IV. Engineering Plasticity

4.1 Engineering plasticity4.2 Elastoplaticity

References1. Forging simulation (M. S. Joun, 2013, Jinseam Media)2. Advanced solid mechanics and finite element method

(M. S. Joun, 2009, Jinseam Media)

4.1 Engineering Plasticity2

1

Tensor quantities and their indicial notationCoordinate axisCoordinate axis

Mechanical quantitiesMechanical quantities

Unit vectorUnit vector

Permutation symbolPermutation symbol ijk

SummationSummation

Partial differentiationPartial differentiation

Kronecker deltaKronecker delta ijδ

First and second order First and second order

ExamplesExamples

Divergence theoremDivergence theorem

ij

k1e

2e

3e

1 2 3, , axis , , axisx y z x x x g

1 2 12, , , orx y xy xyu u u u g

1 2 3, , , ,i j k e e eg

11 12 13

21 22 23

31 32 33

ori

xx xy xz

ij ij yx yy yz

zx zy zz

u u

u rg

g

0 if1 ifij

i ji j

g

2

, , , ,, , , ijii ij i j ij j

i i j i jx x x x x

g

3

, ,1

0 0

Free index: once in a termDummy index: twicein a term

ij j i ij j ij

f f

g

g

g

01 , , 1, 2,3 2,3,1 3,1,21 , , 1,3, 2 2,1,3 3,2,1

ijk

if i j or j k or k iif i j k or orif i j k or or

g

,

,

,

2,

i i

i ijk j k

i

i i

ijk k j

ii

W W a bc c a b

graddivcurl

a ba b

vv

g

g

g

g

g

g

.. , ..

: Outwardly directed unit normal vectorjk m i jk m iV S

i

Q dV Q n dS

n

g

Tangential plane

n

S

Outwardly directed unit normal vector

S

Mechanical quantities and their correlation

0th ordertensor(scalar)

Temperature,Effective strain,Effective strain rate, Energy, Power

1th order tensor

(vector)

Displacement, Velocity, Force

2th ordertensor

(Dyadic)

Stress,Strain,Strain rate

Displacement

Strain-displacement relationVelocity

Strain rate-velocity relation

Acceleration

Strain

Strain rate

Temperature

• Damage D• Microstructure M

Thermodynamics • First law • Second law

Stress

Constitutive law• Isothermal• Nonisothermal

Newton’ s law of motion• Equation of motion• Equation of equilibrium

Virtual work principleVirtual work-rate principle

• Minimum total potential theorem• Hamilton’ s principle• etc.

Coupled analysis

, ,

1 ( )2ij i j j iv v &

ij ij &

T

,ij j i if v &

, 0ij j if

( , , , ), .ij ij ij ijT ect &

( , , , ), .ij ij ij ij D M ect &ij

T

ij&

ij

ii

dvadt

ii

duvdt

, ,

1 ( )2ij i j j iu u

( , )i i ju u x t

Tensile testTensile test

0 0

,e eP

L A g

Predictions of tensile test

Hooke's law in uniaxial loadingHooke's law in uniaxial loading

,t tE E g

0L

0LLf

P

Current area A

P

Initial area A0

,x x xx xxE E g

: engineeringe

Engineering strain, Engineering stress (Engineering = Conventional=Nominal)

Before necking occurs

Tensile test

Engineering stress-engineering strain curve

Definition of and Definition of and ee

: Yield strength: Tensile strength: Elongation

YU

max 100(%)

lat t

long a

Poisson’s ratio

True strainTrue strain

0

0

ln ln 1

e

ft e

LLL

Engineering strain

True strain

0L0fL L

P

Current area A

P

Initial area A0

0

0 0 0

1 0

0

0 0 0

( 1)

