Symbol Technologies by mikesanye

VIEWS: 8 PAGES: 27

									         Transformation of stress components
                 and of elastic constants


Principal material coordinates are ‘natural’ coordinates:
Sign convention:




It is often necessary to know the stress-strain relationships in non-
principal coordinates (‘off-axis’) such as x and y.
Therefore:
How do we transform stress and strain?
How do we transform the elastic constants?
   Transformation of stress components between
                 coordinate axes

 This is obtained by writing a force balance equation in a
 given direction. For example, in the x direction:




Fx   x dA   1 dA cos 2    2 dA sin 2   2 12 dA sin  cos  0
            x   1 cos 2    2 sin 2   2 12 sin  cos

 By repeating this, the complete set of stress transformations in
 xy coordinates can be obtained:

   x  c 2 s 2     2cs   1 
     2                              (*)
   y   s     c2  2cs   2 
     cs  cs c 2  s 2                  where c = cos and s = sin
   xy                     12 
            1 
        1     
   T   2 
            
            12 

And in the 12 system       1         x                  c2 s2       2cs 
we have:                              
                           2   T  y     with T    s 2 c 2
                                                                          2cs 
                                                                                 

                                                        cs cs    c2  s2 
                           12         xy                                    
                                             
  Similarly, we have:      1            x    
                                                 (**)
                             2   T   y   
                           12          xy   
                           2
                                        2     
                                                

Now, remember that for a 2-dimensional lamina in its principal
coordinates we showed that:

             1  Q11   Q12    0    1   1            
               Q                                    1 
             2   12    Q22    0    2    2   Q   2 
                                     
                                                                      (***)
             12   0
                        0    2Q66   12   12 
                                      2   
                                                             12 
                                                             2
                                                               


   (remember: only 4 independent constants).
Then, substituting (**) into (***), and then into (*), we find


                    x                    x 
                                          
                    y   T 1 Q T   y 
                                         
                    xy                    xy 
                     Q11     Q12    Q16    x 
                                         
                    Q12     Q22    Q26    y 
                     Q16
                             Q26    Q66   xy 
                                          
                        x 
                      
                    Q  y 
                        
                        xy 
   and the components of the ‘transformed reduced
   stiffness matrix’ are as follows:



                                                    (How many
                                                    independent
                                                    constants among
                                                    these
                                                    transformed
                                                    components?)




In terms of
engineering
constants, we
have:
EXAMPLES:
            Nylon-reinforced elastomer
                    composite
Zinc (hexagonal   Carbon AS4/epoxy
  symmetry)           composite
Anisotropy of bone (J.   Biomechanics 1992)
  Therefore, from the relationships




it is clear that we must know (and find ways to measure and/or
predict) the principal elastic constants E 1, E2, G12, n12 !


This is the topic of micromechanics models
    Micromechanics models for elastic constants


1. Microstructural aspects

   The elastic constants can be measured through careful
   experiments. However, it is desirable to be able to reliably
   predict lamina properties as a function of constituent
   properties and geometric characteristics (such as fiber
   volume fraction and geometric packing characteristics).

   Thus we will be looking for a functional relationship of the
   following type:
             C ij  f E f , E m ,n f ,n m , V f , Vm , Vvoids , S , A) 
Typical transverse cross-sections of unidirectional composites:
Composites with low volume fractions tend to have a random
fiber distribution, whereas with high volume fractions the fibers
tend to pack hexagonally.



