review (2 nd order tensors): tensor – linear mapping of a vector onto another vector tensor...
TRANSCRIPT
Review (2nd order tensors):
Tensor – Linear mapping of a vector onto another vector
d d u G x
e1
e2
e3
a1a2
a3Tensor components in a Cartesian basis (3x3 matrix):
11 12 13
21 22 23
31 32 33
U U U
U U U
U U U
( ) ( )ik jl ij i jij klS Q S Q Q m e m e
Basis change formula for tensor components
e1
e2
e3
m1
m2
m3
c
O
Pr
p
Dyadic vector product
ij i jS a b S a b ( ) ( ) ( )ij j i k k i k kS u a b u a b u S u a b u a b u
3 3
1 1ij i j
j i
S
S e eGeneral dyadic expansion
Routine tensor operations
11 12 13 11 11 12 12 13 13
21 22 23 21 21 22 22 23 23
31 32 33 31 31 32 32 33 33
U U U S T S T S T
U U U S T S T S T
U U U S T S T S T
Addition U S T
Vector/Tensor product v S u1 11 12 13 1 11 1 12 2 13 3
2 21 22 23 2 21 1 22 2 23 3
3 31 32 33 3 31 1 32 2 33 3
v S S S u S u S u S u
v S S S u S u S u S u
v S S S u S u S u S u
v u S 11 12 13 1 11 2 21 3 31
1 2 3 1 2 3 21 22 23 1 12 2 22 3 32
31 32 33 1 13 2 23 3 33
S S S u S u S u S
v v v u u u S S S u S u S u S
S S S u S u S u S
11 12 13 11 12 13 11 12 13
21 22 23 21 22 23 21 22 23
31 32 33 31 32 33 31 32 33
11 11 12 21 13 31 11 12 12 22 13 32 11 13 12 23 13 33
21 11 22 21 23 31 21 12
U U U T T T S S S
U U U T T T S S S
U U U T T T S S S
T S T S T S T S T S T S T S T S T S
T S T S T S T S
22 22 23 32 21 12 22 22 23 32
31 11 32 21 33 31 31 12 32 22 33 32 31 13 32 23 33 33
T S T S T S T S T S
T S T S T S T S T S T S T S T S T S
Tensor product U T S
Routine tensor operations
Transpose11 21 31
12 22 32
13 23 33
TS S S
S S S
S S S
ST u S S u T T T A B B ATS
TraceInner productOuter productDeterminant
Inverse
Invariants (remain constant under basis change)
Eigenvalues, Eigenvectors (Characteristic Equation – Cayley-Hamilton Theorem)
11 22 33trace S S S S: ij ijS TS T
ij jiS T S T1
det( )6 ijk lmn li mj nkS S S S
1 1
2det( )ji ipq jkl pk qlS S S S
1 S S I
21 2 3
1 1 1trace( ) (trace( ) ) ( ) det( )
2 2 6kk ii jj ij ji ijk pqr ip jq krI S I S S S S I S S S S S S S S
S m m3 2 2
1 2 3 1 2 1 30 ( ), ( ) / 2 det( )I I I I trace I I I S S S S
3 21 2 3I I I S S S 0
Recipe for computing eigenvalues of symmetric tensor
11
32 1 1 2 3
2 1
21 2 1 3
1 3 3 2( 1)2 cos cos 1,2,3
3 3 3 2 3
2 9 271
3 271
( ) : det( )2
kI p q k
kp p
I I I Ip I I q
I trace I I I
S S S S
Special Tensors
Symmetric tensorsTS S
Have real eigenvalues, and orthogonal eigenvectors
Skew tensors T S S
S u w uHave dual vectors satisfying
Proper orthogonal tensors1
T T
T
R R R R I
R R
det( ) 1R
Represent rotations – have Rodriguez representation
cos (1 cos ) sinij ij i j ijk kR n n n
Polar decomposition theorem T T T A R U V R RR I U U V V
Polar Coordinates
R
P
i
k
j
sin cos cos cos sin
sin sin cos sin cos
cos sin 0
x R
y
z
a a
a a
aa
sin cos sin sin cos
