复合材料力学-2

1、Center for Composite Materials, Harbin Institute of Technology2Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology简单层板的简单层板的宏观力学性宏观力学性能能简单层板的简单层板的微观力学性微观力学性能能简单层板的简单层板的应力应力-应变关应变关系系简单层板的简单层板的强度问题强度问题刚度的弹性刚度的弹性力学分析方力学分析方法法刚度的材

2、料刚度的材料力学分析方力学分析方法法强度的材料强度的材料力学分析方力学分析方法法简单层板的简单层板的力学性能力学性能Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology经典经典层合理论层合理论层合板的层合板的强度问题强度问题层合板的应层合板的应力应变关系力应变关系刚度的刚度的特殊情况特殊情况层间应力层间应力强度分析方强度分析方法法层合板设计层合板设计层合板的宏层合板的宏观力学性能观力学性能层合板弯

3、曲层合板弯曲振动与屈曲振动与屈曲Mechanics 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,

4、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 TechnologyMechanics of Mecha

5、nics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology) )( (/ /E EG G 12独立常数只有独立常数只有2个个Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology621 ,.,., ,j j, , i iC Cj jijiji

6、i 应力分量，刚度矩阵，应变分量应力分量，刚度矩阵，应变分量621 , ,. . . . . ., , ,j j, , i iS Sj ji ij ji i 柔度矩阵柔度矩阵Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 12312332166656463626156555453525146454443424136353433323126252423222116151413121112312

7、3321C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC C各向异性线弹性材料最通用的定律，各向异性线弹性材料最通用的定律，要完整描述这种材料需要要完整描述这种材料需要36个分量或常数，该类材料个分量或常数，该类材料没有材料对称性，这种材料也叫做三斜晶系材料没有材料对称性，这种材料也叫做三斜晶系材料Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCent

8、er for Composite Materials, Harbin Institute of Technology 123123321666564636261565554535251464544434241363534333231262524232221161514131211123123321C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC Cz zw wy yv vx xu u 321x xv vy yu uz zu ux x

9、w wz zv vy yw w 123123简写了表简写了表达符号达符号几何方程几何方程Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technologyxyzxzxz yzyz zdydyy yy yy y dydyy yzyzyzyzy d dy yy yx xy yx xy y x xy yz zx xy yz zz zy yx x, , , , , , 六个应力分量六个应力分量主应力和主方向主应力和主方

10、向材料往往在受力最大的面发生破坏，材料往往在受力最大的面发生破坏，物体内每一点都有无穷多个微面通物体内每一点都有无穷多个微面通过，斜面上剪应力为零的面为主平过，斜面上剪应力为零的面为主平面，其法线方向为主方向，应力为面，其法线方向为主方向，应力为主应力，三个主应力，包括最大和主应力，三个主应力，包括最大和最小应力最小应力Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology000 z zy yx x

11、z zy yx xz zy yx xz zyzyzzxzxyzyzy yxyxyxzxzxyxyx x x xy yz zx xy yz zz zy yx xx xy yz zx xy yz zz zy yx xS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS S66646463626151413121161514131211 j ji ij ji iC C 柔度分量、模量分量柔度分量、模量分量各向异性体弹性各向异性体弹性力学基本方程力学基本方程平衡方程平衡方程弹性体受力变形的弹性体受力变形的应力与应变关系应力与应变关系本构方程本构方程36Mecha

12、nics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology z zy yx xz zy yx xz zy yx xy yx xz zz zy yx xx xz zy yxyxyzxzxyzyzz zxyxyzxzxyzyzy yxyxyzxzxyzyzx x222222222222222222222y yz zz zy yx xz zz zx xx xy yy yx xz zy yy yz zz zx xz zx

13、 xy yx xx xy y zwyvxuzyx xvyuzuxwzvywxyzxyz 几何方程消除位移分量几何方程消除位移分量连续性方程或变形协调方程连续性方程或变形协调方程6几何方程几何方程Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology弹性力学问题的一般解法弹性力学问题的一般解法6个应力分量个应力分量6个应变分量个应变分量3个位移分量个位移分量w w, ,v v, ,u u, , , ,

