Micromechanics
Macromechanics
Fibers
Lamina
Laminate
Structure
Matrix
Macromechanics
Study of stress-strain behavior of composites
using effective properties of an equivalent
homogeneous material. Only the globally
averaged stresses and strains are considered,
not the local fiber and matrix values.
Stress-Strain Relationships for
Anisotropic Materials
First, we discuss the form of the stress-strain
relationships at a point within the material,
then discuss the concept of effective moduli
for heterogeneous materials where properties
may vary from point-to-point.
General Form of Elastic - Relationships
for Constant Environmental Conditions
 ij  Fij (11 , 12 , 13 ,...),
i, j  1, 2,3... (2.1)
Each component of stress, ij, is related to
each of nine strain components, ij
(Note: These relationships may be
nonlinear)
Expanding Fij in a Taylor’s series and
Retaining only the first order terms,
 ij  Cijkl kl ,
i, j, k , l  1,2,3
for a linear elastic material
 ij  9 com ponents
 kl  9 com ponents
Cijkl  81 com ponents
3D state of stress
Generalized Hooke’s Law for
Anisotropic Material
 11 
  
 22  C1111
 33  C 2211

 C 3311
 23  C 2311
 31  = C 3111

 C1211
 12   ...
  C 3211
 32  C
 13   1311
  C 2111
 21 
C
C
C
C
C
C
1122
2222
3322
2322
3122
1222
...
C
C
C
3222
1322
2122
C
C
C
C
C
C
1133
2233
3333
2333
3133
1233
...
C
C
C
3233
1333
2133
C
C
C
C
C
C
1123
2223
3323
2323
3123
1223
...
C
C
C
3223
1323
2123
C
C
C
C
C
C
1131
2231
3331
2331
3131
1231
...
C
C
C
3231
1331
2131
C
C
C
C
C
C
1112
2212
3312
2312
3112
1212
...
C
C
C
3212
1312
2112






..



C
C
C
C
C
C
1132
2232
3332
2332
3132
1232
...
C
C
C
3232
1332
2132
C
C
C
C
C
C
1113
2213
3313
2313
3113
1213
...
C
C
C
3213
1313
2113


2221

3321

2321

3121

1221 
... 

C 3221
C1321
C 2121
C
C
C
C
C
C
1121
  11 
 
 22 
 33 



 23 
  31 


  12 
 
 32 
  13 
 
 21 
(2.2)
Symmetry Simplifies the Generalized
Hooke’s Law
1. Symmetry of shear stresses and strains:
 21
2
1
O
 12
Static Equlibrium
M
 0 im plies 12   21
or in general,
0
 ij   ji or  ij   ji
Same condition for shear strains,  ij
  ji
2. Material property symmetry – several types will
be discussed.
Symmetry of shear stresses and shear strains:
 ij   ji and  ij   ji
Thus, only 6 components of ij are
independent, and likewise for ij.
This leads to a contracted notation.
Stresses
Tensor Notation
Contracted
Notation
11
1
22
2
33
3
23= 32
4
13= 31
5
12= 21
6
Strains
Tensor Notation
Contracted
Notation
 11
1
 22
2
 33
3
2 23= 2 32=  23=  32
4
2 13= 2 31=  13=  31
5
2 12= 2 21=  12= 21
6
Geometry of Shear Strain
 xy 
 xy
2
 xy = Engineering Strain
 xy = Tensor Strain
 xy 
 xy
2
Total change in original angle =  xy
Amount each edge rotates =  xy/2 =  xy
Using contracted notation
 i  Cij j
i, j  1,2,...,6
or in matrix form    C  
(2.3)
(2.4)
where  and   are column vectors
and [C] is a 6x6 matrix (the stiffness
matrix)
Alternatively,
 i  Sij j
i, j  1,2,...,6
or    S  
(2.5)
(2.6)
where [S] = compliance matrix
and
S   C
1
Expanding:
1   S11
   S
 2   21
 3   S31
 
 4   S 41
 5   S51
  
 6   S 61
S12
S13
S14
S15
S 22
S 23
S 24
S 25
S32
S 42
S33
S 43
S34
S 44
S35
S 45
S52
S53
S54
S55
S 62
S 63
S 64
S 65
S16   1 



