3 MATHEMATICAL PRELIMINARIES
3.1 Vector Representation
Orthogonal set of base vectors e1, e2 , e3
vector x = x1e1 + x2e2 + x3e3
3
x =  xi ei
i =1
3.2 Indicial/Tensor Notation
➢ Summation convention
• Repeated indices indicate summation
x = xi e i
through 1, 2, 3
u = ui e i
i = 1,2,3
➢ Indices can only appear twice per term
3.2.1
Define Kronecker Delta
0
ei  e j = ij = 
1
3.2.2
1 0 0
i j
 I = 0 1 0


i= j
0 0 1
Dot Product Between Two Vectors
ab =
a = a1e1 + a2e 2 + a3e 3 = ai ei
b = b1e1 + b2e 2 + b3e 3 = bi ei
1
a  b = ( ai ei )  ( b j e j )
= ai b j ( ei  e j )
= ai b j ij = a1b111 + a1b212 + a1b313
 __ 
i, j = 1,2,3
+ a 2b121 + a2b222 + a2b323
+ a3b13 + a3b232 + a3b333
= ai bi = bi ai
3.2.3
Simplified Representation
We can drop the base vectors for vector representation:
ai
 a1 
 
a = a2  , bi
a 
 3
 b1 
 
b = b2  , cij
b 
 3
a  b = a b = a1 a2
bT ca = b1 b2
c13 
c23  , dij  d
c33 
 b1 
 
a3  b2  = aibi = a1b1 + a2b2 + a3b3
b 
 3
T
 c11 c12
b = ca =  c21 c22

 c31 c32
 c11 c12
c = c21 c22
c31 c32
c13   a1 
 
c23  a2  = cij a j = bi 

c33   a3  = a j cij = bi  all equivalent
= ck a = bk 
 c11 c12
b3  c21 c22
c31 c32
c13   a1 
 
c23  a2  bi cij a j
c33   a3 
The order of the terms in the indicial notation representation is irrelevant.
3.2.4
Matrix Multiplication
Some common operations:
cd cij d jk
cT d cij dik
cdT cij d kj
summation on middle
summation on first
summation on last
2
3.3 Transformation of Vector and Tensor Coordinate Reference Frame
v = vi e i
v = vj ej
v j e j = viei
v j e j = viei
dot with ek
v j ( e j  ek ) = vi ( ei  ek )
( e  e ) v
k
j
j
= viik
akj v j = vk
vk = akj v j
akj = ek  e j

v = av ,
a is
direction cosines
Similarly,
v j e j = viei
dot with e k
v j ( e j  e k ) = vi ( ei  e k )
vk = viaik
vk = aik vi
,
aik = ei  e k
v = aT v 
/
v = aT v
v = av
v = aT av
only true if aT a=I
Thus,
aT = a−1
and a is orthonormal.
3
An Inplane Example:
vi = aij v j
v = av
a = ei  e j
a11 = e1  e1 = e1 e1 cos  = cos 


a12 = e1  e 2 = cos  −   = sin 
2

a21 = − sin 
a22 = cos 
a13 = 0
 cos  sin  0
a =  − sin  cos  0


0
1 
 0
a23 = 0
a33 = 1
a31 = 0
a32 = 0
 v1  
  v1 
  
 
v2  =  a  v2 
 v  
  v3 
 3 
3.4 Transformation of Matrix/Higher-Order Tensors
Tensor: A mathematical representation of a physical quantity. They have components that
transform from one coordinate system to another according to “transformation equations” (rules),
e.g.,
v = av
vector v is a first-order tensor
u = au
vector u is a first-order tensor
4
 u1 
 
w = uv = u2  v1 v2
u 
 3
T
v3  = ui v j = wij
Matrix w is a second-order tensor because it is the product of two first-order tensors.
w = uvT ,
u = au ,
v = av ,
vT = v T aT
w = uvT = a uvT aT
w = awaT
rule for transformation of a second-order tensor
wij = air a js wrs
Similarly,
w = aT wa
wij = ari asj wrs
where aij = ei  e j
For example, stress is a second-order tensor:
ij = air a js rs ,
s  = as aT
Later we’ll see the fourth-order tensor, Cijk , a generalized Hookean stiffness tensor
 = air a js akt a uCrstu
Cijk
Four dimensions—can no longer be written in vector format.
5
3.5 Stress
Matrix/tensor representation
  xx  xy

s =   yx  yy
  zx  zy
x 1
 xz 

 yz  = ij , i = 1, 2,3
 zz 
y2
12 = 12
z3
23 = 23
31 = 31
Equilibrium of moments and forces:
 xy =  yx
 yz =  zy
 zx =  xz
Six independent stresses in compact notation:
11  1 
   
 22   2 

   
s = i =  33  =  3 
23  4 
31  5 
   
12  6 
6
3.5.1
Transformation of Coordinate Reference Frame for Stress
➢ For laminates, we are usually concerned with rotations in the x,y plane
 cos  sin  0
aij = ei  e j =  − sin  cos  0


0
1
 0
 c s 0  11 12 13   c − s 0
s  = as a =  − s c 0  12 22 23   s c 0




 0 0 1  13 23 33  0 0 1
T
Expand and put in compact form:
 1   c 2 s 2
   2
c2
 2  s