cos cos cos sin sin
sin cos 0
R x
y
z
a a
a a
a a
Basis change formulas
0 0
sin cos sin cos
R RR
RR
R R R
e ee ee ee e
eeee e e e
1 1
sinR R R R
e e e
1 1
sinR R Rv v vR R R
v e e e e e e
1 1
sin
1 1cot
sin
1 1cot
sin
R R R
R
R
vvv v v
R R R R R
vv v vv
R R R R R
v v v v v
R R R R R
v
R
P
i
k
j
Gradient operator
Review:
Deformation Mapping
Eulerian/Lagrangian descriptions of motion
Deformation Gradient
x
u(x)
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
y
1 2 3( , , , )i i iy x u x x x t
const( , ) ( , )
i
i ii i i j i j
x
y uy x u x t v x t
t t
2
2const const
( , ) ( , ) ( , )i
i
i ii i i j i j i j
x x
y yy x u y t v y t a y t
t t
2
2const constconst const
( , ) ( , )i ii i
i k i i i iik i j k j
k ky yx x
u y u y v va y t v y t
y t t t yt
( )
or i ii i ij ij
j j j
y ux u F
x x x
y x u x F
x u(x)
e3
e1
e2
dx dy
u(x+dx)
OriginalConfiguration
DeformedConfiguration
i ik k
d d
dy F dx
y F x
Review
Sequence of deformations
x
u(1)(x)
e3
e1
e2
dx dy
OriginalConfiguration
After firstdeformation
dzu(2)(y)
y z
After seconddeformation
(2) (1)(2) (1)with or i ij j ij ik kjd d dz F dx F F F z F x F F F
Lagrange Strain
e3
e1
e2
l0
OriginalConfiguration
DeformedConfiguration
ml
1 1( ) or ( )
2 2T
ij ki kj ijE F F E F F I
22 202 2
00 02 2ij i j
ll l lE m m
ll l
m E m
Review
0
detdV
JdV
FVolume Changes
dxdy dz dr
dv
dw
e3
e1
e2
Undeformed
Deformed
dVdV0
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
t
dA0dA
n
n0
x
u(x)
dP(n)dP(n)0
Area Elements1 0
0 0 0T
i ki kdA J dA dAn JF n dA n F n
1 1or
2 2jT i
ijj i
uu
x x
ε u uInfinitesimal Strain
y
x x
y
l0
l0
xx
yy
l0
l0
e1
e2
xy
O A
BC
OA
B C
1 1
2 2j ji k k i
ij ijj i j i j i
u uu u u uE
x x x x x x
Approximates L-strain
Related to ‘Engineering Strains’
11
22
12 21 / 2 / 2
xx
yy
xy yx
Review
Principal values/directions of Infinitesimal Strain
e3
e1
e2
OriginalConfiguration
n(1)
n(2)n(2)
n(1)
( ) ( )
( ) ( )or
i ii
i ikl il l
e
n e n
ε n n
1 1or
2 2jT i
ijj i
uuw
x x
w u uInfinitesimal rotation
e3
e1
e2
OriginalConfiguration Deformed
Configuration
n(1)
n(2)I+w
n(2)
n(1)
Decomposition of infinitesimal motion
iij ij
j
uw
x
Left and Right stretch tensors, rotation tensor F R U F V R
(1) (1) (2) (2) (3) (3)1 2 3 U u u u u u u
(1) (1) (2) (2) (3) (3)1 2 3 V v v v v v v
U,V symmetric, so
i principal stretches
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
u(1)
u(2) v(2)
v(1)
U Ru(2)
u(1)
u(2)
u(1)
u(3)
u(1)1 1
1
u(2)
u(3)
Review
Left and Right Cauchy-Green Tensors 2
2
T
T
C F F U
B F F V
Review
Generalized strain measures3
( ) ( )
i=13
( ) ( )
i=1
Lagrangian Nominal strain: ( 1)
Lagrangian Logarithmic strain: log( )
i ii
i ii
u u
u u
3( ) ( )
i=13
( ) ( )
i=1
Eulerian Nominal strain: ( 1)
Eulerian Logarithmic strain: log( )
i ii
i ii
v v
v v
* 1 * 1 11 1( ) or ( )
2 2T
ij ij ki kjE F F E I F FEulerian strain