14、 , , , , , , ,xyxyzxzxyzyzz zy yx xxyxyzxzxyzyzz zy yx x 几何关系（位移和应变关系）：几何关系（位移和应变关系）：6物理关系（应力和应变关系）：物理关系（应力和应变关系）：6平衡方程（应力之间的关系）：平衡方程（应力之间的关系）：315个方程求个方程求15个未知数个未知数可解可解(材料性质已知材料性质已知)难以实现难以实现简化或数值解法简化或数值解法Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials,

15、Harbin Institute of TechnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology000 z zy yx xz zy yx xz zy yx xz zyzyzzxzxyzyzy yxyxyxzxzxyxyx xz zw wy yv vx xu u 321x xv vy yu uz zu ux xw wz zv vy yw w 123123 12312332166656

16、4636261565554535251464544434241363534333231262524232221161514131211123123321C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of

17、Technologyn nz zy yz zx xz zn nz zy yy yx xy yn nz zx xy yx xx xZ Z) )z z, ,n nc co os s( () )y y, ,n nc co os s( () )x x, ,n nc co os s( (Y Y) )z z, ,n nc co os s( () )y y, ,n nc co os s( () )x x, ,n nc co os s( (X X) )z z, ,n nc co os s( () )y y, ,n nc co os s( () )x x, ,n nc co os s( ( Mechanics

18、of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology* * * *w ww wv vv vu uu u: :S Sonon sMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyS SS SS Sw ww w, ,v v

19、v v, ,u uu u: :S SononX Xn n: :S Sononu u* * * *u ui ij jijij SuS Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology离散替代连续离散替代连续Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials,

20、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 Technology 12312332166656463

21、6261565554535251464544434241363534333231262524232221161514131211123123321C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC C回来继续关注刚度矩阵回来继续关注刚度矩阵3636个分量个分量Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Mate

22、rials, Harbin Institute of Technology 在刚度矩阵在刚度矩阵Cij中有中有36个常数，但在材料中，实际常数个常数，但在材料中，实际常数小于小于36个。首先证明个。首先证明Cij的对称性：的对称性： 存在有弹性位能或应变能密度函数的弹性材料，当应存在有弹性位能或应变能密度函数的弹性材料，当应力力 i作用产生作用产生d i的增量时，单位体积的功的增量为：的增量时，单位体积的功的增量为： dW= i d i 由应力由应力-应变关系应变关系 i= Cij d j，功的增量为：，功的增量为： dW= Cij d j d i 沿整个应变积分，单位体积的功为：沿整个应变积

23、分，单位体积的功为： W=1/2 Cij j i Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technologyijijj ji ij jijiji iC Cw wC Cw w 2jijii ij jC Cw w 2Cij的脚标与微分次序无关：的脚标与微分次序无关： Cij=Cji同理同理广义胡克定律关系式可由下式导出：广义胡克定律关系式可由下式导出：W=1/2 Cij j i 621, ,. . . .

24、. ., , ,j j, , i iS Sj ji ij ji i Sij=SjiMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123321665646362616565545352515464544342414363534332313262524232212161514131211123123321C CC CC CC CC CC CC CC CC CC CC CC CC CC C

25、C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC C刚度矩阵是对称的，只有刚度矩阵是对称的，只有21个常数是独立的个常数是独立的Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123321663626165545454436332313262322121613121112312332100000000000

26、00000C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 12312332166465535251546443533231325232212151312111231233210000000000000000C CC CC CC CC CC CC CC CC CC CC CC

27、 CC CC CC CC CC CC CC CC Cy=0Mechanics 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 123123321665544332

28、331232221131211123123321000000000000000000000000C CC CC CC CC CC CC CC CC CC CC CC C正应力与剪应变之间没有耦合，剪应力与正应变之间没有耦合正应力与剪应变之间没有耦合，剪应力与正应变之间没有耦合不同平面内的剪应力和剪应变之间也没有相互作用不同平面内的剪应力和剪应变之间也没有相互作用Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Te