0
 3   0
 =
0
4   0
 5   0
0
  
 6   − sc sc
s  = Ts
0
0
1
0
0
0
0 0
sin 2
0 0 − sin 2
0 0
0
c −s
0
s
c
0
2
0 0 c − s2
  1 
  
 2
 3 
 
 4 
  5 
 
  6 
s = T −1s 
3.5.2
Simplify 3D Transformation for 2D Transformation
 1   11 
   
 2   22   1   11    xx 

 3  33  2D      
  =   2  =  22  =  yy 
4   23           
 5   31   6   12   xy 
   
 6   12 
 c2 s2
 2
c2
s
 0
0
T=
0
 0
 0
0

 − sc sc
0
0
1
0
0
0
0 0
sin 2

0 0 − sin 2
0 0
0 

c −s
0 
s
c
0 

0 0 c 2 − s 2 
7
T for 2D plane stress or plane strain s  = Ts
   c 2 s 2 sin 2   11 
 11
   2
 
c 2 − sin 2 22 
 22  =  s
    − cs sc c 2 − s 2    
 12  
  12 
 = c211 + s 222 + sin 212
11
1
(1 − cos 2 ) 22
2
1
11 cos2  = (1 + cos 2 ) 11
2
22 sin 2  =
For Mohr’s Circle, from your “strength of materials” class:
 =
11
11 + 22 11 − 22
+
cos 2 + 12 sin 2
2
2
22 =
11 + 22 11 − 22
−
cos 2 − 12 sin 2
2
2
 − 22 
 = −  11
12
 sin 2 + 12 cos 2
2


3.6 Strain
Displacement Field
u = ui e i
u = u1 = u1 ( x, y , z )
v = u2 = u2 ( x, y , z )
w = u3 = u3 ( x, y , z )
From strength of materials, normal strains
 xx =
u
x
8
 yy =
v
y
 zz =
w
z
Engineering shear strain:
 xy =
u v
+
y x
 yz =
v w
+
z y
 zx =
w u
+
x z
 xy =
dv du
+
dx dy
Mathematical/tensorial shear strain:
ij =
1  ui u j 
+


2  x j xi 
 xy =
1  u v 
+
2  y x 
 yz =
1  v w 
+
2  z y 
 zx =
1  w u 
+ 

2  x z 
so
12 = 212
 23 = 223
 31 = 231
9
Strain tensor:
 11 12
ij = 
22

sym
13   11 12 2 13 2 
23  = 
22
 23 2
 

33  
33 
but, in compact form,
 1    xx    xx 
      
 2   yy   yy 

    zz    zz 
i =  3  = 
= 
4   2 yz    yz 
 5   2 zx    zx 
  
  
 6  2 xy    xy 
3.6.1
Coordinate Transformation of a Second-Order Strain Tensor
Rules for transformation only apply to tensorial strain, NOT engineering strain, i.e.,
 11 12 13 
e =  12 22 23 


 13 23 33 
(tensorial strain matrix)
From the rules for transformation,
e = aeaT
or
ij = air a js  rs
where
aij = ei  e j
For a rotation about x3 of  , one needs to expand this relation for all 6
components and put into compact form, being very careful of the
factors of 2!
Multiplying out e = aeaT , where we have substituted the engineering
shear strains for the tensorial shear strains, as shown by
10
 12
 13
   c s 0  11
12 / 2 13 / 2   c − s 0
 11





 22 23 = − s c 0 12 / 2 22
e = 12
 23 / 2  s c 0

 



 23 33   0 0 1  13 / 2  23 / 2 33  0 0 1
 13
yields 6 equations for the tensorial shear strains in the primed coordinate system
 = 11 cos2 ( ) + 22 sin 2 (  ) + 12 cos (  ) sin (  )
11
22 = 11 sin 2 (  ) +  22 cos2 (  ) − 12 cos (  ) sin (  )
33 = 33
 = −11 cos ( ) sin (  ) +  22 cos (  ) sin (  ) + 12 12 cos ( 2 )
12
 =
13
1
2
23 =
1
2
(
(
13
cos ( ) +  23 sin (  ) )
23
cos ( ) − 13 sin (  ) )
However, we know that
 = 212

12
23 = 223
31 = 231
and with further simplifying this yields
 = −11 sin ( 2 ) +  22 sin ( 2 ) + 12 ( cos 2 (  ) − sin 2 (  ) )
12
 = 13 cos ( ) +  23 sin (  )
13
23 =  23 cos ( ) − 13 sin (  )
Taking those 6 equations and rewriting them results in the following matrix system
   c2
s2
 1   11
     
2
c2
 2   22   s

0
 3   33   0
 =
=
0
4  223   0
 5   231   0
0
  
 
   − sin 2 sin 2
 6   212
0
0
1
0
0
0
0 0
sc   1 
 
0 0
− sc  2 
0 0
0  
 3
 
c −s
0  4 
s c
0   5 
 
0 0 c 2 − s 2   6 
which can be expressed as
e = ( T −1 ) e
T
11
Inverting that system results in
e = TT e
Again, this representation could be simplified for two-dimensions similar to Mohr’s circle for
strains.
12
1