Review (2nd order tensors):
Tensor – Linear mapping of a vector onto another vector
du  G  dx
e3
Tensor components in a Cartesian basis (3x3 matrix):
a3
U11 U12 U13 
U

U
U
21
22
23


U 31 U 32 U 33 
a2
a1
e1

Basis change formula for tensor components
e2
(e )
Sij(m)  Qik Skl
Q jl
Qij  mi  e j
r
m1
p
P
e1
m3
e3
Dyadic vector product
Sij  ai b j
m2
c
O
S  a b
e2
S  u   a  b   u  a(b  u)
General dyadic expansion
S
3
3
Sij u j  (ai bk )uk  ai (bk uk )
 Sij ei  e j
j 1 i 1
Routine tensor operations
Addition U  S  T
Vector/Tensor product
U11 U12 U13   S11  T11 S12  T12
U
  S  T
U
U
21
22
23

  21 21 S22  T22
U 31 U 32 U 33   S31  T31 S32  T32
v  S u
v  u  S v1
Tensor product U  T  S
 v1   S11 S12
v    S
 2   21 S22
 v3   S31 S32
v2
S13  T13 
S23  T23 
S33  T33 
S13   u1   S11u1  S12u2  S13u3 
S23  u2    S21u1  S22u2  S23u3 
S33   u3   S31u1  S32u2  S33u3 
 S11 S12
v3   u1 u2 u3   S21 S22
 S31 S32
S13   u1S11  u2 S21  u3S31 
S23   u1S12  u2 S22  u3S32 
S33   u1S13  u2 S23  u3S33 
U11 U12 U13  T11 T12 T13   S11 S12 S13 
U
  T
 S

U
U
T
T
S
S
21
22
23
21
22
23
21
22
23

 


U 31 U 32 U 33  T31 T32 T33   S31 S32 S33 
 T11S11  T12 S21  T13S31 T11S12  T12 S 22  T13S32 T11S13  T12 S 23  T13S33 
 T21S11  T22 S 21  T23S31 T21S12  T22 S22  T23S32 T21S12  T22 S22  T23S32 
T31S11  T32 S21  T33S31 T31S12  T32 S 22  T33S32 T31S13  T32 S 23  T33S33 
Routine tensor operations
Transpose
T
S
 S11
ST   S12
 S13
T
u S  S u
S21
S22
S23
S31 
S32 
S33 
 A  B T
 BT  AT
Trace trace  S   S11  S22  S33
Inner product S : T  SijTij
Outer product S T  SijT ji
1
Determinant det(S)  ijk lmn Sli Smj Snk
6
Inverse
S1  S  I
S ji1 
1
ipq jkl S pk Sql
2det(S)
Invariants (remain constant under basis change)
I1  trace(S)  Skk
1
1
I 2  (trace(S) 2  S S)  ( Sii S jj  Sij S ji )
2
2
1
I3  det(S)  ijk pqr Sip S jq Skr
6
Eigenvalues, Eigenvectors (Characteristic Equation – Cayley-Hamilton
Theorem)
S  m  m
 3  I1 2  I 2  I3  0
I1  trace(S), I 2  ( I12  S S) / 2 I3  det(S)
S3  I1S 2  I 2S  I3  0
Recipe for computing eigenvalues of symmetric tensor
 1
I1
p

1  3q 3  2( k  1) 
k   2
cos  cos 



3
3
3
2
p
p
3




1
p  I 2  I12
3
I1  trace(S)
2 I13  9 I1I 2  27 I 3
q
27
1
I 2  I12  S : S
I 3  det(S)
2


k  1, 2,3
Special Tensors
T
Symmetric tensors S  S
Have real eigenvalues, and orthogonal eigenvectors
Skew tensors
ST  S
Have dual vectors satisfying
Proper orthogonal tensors
S u  w u
R  RT  RT  R  I
R 1  R T
Represent rotations – have Rodriguez
representation
Polar decomposition theorem
det(R)  1
Rij  cosij  ni n j (1  cos )  sin  ijk nk
A  R  U  V  R RRT  I
U  UT
V  VT
Polar Coordinates
k
P

