# Mechanics of Composite Materials (PowerPoint)

Document Sample

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

  E1
  G
 2   1
E
G
21   
 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:

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
 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                 

 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-

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

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