Review
iij
j
vL
y
yL vVelocity Gradient
( ) ( ) ii i i j
j
vdv v d v dy
y
y y y
x
u(x)
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
y
1i ij jk kdv F F dy i i ij j ij j
d ddv dy F dx F dx
dt dt
1d
dt
FL F
Stretch rate and spin tensors1 1
( ) ( )2 2
T T D L L W L L
e3
e1
e2 l0
OriginalConfiguration
DeformedConfiguration
n
l1
i ij jdl
n D nl dt
n D n
Vorticity vector ( ) ki ijk
j
vcurl w
y
w v 2 ( ) i ijk jkdual W ω W
1( ) 2
2k k
i ii k k ik k
ix const y const
v va v v W v
t t y
1( )
2k k
i ii k k ijk j k
ix const y const
v va v v v
t t y
k i kijk ij j i
j kconst
a vD
y t y
x
Spin-acceleration-vorticity relations
Review
dA
dP
0limdA
d
dA
Pt
dV
dP
0limdV
d
dV
Pb
t
R
t
e3
e2
e1
n T(n)
T(-n)-n( )
0( ) lim
dA
d
dA
nPT n
dA
dP
Review: Kinetics
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
t
Surface tractionBody Force
Internal Traction
( )A V
dA dV P T n b
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0b
t
n
V T(n)
Resultant force on a volume
Restrictions on internal traction vector
e1
e3
e2
T(n)
n
T(-e1)
-e1
dA1
dA(n)
dA1 dA(n)
Review: Kinetics
n-n
T(n)
T(-n)( ) ( ) T n T nNewton II
( ) or ( )i j jiT n T n n σ n
Newton II&III
Cauchy Stress Tensor
e1
e3
e2
11
12
13
21
22
23
31
32
33
Other Stress Measures
ij ijJ J τ σ
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
ti
ij ijj
uF
x
F I u
det( )J F
Kirchhoff
Nominal/ 1st Piola-Kirchhoff
Material/2nd Piola-Kirchhoff
1 1ij ik kjJ JFS S F σ
1 1 1Tij ik kl jlJ JF F Σ F Fσ
( ) 00 i ijjdP dA n Sn
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
t
dA0dA
n
n0
x
u(x)
dP(n)dP(n)0
( 0) ( )ij j iF dP dPn n ( 0) 0
0 j jiidP dA n n
Review – Reynolds Transport Relation
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
R
R0n
S
yx
u(x)S0
C0
C
i i
j jconst constV V V
v vddV dV dV
dt t y t y
x y
Review –
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0b
t
n
V T(n)
yx
u(x)
or ij
j ji
b ay
y σ b a
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0n
V
yx
u(x)V0
Mass Conservation
Linear Momentum Conservation
0 0i i
i iconst constV
v vdV
t y t y
x x
0 ( ) 0i
iconst const
v
t y t
yy y
v
Angular Momentum Conservation ij ji
( ) 1
2i i i ij ij i iiA V V V
dr T v dA b v dV D dV v v dV
dt
n
Rate of mechanical work done ona material volume
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0b
t
n
V T(n)
yx
u(x)
Conservation laws in terms of other stresses
0 0 0 0ij
j ji
Sb a
x
S b a
0 0 0 0ik jk
j ji
Fb a
x
TF b a
Mechanical work in terms of other stresses
0 0
( )0 0 0
1
2i i i ij ji i iiA V V V
dr T v dA b v dV S F dV v v dV
dt
n
0 0
( )0 0 0
1
2i i i ij ij i iiA V V V
dr T v dA b v dV E dV v v dV
dt
n
Review: Thermodynamics
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
t
Specific Internal EnergySpecific Helmholtz free energy s
Temperature
Heat flux vector q
External heat flux q