29、chnology 123123321121144443313131311121312111231233212000000000000000000000000C CC CC CC CC CC CC CC CC CC CC CC CC C2121166C CC CC C 根据纯剪切和拉伸与压缩组合之间的等根据纯剪切和拉伸与压缩组合之间的等效推导而出效推导而出1-2平面平面1，2可互换可互换Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Ins

30、titute of Technology21211665544312312332211/ /) )C CC C( (C CC CC CC CC CC CC CC CC C 123123321121112111211111212121112121211123123321200000020000002000000000000C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials

31、, Harbin Institute of Technology 123123321665646362616565545352515464544342414363534332313262524232212161514131211123123321S SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS S与刚度矩阵一样有相似的性质与刚度矩阵一样有相似的性质刚度矩阵与柔度矩阵互为逆矩阵刚度矩阵与柔度矩阵互为逆矩阵Mechanics of Me

32、chanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 12312332166362616554545443633231326232212161312111231233210000000000000000S SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SZ=0的平面对称，的平面对称，13个独立常数个独立常数Mechanics of Mechanics of c

33、oMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123321665544332313232212131211123123321000000000000000000000000S SS SS SS SS SS SS SS SS SS SS SS S正交各向异性，正交各向异性，9个独立常数个独立常数Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter fo

34、r Composite Materials, Harbin Institute of Technology 123123321121144443323132322121312111231233212000000000000000000000000) )S SS S( (S SS SS SS SS SS SS SS SS SS SS S横观各向同性（横观各向同性（1-2平面是各向同平面是各向同性面），性面），5个独立常数个独立常数Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite

35、 Materials, Harbin Institute of Technology 123123321121112111211221212121112121211123123321200000020000002000000000000) )S SS S( () )S SS S( () )S SS S( (S SS SS SS SS SS SS SS SS S各向同性，各向同性，2个独立常数个独立常数Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Ha

36、rbin Institute of Technology对称性对称性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 Technology各向异性材料的性质更多地取决

37、于非零分量的个数各向异性材料的性质更多地取决于非零分量的个数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 Technology 12312332166564636

38、2616565545352515464544342414363534332313262524232212161514131211123123321S SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Tec

39、hnologyMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123322311333221123312211100000010000001000000100010001G GG GG GE EE EE EE EE EE EE EE EE ES Si ij jE1、E2、E3为为1，2，3方向上的弹性模量方向上的弹性模量 ij为应力在为应力在i方向上作用时方向上作用时j方向的横向应变的

40、泊松比方向的横向应变的泊松比G23,G31,G12为为2-3，3-1，1-2平面的剪切应变平面的剪切应变Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technologyi ij jijij 321 , , ,j j, , i iE EE Ej jjijii iijij ij为应力在为应力在i方向上作用力时引方向上作用力时引起起j方向的横向应变的泊松比方向的横向应变的泊松比正交各向异性材料只有九个独立常数，现在

41、有正交各向异性材料只有九个独立常数，现在有12个常数个常数根据根据S矩阵的对称性，有：矩阵的对称性，有： 123123322311333221123312211100000010000001000000100010001G GG GG GE EE EE EE EE EE EE EE EE ES SijijMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology12LLL LE EE EL L 1122

42、111112LL不管不管E1和和E2如何，应力作用在如何，应力作用在2方向引起的横向变形和方向引起的横向变形和应力作用在应力作用在1方向引起的相同方向引起的相同L LE EE EL L 22112222Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123321665646362616565545352515464544342414363534332313262524232212161

43、514131211123123321S SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS SS S621, ,. . . . . ., , ,j j, ,i iC Cj ji ij ji i 621, ,. . . . . ., , ,j j, , i iS Sj ji ij ji i Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Comp

44、osite Materials, Harbin Institute of Technology2323122123321322223113322116666555544441123131223212221133221323121321311332233122313122233322112111S SS SS SS SS SS SS SS SS SS SS SS SS SC CS SC CS SC CS SS SS SS SC CS SS SS SC CS SS SS SS SC CS SS SS SC CS SS SS SS SC CS SS SS SC C 在此方程中，符号在此方程中，符号C

