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1、Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology简单层板的宏简单层板的宏观力学性能观力学性能简单层板的微简单层板的微观力学性能观力学性能简单层板的应简单层板的应力应变关系力应变关系简单层板的强简单层板的强度问题度问题刚度的弹性力刚度的弹性力学分析方法学分析方法刚度的材料力刚度的材料力学分析方法学分析方法强度的材料力强度的材料力学分析方法学分析方法简单层板的宏简单层板的宏观力学性能观力学性能Me
2、chanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute o
3、f TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite
4、MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite
5、 Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechan
6、ics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite
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8、 Institute of Technology 1221662221121112210000Q QQ QQ QQ QQ Q 1221662221121112210000S SS SS SS SS S x xy yy yx xx xy yy yx xQ QQ QQ QQ QQ QQ QQ QQ QQ Q662616262221161211 kkkQ 第第k k层的应力层的应力- -应变关系应变关系Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harb
9、in Institute of Technology0 z 0, 0 zyzx Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technologyx,uy,vz,
10、w变形前的横截面变形前的横截面变形后的横截面变形后的横截面XZXZ平面内的变形几何平面内的变形几何z zx xz zc cA AB BC CD Du u0 0 ww0 0 A AB BC CD Dz zc c Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyB B：中面上一点：中面上一点C C：任意点：任意点 c cC Cz zu uu u0 是层合板中面在是层合板中面在X X方向上的斜率方向
11、上的斜率x xww 0层合板厚度上任意一点层合板厚度上任意一点z z的位移的位移u u为：为：x xwwz zu uu u 00同样，在同样，在yzyz平面内，平面内，y y方向上的位移方向上的位移v v为：为：y ywwz zv vv v 00z zx xz zc cA AB BC CD Du u0 0 ww0 0 A AB BC CD Dz zc c Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Techn
12、ology板内任一点的位移分量可表示为：板内任一点的位移分量可表示为：) )z z, ,y y, ,x x( (wwww) )z z, ,y y, ,x x( (v vv v) )z z, ,y y, ,x x( (u uu u 由直法线不变假设，得由直法线不变假设，得00 z zy yz zx xz zy ywwz zv vv vx xwwz zu uu u) )y y, ,x x( (wwww 00000Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials
13、, Harbin Institute of TechnologyUndeformedUndeformedClassical plate theoryClassical plate theoryFirst-order plate theoryFirst-order plate theoryThird-order plate theoryThird-order plate theoryMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Insti
14、tute of TechnologyClassical plate theoryClassical plate theoryFirst-order plate theoryFirst-order plate theoryThird-order plate theoryThird-order plate theoryMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyClassical plate
15、theoryClassical plate theoryy ywwz zv vv vx xwwz zu uu u) )y y, ,x x( (wwww 00000Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology2020 x xwwz zx xu ux xu ux x 2020y ywwz zy yv vy yv vy y y yx xwwz zx xv vy yu ux xv vy yu ux
16、 xy y 02002 k k z z 0应变由位移确定如下应变由位移确定如下: :若用矩阵形式表示若用矩阵形式表示 x xv vy yu uy yv vx xu u 00000 y yx xwwy ywwx xww k k 222222Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology) )x xww( (k kx x22 ) )y yww( (k ky y22 ) )y yx xww( (k
17、 kxyxy 22T T) )x xv vy yu u( ( , ,y yv v, ,x xu u 00000T T y yx xww, ,y yww, ,x xww k k 222222Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technologyy ywwz zv vx xwwz zu uz zwwz zy yz zx xz z 0y yx xz zv vv vz zu uu u) )y y, ,x x
18、( (wwww 000 x xz zx xu ux xu ux xx x 0y yz zy yv vy yv vy yy y 0) )x xy y( (z zx xv vy yu ux xv vy yu uy yx xx xy y 00 ) )x xy y( (y yx x k k y yx xy yx xMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology k k z z 0 k kk kk k