S 26   2 
S36   3 
 
S 46   4 
S56   5 
 
S 66   6 
• Up to now, we only considered the stresses
and strains at a point within the material,
and the corresponding elastic constants at a
point.
• What do we do in the case of a composite
material, where the properties may vary
from point to point?
• Use the concept of effective moduli of an
equivalent homogeneous material.
Concept of an Effective Modulus of an Equivalent
Homogeneous Material
Heterogeneous composite
under varying stresses and
strains
x3
d
2 L
x3
2
2
Stress, 2
Equivalent homogeneous x
3
material under average
stresses and strains
2
2
Strain,  2
x3
2
2
Stress
Strain
2
Effective moduli, Cij
 i  Cij  j
where,
 i  average stress 
(2.9)

 i dv
v
(2.7)

dv
v
 i  average strain 

 i dv
v
(2.8)

v
dv
3-D Case
General Anisotropic Material
• [C] and [S] each have 36 coefficients, but
only 21 are independent due to symmetry.
• Symmetry shown by consideration of strain
energy.
• Proof of symmetry:
Define strain energy density
1
W   i i i  1,2,...,6
2
1
1
1
W   11   11  ...   6 6
2
2
2
but
 i  Cij j
1
W  Cij i j
2
(2.12)
Now, differentiate:
but
 j
 j
W 1
1
 Cij j  Cij i
 i 2
2
 i
  ij  Kronecker delta  1 if i=j
 i
0 if ij
 ij i   j
(show)
W

 Cij j
 i
 2W

 Cij
 i  j
(2.11)
(2.13)
But if the order of differentiation is reversed,
W

 C ji
 j  i
2
(2.14)
Since order of differentiation is immaterial,
Cij  C ji (Symmetry)
Similarly,
1
W  S ij i j
2
and
Sij  S ji
 Only 21 of 36 coefficients are
independent for anisotropic material.
Stiffness matrix for linear elastic
anisotropic material with no material
property symmetry
(2.15)
3-D Case, Specially Orthotropic
3
2
1
1, 2 , 3 principal
material coordinates
1
 12
2
2
1
(a)
 12
(b)
(c)
Simple states of stress used to define lamina
engineering constants for specially orthotropic
lamina.
Consider normal stress 1 alone:
3
1
1
1
2
Resulting strains,
1 
1
E1
;  2  121  12
1
E1
(2.19)
Typical stress-strain curves from
ASTM D3039 tensile tests
Stress-strain data from longitudinal tensile test of carbon/epoxy composite.
Reprinted from ref. [8] with permission from CRC Press.
Similarly,
 3  131  13
1
E1
where E1 = longitudinal modulus
ij = Poisson’s ratio for strain
along j direction due to
loading along i direction
Now consider normal stress 2 alone:
Strains:
2 
2
E2
2
3
;
2
1  21 2  21
E2
2
3  23  2  23
E2
Where E2 = transverse modulus
Similar result for 3 alone
1
2
2
(2.20)
• Observation:
All shear strains are zero under pure
normal stress (no shear coupling).
 12   13   23  0
For
1,  2 ,  3 alone
Now, consider shear stress
 12 alone,
3
1
Strain
 12 
 12
2
 12   12
G12
Where G12 = Shear modulus in 1-2 plane
1   2   3   13   23  0
(No shear coupling)
(2.21)
Similarly, for  13 alone
 13 
 13
G13
;
1   2   3   12   23  0
and for 23 alone
 23 
 23
G23
;
1   2   3   13   12  0
Now add strains due to all stresses using
superposition
Specially Orthotropic 3D Case
 1
 E
 1
 12

 1   E1
  
 2    13
 3   E1
 
  23   0
  31  
  
 12  
0


 0


 21
E2
31

E3
0
0
1
E2
32

E3
0
0
23

E2
1
E3
0
0
0
0
1
G23
0
0
0
0
1
G31
0
0
0
0

0 


0 
  1 
  2 
0  
  3 
  
0   23 
  31 
  
 
0   12 


1 
G12 
12 coefficients, but only are 9 independent
(2.22)
Symmetry:
Sij  S ji

ij
Ei

 ji
Ej
 Only 9 independent coefficients.
Generally orthotropic 3-D case –
similar to anisotropic with 36 nonzero
coefficients, but 9 are independent as with
specially orthotropic case
Specially Orthotropic – Transversely Isotropic
3
2
1
Fibers randomly packed in 2-3
plane, so properties are invariant to
rotation about 1-axis (2 same as 3)
Specially orthotropic, transversely isotropic
(2 and 3 interchangeable)
G13  G12 , E2  E3 , 21  31
E2
G23 
2(1  32 )
(2.23)
Now, only 5 coefficients are independent.
Isotropic
G13  G23  G12  G
E1  E2  E3  E
12  23  13  
E
G
2(1  )
2 independent coefficients
Usually measure E, υ – calculate G
Isotropic – 3D case
 1
 E