R

j
i
Basis change formulas
 ax  sin  cos 
  
 a y    sin  sin 
 a   cos 
 z 
cos cos 
cos sin 
 sin 
 sin    aR 
 
cos    a 
0   a 
 aR   sin  cos 
  
 a   cos cos 
 a    sin 
 
sin  sin 
cos sin 
cos 
cos    ax 
 
 sin    a y 
0   az 
Gradient operator
k
P

R


1 
1
 
   eR
 e
 e

R 
R sin   
 R
e R e e
e R


0
 e
R
R
R

e
e R
 sin  e
 cos e


e
e
 e R
0


e
  sin  e R  cos e


i


1 
1
 
v    vR e R  v e  v e   e R
 e
 e


R
R


R
sin







 vR

 R
 v
v    
 R
 v
 
 R
1 vR v

R 
R
1 v vR

R 
R
1 v
R 




v

v
1
  cot  

R sin  
R 
v vR 
1 v
 cot 

R sin  
R
R 
1 vR v

R sin  
R
j
Review:
Deformation Mapping
yi  xi  ui ( x1, x2 , x3 , t )
u(x)
e2
x
y
Original
Configuration
e1
e3
Eulerian/Lagrangian descriptions of motion
yi
u
 i  vi ( x j , t )
t x const t
i
yi  xi  ui ( x j , t )
yi  xi  ui ( y j , t )
Deformed
Configuration
 2 yi
yi
 vi ( y j , t )
t x const
i

ui  yk
u
 i
  ik 

yk  t
t y const

i
xi const
 2 yi
t 2 x const
i
t
2
 ai ( y j , t )
xi const
 ai ( y j , t ) 
vi
v
 vk ( y j , t ) i
t y const
yk
i
Deformation Gradient
y    x  u(x)   F
yi
u

or

 xi  ui   ij  i  Fij
x j x j
x j
u(x+dx)
dy  F  dx
dyi  Fik dxk
dx
e2
x
e1
e3
Original
Configuration
dy
u(x)
Deformed
Configuration
Review
Sequence of deformations
e2
dz  F  dx with F  F(2)  F(1) or dzi  Fij dx j
Lagrange Strain
After second
deformation
Fij  Fik(2) Fkj(1)
m
l0
1
1
E  (FT  F  I) or Eij  ( Fki Fkj  ij )
2
2
l  l0
 l  l 
 
l0
2l02
2l02
2
2
l
e2
e1
m  E  m  Eij mi m j 
z
After first
deformation
Original
Configuration
e3
dz
dy
y
x
e1
u(2)(y)
u(1)(x)
dx
2
e3
Original
Configuration
Deformed
Configuration
Review
e2
dV
 det  F   J
dV0
Volume Changes
dv
dV0
dy
dx
e1
e3
dV
dw
dr
dz
Deformed
Undeformed
dP0(n)
n0
dP(n)
Area Elements
dAn  JF T  dA0n0
n
dA0
dAni  JFki1nk0dA0
e2
x
e1
e3
Infinitesimal Strain
Approximates L-strain


or
1  u u j
 ij   i 
2  x j xi




1   u  u j  uk  uk  1   ui  u j 
Eij   i 


 
   ij
2   x j  xi  x j  xi  2   x j  xi 
Original
Configuration
t
Deformed
Configuration
e2
y
y
1
T
ε  u   u 
2
dA
u(x)
l0
B
C
e1
l0yy
l0
O
A
x
O
C
B
xy
l0 xx
Related to ‘Engineering Strains’
11   xx
 22   yy
12   21   xy / 2   yx / 2
A
x
Review

n(2)
Principal values/directions of
Infinitesimal Strain
ε  n (i )  e n ( i )
e2
n(2)
n(1)
n(1)
i
or  kl nl(i )  ei nl(i )
Infinitesimal rotation
w

1
T
u   u 
2
e1
e3

1  u u j 
or wij   i 

2  x j xi 
Decomposition of infinitesimal
motion

n(2)
e2
ui
  ij  wij
x j
Original
Configuration
n(1)
e1
e3
I+w
n(2)
Original
Configuration
n(1)
Deformed
Configuration
Review
F  RU
Left and Right stretch tensors, rotation tensor
F  VR
U  1u (1)  u (1)  2u (2)  u(2)  3u(3)  u(3)
U,V symmetric, so
V  1 v (1)  v (1)  2 v (2)  v (2)  3 v (3)  v (3)
i principal stretches
U
(2)
u
R
u(2)
u(2)
v
e2
u(3)
u(1)
e1
e3
Original
Configuration
u(1)
u(2)
(2)
v(1)
Deformed
Configuration
Left and Right Cauchy-Green Tensors
1
1
1

u(3)

(1)
u
C  FT  F  U 2
B  F  FT  V 2

u(1)
Review
Generalized strain measures
Lagrangian Nominal strain:
3
 (i  1)u(i)  u(i)
i=1
3
 log(i )u(i)  u(i)
Lagrangian Logarithmic strain:
i=1
3
Eulerian Nominal strain:
 (i  1) v(i)  v(i)
i=1
3
Eulerian Logarithmic strain:
 log(i ) v(i)  v(i)
i=1
Eulerian strain
1
1
E*  (I  FT  F1) or Eij*  (ij  Fki1Fkj1)
2
2
Review
u(x)
Velocity Gradient
v
L   y v  Lij  i
y j
dvi  vi (y  dy )  vi (y ) 

e2

d
d
dvi  dyi 
Fij dx j  Fij dx j
dt
dt
Stretch rate and spin tensors
x
y
e1
e3
Deformed
Configuration
Original
Configuration
vi
dy j
y j
L
1
dvi  Fij F jk
dyk
1
D  (L  LT )
2
dF 1
F
dt
1
W  (L  LT )
2
n
1 dl
 n  D  n  ni Dij n j
l dt
l
l0
e2
e1
e3
Original
Configuration
Deformed
Configuration
Review
Vorticity vector
w  curl ( v)
wi ijk
vk
y j
ω  2dual ( W)
Spin-acceleration-vorticity relations
ai 
vi
v
1 
 i