First Law of Thermodynamics ( )d
KE Q Wdt
iij ij
iconst
qD q
t y
x
Second Law of Thermodynamics 0dS d
dt dt
Specific entropy s
( / )0i
i
qs q
t y
10ij ij i
iD q s
y t t
Conservation Laws for a Control Volume
R is a fixed spatial region –material flows across boundary B
e3
e1
e2
DeformedConfiguration
B
b
n
R T(n)
y
0
R B
ddV dA
dt v nMass Conservation
Linear Momentum Conservation ( )B R R B
ddA dV dV dA
dt n σ b v v v n
Angular Momentum Conservation ( ) ( ) ( )
B R R B
ddA dA dV dA
dt y n σ y b y v y v v n
1 1( ) : ( ) ( )
2 2B R R R B
ddA dV dV dV dA
dt n σ v b v σ D v v v v v nMechanical Power Balance
First Law1 1
( )2 2
B R B V R B
ddA dV dA qdV dV dA
dt
n σ v b v q n v v v v v n
Second Law ( ) 0
R B B R
d qsdV s dA dA dV
dt
q n
v n
Review: Transformations under observer changes
Transformation of space under a change of observer
e3
e1
e2
DeformedConfiguration
b
n
y
DeformedConfiguration
b*
n*
y*
e2*
e3*
e2*
Inertial frame
Observer frame
* *0 0( ) ( )( )t t y y Q y y
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
* * * * 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
*** * * * *0
0 0
2 * *2 2 2 * ** * * 2 * *0 0
0 02 2 2 2
( ( )) ( ( ))
( )( ( )) ( ( )) 2 ( )
T
T
dd d dt t
dt dt dt dt
d d td d d d dt t
dt dt dtdt dt dt dt
yy yv Qv Q Q Q y y Ω y y
y yy y Ω ya Qa Q Q Q y y Ω y y Ω
Td
dt
QΩ Q
The deformation mapping transforms as * *0 0( , ) ( ) ( ) ( , )t t t t y X y Q y X y
The deformation gradient transforms as *
*
y y
F Q QFX X
The right Cauchy Green strain Lagrange strain, the right stretch tensor are invariant * * * * *T T T C F F F Q QF C E E U U
The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent * * * *T T T T T B F F QFF Q QCQ V QVQ
The velocity gradient and spin tensor transform as
* * * 1 1
* * *( ) / 2
T T
T T
L F F QF QF F Q QLQ Ω
W L L QWQ Ω
The velocity and acceleration vectors transform as **
* * * * *00 0
2 * *2 2 2 * ** * * 2 * *0 0
0 02 2 2 2
( ( )) ( ( ))
( )( ( )) ( ( )) 2 ( )
T
T
dd d dt t
dt dt dt dt
d d td d d d dt t
dt dt dtdt dt dt dt
yy yv Qv Q Q Q y y Ω y y
y yy y Ω ya Qa Q Q Q y y Ω y y Ω
(the additional terms in the acceleration can be interpreted as the centripetal and coriolis accelerations)
The Cauchy stress is frame indifferent * Tσ QσQ (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 * 1 1( ) T T TJ J S QF Q Q F Q SQσ σ (note that this transformation rule will differ if the nominal stress is defined as the transpose of the measure used here…)
Some Transformations under observer changes
Constitutive Laws
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)
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.
Equations relating internal force measures to deformation measures are knownas Constitutive Relations
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
10ij ij i
iD q s
y t t