EM 424 Stress-Strain Relations
Generalized Hooke’s Law for Anisotropic materials
In the most general case, if we assume that the stresses are linearly related to the strains,
we can write this relationship in matrix form as
σ xx   C11
σ  
 yy  C21
σ zz  C31
 =
σ yz  C41
σ xz  C51
  
σ xy  C61
C12
C13
C14
C15
C22
C23
C24
C25
C32
C42
C33
C43
C34
C44
C35
C45
C52
C53
C54
C55
C62
C63
C64
C65
C16   exx 
 
C26   eyy 
C36   ezz 
 
C46  γ yz 
C56  γ xz 
 
C66  γ xy 
where, in the 6x6 stiffness C matrix there are a total of 21 elastic constants since the
matrix is symmetric, i.e. C12 = C21 , C23 = C32 , etc. Most anisotropic materials used in
engineering practice, however, have certain directional symmetries that considerably
reduce the number of independent constants. Below, we list some of the important special
cases.
Orthotropic material (9 constants)
σ xx   C11
σ  
 yy  C12
σ zz  C13
 =
σ yz   0
σ xz   0
  
σ xy   0
C12
C22
C13
C23
0
0
0
0
C23
C33
0
0
0
0
C44
0
0
0
0
0
0
0
C55
0
0   exx 
 
0   eyy 
0   ezz 
 
0  γ yz 
0  γ xz 
 
C66  γ xy 
This can also be turned around to write the strains in terms of the stresses as
 exx   D11
e  
 yy   D12
 ezz   D13
 =
γ yz   0
γ xz   0
  
γ xy   0
D12
D13
0
0
D22
D23
0
0
D23
0
D33
0
0
D44
0
0
0
0
0
D55
0
0
0
0
which is usually written in the form
0  σ xx 
 
0  σ yy 
0  σ zz 
 
0  σ yz 
0  σ xz 
 
D66  σ xy 
EM 424 Stress-Strain Relations
exx =
ν
ν
1
σ xx − yx σ yy − zx σ zz
Ex
Ey
Ez
eyy =
ν
ν
1
σ yy − xy σ xx − zy σ zz
Ey
Ex
Ez
ezz =
ν
ν
1
σ zz − xz σ xx − yz σ yy
Ez
Ex
Ey
γ yz =
1
σ yx
G yz
γ xz =
1
σ xz
Gxz
γ xy =
1
σ xy
Gxy
where
ν xy
Ex
=
ν yx ν xz
,
E y Ex
=
ν zx ν yz
=
,
Ez Ey
ν zy
Ez
For plane stress (σ zz = σ xz = σ yz = 0) the stress-strain relations for an orthotropic solid
reduce to the form
σ xx =
σ yy =
Ex
1− ν xyν yx
Ex
1 − ν xyν yx
(e
+ ν yx e yy )
(e
+ ν xy exx )
xx
yy
σ xy = Gxyγ xy
Cubic material (3 constants)
σ xx   C11
σ  
 yy  C12
σ zz  C12
 =
σ yz   0
σ xz   0
  
σ xy   0
C12
C12
0
0
C11
C12
0
0
C12
C11
0
0
0
0
0
0
C44
0
0
C44
0
0
0
0
0   exx 
 
0   e yy 
0   ezz 
 
0  γ yz 
0  γ xz 
 
C44  γ xy 
EM 424 Stress-Strain Relations
Isotropic material ( 2 constants)
σ xx   C11
σ  C
 yy   12
σ zz  C12
 =
σ yz   0
σ xz   0
  
σ xy   0
where
C12
C12
0
0
C11 C12
0
0
0
0
( C11 − C12 )
0
C12
C11
0
0
0
0
0
0
0
0
1
2
1
2
( C11 − C12 )
0
  exx 
 e 
0
  yy 
  ezz 
0
 
0
 γ yz 
 γ xz 
0

 
1
 γ xy 
2 ( C11 − C12 ) 
0
E (1− ν )
Eν
, C12 =
(1 + ν )(1− 2ν )
(1 + ν )(1 − 2ν )
1
(C11 − C12 ) = E = G
2
2(1 + ν )
C11 =
If these stress-strain relations are written out explicitly, we have
σ xx =
E
(1 −ν ) exx + ν ( e yy + ezz ) 