( 1)

ln ln ln 1 ln(1 )

f

t

n L

Li

fe

L L L ndL

L i LL LL L L

0L 0L 0 2L 0 ( 1)L n

0 ( 2)L n

True strain and True stress

True stressTrue stress

0

0

0 0

(1 )

e

t e e e

PA

AP P lA A A l

Engineering stress

True stress

0 0

0

0

0

0 0

,

ln ln 1

1

e e

ft e

t

fe e

PL ALL

AP PA A ALP

A L

g

g

g

True and engineering stress-strain curves

Specimen Yield Point

Necking Point

Fracture Point

O Y N F

Engineering

True

ueln(1 )u u

t e

NF

Y

O

F

Y

N

Y

U

max 100(%)

P

E

Internal force in tensile test before necking

(X) (X) (X) (O)P/n n 개

UP

Down UP

DownP

P

P

P

P

P

P

P

P

P P

P

P

P

Saint-Venant’s principle

=

=

●●●

Principle of symmetry

⊙ Two statically equivalent loads have nearly the same influence on the material except the region near to the load exerting area.

End Effect

End-Effect

Non-end Effect

0

03

2

02

2 0

12

Internal force, stress vector in tensile test

θ 60 o θ 45 o θ 30 o

θ 90 o

θθ

θ 0

θ=0o

nn

nnn

0

0A

0

θo 0

sinAA

cos sini j nr r

0 0P A iuur v

θ

( )

0 sinP iA

n

(n)T tur

v

θ

=

o o o o o

o o0 sin

0 cos sin nt @

20 sin nn @

x

y

03

4

014

ir

jr

0 0P A

nrtr

Stress vector

( ) 20 0sin cos sinn t nt

v v

0 0P A

θ

0

034n

012n

014n

Internal force and stress components

θ 60 oθ 45 o θ 30 o

θ 90 o

θθ

θ

θ = 0o

nn

nnn

0

0A

0

o

o o o oo

θ

03

4nt 012nt

03

4nt

= sinN P

= cosT P

0

sinAA

2 20

0

sin sinnnN PA A

00

cos sin cos sinntT PA A

t

tt

P P P P

θ

n

o

tNT

P

0

Normal stress

Shear stress

0 0P A

( ) 20 0sin cos sin nn ntn t n t nt

v v v v

Stress vector and stress

F1

2F

F3

F

F4

F

Fn

F

F

F

A

n t

t(n) n

(-n)

t(n)

n

( )nt%

N n

( ) ( )( ) ,

n

i i iT T t n (n)

n (n)T = tr

( ) ( )i it t

( n) (n)

n nt = t

Stress vector

( )ij ijt e

1 1 2 2 3 3i

i i i (e )t e e e

A point in 3D mechanics

y face

x face

zz

zyzxyz

yyyxxy

xx

xz

z

x y

z face

1e 2e

3e

Stress

xx xy xz

ij yx yy yz

zx zy zz

Point

x

yInfinitesimal area

A point in 2D mechanics = Square

Stresses in 2D

Stress : Force exerting on unit area1

x xyy

(i)F t i j

1F

2F

3F

4Fxy

y

x

y( )yx yx

x

( )xy xy

x

y

Thickness=1

Upper Die

Lower Die

Material

( ) ( ) 0( ) ( ) 00 0 0

xx x xy xy

yx yx yy y

Plane stress

Body force(weight) was neglected

x

y

0yxxx

x y

( , )( , )( , )( , )

xx xx

yy yy

xy xy

yx yx

x yx yx yx y

1 3 2 40 ; 0x x x x xF F F F F g

0 ; 0yx yyyF x y

g

( , )xy x x y

2yF 2

xF

1xF

1yF

4yF

4xF

3xF

3xF

y

x

1F

Infinitesimal area

2F

3F

4Fxy

yx

xy yx

, , .xx x xy xy etc

0 ;AM g

( , ) ( , )( , ) ( , ) 0yx yxxx xx x y y x yx x y x yx y

( , )yx x y y

( , )xx x x y

( , )yy x y y

xy

y

x

( , )xx x y

( , )xy x y

( , )yx x y ( , )yy x y

A

Equation of equilibrium in 2D

Equation of equilibrium in 2D (Plane stress, Plane strain, Axis-symmetric)