45、和和S在每一处都可以互换的在每一处都可以互换的Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 123123322311333221123312211100000010000001000000100010001G GG GG GE EE EE EE EE EE EE EE EE ES Si ij jMechanics of Mechanics of coMposite MaterialscoM

46、posite MaterialsCenter for Composite Materials, Harbin Institute of Technology32113322131133223211221211233442113212331311232233131132255212312133232213113663113321232233121123232231121111E EE EE EE EE EC CC CE EE EE EE EC CE EE EC CC CE EE EE EE EC CC CE EE EE EE EC CE EE EC C Mechanics of Mechanic

47、s of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology) )( (/ /E EG G 121 为保证为保证E和和G为正值，即正应力或剪应力乘为正值，即正应力或剪应力乘以正应变或剪应变产生正功以正应变或剪应变产生正功 各向同性材料，弹性常数满足某些关系式，各向同性材料，弹性常数满足某些关系式，如剪切模量如剪切模量G可以有弹性模量可以有弹性模量E和泊松比和泊松比v给出给出Mechanics of Mechanics of coMposite M

48、aterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology 213/ /E EK K同样对于各向同性体承受静压力同样对于各向同性体承受静压力P的作用，体积的作用，体积应变（三个正应变或拉伸应变之和）可定义为：应变（三个正应变或拉伸应变之和）可定义为： K KP P/ /E EP Pz zy yx x 213) )( (E EP PE EE EE E) )( (E EP PE EE EE E) )( (E EP PE EE EE EP Px xy yz zz zz zx xy

49、yy yz zy yx xx xz zy yx x 21212121121/ / / K为正值（如果为正值（如果K为负，为负，静压力将引起体积膨胀）静压力将引起体积膨胀）Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology621 ,.,., ,j j, , i iC Cj jijiji i 621 , ,. . . . . ., , ,j j, , i iS Sj ji ij ji i Mecha

50、nics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology0665544332211 S S, ,S S, ,S S, ,S S, ,S S, ,S S0121323321 G G, ,G G, ,G G, ,E E, ,E E, ,E E 123123322311333221123312211100000010000001000000100010001G GG GG GE EE EE EE EE EE EE E

51、E EE ES Si ij jMechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology0665544332211 C C, ,C C, ,C C, ,C C, ,C C, ,C C010101211231133223 ) )( () )( () )( (021133221311332232112 由于正定矩阵的行列由于正定矩阵的行列式必须为正式必须为正3211332213113322321122121

52、1233442113212331311232233131132255212312133232213113663113321232233121123232231121111E EE EE EE EE EC CC CE EE EE EE EC CE EE EC CC CE EE EE EE EC CC CE EE EE EE EC CE EE EC C Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technolo

53、gy211211122133111321332223/ / / /) )S SS S( (S S) )S SS S( (S S) )S SS S( (S S 666655554444112313122321222113322132312132131133223312231312223332211111S SC CS SC CS SC CS SS SS SS SC CS SS SS SC CS SS SS SS SC CS SS SS SC CS SS SS SS SC CS SS SS SC C C为正定为正定Mechanics of Mechanics of coMposite Materi

54、alscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology321 , , ,j j, , i iE EE Ej jjijii iijij 211331213113213223212332212112211221/ / / / / / /E EE EE EE EE EE EE EE EE EE EE EE E 010101211231133223 ) )( () )( () )( (211211122133111321332223/ / / /) )S SS S( (S S) )S SS

55、 S( (S S) )S SS S( (S S 代入工程常数也可得到代入工程常数也可得到Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technology2121132133223221221133221/ /E EE EE EE EE EE E 021133221311332232112 0112211321322113211321332232 / / /E EE EE EE EE EE EE EE E 21

56、1221132132132232121332212112211321321322321213321111/ / / / / / /E EE EE EE EE EE EE EE EE EE EE EE EE EE EE EE E为了用另外两个泊松比表达为了用另外两个泊松比表达 2121的界限，继续转化的界限，继续转化对对 3232 1313可得可得相似的表达相似的表达式式Mechanics of Mechanics of coMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute o

57、f Technology992971212112. .E EE E. ./ / 是合理的数据是合理的数据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 Techno

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