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Mechanics of Composite Materials (PowerPoint)

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					Mechanics of Composite
      Materials
     Constitutive Relationships for
        Composite Materials
Ⅰ. Material Behavior in Principal Material Axes
     • Isotropic materials
          –   uniaxial loading




                  E1
                                                    G
                 2   1
                                         E
                                 G
                                      21   
           –    2-D loading
                                                      1            
                                                            , , 0 
                                             x   E E               x 
                                               
                                                             1       
                                                                        
                                            y   ,           ,0  y 
                                                                    
                                               E E                  xy 
                                             xy 
                                                                1  
                                                                        
                                                          0 , 0 ,
                                                     
                                                                 G 
                                                S   Where [ S ]: compliance matrix
          E            E       
                   ,        ,0 
 x  1           1  2
                2
                                   x 
   E
                       E        
                                    
 y             ,        , 0   y 
  1              1  2
                2
 xy                             
   0             , 0 , G   xy 
                                    
         
                                
                                              Where [Q]: stiffness matrix
    Q  
            Isotropic Materials
Note:
1. Only two independent material constants in the
  constitutive equation.
2. No normal stress and shear strain coupling, or no
  shear stress and normal strain coupling.

Examples:   polycrystalline metals,
            Polymers
            Randomly oriented fiber-reinforced composites
            Particulate-reinforced composites
Transversely isotropic materials
                            In L–T plane
                                                1                    
                                                E   , LT , 0         
                                                        EL
                                       L       L                     L 
                                             TL      1                  
                                       T       ,     , 0           T 
                                              ET    ET               
                                       LT                             LT 
                                                             1
                                                0 , 0 ,               
                                                           G LT       

                                                   EL             TL E L       
                                              1         ,                , 0 
                                                                1   LT TL
Principal material axes               L         LT TL                           L 
                                            LT ET              ET                  
L: longitudinal direction             T                ,              , 0   T 
T: transverse direction                1   LT TL       1   LT TL          
                                      LT                                          LT 
                                                     0      ,       0      , G LT 
                                                                                 
                                                                                 
Transversely isotropic materials
                            In T1, T2 plane

                                                1                  
                                                E   , TT , 0       
                                      T   T          ET            T1 
                                      1    TT
                                                        1                 
                                                                               
                                      T2        ,     , 0        T2 
                                               ET     ET                  
                                      T1T2  
                                                            1       T1T2 
                                                                               
                                                0 , 0 ,             
                                                            GTT     

                               Same as those for isotropic materials:
Principal material axes
L: longitudinal direction                                 ET
                                              GTT 
T: transverse direction                               21   TT 
Transversely isotropic materials
Where EL: elastic modulus in longitudinal direction
      ET: elastic modulus in transverse direction
      GLT: shear modulus in L – T plane
      GTT: shear modulus in transverse plane
      LT: major Poisson’s ratio
               (strain in T – direction caused by stress in L – direction)
      TL : minor Poisson’s ratio
                                            LT  TL
                                               
                                 And        EL   ET
Note:   1. 4 independent material constants (EL, ET, GLT, LT ) in L – T
        plane while 5 (EL, ET, GLT, LT, GTT) for 3-D state.
        2. No normal stress and shear strain coupling in L – T axes or
        no shear stress and normal strain coupling in L – T axes
             Orthotropic materials
                        For example in 1-2 plane

                                      1  12        
                                      E ,    , 0 
                                            E1
                            1   1                  1 
                                21       1        
                             2      ,    , 0   2 
                               E 2 E 2             
                             12                      12 
                                                  1 
                                      0 , 0 ,       
                                                G12 

1.2.3: principal                    E1             E               
material axes                                  , 21 1 , 0
                                   1    1                     
                             1       12 21       12 21
                                                                       1 
                                12 E 2          E2                
                            2             ,            , 0        2 
                              1   12 21 1   12 21            
       12        21        12                                      12 
                                          0      ,     0       , G12 
                                                                   
       E1        E2                                                 
            Orthotropic Materials
Note:
1. 4 independent constants in 2-D state (e.g. 1-2 plane, E1,
   E2, G12, 12 )while 9 in 3-D state (E1, E2, E3, G12, G13, G23, 12 ,
   13 , 23 )
2. No coupling between normal stress and shear strain or
   no coupling between shear stress and normal strain
                    Question
Ex.   Find the deformed shape of the following composite:




  Possible answers?
 Off-axis loading of unidirectional
             composite
For orthotropic material in principal material axes (1-2 axes)


                       1  Q11 Q12 0   1 
                                        
                       2   Q21 Q22 0   2 
                         0      0 Q66   12 
                       12               

By coordinate transformation
          x  cos2          sin 2       - 2cossin          1        1 
            2                                                        1     
          y   sin          cos2            2cossin   2   T   2 
           
          xy  cossin , - cossin ,   cos2  - sin 2     12 
                                                                  
                                                                               
                                                                               12 

          x            1 
          
                    1     
          y   T   2        , xyxy   are tensorial shear strains
                        
          xy 
                        12 
Let
         1  1 0            0  1         1 
                                           
         2   0 1          0  2   R  2 
                                 
          0 0              2   12       
         12                               12 
Then

        x            1                1                    1 
                                                              
        y   T 1  2   T 1 Q  2   T 1 Q R  2 
                                                            
        xy           12                12                    12 
                                   x                             x     x 
                                                              1         
                                                                             
                T  Q R T  y   T  Q R T R   y   Q  y 
                     1                        1

                                                                         
                                   xy                            xy    xy 
   Transformed stiffness matrix
Where   Q   T 1QRT R1   = transformed stiffness matrix



   Q11  Q11m 4  2Q12  2Q66 m 2 n 2  Q22 n 4
   Q22  Q11n 4  2Q12  2Q66 m 2 n 2  Q22 m 4
                                            
   Q12  Q11  Q22  4Q66 m 2 n 2  Q12 m 4  n 4     
                                                    
   Q66  Q11  Q22  2Q12  2Q66 m 2 n 2  Q66 m 4  n 4   
   Q16  Q11  Q12  2Q66 m 3 n  Q12  Q22  2Q66 mn3
   Q26  Q11  Q12  2Q66 mn3  Q12  Q22  2Q66 m 3 n
                                           m  cos , n  sin 
Transformed compliance matrix
   x             x       x 
                  
                1            
   
          
   y   Q
                         
                             
                    y   S  y 
                           
   xy 
                  xy 
                             xy 
                               

               S    : transformed compliance matrix
Off-axis loading - deformation
               x  Q11 Q12 Q16   x 
                
                                   
                                       
               y   Q12 Q22 Q26   y 
                                    
                xy  Q16 Q26 Q66   xy 
               
                                    
1. 4 material constants in 1-2 plane.
2. There is normal stress and shear strain coupling (forθ≠0, 90˚ ), or
   shear stress and normal strain coupling.
        Transformation of engineering
                 constants
      For uni-axial tensile testing in x-direction          x  0,  y   xy  0

                         ∴ stresses in L – T axes
                                                                           
                                                       x   x cos 
                                                                      2
                                          L                              
                                                                       
                                          T   T  0    x sin 
                                                                      2
                                                                            
                                                    0    sin  cos 
                                          LT           x               
                                                                           
                         L   S11 S12 0   L 
Strains in L – T axes                        
                         T    S12 S 22 0   T 
                           0        0 S 66   LT 
                         LT                  
                                 1         LT                              cos2         sin 2  
                                       ,      , 0                                   TL         
                                   EL       EL                                  EL            ET 
                                                       x cos 
                                                                2
                                                                        
                                   TL      1                             sin 2 
                                                                                            cos2  
                                     ,      , 0   x sin 2
                                                                         x          LT         
                                   ET      ET           sin  cos        ET            EL 
                                                  1   x                   sin  cos            
                                     0 , 0 ,                                                     
                                                 GLT                       
                                                                                     GLT            
                                                                                                     
And strains in x – y axes
                                           cos2          sin 2  
                                                   TL         
                                            EL             ET 
                       L            
           x
                                          sin 2         cos2  
                    1                                        
          y   T    T   T   x 
                                     1
                                                      LT         
                       1                 ET             EL 
         1
         xy             LT                                  
                                                  sin  cos
       2
             
                         2                                    
                                          
                                          
                                                    2G LT          
                                                                   
                      cos4  sin 4  1  1           2                             
                                                LT  sin 2 2                    
                    EL          ET     4  G LT
                                                      EL                           
                                                                                  
