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1、Center for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology2Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite Materi

2、alscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology簡單層板的簡單層板的宏觀力學性宏觀力學性能能簡單層板的簡單層板的微觀力

3、學性微觀力學性能能簡單層板的簡單層板的應力應力-應變關(guān)應變關(guān)系系簡單層板的簡單層板的強度問題強度問題剛度的彈性剛度的彈性力學分析方力學分析方法法剛度的材料剛度的材料力學分析方力學分析方法法強度的材料強度的材料力學分析方力學分析方法法簡單層板的簡單層板的力學性能力學性能Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite Ma

4、terialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology經(jīng)典經(jīng)典層合理論層合理論層合板的層合板的強度問題強度問題層合板的應層合板的應力應變關(guān)系力應變關(guān)系剛度的剛度的特殊情況特殊情況層間應力層間應力強度分析方強度分析方法法層合板設(shè)計層

5、合板設(shè)計層合板的宏層合板的宏觀力學性能觀力學性能層合板彎曲層合板彎曲振動與屈曲振動與屈曲Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCe

6、nter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Te

7、chnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin

8、 Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite M

9、aterials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite Materialsc

10、oMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechan

11、ics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Ins

12、titute of Technology) )( (/ /E EG G 12獨立常數(shù)只有獨立常數(shù)只有2個個Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Tec

13、hnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology621 ,.,., ,j j, , i iC Cj jijiji i 應力分量，剛度矩陣，應變分量應力分量，剛度矩陣，應變分量621 ,.,., ,j j, , i iS Sj jijiji i 柔度矩陣柔度矩陣Mechanics of Mechanics of Mechanics of Mechanics of Mechanics o

14、f Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institut

15、e 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 C各向異性線彈性材料最通用的定律，各向異性線彈性材料最通用的定律，要完整描述這種材料需要要完整描述這種材料需要36個分量或常數(shù)，該類材料個分量或常數(shù)，該類材料沒有材料對稱性，

16、這種材料也叫做三斜晶系材料沒有材料對稱性，這種材料也叫做三斜晶系材料Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for C

17、omposite Materials, Harbin Institute of TechnologyCenter 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

18、 CC CC CC Cz zw wy yv vx xu u 321x xv vy yu uz zu ux xw wz zv vy yw w 123123簡寫了表簡寫了表達符號達符號幾何方程幾何方程Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for

19、 Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technologyxyzxzxz yzyz zdydyy yy yy y dydyy yzyzyzyzy dydyy yxyxyxyxy x xy yz zx xy yz zz zy yx x, , , , , , 六個應力分量六個應力分量主應力和主方向主應力和主方

20、向材料往往在受力最大的面發(fā)生破壞，材料往往在受力最大的面發(fā)生破壞，物體內(nèi)每一點都有無窮多個微面通物體內(nèi)每一點都有無窮多個微面通過，斜面上剪應力為零的面為主平過，斜面上剪應力為零的面為主平面，其法線方向為主方向，應力為面，其法線方向為主方向，應力為主應力，三個主應力，包括最大和主應力，三個主應力，包括最大和最小應力最小應力Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMp

21、osite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology000 z zy yx xz zy yx xz zy yx xz zyzyzzxzxyzyzy yxyxyxzxzxyxy

22、x x xyxyzxzxyzyzz zy yx xxyxyzxzxyzyzz 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、模量分量各向異性體彈性各向異性體彈性力學基本方程力學基本方程平衡方程平衡方程彈性體受力變形的彈性體受力變形的應力與應變關(guān)系應力與應變關(guān)系本構(gòu)方程本構(gòu)方程36Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of

23、Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute

24、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 xy yx xx xy y zwyvxuzyx xvyuzuxwzvywxyzxyz 幾何方程消除位移分量幾何方程消除位移分量連續(xù)性方程或變形協(xié)調(diào)方程連續(xù)性方程或變形協(xié)調(diào)方程6幾何方程幾何方程Mechanics o

25、f Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Techn

26、ologyCenter for Composite Materials, Harbin Institute of Technology彈性力學問題的一般解法彈性力學問題的一般解法6個應力分量個應力分量6個應變分量個應變分量3個位移分量個位移分量w w, ,v v, ,u u, , , , , , , , , , ,xyxyzxzxyzyzz zy yx xxyxyzxzxyzyzz zy yx x 幾何關(guān)系（位移和應變關(guān)系）：幾何關(guān)系（位移和應變關(guān)系）：6物理關(guān)系（應力和應變關(guān)系）：物理關(guān)系（應力和應變關(guān)系）：6平衡方程（應力之間的關(guān)系）：平衡方程（應力之間的關(guān)系）：315個方程求個方程求15