(1 +ν )(1 − 2ν ) 
σ yy =
E
(1 −ν ) eyy + ν ( exx + ezz ) 
(1 +ν )(1 − 2ν ) 
σ zz =
E
(1 −ν ) ezz +ν ( exx + eyy ) 

(1 +ν )(1 − 2ν ) 
σ xy = Gγ xy
σ xz = Gγ xz
σ yz = Gγ yz
or, if we write instead the strains in terms of the stresses
EM 424 Stress-Strain Relations
1
σ xx −ν (σ yy + σ zz ) 

E
1
eyy = σ yy −ν (σ xx + σ zz ) 
E
1
ezz = σ zz −ν (σ xx + σ yy ) 
E
exx =
σ xy
γ xy =
G
σ xz
γ xz =
G
σ yz
γ yz =
G
In the case of plane stress, σ zz = σ xz = σ yz = 0 . Thus, setting the equation for σ zz equal to
zero shows that
ezz =
−ν
e + e 
(1 −ν )  xx yy 
so that placing thus expression into the other two normal stress relations allows us to
write all the stresses in terms of exx , eyy , γ xy only. Thus for plane stress
E
 exx +ν eyy 
1 − v2 
E
 eyy + ν exx 
σ yy =
1 − v2 
σ xy = Gγ xy
σ xx =
In the case of plane strain, we can set ezz = γ xz = γ yz = 0 . In this case we have
σ xx =
E
(1 −ν ) exx + ν eyy 
(1 +ν )(1 − 2ν ) 
σ yy =
E
(1 −ν ) eyy +ν exx 
(1 +ν )(1 − 2ν ) 
σ xy = Gγ xy
EM 424 Stress-Strain Relations
but note that there is still a normal stress, σ zz , given by
σ zz =
Eν
e + e 
(1 +ν )(1 − 2ν )  xx yy 
Transformation of the elastic constants C matrix
One important difference between the isotropic case and the anisotropic cases listed
above is that while the matrix of coefficients for the isotropic case is the same for any
orientation of the x,y, z axis this is not true in general for the other materials and the
expressions given above are only valid for a particular set of material axes. We can
obtain the transformation equations for these coefficients, which like the stress and strain
transformations involve the direction cosines relating a pair of axes, as will be shown
below.
Another important difference between isotopic and anisotropic media is that the principal
stress and principal strain directions do not coincide in general for anisotropic materials
so that we need to calculate those directions (and the corresponding principal stress and
strain values) separately for the stress and strain.
To obtain the transformation relations for the elastic constants, recall that we found that
the stresses and (tensor) starins both transformed from one coordinate system to another
according to the same rules, namely
[σ '] = [l ] [σ ][l ]
T
[e '] = [l ] [ e][l ]
T
where [l ] is the 3x3 matrix of direction cosines ( see figure below):
 nx
[l ] =  ny
 nz
tx
ty
tz
vx   l11 l12
v y  = l21 l22
vz  l31 l32
l13 
l23 
l33 
EM 424 Stress-Strain Relations
v (z')
t (y')
z
n (x')
y
x
If we write the stresses and strains in the original (unprimed) coordinates and in the
rotated (primed) coordinates, we have
{σ } = [C ]{e}
{σ '} = [C ']{e '}
where here the stresses and strains are the column vectors shown previously. Using the
stress and strain transformation relations and these stress-strain relations, after some
considerable algebra one can show that
[C '] = [ M ][C ][ M ]
T
where the 6x6 M matrix is given by
 l112
 2
 l12
 l132
[M ] = 
l12l13
l l
 11 13
l11l12
l212
l312
2l21l31
2l31l11
l222
l322
2l22l32
2l12l32
2
23
2
33
2l13l33
l12l33 + l32l13
l
l
l22l23
l32l33
2l23l33
l22l33 + l32l23
l21l23
l31l33
l21l33 + l31l23
l11l33 + l31l13
l21l22
l31l32
l21l32 + l31l22
l11l32 + l31l12


2l12l22 
2l13l23 

l12l23 + l22l13 
l11l23 + l21l13 

l11l22 + l21l12 
2l21l11