StressStress

Equation of equilibriumEquation of equilibrium

Symmetry of stress tensorSymmetry of stress tensor

xx xy xz

ij yx yy yz

zx zy zz

, ,xy yx yz zy zx xz ij ji

( )

' ' ' '

, ,'

0

( ) 0 0

i i ij j iS V S V

ij j i ij j iV

t dS f dV n dS f dV

f dV f

n

A point in 3D mechanics

y face

x face

zz

zyzxyz

yyyxxy

xx

xz

z

x y

z face

Stresses in 3D and equation of equilibrium

S

P

S

V0

0

0

yxxx zxx

xy yy zyy

yzxz zzz

fx y z

fx y z

fx y z

' '

0 0ijk jk jk kjS V

dS dV (n)x t x f

, 0ij j if

Cauchy's formula and coordinate transformation

y

yx

xxy

x

y

( )

( )0 ; cos sin 0

cos sin

nx x x yx

nx x yx

F t l l lt

r

r

( )

( )0 ; cos sin 0

cos sin

ny y xy y

ny xy y

F t l l lt

r

r

x

cos sinn i j 　

1

length l

width

sinl

yx

cosl

xy

( )( ),

nnT t

rr

xn ynO

= =

ys

( )

( )

[cos , sin ] nx x x yx y ix iny xy x y y iy i

n t n n nt n n n

r

r

r

( )ni ji j ji j

jt n n r( )

( )

nx xy xx

nyx y yy

ntnt

r

r

Cauchy’s formula

x

length l

sinl yyxcosl

x

xy

x

yy

x x y

2 2

2 2

2 2

cos sin 2 sin cos( 90 ) sin cos 2 sin cos

( )cos sin (cos sin )

x x y xy

y x x y xy

x y x y xy

o

cos sin cos sinsin cos sin cos

x x y x xy

y x y yx y

Cauchy’s formula in 2DCauchy’s formula in 2D

CoordinatetransformationCoordinatetransformation

Since two transformation matrixes are used, stress is a tensor of order two!i j i p j q pqT T

ABC S g

1 2 3, ,OBC n S OAC n S OAB n S g

body forceif

( )( ) ( )( )1 2 3

10 : 03i i i i i iF t S t n S t n S t n S f hS 31 2 ee eng

( )1 11 21 31 1

( )( ) ( )2 12 22 32 2( )3 13 23 33 3

ji i j ji j

t nt t n n t n

t n

n

en n

n

g

00

Sh

(n)tn

Tangential plane on the surface

Outwardly directed unit normal vector

Traction vector

Cauchy's formula in 3D

Cauchy’s formula in 3DCauchy’s formula in 3D

Stress vector and tractionStress vector and traction

1

2

3

nnn

ng

-Traction = stress vector normal to the tangential plane on the surface -Force or moment prescribed boundary = traction prescribed boundary

( ) ( )( ) ,

n

i i iT T t n (n)

n (n)T = tr

Stress vector

cos( , )i in x n

Characteristic equationCharacteristic equationNormal comp. of stress vectorNormal comp. of stress vector

Principal stress

( ) ( )i ji jt n n nt Cauchy's formulag

( )i ji j N i

xx xy xz x x

yx yy yz y N y

zx zy zz z z

t n n

n nn nn n

n

g

Eigenvalue problemEigenvalue problem

Stress invariantsStress invariants

1 1 2 3

2

2 2 2

1 2 2 3 3 1

2 2 23

1 2 3

12

2

ii xx yy zz

ij ij ii jj

xx yy yy zz zz xx xy yz zx

ij xx yy zz xy yz zx xx yz yy zx zz xy

I

I

I

g

g

g

t(n)