27、個未知數(shù)個未知數(shù)可解可解(材料性質(zhì)已知材料性質(zhì)已知)難以實現(xiàn)難以實現(xiàn)簡化或數(shù)值解法簡化或數(shù)值解法Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Technolo

28、gyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite Materialsc

29、oMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter 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

30、 uz zu ux xw wz zv vy yw w 123123 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 CMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mec

31、hanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of

32、Technologyn nz zyzyzxzxzn nzyzyy yxyxyn nzxzxyxyxx xZ Z) )z z, ,n ncos(cos() )y y, ,n ncos(cos() )x x, ,n ncos(cos(Y Y) )z z, ,n ncos(cos() )y y, ,n ncos(cos() )x x, ,n ncos(cos(X X) )z z, ,n ncos(cos() )y y, ,n ncos(cos() )x x, ,n ncos(cos( Mechanics of Mechanics of Mechanics of Mechanics of Mechan

33、ics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Ins

34、titute of Technology* * * *w ww wv vv vu uu u: :S Sonon sMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of

35、 TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyS SS SS Sw ww w, ,v vv v, ,u uu u: :S SononX Xn n: :S Sononu u* * * *u ui ij jijij SuS Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of

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37、離散替代連續(xù)離散替代連續(xù)Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Ha

38、rbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composi

39、te Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite Materi

40、alscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 12312332166656463626156555453525146454443424136353433

41、3231262524232221161514131211123123321C 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回來繼續(xù)關(guān)注剛度矩陣回來繼續(xù)關(guān)注剛度矩陣3636個分量個分量Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite

42、 MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 在剛度矩陣在剛度矩陣Cij中有中有36個常數(shù)，但在材料中，實際常數(shù)個常數(shù)，但在材料中，實際常

43、數(shù)小于小于36個。首先證明個。首先證明Cij的對稱性：的對稱性： 存在有彈性位能或應變能密度函數(shù)的彈性材料，當應存在有彈性位能或應變能密度函數(shù)的彈性材料，當應力力 i作用產(chǎn)生作用產(chǎn)生d i的增量時，單位體積的功的增量為：的增量時，單位體積的功的增量為： dW= i d i 由應力由應力-應變關(guān)系應變關(guān)系 i= Cij d j，功的增量為：，功的增量為： dW= Cij d j d i 沿整個應變積分，單位體積的功為：沿整個應變積分，單位體積的功為： W=1/2 Cij j i Mechanics of Mechanics of Mechanics of Mechanics of Mechani

44、cs of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Inst

45、itute of Technologyijijj ji ij jijiji iC Cw wC Cw w 2jijii ij jC Cw w 2Cij的腳標與微分次序無關(guān)：的腳標與微分次序無關(guān)： Cij=Cji同理同理廣義胡克定律關(guān)系式可由下式導出：廣義胡克定律關(guān)系式可由下式導出：W=1/2 Cij j i 621, ,. . . . . ., , ,j j, , i iS Sj ji ij ji i Sij=SjiMechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite Mat

46、erialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 123123321665

47、646362616565545352515464544342414363534332313262524232212161514131211123123321C 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剛度矩陣是對稱的，只有剛度矩陣是對稱的，只有21個常數(shù)是獨立的個常數(shù)是獨立的Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics o

48、f coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technolo

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50、 MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 12312332166465535251546443533231325232212151312111231233210000000000000000C CC CC CC

51、 CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC Cy=0Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of Tec

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53、ialscoMposite MaterialsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 123123321665544332331232221131211123123321000000000000000000000000C CC CC CC CC CC CC CC C

54、C CC CC CC C正應力與剪應變之間沒有耦合，剪應力與正應變之間沒有耦合正應力與剪應變之間沒有耦合，剪應力與正應變之間沒有耦合不同平面內(nèi)的剪應力和剪應變之間也沒有相互作用不同平面內(nèi)的剪應力和剪應變之間也沒有相互作用Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite Materia

55、lsCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, Harbin Institute of Technology 123123321121144443313131311121312111231233212000000000000000000000000C CC CC CC CC CC CC CC CC CC CC CC CC C212

56、1166C CC CC C 根據(jù)純剪切和拉伸與壓縮組合之間的等根據(jù)純剪切和拉伸與壓縮組合之間的等效推導而出效推導而出1-2平面平面1，2可互換可互換Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Ha

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59、or Composite Materials, Harbin Institute of TechnologyCenter for Composite Materials, 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 S

60、S SS SS SS SS S與剛度矩陣一樣有相似的性質(zhì)與剛度矩陣一樣有相似的性質(zhì)剛度矩陣與柔度矩陣互為逆矩陣剛度矩陣與柔度矩陣互為逆矩陣Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of Mechanics of coMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialscoMposite MaterialsCenter for Composite Materials, Harbin