(vk vk )  2Wik vk
t x const t y const 2 yi
k
k
ai 
vi
v
1 
 i

(vk vk ) ijk  j vk
t x const t y const 2 yi
k
k
ijk
ak i
v

 Dij j  k i
y j
t xconst
yk
i  ijk W jk
Review: Kinetics
S
S0
b
R0
e2
R
e1
e3
t
Surface traction
Body Force
dP
dP
dA0 dA
Deformed
Configuration
Original
Configuration
dP
dP
 b  lim
dV 0 dV
t  lim
dA
dV
t
Internal Traction
T(n)
n
dP
dP(n)
T(n)  lim
dA0 dA
e2
R
-n
T(-n)
t
e1
dA
e3
Resultant force on a volume
P   T(n)dA    b dV
A
n
S
S0
R0
e2
V
b
T(n)
t
e1
e3
V
R
Original
Configuration
Deformed
Configuration
Review: Kinetics
Restrictions on internal traction vector
T(n)
T(n)  T(n)
Newton II
T(-n)
n
-n
e2
dA(n)
Newton II&III
T(n)  n  σ
n
dA1
or
Ti (n)  n j ji
T(n) dA(n)
e1
-e1
T(-e1) dA1
e3
22 e2
Cauchy Stress Tensor
32 23
33 
31
e3
21
13
12
11
e1
Other Stress Measures
F  I  u
Fij   ij 
S
S0
ui
x j
J  det(F)
b
R0
e2
R
e1
e3
t
Deformed
Configuration
Original
Configuration
n0
dP0(n)
dP(n)
n
dA0
e2
x
e1
Kirchhoff
τ  Jσ  ij  J ij
Nominal/ 1st Piola-Kirchhoff
S  JF 1  σ
e3
dA
u(x)
Original
Configuration
t
Deformed
Configuration
Sij  JFik1 kj
dPj(n)  dA0ni0 Sij
Material/2nd Piola-Kirchhoff
Fij dPj (n0)  dPi (n)
Σ  JF 1  σ  F T ij  JFik1 kl F jl1
dPi(n 0)  dA0 n0j  ji
Review – Reynolds Transport Relation
R0
e2
 
v
d
 dV  
 i
 t xconst
dt
y j
V
V


u(x)
y
Original
Configuration
 

 vi

dV  

 t y const y j

V

n
S0
x
e1
e3
C0

dV


S
C
R
Deformed
Configuration
Review –
Mass Conservation
 
vi



t xconst
yi
V


vi

dV

0



0

t xconst
yi

 vi


0
t y const yi
e2
e3

  y  (  v)  0
t y const
or
 ij
 yi
e2
  bj   a j
e1
e3
Angular Momentum Conservation
x
 ij   ji
u(x)
V
R
y
Deformed
Configuration
Original
Configuration
n
S
R0
S0
n
S
V0
x
e1
Linear Momentum Conservation
y  σ  b  a
R0
S0
u(x)
y
Original
Configuration
V
R
b
T(n)
t
Deformed
Configuration
Rate of mechanical work done on
a material volume
e2

d
 1

(n )
r   Ti vi dA   bivi dV    ij Dij dV     vivi dV 
dt  2

A
V
V
V

u(x)
y
x
e1
e3
n
S
R0
S0
Original
Configuration
V
R
b
T(n)
t
Deformed
Configuration
Conservation laws in terms of other stresses
  S   0b   0a
Sij
xi
 0b j  0 a j
     F T   0b  0a


Mechanical work in terms of other stresses


d  1

r

b
v
dV

S
F
dV


v
v
dV


i
i
ij
ji
0
0
i
i
0


dt   2

A
V
V0
V0



d  1

(n )
r   Ti vi dA    bi vi dV   ij Eij dV0    0vi vi dV0 
dt  2

A
V
V0
V0

Ti(n )vi dA 

 ik F jk
xi
 b
0 j
 0 a j
Review: Thermodynamics
S
S0
Temperature 
Specific Internal Energy 
Specific Helmholtz free energy      s
Heat flux vector q
External heat flux q
Specific entropy s
First Law of Thermodynamics
Second Law of Thermodynamics
e2
R
e1
e3
t
Original
Configuration
d
(  KE )  Q  W
dt

b
R0
q

  ij Dij  i  q
t xconst
yi
dS d 

0
dt dt
s (q /  ) q
  i
 0
t
yi

1 
 
 
 ij Dij  qi
 
s
0
 yi
t 
 t
Deformed
Configuration
Conservation Laws for a Control Volume
R is a fixed spatial region –
material flows across boundary B
n
B
R
e2
b
T(n)
y
Mass Conservation
e1
d
 dV   v  ndA  0
dt