 
 1   E
  
 2   
 3   E
 
 4   0
 5  
  
6   0


 0



E
1
E


E


E


E
1
E
0
0
0
0
0
0
0
0
1
G
0
0
0
0
1
G
0
0
0
0

0

0   
 1
  
2


0  
  3 
  
0  4
  5 
 
0  6 

1

G
Same form for any set of coordinate axes
3-D Isotropic – stresses in terms of strains
E
(1  ) x  ( y   z ) 
x 
(1  )(1  2)
E
(1  ) y  ( x   z ) 
y 
(1  )(1  2)
E
(1  ) z  ( x   y ) 
z 
(1  )(1  2)
 xy  G  xy
E

 xy
2(1  )
 xy  G  xy
E

 xy
2(1  )
 xy  G  xy
E

 xy
2(1  )
3-D Case, Generally Orthotropic
z
y
x
Material is still orthotropic,
but stress-strain relations
are expressed in terms of
non-principal xyz axes
Generally Orthotropic
  x   S 11
  
 y   S 21
  z   S 31
 

 yz   S 41
 zx   S 51
  
 xy   S 61
S 12
S 22
S 32
S 42
S 52
S 62
S 13
S 23
S 33
S 43
S 53
S 63
S 14
S 24
S 34
S 44
S 54
S 64
S 15
S 25
S 35
S 45
S 55
S 65
S 16   x 
  
S 26   y 
S 36   z 
 
S 46   yz 
S 56   zx 
 
S 66   xy 
Same form as anisotropic, with 36 coefficients, but 9
are independent as with specially orthotropic case
Elastic coefficients in the stress-strain relationship for different
materials and coordinate systems
Material and coordinate system
Number of nonzero
coefficients
Number of
independent coefficients
Anisotropic
36
21
Generally Orthotropic
(nonprincipal coordinates)
36
9
Specially Orthotropic (Principal coordinates)
12
9
Specially Orthotropic, transversely isotropic
12
5
Isotropic
12
2
Anisotropic
9
6
Generally Orthotropic
(nonprincipal coordinates)
9
4
Specially Orthotropic (Principal coordinates)
5
4
Balanced orthotropic, or square symmetric
(principal coordinates)
5
3
Isotropic
5
2
Three – dimensional case
Two – dimensional case (lamina)
2-D Cases
Use 3-D equations with,
 3  13   23  0
Plane stress,
 1 ,  2 , 12 ,  0
Or
 x ,  y ,  xy ,  0
Specially
Orthotropic
Lamina
 1   S11
  
  2    S 21
   0
 12  
S12
S 22
0
0   1 
 

0   2  (2.24)
S 66   12 
Or
2
1
 1  Q11 Q12
  
 2   Q21 Q22
   0
0
 12  
0   1 
  (2.26)

0   2 
Q66   12 
5 Coefficients - 4 independent
Specially Orthotropic Lamina in Plane Stress
 1   S11
  
  2    S 21
   0
 12  
S12
S 22
0
0   1 
 

0   2 
S 66   12 
5 nonzero coefficients
4 independent coefficients
(2.24)
Or in terms of ‘engineering constants’
1
S11 
E1
1
S 22 
E2
21
12
S12  S 21  

E2
E1
1
S66 
G12
(2.25)
Experimental Characterization of
Orthotropic Lamina
• Need to measure 4 independent elastic
constants
• Usually measure E1, E2, υ12, G12
(see ASTM test standards later in Chap. 10)
Stresses in terms of tensor strains,
 1  Q11 Q12
  
 2   Q21 Q22
   0
0
 12  
0   1 



0    2  (2.26)



2Q66   12 / 2
where Q  S 
1
Inverting [S]:
S 22
E1
Q11 

2
1  12 21
S11 S 22  S12
S12
12 E2
Q12  

2
1  12 21
S11 S 22  S12
S11
E2
Q22 

2
1  12 21
S11 S 22  S12
1
Q66 
 G12
S66
Off – Axis Compliances:

S ij  f ij all Sij and angle 

Off – Axis Stiffnesses:

Q ij  f ij ' all Qij and angle 

Where fij and fij’ are found from transformations
of stress and strain components from 1,2 axes to
x, y axes
Sign convention for lamina orientation
y
y
1
2
2

x
x

Positive 
Negative 
1
Stress Transformation:
 2 dAsin 
Y
2
1

 12 dAsin 
X
12dAcos
 1dAcos
F
x
 xydA
 0 and  Fy  0
 x dA

dA
F
x
  x dA   1dAcos    2 dAsin 
2
2
 2 12 dAsin  cos  0
 x  1 cos    2 sin   212 sin  cos
2
F
y
2
0
 xy   1 cos sin    2 cos sin 
  12 (cos   sin  )
2
2
Equations used to generate Mohr’s circle.
(2.29)
Resulting stress Transformation:
2
2


c
s
 2cs   1 
 X
 1 
 
   2

1 
2
c
2cs   2   T   2 
 Y   s
   cs  cs c 2  s 2   
 
 XY  
 12 
  12 
Where c  cos , s  sin 
or
 1 
 X 
 
 
 2   T   Y 
 
 
 12 
 XY 
(2.31)
(2.30)
Where
 c2 s2
2cs 
 2

2
[T ]   s
c
 2cs 
 cs cs c 2  s 2 


(2.32)
Strain Transformation:
 x 
 1 




  2   T   y 
 / 2
 / 2
 12 
 xy 
(2.33)
Recall: Tensor shear strain
 xy
1
  xy
2
Where xy = engineering shear strain
or
 x 
 1 



1 
  y   T    2 
 / 2
 / 2
 12 
 xy 
Substituting (2.33) into (2.26), then substituting
the resulting equations into (2.30)
 x 
 x 
 

 (2.34)
1
 y   [T ] [Q][T ]  y 
 
 / 2
 xy 
 xy 
Carrying out matrix multiplications and
converting back to engineering strains,
 x  Q11 Q12 Q16    x 
 
  
 y   Q12 Q 22 Q 26    y 
  Q
 

Q
Q
26
66   xy 
 xy   16
(2.35)
Where
Q11  Q11c  Q22 s  2(Q12  2Q66 )s c
4
4
2 2
Q11  

(2.36)
Q 66  
Alternatively
  x   S 11
  
  y    S 12
   S 16
 xy  
Where
S 
1
S 12
S 22
S 26
 [Q]
S 16   x 
 
S 26   y 
S 66   xy 
(2.37)
Generally
Orthotropic
Lamina (Off
Axis)
  x   S 11
  
  y    S 12
   S 16
 xy  
S 12
S 22
S 26
S 16   x 
 
S 26   y 
S 66   xy 
(2.37)
Or
x
2

1
y
 x  Q11 Q12
  
 y   Q12 Q 22
  Q
 xy   16 Q 26
Q16    x 
 
Q 26    y 
Q 66   xy 
9 Coefficients - 6 independent
In expanded form:
S11  S11c 4   2S12  S66  s 2c 2  S 22 s 4


S12  S12 s 4  c 4   S11  S 22  S66  s 2c 2
S22  S11s 4   2S12  S66  s 2c 2  S 22c 4
(2.38)
S16   2S11  2S12  S66  sc3   2S 22  2S12  S66  s 3c
S26   2S11  2S12  S66  s 3c   2S 22  2S12  S66  sc 3

S66  2  2S11  2S22  4S12  S66  s 2c 2  S66 s 4  c 4

Off-axis lamina engineering constants
Young’s modulus, Ex
2
y
x
x
1
or
Ex 
x
Ex 
x
When  x  0,  y   xy  0
x
1
 Ex 

