrical average by the analysis of a unit cell. These properties are then employed in the Timoshenko beam theory to obtain the dynamics di erential equations. Comparison is made with the prediction of a finite element analysis of the complete cellular structure, which is taken as the benchmark for accuracy. Good agreements both on the frequencies and mode shapes are found by a variety of examples where the method of modal assurance criterion (MAC) is employed for the comparison of the mode shapes. The emphasis is put on the influence of the relative size of the void diameter, the size ratio of the beam height to the cell size, and the aspect ratio of the beam-like structure on the accuracy of the e ective beam model. A series of methods on the natural vibration analysis of a cellular solid beam-like structure is recommended conservatively based on these analyses.
Key words: cellular solid    dynamics    couple-stress    Timoshenko beam    size e ect