R
B
Linear Momentum Conservation

n  σdA   bdV 
B
R

Mechanical Power Balance
d
 vdV  (  v) v  ndA
dt

Angular Momentum Conservation

R
B

B
R



R
R
d
y   vdV  (y   v) v  ndA
dt
(n  σ)  vdA   b  vdV  σ : DdV 

d
1


R
B
d 1
1
 ( v  v)dV 
 ( v  v) v  ndA
dt 2
2


R
B


1

 (n  σ)  vdA   b  vdV   q  ndA   qdV  dt      2 v  v  dV       2 v  v  v  ndA
B
Second Law

y  (n  σ)dA  y  (  b)dA 
B
First Law
Deformed
Configuration
e3
R
B
V
R
d
qn
q
 sdV   s( v  n)dA 
dA 
dV  0
dt






R
B
B
R
B
Review: Transformations under observer changes
Transformation of space under
a change of observer
y*  y*0 (t )  Q(t )(y  y 0 )
n
Inertial frame
e2
b
y
e1
dQ T
Ω
Q
dt
e3
All physically measurable vectors can be
regarded as connecting two points in the
inertial frame
These must therefore transform like
vectors connecting two points under a
change of observer
Deformed
Configuration
e2*
n*
e3*
e2*
Observer frame
b*
y*
Deformed
Configuration
b*  Qb n*  Qn v*  Qv a*  Qa
Note that time derivatives in the observer’s reference frame have to
account for rotation of the reference frame
dy
d T * *
dy* dy*0
v  Qv  Q
 Q Q (y  y 0 (t )) 

 Ω(y*  y*0 (t ))
dt
dt
dt
dt
*
*
a  Qa  Q
d 2y
dt
2
Q
d2
dt
2
T
Q (y
*
 y*0 (t )) 
dy* dy*0 (t )
 2 dΩ  * *

 Ω 

)
 (y  y 0 (t ))  2Ω(
2
2
dt
dt
dt


dt
dt
d 2 y*
d 2 y*0
Some Transformations under observer changes
The deformation mapping transforms as y* (X, t )  y*0 (t )  Q(t )  y(X, t )  y 0 
y*
y
The deformation gradient transforms as F 
Q
 QF
X
X
The right Cauchy Green strain Lagrange strain, the right stretch tensor are invariant
C*  F*T F*  FT QT QF  C E*  E U*  U
The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent
B*  F*F*T  QFFT QT  QCQT
V*  QVQT
The velocity gradient and spin tensor transform as
*


L*  F*F*1  QF  QF F 1QT  QLQT  Ω
W*  (L*  L*T ) / 2  QWQT  Ω
The velocity and acceleration vectors transform as
dy
d T * *
dy* dy*0
v  Qv  Q
 Q Q (y  y 0 (t )) 

 Ω(y*  y*0 (t ))
dt
dt
dt
dt
*
dy* dy*0 (t )
 2 dΩ  * *
a  Qa  Q
Q
Q (y

 Ω 

)
 (y  y 0 (t ))  2Ω(
2
2
2
2
dt
dt
dt


dt
dt
dt
dt
(the additional terms in the acceleration can be interpreted as the centripetal and coriolis accelerations)
*
d 2y
d2
T
*
 y*0 (t )) 
d 2 y*
d 2 y*0
The Cauchy stress is frame indifferent σ*  QσQT (you can see this from the formal definition, or use
the fact that the virtual power must be invariant under a frame change)
The material stress is frame invariant Σ*  Σ
The nominal stress transforms as S*  J (QF)1  QσQT  JF 1  σQT  SQT (note that this
transformation rule will differ if the nominal stress is defined as the transpose of the measure used here…)
Constitutive Laws
Equations relating internal force measures to deformation measures are known
as Constitutive Relations
General Assumptions:
1. Local homogeneity of deformation
(a deformation gradient can always be calculated)
2. Principle of local action
(stress at a point depends on deformation in
a vanishingly small material element surrounding
the point)
e2
e1
e3
Original
Configuration
Restrictions on constitutive relations:
1. Material Frame Indifference – stress-strain relations must transform
consistently under a change of observer
2. Constitutive law must always satisfy the second law of
thermodynamics for any possible deformation/temperature history.
1 
 
 
 ij Dij  qi
 
s
0
 yi

t

t


Deformed
Configuration