       
       
           x
              
                             1  1 2 LT         1     1  2                       
        y    x  LT                               sin 2                 
                      EL 4  EL        EL      ET G LT                          
         1
         xy                                                                       
                                                                                   
       2
             
                     1 sin 2   LT  1  1  cos2   1  2 LT  1  1
                                                                                  
                      2                                         E                 
                                   E L ET 2G LT                 L  EL ET G LT    
Recall for uni-axial tensile testing

                      x
               Ex 
                      x
              1   cos4  sin 4  1  1      2             2
                                       LT            sin 2
             Ex    EL         ET  4  G LT
                                            EL           
                                                          
           and  y   xy  x
                       y             y
            xy          
                       x           x
                                         Ex
                xy        y        LT    1 1    2     1   1      2
                                              LT             sin 2
               Ex          x       EL      4  EL
                                                    EL   ET G LT    
                                                                     
Define cross-coefficient, mx
                      
          xy  m x x
                      EL
                  xy E L
        mx  
                   x
                           E    EL                      E   E          
             sin 2  LT  L       cos2  1  2 LT  L  L
                                                                        
                                                                         
                           ET 2G LT                     ET G LT        
Similarly, for uni-axial tensile testing in y-direction
             1    sin 4  cos4  1  1     2           2
                                       LT          sin 2
             Ey     EL     ET    4  G LT
                                           EL         
                                                       
             yx        TL    1 1    2     1   1     2         xy
                                    TL            sin 2 
             Ey        ET      4  EL
                                       ET   ET G LT   
                                                                  Ex
                               E    EL                       E   E          
            m y  sin 2  LT  L       sin 2  1  2 LT  L  L
                                                                             
                                                                              
                               ET 2G LT                      ET G LT        
 For simple shear testing in x – y plane
                           x   y  0,  xy  0
 stresses in L – T axes
                           L         0  2 xy sin  cos               
                                                                       
                                                                             
                           T   T  0    2 xy sin  cos           
                           
                           LT 
                                         
                                                        
                                        xy   xy cos2   sin 2         
                                                                             
                                                                             
   Strains in L – T axes

          1                                                                                                 1  TL         
                                                                                                    2 sin  cos 
                                                                                                                  E  E 
                                               1
              ,  LT , 0                        , LT , 0                                                                      
            EL     EL                       EL       EL          2 xy sin  cos                             L       T      
 L                          L 
                                                                                                    
       TL      1                      TL     1                               
                                                                                                                   1  LT      
 T       ,       , 0      T          ,      , 0       2 xy sin  cos        xy  2 sin  cos 
                                                                                                                    E  E        
                                                                                                                                  
   ET         ET                    ET      ET                                                          T           
 LT                          LT                               
                                                                   xy cos   sin 
                                                                             2       2
                                                                                         
                                                                                         
                                                                                                    1
                                                                                                                              L
                                                                                                                                    
          0 ,      0 ,
                          1
                                           0 , 0 ,
                                                            1
                                                                                                                       
                                                                                                           cos2   sin 2          
                        GLT                             GLT                                      GLT
                                                                                                                                   
                                                                                                                                    
Strains in x – y axes
                               
                        L    
       x      
                     1        
       y       T   T      
                              
        xy              LT    
       2
              
                          2     
                                  xy
      where  x  m x
                                 EL
                                  xy
                   y  m y
                                 EL
                                1   2     1  1    2     1   1             
                  xy   xy        LT          LT            cos2 2 
                                EL   EL   ET  E L
                                                     EL   ET G LT   
                                                                              
                         xy
        G xy 
                         xy
                1    1   2     1  1    2     1   1                   
                       LT          LT                           cos2 2
               G xy E L   EL   ET  E L
                                         EL   E L G LT                 
                                                                        
In summary, for a general planar loading, by principle of superposition
                            1        xy m x 
                                 ,      ,    
                    x  
                              Ex      Ex EL 
                            
                      yx 1                     x 
                                             my  
                                                
                   y        ,       ,        y
                         Ey Ey           EL  
                    xy 
                                               xy 
                                                   