n

( )nt%

N n

ij

S

N

( )nt

z

x y

2

3

1

=

Principal stresses and stress invariants

y face

x face

zz

zyzxyz

yyyxxy

xx

xz

z

xy

z face

Direction of principal stressassociated with k

( )N

nt n

Examples of stress invariants

20

60

80

103.9

40.0

a

b

3.88

ο40

117.1

17.1

12 o31.7

67.1R

2 1c

2 P

yb

60xy

x

80o

a

20y

80x

1

2

80 20 10 11080 20 20 10 10 80 60 60 1000

3 80 20 10 2 80 0 80 0 20 0 10 60 60 20000

III

10, 0z zx zx

1

2

3

117.1 17.1 10 110117.1 ( 17.1) ( 17.1) 10 10 117.1 1000

117.1 ( 17.1) 10 20000

III

1

2

3

103.9 3.9 10 110103.9 ( 3.9) ( 3.9) 10 10 103.9 40 40 1000

103.9 ( 3.9) 10 2 40 0 103.9 0 ( 3.9) 010 40 40 20000

III

1 1 2 3

2

2 2 2

1 2 2 3 3 1

2 2 23

1 2 3

12

2

ii xx yy zz

ij ij ii jj

xx yy yy zz zz xx xy yz zx

ij xx yy zz xy yz zx xx yz yy zx zz xy

I

I

I

g

g

g

80 60 060 20 00 0 10

1/ 3 / 3m ii I p g

2 2 22 2 2 22 1

2 2 221 1 2 2 3 3 1

21 2

2 2 22 1 2 2 3 3 1

1 1 6( )3 61 13 613

1 ( ) ( ) ( )6

xx yy yy zz zz xx xy yx zxI I

I

I J

J

g

2 2 2 2 2 22

2 2 21 2 2 3 3 1

3 13 6 6 62 2

12

ij ij xx yy yy zz zz xx xy yz zxJ

g

Second invariant of stress and deviatoric stress tensorsSecond invariant of stress and deviatoric stress tensors

Effective (equivalent) stressEffective (equivalent) stress

Mean stress and hydrostatic pressureMean stress and hydrostatic pressure

Second invariant of ij

Deviatoric stress Deviatoric stress ij

xx m xy xz

ij yx yy m yz

zx zy zz m

g

=

Effective (equivalent) stress

Tensile test

y face

x face

zz

zyzxyz

yyyxxy

xx

xz

z

xy

z face

60 50 10 30 50 1050 20 0 , 30, 50 10 010 0 10 10 0 20

ij m ijp

Displacement and deformation, velocity and rate of deformation

A velocity field in cross wedge rollingA velocity field in cross wedge rolling

A displacement fieldA displacement field

2

,1, 12

x

y

u x y xy

u x y x y

2

8

1 6 x

y

5 76

3

4 37

2

4

85

1'

(1, 1) 1

1

Deformation of die, exaggeratedDeformation of die, exaggerated

Deformation = displacement – rigid-body motion

Velocity field Effective strain-rate

lim

lim

22

xx C O

yy C O

xy xy

O C OCOC

O E OEOE

E O C

Undeformed

Deformedxy

Strain tensor

00

0 0 0

xx xy

ij yx yy

xx xy xz

ij yx yy yz

zx zy zz

1 ( )2

jiij

j i

uux x

Displacement-strain relationDisplacement-strain relationDeformation = Displacement – Rigid-body motion