19、Q Q x xy yy yx xx xy yy yx xx xy yy yx xk kk kk kz zQ QQ QQ QQ QQ QQ QQ QQ QQ Q000662616262221161211Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyLaminateStraindistributionStressdistributionMechanics of Mechanics of coMp
20、osite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Com
21、posite Materials, Harbin Institute of Technology定义作用在单位宽度上层合板的平均内力定义作用在单位宽度上层合板的平均内力 N Ni i 和内力矩和内力矩MMi i为为 2h2hiidzN/ 22/ /h h/ /h hi ii iz zd dz zMM（i=xi=x，y y，xyxy）xyzNyxNyNxyNxxyz层合平板的力矩层合平板的力矩MyMyxMxyMxMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Material
22、s, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology N1kzzkxyyx2t2txyyxxyyxdzdzNNNk1k/ N Nk kz zz zk kx xy yy yx x/ /t t/ /t tx xy yy yx xx xy yy yx xd dz zz zz zd dz zMMMMMMk kk k1221N N层层合板上作
23、用的全部合力和力矩为：层层合板上作用的全部合力和力矩为： 2t2tiidzN/ 2t2tiizdzM/Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology k kk kk kk kz zz zx xy yy yx xz zz zx xy yy yx xN Nk kk kx xy yy yx xz zd dz zk kk kk kd dz zQ QQ QQ QQ QQ QQ QQ QQ QQ QN
24、NN NN N110001662616262221161211 k kk kk kk kz zz zx xy yy yx xz zz zx xy yy yx xN Nk kk kx xy yy yx xd dz zz zk kk kk kz zd dz zQ QQ QQ QQ QQ QQ QQ QQ QQ QMMMMMM1120001662616262221161211不是不是z z的函数而是中面值可以从的函数而是中面值可以从积分中提出积分中提出Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for
25、Composite Materials, Harbin Institute of Technology x xy yy yx xx xy yy yx xx xy yy yx xk kk kk kB BB BB BB BB BB BB BB BB BA AA AA AA AA AA AA AA AA AN NN NN N662616262221161211000662616262221161211 k kk kk kk kz zz zx xy yy yx xz zz zx xy yy yx xN Nk kk kx xy yy yx xz zd dz zk kk kk kd dz zQ QQ QQ
26、 QQ QQ QQ QQ QQ QQ QN NN NN N110001662616262221161211Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology x xy yy yx xx xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DD DD DD DD DB BB BB BB BB BB BB BB BB BMMMMMM662616262221161
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28、ology N Nk kk kk kk ki ij ji ij jN Nk kk kk kk ki ij ji ij jN Nk kk kk kk ki ij ji ij j) )z zz z( (Q QD D) )z zz z( (Q QB B) )z zz z( (Q QA A13131212113121子矩阵子矩阵AA、BB和和 DD面内刚度矩阵面内刚度矩阵耦合刚度矩阵耦合刚度矩阵弯曲刚度矩阵弯曲刚度矩阵都是都是3 33 3对称矩阵对称矩阵Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for
29、Composite Materials, Harbin Institute of Technology) )s si in n( (c co os sQ Qc co os ss si in n) )Q QQ QQ QQ Q( (Q Qc co os ss si in n) )Q QQ QQ Q( (c co os ss si in n) )Q QQ QQ Q( (Q Qc co os ss si in n) )Q QQ QQ Q( (c co os ss si in n) )Q QQ QQ Q( (Q Qc co os sQ Qc co os ss si in n) )Q QQ Q( (s s
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31、kk ki ij ji ij j) )z zz z( (Q QD D) )z zz z( (Q QB B) )z zz z( (Q QA A13131212113121Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Mater
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33、 Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite Mater
34、ialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Insti
35、tute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology) )s si in n( (c co os sQ Qc c
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37、n nQ Qc co os ss si in n) )Q QQ Q( (c co os sQ QQ Q 44662266122211663662212366121126366221236612111642222661241122441222662211124222266124111122222222422 N Nk kk kk kk ki ij ji ij jN Nk kk kk kk ki ij ji ij jN Nk kk kk kk ki ij ji ij j) )z zz z( (Q QD D) )z zz z( (Q QB B) )z zz z( (Q QA A13131212113