                              mx     my      1 
                               ,      ,      
                             E L E L G xy 
    Micromechanics of Unidirectional
             Composites
•       Properties of unidirectional lamina is
        determined by
    –     volume fraction of constituent materials
          (fiber, matrix, void, etc.)
    –     form of the reinforcement (fiber, particle, …)
    –     orientation of fibers
 Volume fraction & Weight fraction
                                   Vi   Vi
• Vi=volume, vi=volume fraction=      
                                   Vi Vc
                                   Wi  W
• Wi=weight, wi=weight fraction=       i
                                   Wi Wc
  Where
     subscripts i = c: composite
                    f: fiber
                    m: matrix
Conservation of mass: Wc  W f  Wm
                          Wf Wm
                              1
                         Wc Wc
                        w f  wm  1

Assume composite is void-free:
                       Vc  V f  Vm
                           Vf Vm
                               1
                          Vc Vc
                         v f  vm  1
               Density of composite
            Wc W f  Wm  f V f g   mVm g
       c             
            Vc g   Vc g         Vc g
          c   f v f   m vm
 or    Vc  V f  Vm
        Wc    Wf    Wm
                
        c g  f g m g
           1       wf       wm
                      
          c       f       m
Generalized equations for n – constituent composite
                                   n
                                                    1
                             c   i vi      n
                                                    wi 
                                 i 1
                                                
                                              i 1  i 
     Void content determination
Experimental result (with voids):    ce   f v f   m vm   v vv
                                           f v f   m vm      

Theoretical calculation (excluding voids): W  W  W
                                            c   f   m

                                            ct 1  vv    f v f   m vm
                                             ct   f v f   m vm   ct vv   

                                        ct   ce
     void content :      vv 
                                            ct
In general, void content    < 1%  Good composite
                            > 5%  Poor composite
  Burnout test of glass/epoxy composite
   Weight of empty crucible = 47.6504 g
   Weight of crucible +composite = 50.1817 g
   Weight of crucible +glass fibers = 49.4476 g
    f  2.5 g 3 ,  m  1.2 g 3
              cm              cm
   Find vv if  ce  1.86 g 3
                                         cm
Sol:
              Wf       49.4476  47.6504
       wf                               0.71
              Wc       50.1817  47.6504
       wm  1  w f  1  0.71  0.29

        c   ct 
                       wf
                            1
                            
                                wm
                                     
                                         0.71 0.29
                                             
                                              1
                                                           
                                                      1.902 g
                                                                 cm 3   
                       f       m       2.5 1.2
              c      1.902                                                         ct   ce 1.902  1.86
       vf       wf         0.71  0.54                                   vv               
              f       2.5                                                              ct       1.902
              c      1.902
                                                                               0.0221  2.21%
       vm       wm         0.29  0.46
              m       1.2
         Longitudinal Stiffness



 For linear fiber and matrix:
                        Ec  E f v f  E m vm  E L

Generalized equation for composites with n constituents:
               n
        E c   E i vi         Rule-of-mixture
              i 1
Longitudinal Strength




    c   f v f   m vm
                     
        f v f  m 1 v f   
                     Modes of Failure
                                 cu   mu 1  v f  1
matrix-controlled failure:



fiber-controlled failure:     cu   fu v f   m  1  v f 
                                                     fu


                                    [ fu   m  ]v f   m          2
                                                     fu               fu




                
    cu  max  cu 1 , cu 2   
    Critical fiber volume fraction
For fiber-controlled failure to be valid:           cu 2    cu 1

                                               
                                              fu   m 
                                                                           fu
                                                                                v   f     m    mu 1  v f
                                                                                                        fu
                                                                                                                     
                                                               mu   m 
                                             vf                                                        v min
                                                                                              fu

                                                         fu   mu   m 
                                                                                                   fu

 For matrix is to be reinforced:
                                         cu   mu

                                    
                                   fu   m 
                                                        fu
                                                             v   f     m    mu
                                                                                         fu

                                             mu   m 
                                  vf                                      vcrit
                                                                      fu

                                             fu   m 
                                                                      fu
    Factors influencing EL and cu
•    mis-orientation of fibers
•    fibers of non-uniform strength due to
     variations in diameter, handling and
     surface treatment, fiber length
•    stress concentration at fiber ends
     (discontinuous fibers)
•    interfacial conditions
•    residual stresses
            Transverse Stiffness, ET

Assume all constituents are in linear elastic range:

                            1   vf  v
                                   m
                            Ec E f Em

Generalized equation for n – constituent composite:

                              1    n
                                       vi       
                                              
                              Ec i 1  Ei
                                      
                                                 
                                                 
                                           1
                       or     Ec                     ET (transvers modulus)
                                                                   e
                                       n
                                           vi   
                                      E 
                                     i 1  i
                                                 
                                                 
                                                 
         Transverse Strength
                     Due to stress (strain) concentration
                               cu   mu


Factors influence cu:
• properties of fiber and matrix
• the interface bond strength
• the presence and distribution of voids (flaws)
• internal stress and strain distribution (shape of
  fiber, arrangement of fibers)
        In-plane Shear Modulus



For linearly elastic fiber and matrix:   1   vf  v
                                                m
                                         Gc G f Gm
                                                            G f Gm
                                         or Gc  GLT 
                                                         Gmv f  G f vm
Major Poisson’s Ratio

         LT   f v f   m vm
Analysis of Laminated Composites
• Classical Laminate Theory (CLT)

                 Displacement field:
                                                     w0
                   u  x, y , z   u 0  x, y   z
                                                      x
                                                     w
                   v  x , y , z   v 0  x, y   z 0
                                                      y
                   wx, y, z   w0 x, y 
Resultant Forces and Moments
                                   Nx            x                 x             k x 
                                             h
                                                                     
                                                                              
                                                                                         
  Resultant forces:                 N y    h  y  dz  1   A  y
                                               2
                                                                                  B  k y 
                                   N          2
                                                    xy                               k 
                                      xy                            xy
                                                                              
                                                                                         xy 

                   M x           x                                k x 
                             h
                                                          x 
                                                                         
Resultant moments:  M y    2h  y z  dz  1   B   y    D  k y 
                              
                   M          2
                                    xy                                k 
                      xy                                xy 
                                                                         xy 


                                Qij k dz   Qij k hk  hk 1 
                  n                          n
           Aij   
                         hk
                        hk 1
                 k 1                       k 1                       [A]: extensional stiffness matrix
                                                 h2  h2          
           Bij                                   
                 n                            n
                                Qij zdz   Qij  k                
                        hk                              k 1
                                               k
                                                                       [B]: coupling stiffness matrix
                      h  k 1      k                  2            
                 k 1                     k 1                    
                                                  h3  h3            [D]: bending stiffness matrix
                                                    
                 n                               n
                                                                  
           Dij               Qij z dz   Qij  k
                        hk             2                 k 1
                        hk 1      k            k     3           
                                                                   
                k 1                       k 1
                                                                  
Laminates of Special Configurations
•   Symmetric laminates
•   Unidirectional (UD) laminates
    – specially orthotropic
    – off-axis
•   Cross-ply laminates
•   Angle-ply laminates
•   Quasi-isotropic laminates
Strength of Laminates
    Maximum Stress Criterion
• Lamina fails if one of the following
  inequalities is satisfied:
                 L     Lt
                         ˆ
                 L     Lc
                           ˆ
                 T     Tt
                         ˆ
                 T     Tc
                           ˆ
                  LT  ˆLT
     Maximum Strain Criterion
• Lamina fails if one of the following
  inequalities is satisfied:
                  L     Lt
                         ˆ
                  L     Lc
                           ˆ
                  T     Tt
                         ˆ
                  T     Tc
                            ˆ
                   LT  ˆ LT
          Tsai – Hill Criterion
• Lamina fails if the following inequality is
  satisfied:
                    2                 2           2
              L                         
             
                    L T   T    LT
                                   ˆ         1
                                                  
              ˆL      L
                         ˆ     ˆT    LT        


                                if  L  0
                                 ˆ
        Where :         ˆ L   Lt
                                Lc if  L  0
                                 ˆ


                              Tt if  T  0
                               ˆ
                        T  
                        ˆ
                              Tc if  T  0
                               ˆ
   Comparison among Criteria
• Maximum stress and strain criteria can tell
  the mode of failure
• Tsai-Hill criterion includes the interaction
  among stress components
Strength of Off-Axis Lamina in Uni-
           axial Loading




   Maximum stress criterion
                              Tsai-Hill criterion
       Strength of a Laminate
• First-ply failure
• Last-ply failure