, ,yx zxx yy zz

uu ux y z

1 ( ), .2

yxxy

uu etcy x

Strain tensor in 3DStrain tensor in 3DPlane strainPlane strain

fixed0lim |x x

xy

u uy y

, ( , ), 0x y zu u f x y u Ex.: Dam, Strip rolling

, ( , ), 0x y zu u f x y u

( , )xu x y y

( , )xu x y

fixed|x xu

y

tan , 1

Definition of strain rateDefinition of strain rate

t t

t t t

L

L L

10.000 ( )t s

10.001 ( )st

1.0 0.01100.0xx 0.01 110.0

0.001xx

xx t s

&

,

, ,

xxxx

xx xx

yy yy zz zz

xy xy yz yz zx zx

L LL t Lt t tL L t tLd dt dtL

d dt d dt

d dt d dt d dt

&&

& &

& & &

100.0mm

101.0mm

xx xy xz

ij yx yy yz

zx zy zz

& & &

& & & &

& & &

Strain rateStrain rate ij&Example

t t

0

t

ij ij

ijij

dt

ddt

&

&

Quantification of rate of deformation-strain rate

1 ( )2

jiij

j j

vvx x

&

xx xx t &

Strain invariantsStrain invariants

Effective strainEffective strain

xx xy xz x x

yx yy yz y y

zx zy zz z z

n nn nn n

g

iL

1

2 2 22

2 2 23

12

2

ii xx yy zz

ij ij ii jj xx yy yy zz zz xx xy yz zx

ij xx yy zz xy yz zx xx yz yy zx zz xy

L

L

L

g

g

g

1

2 2 2 22 2 2

23

2 63

ij ij

xx yy yy zz zz xx xy yz xy

g

3 21 2 3

1 2 3

0

0

, , Principal strain

ij ij

L L L

g

g

g

( )

( ) ( )

( ) ( )

; Direction of principal strain

Directions of principal strains are orthogonal

kk ij j k i i i

i

k kij j k i

k li i kl

For n n n n

n

n n

n n

g

g

g

:

1 03 3

iivL g

Eigenvalue problemEigenvalue problem

Characteristic equationCharacteristic equation

EigenvectorEigenvector

IncompressibilityIncompressibility Deviatoric strainDeviatoric strain ij

xx m xy xz

ij yx yy m yz

zx zy zz m

g

Principal strain and effective strain

Volumetric strain

1

2 2 2 22 2 2

23

2 63

ij ij

xx yy yy zz zz xx xy yz xy

& & &g

& & & & & & & & &

Eigenvalue problemEigenvalue problem

Characteristic eq’nCharacteristic eq’n

Eigen vectorEigen vector

Strainrate invariantsStrainrate invariants

IncompressibilityIncompressibility

Effective strain rateEffective strain rate

iL&

xx xy xz x x

yx yy yz y y

zx zy zz z z

n nn nn n

& & &

& & & &g

& & &

1

2 2 22

2 2 23

12

2

ii xx yy zz

ij ij ii jj xx yy yy zz zz xx xy yz zx

ij xx yy zz xy yz zx xx yz yy zx zz xy

L

L

L

& & & & &g

& & & & & & & & & & & & & &g

& & & & & & & & & & & & & &g

3 21 2 3

1 2 3

0

0

, ,

ij ij

L L L

& &g

& & && & &g

& & & &g Principal strain rate

1 03 3

iiv m

L & &

& &g

Effective strain ( ) and effective strain rate ( ) Effective strain ( ) and effective strain rate ( ) &

&

Deviatoric strain rateDeviatoric strain rate ij &

xx m xy xz

ij yx yy m yz

zx zy zz m

& & & &

& & & & &g

& & & &

1

0 0

t

ij ijdt dt & &g

Principal strain rate and effective strain rate

( )

( ) ( )

( ) ( )

;

kk ij j k i i i

i

k kij j k i

k li i kl

For n n n n

n

n n

n n

& & & &g

& &g

g

Direction of principal strain rate

Directions of principal strain rates are orthogonal

:

03

2 2 21 1 2 2 0Y

1 2max 2 2

Y

1 20

von Mises yield function in 3D problemvon Mises yield function in 3D problem

Huber-von Mises yield criterionHuber-von Mises yield criterion

Yield function of plane stress problemYield function of plane stress problem

2 22

2 2 21 2 2 3 3 1

2 2 2 2 2 2

1 0,2 3

121 62

, , or , ,

ij ij

xx yy yy zz zz xx xy yz zx

p p

Yf J k k k

Y

Y

Y Y T Y T

g

g

g

& &g

2 2 2 21 2 3 1 2 3

1, , , , 02

f k g

1 2 3) , , coordinate systema 1 2 3) , , coordinate systemb

Tresca yield criterionTresca yield criterion

1 2 3 0

max 0

2

f kYk

g

g

2 2 2 21 2 3 1 2 2 3 3 1

1, , 06

f k g

<Yield locus> <Yield surface>

Y

Y

von MisesTresca

Torsion test

Tensile test

Yield criterion for isotropic hardening material

2 2 2 2 21 2 2 1

2 2 21 1 2 2

12

Y

Y

ⅰ)

ⅱ)

ⅲ) ~~ omitted.

11 2 max 1

00;2 2

Y Y

1 21 2 max 1 20 ;

2 2Y Y

von Mises

Tresca

Stress at the initial yielding point in tensile test

Direction of increase in principal stressesduring torsional test

1

2

Y

1

uY

E

•

•

⊙ Yield locus in the case of plane stress ; 3 0

A: ElasticB: Tresca impossible

Mises elasticC: Mises plasticD: Both impossibleE: Tresca plastic

Mises elastic

①: Tensile test, uniaxial loading②: Tortional test, torsion

1

Y

2

1

Isotropic hardening

Y = Y(

1

2

Y

Y

Y

Y

AB

DC

E1

10

2 2Y Y

2 1

2 2Y

102 2

Y

2 2 21 1 2 2 Y

①

②

⊙ Causes of strain hardening-Plastic deformation accompanies dislocation.

-Dislocation is a kind of defects occurring due to slip or

twist of atomic structures.

-Existing dislocations play a role of prevention of generation

of new dislocations which is called strain hardening2

1

Kinematic hardening

Strain hardening

0

xx

y z x

xy yz zx

Ev

0

yy

x z y

xy yz zx

Ev

0

zz

x y z

xy yz zx

Ev

1

1

1

1 1 1, ,

x x y z

y y x z

z z x y

xy xy yx yz zx zx

T

T

T

vE

vE

vE

G G G

Small deformationPrinciple of superposition

Coefficient of thermal expansion

x

y

z

xy

z x

y

zisotropic

0i

Hooke’s law for an isotropic material

xx xx

xx1

Poisson’s ratio1

1 1 ( )xx

yy zz xx xxE

Generalized Hooke's law for an isotropic material

2(1 )EG

x

y

z

x23 xx

isotropic

3x

m

1

1

1 xx

112 xx

3x

20 0 0 03 30 0

0 0 0 0 0 0 03 3

0 0 00 0 0 0

3 3

xxxx

xxx xx

ij

xx xx

0 0

0 02

0 02

xx

xxij

xx

: : : :xx yy zz xx yy zz

: :yyxx zz

xx yy zz

constant

VV V

3x

3x

3x

3x

3x

m ij ij

ij

Plastic deformation owing to stress

• Normality due to Drucker’s postulate: Strain rate tensor is normal to yield surface for incompressible materials

• For von Mises yield criterion

212

1,

pq ij ij

ij ij ij ij ijij ij

f k

f f

& && &&

<Normality>

ij ijij ij

f f

&& 또는

32

&

&

21 1 1( ) ( )2 2 2pq pq ip jq pq pq ip iq ij ij ij

ij ij

f k

22 3, 33 2ij ij ij ijJ & & &

:( ) 0ijf

Yield surface

ij&33

23

22 13

12

11

2 23 3 2

3

ij ij ij

kl kl

& &

&& &

yy xy yzxx zxzz

xx yy zz xy yz zx

& & && && &

Associated flow rule

Flow stress given by user

Zero in the elastic region

Normality

Idealization of deformationIdealization of deformation

: Plastic strain- rate

: Difference strain- rate

: Elastic strain rate

0

p dij ij ij

pij

dij

d eij ij

eij

dij

Elastoplastic

Rigid plastic

& & &g

&

&

& &g

&

&g

:

:

Idealization of deformation and flow stress

Idealization of flow stressIdealization of flow stress

Perfectly plastic constant

Elastoplastic ,

Rigid plastic

Rigid- viscoplastic ,

Rigid- thermoviscoplstic , ,

e p

p

p p

p p T

g

g

g

&g

&g

: =

: =

: =

: =

: =

Example of flow stress model at room temperatureExample of flow stress model at room temperature

Example of flow stress model at elevated temper.Example of flow stress model at elevated temper.

1 , ,

: Initial yield stress: Strength coefficient

: Strain hardening exponent

nn n

o o

o

Y K Y Kb

YKn

g

,: Strain hardening exponent: Strain rate dependency: Strength constant of hot material

n m mC CnmC

& &g

0

T/T

Flo

w s

tress

m

0.5

1 2,.

,2 1

.

1, 1

.

2 > 1

2 > 1

..

Special boundary conditions: FrictionSpecial boundary conditions: Friction

t

t n

n

g :

:

:

:

Low of Coulomb frictionCoefficient of Coulomb frictionFrictional stressNormal stress

3

t

t n t

mkm

k Y

mk

g

g

:

:

:

: ,

The Tangential component of displacement or velocity

sho

Friction factor

Shear yield stressIndependent of normal stress

When sticking occurs or

Low of constant shear friction

uld be same in the contact surface of the two bodies

Law of mass conservationLaw of mass conservation

,

,

,

00

( ) 0

ii i i

ii i i

i i

uv

vt

g

&

g

Incompressibility condition

Law of mass conservationg Direction of friction and sticking condition

1

( ) on( ) on

( )2( ) tan

t n t C

t t C

t tt

g v Smkg v S

v vg va

Strong for stickingVery weak to sticking

Laws of friction and incompressibility

g Hybrid frictional law( ) on '

' ( ) on 't n t C n

t t C n

g v S when m km k g v S when m k

Heat transfer phenomena in solid

x x x T T T

xq

Heat fluxCoefficient of heat conduction

• One-dimensional • 2 or 3 dimensional

Fourier’s law of heat conductionFourier’s law of heat conduction

Law of heat convectionLaw of heat convection

( )c qq h T T

Heat transfer coefficient

4 4( )r qq T T

Law of radiationLaw of radiation

Stefan-Boltzmann constant

Boundary conditionsBoundary conditions

TT T on S

4 4, ( ) ( )i i q q q qk T kT n T T h T T n on S

, .xx x x

qq q Δx etcx

Equation of heat conductionEquation of heat conduction( ) ( )

( )

x x x y y y

z z z g

q q y z q q z x

Tq q x y q x y z c x y zt

gT T T Tk k k q c

x x y y z z t

Thermal capacity

xx x x

q Tq q Δx kx x x

Theory and law of heat conduction

(0.9 1.0)gq &:

Equation of heat conduction

Rigid-plasticityRigid-plasticity

Heat transferHeat transfer

ElasticityElasticity

S

P

S

V

iuS

itS

V

S

SDie

ti

S it

Si

c

Deformation and heat transfer problems

Input: Young's modulus, Poisson's ratio, Flow stress, Frictional condition, Die velocity, Thermal conditions