        Silicon carbide/glass ceramic,
        fiber diameter ~ 15 mm,
        Vf ~ 0.40




           Carbon/epoxy,
           fiber diameter ~ 8 mm,
           Vf ~ 0.70
 Expected volume fractions in composites using
        representative area elements

  NOTE - It is assumed that the fiber spacing (s) and the fiber
  diameter (d) do not change along the fiber length, so that area
                   fractions = volume fractions


                                                         d 
                                                                    2

                                                 vf        
                                                        2 3 s


                                                    Max(Vf) @ s=d:
                                                    Vf = 0.907
        d 
                2

vf                  Max(Vf) @ s=d:
       4s            Vf = 0.785
                                                   In practice,
                                                max(Vf) = 0.5 – 0.8
      Fiber content, matrix content, void content

            For n constituent materials          n

            we must have for volume             v     i   1
            fractions vi = Vi/Vtot              i 1




 In many cases this simply reduces to     v f  v m  v voids  1



For weight fractions wi = W i/W tot, we have:
               n

             w      i   1             w f  wm  1
              i 1
    Since weight = (density)*(volume), we obtain immediately
                                         n
                                c    i vi
                                     i 1


thus, a Rule of Mixtures, which for 2-phase composites becomes:
                       c   f v f   m vm

  And in terms of weight fractions instead of volume fractions:
                                 1
                   c 
                                    
                         w f  f  w m  m 


   The void fraction can                                         W               Wf   
   be calculated from                           W       f         composite

                                                                           m
                                                     f

   measured weights              vvoids  1 
   (not fractions) and                                   W
                                                          composite    composite 
   densities:
    Other structural aspects to be quantified:




The fiber length
  distribution
The fiber orientation and
  the fiber orientation
       distribution




                         b
                   sin  
                        1

                         a
The same formula
may be used to
calculate the real
depth of a hole from
a (SEM) picture at an
angle
2. Elementary mechanics models


  Longitudinal Young’s modulus - Assume an
  isostrain situation (Voigt 1910) [implicitely:
  perfect interfacial adhesion]:


                                        f   m   composite

                                      Assuming linear elasticity
                                      for the fiber and matrix:
                                            f  E f  composite

                                             m  E m  composite
Assume Ac, Af, Am are the cross-sectional areas of the
composite, the fiber and the matrix.
From balancing the forces in the fiber direction, we have:

 Pcomposite  Pf  Pm                    composite Acomposite   f A f   m Am


 Therefore:       composite Acomposite  ( E f A f  E m Am ) composite

                      composite          Af              Am
 and: Ecomposite                 (E f             Em            )
                      composite        Acomposite      Acomposite

  Thus:
                     E composite  E f v f  E m v m  E11       (Voigt)


  Also:                  composite   f v f   m v m
Transverse Young’s modulus - Assume an
isostress situation (Reuss 1929):



                                        f   m   composite


                           In this case it is easily shown that:


                       1          vf   vm
                                         E22         (Reuss)
                   Ecomposite     E f Em

                     composite   f v f   m v m
Voigt bound




       Reuss bound
                       A few remarks


• The longitudinal modulus is always greater than the transverse
  modulus
• The ratio E11/E22 may be considered as a measure of the degree
  of anisotropy (or orthotropy) of the lamina
• Poisson ratio n12 is also predicted by a rule of mixtures
• The shear modulus G12 is predicted by an inverse rule of
  mixtures
• Improved (tighter) bounds were developed by various authors
  (Hashin & Rosen)
3. Semiempirical models


• The most successful semiempirical model is that of Halpin & Tsai
  (1967):
                            p 1  v f
                              
                           pm   1  v f
                           with
                              p       pm   1
                           
                                   f

                              p   f   pm   

 where p represents composite moduli, for example E11, E22, G12 or
 G23; pf and pm are the corresponding fiber and matrix moduli.  is
 the ‘measure of the degree of reinforcement of the matrix by the
 fibers’, which depends on fiber geometry and distribution, loading
 conditions, etc. It is essentially a fitting factor.
These equations are quite accurate at low volume fractions, a bit
less so at higher volume fractions. They have been modified to
include the maximum packing fraction. Specific values include the
following:

              Modulus                            
                 E11                          2(l/d)
                 E22                           0.5
                 G12                           1.0
                 G21                           0.5
                  K                             0


     Note finally that when 0, the ‘inverse’ rule of mixtures is
     obtained, and when , the direct rule of mixtures is
     obtained.

								
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