(2.39)
S 11 x S 11
1
1 4  212
1  2 2 1 4
c  

s
c s 
E1
G12 
E2
 E1
(2.40)
Complete set of transformation equations for lamina
engineering constants
1
 1
2 
1 4
Ex   c 4  
 12  s 2 c 2 
s 
E1 
E2 
 G12
 E1
1
1 4  1
212  2 2 1 4 
Ey   s  

c 
s c 
E
G
E
E
 1

1 
2
 12
1
 1
 1
1 212
1
Gxy  
s4  c4  4  


E1
2G12
 G12
 E1 E2


 12 4
 1
1
1  2 2
4
 xy  Ex 
s c  

s c 
E
E
E
G
 1

2
12 
 1


(2.40)
 2 2
s c 


1
Variations of off-axis engineering constants with lamina orientation for
unidirectional carbon/epoxy, boron/aluminum and glass/epoxy composites.
(From Sun, C.T. 1998. Mechanics of Aircraft Structures. John Wiley & Sons, New York.
With permission.)
Shear Coupling Ratios, or Mutual
Influence Coefficients
• Quantitative measures of interaction
between normal and shear response.
• Example: when  x  0,
 y   xy  0,
Shear Coupling Ratio
 xy S 16 x S 16
 x, xy 


 x S 11 x S 11
Analogous to Poisson’s Ratio
(2.41)
Example of off-axis strain in terms of
off-axis engineering constants
 yx
xy , x
1
x 
x 
y 
 xy
Ex
Ey
Gxy
(2.43)
Compliance matrix is still symmetric for
off-axis case, so that, for example
S12  S21
and
 yx
Ey

 xy
Ex
Balanced Orthotropic Lamina
(Ex: Woven cloth, cross-ply)
E1  E2
2
Q11  Q22
S11  S22
1
Only 3 independent
coefficients
Lamina Stiffness Transformations
Q11   c 4
   4
Q 22   s
Q  c 2 s 2
12
  2 2
Q 66  c s
Q   c 3 s
 16   3
Q 26   cs
s4
2c 2 s 2
4c 2 s 2 

4
2 2
2 2
Q

c
2c s
4c s
 11 
Q 
2 2
4
4
2 2

c s
c s
 4c s
 22 

2 2
2 2
2
2 2 
c s
 2c s
(c  s )  Q12 
3
3
3
3
3 
 cs cs  c s 2(cs  c s ) Q66 

3
3
3
3
3
 c s c s  cs 2(c s  cs )
Use of Invariants
The lamina stiffness transformations can be
written as:
Q11  U1  U 2 cos 2  U 3 cos 4
Q12  U 4  U 3 cos 4
Q 22  U1  U 2 cos 2  U 3 cos 4
U2
sin 2  U 3 sin 4
Q16 
2
U2
sin 2  U 3 sin 4
Q 26 
2
(2.44)
Where the invariants are
1
U1  (3Q11  3Q22  2Q12  4Q66 )
8
1
U 2  (Q11  Q22 )
8
1
U 3  (Q11  Q22  2Q12  4Q66 )
8
1
U 4  (Q11  Q22  6Q12  4Q66 )
8
1
U 5  (Q11  Q22  4Q66  2Q12 )
8
(2.45)
Alternatively, the off-axis compliances can be
expressed as
S11  V1  V 2 cos 2  V 3 cos 4
S12  V 4  V 3 cos 4
S 22  V1  V 2 cos 2  V 3 cos 4
S16  V 2 sin 2  2V 3 sin 4
S 26  V 2 sin 2  2V 3 sin 4
S 66  2(V1  V 4 )  4V 3 cos 4
(2.46)
where the invariants are
1
V1  (3S11  3S 22  2S12  S 66 )
8
1
V 2  (S11  S 22 )
2
1
V 3  (S11  S 22  2S12  S 66 )
8
1
V 4  (S11  S 22  6S12  S 66 )
8
(2.47)
Example: decomposition of Q11 using invariants
Q11
U
Q11
0

2

1
 0
U3 cos4
U 2 cos2
U1



U2
0
2

U3
 0

2
2
References: Mechanics of Composite Materials, Jones
Introduction to Composite Materials, Tsai
& Hahn

Invariants in transformation of stresses:
Mohr’s circle
Ex :
x
  I  R cos2
R
x
2 p
Where
I
I
 x  y
2
p
(2.48)
 Invariant
  x  y 
2
   xy  Invariant
R  
 2 
2
Similar graphical interpretation of stiffness
transformations
Ex: Q11  U1  U2 cos2  U3 cos4
Isotropic
Part
Orthotropic Part
(2.49)
U1 and U4: First
order invariants
U2 and U3:
Second order
invariants
Radii of circles indicates degree of orthotropy.
(i.e., if U2=U3=0, we have isotropic material)