38、121Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology各向同性单层各向同性单层A A) )( (EtEtA AA AA AA AA AA AA AEtEtA A21120016626162212211 0 i ij jB BD D) )( (EtEtD DD DD DD DD DD DD DD D) )( (EtEtD D211240112366261622122311 Mechanics o
39、f Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 000210000 x xy yy yx xx xy yy yx xA AA AA AA AA AN NN NN N x xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DMMMMMM210000122A At tD D 合力仅仅与层合板中面内的应变有关，合力矩仅与中合力仅仅与层合板中面内的应变有关，合力矩仅与中面的曲率有关
40、面的曲率有关各向同性层板的拉伸与弯曲之间没有耦合影响，面内没有各向同性层板的拉伸与弯曲之间没有耦合影响，面内没有耦合，同时耦合，同时Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology特殊正交各向异性单层特殊正交各向异性单层厚度为厚度为t，单层刚度，单层刚度Qij有如下公式给出的有如下公式给出的正交各向单层正交各向单层12662112222211212121122121221121111111G
41、GQ QE EQ QE EE EQ QE EQ Q 66662122211112221222111212212221122111S SQ QS SS SS SS SQ QS SS SS SS SQ QS SS SS SS SQ Q Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology特殊正交各向异性单层特殊正交各向异性单层t tQ QA AA AA At tQ QA At tQ QA At tQ
42、QA A666626162222121211110 0 ij ijB B120121212366662616322223121231111t tQ QD DD DD Dt tQ QD Dt tQ QD Dt tQ QD D 00066221212110000 x xy yy yx xx xy yy yx xA AA AA AA AA AN NN NN N x xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DMMMMMM66221212110000合力仅仅与层合板中面内的应变有关，合力矩仅与中面的曲率有关合力仅仅与层合板中面内的应变有关，合力矩仅与中面
43、的曲率有关拉伸与弯曲之间没有耦合影响，面内没有耦合拉伸与弯曲之间没有耦合影响，面内没有耦合Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology一般正交各向异性单层一般正交各向异性单层1203t tQ QD DB Bt tQ QA Aij ijij ijij ijij ijij ij 000662616262212161211x xy yy yx xx xy yy yx xA AA AA AA AA
44、 AA AA AA AA AN NN NN N x xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DD DD DD DD DMMMMMM662616262212161211拉伸与弯曲之间没有耦合影响，面内有耦合拉伸与弯曲之间没有耦合影响，面内有耦合Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of co
45、Mposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 000662616262212161211x xy yy yx xx xy yy yx xA AA AA AA AA AA AA AA AA AN NN NN N x xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DD DD DD DD DMMMMMM662616262212161211多层各向同性层多层各向同性层的对称层合板的对称层合板 ) )( (E
46、EQ QE EQ QQ QQ QE EQ QQ Qk kk kk kk kk kk kk kk kk kk kk kk kk k 1210166212261622211对称层合板的合力和合力矩对称层合板的合力和合力矩 00066111212110000 x xy yy yx xx xy yy yx xA AA AA AA AA AN NN NN N x xy yy yx xx xy yy yx xk kk kk kD DD DD DD DD DMMMMMM66111212110000Mechanics of Mechanics of coMposite MaterialscoMposite M
47、aterialsCenter for Composite Materials, Harbin Institute of Technology) )s si in n( (c co os sQ Qc co os ss si in n) )Q QQ QQ QQ Q( (Q Qc co os ss si in n) )Q QQ QQ Q( (c co os ss si in n) )Q QQ QQ Q( (Q Qc co os ss si in n) )Q QQ QQ Q( (c co os ss si in n) )Q QQ QQ Q( (Q Qc co os sQ Qc co os ss si
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49、 ji ij jN Nk kk kk kk ki ij ji ij j) )z zz z( (Q QD D) )z zz z( (Q QB B) )z zz z( (Q QA A13131212113121分析为零的可能性分析为零的可能性Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology NkkkkijijNkkkkijijNkkkkijijzzQDzzQBzzQA1313121211)(31)
50、(21)(Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology多层特殊正交各向异性层组成的对称层合板：材料主方向与多层特殊正交各向异性层组成的对称层合板：材料主方向与层合板轴向一致层合板轴向一致 k kk kk kk kk kk kk kk kk kk kk kk kk kk kk kk kk kG GQ QE EQ QQ QQ QE EQ QE EQ Q126621122222616211211