review (2 nd order tensors):

Review (2nd order tensors):

Tensor – Linear mapping of a vector onto another vectord 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 UU U UU 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

cO

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 u3 3

1 1ij i j

j iS

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 TU U U S T S T S TU 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 uv S S S u S u S u S uv 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 Sv 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 SU U U T T T S S SU 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 ST 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 ST 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 SS S SS 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 T1det( )6 ijk lmn li mj nkS S S S

1 12det( )ji ipq jkl pk qlS S S

S1 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 1,2,33 3 3 2 3

2 9 2713 27

1( ) : det( )2

kI p q k k

p p

I I I Ip I I q

I trace I I I

S S S S

Special Tensors

Symmetric tensors TS 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 Rdet( ) 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 sinsin sin cos sin cos

cos sin 0

x R

y

z

a aa a

aa

sin cos sin sin coscos cos cos sin sin

sin cos 0

R x

y

z

a aa aa a

Basis change formulas

0 0

sin cos sin cos

R RR

RR

R R R

e ee ee e e e

eee e e e e

1 1sinR R R R

e e e

1 1sinR R Rv v v

R R R

v e e e e e e

1 1sin

1 1 cotsin

1 1 cotsin

R R R

R

R

vvv v vR R R R R

vv v vvR R R R Rv v v v vR R R R R

v

R

P

i

k

j

Gradient operator

Review:

Deformation Mapping

Eulerian/Lagrangian descriptions of motion

Deformation Gradient

xu(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)



i ik k

d ddy F dx

y F x

Review

Sequence of deformations

x

u(1)(x)

e3

e1

e2

dx dy


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



ml

1 1( ) or ( )2 2

Tij ki kj ijE F F E F F I

22 202 2

00 02 2ij i j

ll l lE m mll l

m E m

Review

0

detdV JdV

FVolume Changes

dxdy dz dr

dv

dw

e3

e1

e2

Undeformed

Deformed

dVdV0

e3

e1

e2



t

dA0 dA

n

n0

xu(x)

dP(n)dP(n)0

Area Elements 1 00 0 0

Ti ki kdA J dA dAn JF n dA n F n

1 1or 2 2

jT iij

j i

uux 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 12 2

j ji k k iij ij

j i j i j i

u uu u u uEx 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


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 2

jT iij

j i

uuwx x

w u uInfinitesimal rotation

e3

e1

e2

OriginalConfiguration Deformed

Configuration

n(1)

n(2) I+wn(2)

n(1)

Decomposition of infinitesimal motion

iij ij

j

u wx

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



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 2

Tij 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 dyy

y y y

xu(x)

e3

e1

e2



y

1i ij jk kdv F F dy i i ij j ij j

d ddv dy F dx F dxdt dt

1d

dt

FL F

Stretch rate and spin tensors 1 1( ) ( )2 2

T T D L L W L L

e3

e1

e2 l0



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 ( ) 22

k k

i ii k k ik k

ix const y const

v va v v W v

t t y

1 ( )2

k 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

ddA

Pt

dV

dP

0limdV

ddV

Pb

t

R

t

e3

e2

e1

n T(n)

T(-n)-n( )

0( ) lim

dA

ddA

nPT n

dA

dP

Review: Kinetics

e3

e1

e2



S

R

R0

S0 b

t

Surface traction Body Force

Internal Traction

( )A V

dA dV P T n b

e3

e1

e2



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



S

R

R0

S0 b

tiij ij

j

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



t

dA0 dA

n

n0

xu(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



R

R0n

S

yx

u(x)S0

C0

C

i i

j jconst constV V V

v vd dV dV dVdt t y t y

x y

Review –

e3

e1

e2



S

R

R0S0b

t

n

V T(n)yx

u(x)

or ijj j

ib a

y

y σ b a

e3

e1

e2



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

vt y t

yy y

v

Angular Momentum Conservation ij ji

( ) 12i i i ij ij i ii

A V V V

dr T v dA b v dV D dV v v dVdt

n

Rate of mechanical work done ona material volume

e3

e1

e2



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

12i i i ij ji i ii

A V V V

dr T v dA b v dV S F dV v v dVdt

n

0 0

( )0 0 0

12i i i ij ij i ii

A V V V

dr T v dA b v dV E dV v v dVdt

n

Review: Thermodynamics

e3

e1

e2



S

R

R0

S0 b

tSpecific Internal EnergySpecific Helmholtz free energy s

Temperature

Heat flux vector qExternal heat flux q

First Law of Thermodynamics ( )d KE Q Wdt

iij ij

iconst

qD q

t y

x

Second Law of Thermodynamics 0dS ddt dt

Specific entropy s

( / )0i

i

qs qt y

1 0ij ij ii

D q sy t t

Conservation Laws for a Control Volume

R is a fixed spatial region –material flows across boundary B

e3

e1

e2


B

b

n

R T(n)

y

0R B

d dV dAdt

v nMass Conservation

Linear Momentum Conservation ( )B R R B

ddA dV dV dAdt

n σ b v v v n

Angular Momentum Conservation ( ) ( ) ( )B R R B

ddA dA dV dAdt

y n σ y b y v y v v n

1 1( ) : ( ) ( )2 2

B R R R B

ddA dV dV dV dAdt

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 dAdt

n σ v b v q n v v v v v n

Second Law ( ) 0R B B R

d qsdV s dA dA dVdt

q nv n

Review: Transformations under observer changesTransformation of space under a change of observer

e3

e1

e2


b

n

y


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 tdt dt dt dt

d d td d d d dt tdt 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

Tddt

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 yF 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 tdt dt dt dt

d d td d d d dt tdt 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



1 0ij ij ii

D q sy t t

review (2 nd order tensors):

Documents

change of observernote

change of observerall

basis changeeigenvalues

tensor linear mapping

small material element

measurable vectors

right stretch tensors

observers reference