eS

TS

qS

V

ivS

itS

itS

cS

V

V

0

0

0

, ,

yxx zxx

xy y zyy

yzxz zz

xy yx xz zy zx xz

fx y z

fx y z

fx y z

1

1

1

, ,

x x y z

y y x z

z z x y

xy yz zxxy yz zx

v TE

v TE

v TE

G G G

V

V

V

, ,

2 ,

x y z

xy xy

u v wx y z

u vy x

Detailed description of an elastic problem

,

,

0

0

0

iji

i i

ij i ii

ij i i

fx

f

f

Indicial notation

2ij ij kk ij

, ,12ij i j j iu u

Equation of

equilibrium

Stress-strain

relation

Displacement-

strain relation

,yz zxv w w uz y x z

1 1 1 2 1 2 2 1 2 1 2 2 1( ) ( )m x c c x c x k k x k x F && & & & &

2 2 2 1 2 3 2 2 1 2 3 2 1( ) ( )m x c x c c x k x k k x F && & &

[ ] ( ) [ ] ( ) [ ] ( ) ( )m x t c x t k x t F t rr r r&& &

1

2

1 2 2

2 2 3

1 2 2

2 2 3

0[ ]

0

[ ]

[ ]

mm

m

c c cc

c c c

k k kk

k k k

Vibration of discrete system

Vibration of string

⊙ Newton’s equation of motion

2 2

2 2

( ) ( , ) ( , ) ( , ) ( , )( ) ( , ) ( ) ( )T x y x t y x t y x t y x tT x dx dx p x t dx T x x dxx x x x t

2

2

( , ) ( , )( ) ( , ) ( )y x t y x tT x p x t x dxx x t

(0, ) 0,y t ( , )( ) 0x L

y x tT xx

⊙ Boundary condition

Torsional or bending vibration of bar or beam

( , )( , ) ( )Tx tM x t GJ xx

2

2

( , ) ( , )( ) ( )x t x tGJ x I xx x t

⊙ Boundary condition

⊙ Differential equation

(0, ) 0,t ( , )( ) 0x L

y x tGJ xx

⊙ Differential equation2 2 2

2 2 2

( , ) ( , )( ) ( )y x t y x tEI x m xx x t

2 2

22 2

( )( ) ( ) ( ) 0Y xEI x m x Y xx x

⊙ Boundary condition(a) Clamped End at x=0 or x=L

0

( ) 0x

dY xx

( ) 0x L

dY xx

(b) Hinged End at x=0 or x=L2

20

( )( ) 0x

d Y xEI xx

2

2

( )( ) 0x L

d Y xEI xx

(c) Free End at x=0 or x=L2

20

( )( ) 0x

d d Y xEI xdx x

2

2

( )( ) 0x L

d d Y xEI xdx x

Torsional vibrationTorsional vibration Bending vibrationBending vibration

Vibration of membrane and plate

Differential equation2

22

wT w pt

22

2

wT wt

Boundary condition

0w 1 1( , )a bat

2 2( , )a bat0wTn

⊙ Vibration of plate

Differential equation2

42EwD w pt

3

212(1 )EEhDv

24

2EwD wt

Boundary condition

(a) Clamped edge

0w

0w

0wn

0nM

0nM 0nsn n

MV Qs

(b) Simply supported edge

⊙ Vibration of membrane

Summary – Natural phenomena and mechanics

Law Details Equation Related contents

Newton

Requirement on equilibrium

Equation of equilibrium Navier-Cauchy equation

Equation of motion Navier-Stokes equation

Energy

Equation of heat conduction For solid

Law of energy conservation For fluid

Mass Equation of continuityIncompressible material

Compressible material

Constitu-tive

Hooke’s law

They satisfy the second law of thermodynamics.

Plastic flow rule

Fourier of heat conduction

‥‥‥

Miscell-aneous

Displacement- strain relation

Velocity-strain rate relation

Essential BC

Natural BC

,i i i iF ma M I , 0,j i j i i j j if

,j i j i if v &

, ,i gik q c

t

, ,,i g j jik q c v

t

, 0i iv

,0i i

vt

12 , 1i j i j k k i j i j i j k k i jE

23i j i j

&&

,i iq k

, ,12i j i j j iu u

, ,12i j i j j iv v &

, ,i i i iu u v v

,,i j i j i i i it n t q k q n

Summary – Solid mechanics

Item Elasticity Plasticity Constants

Definition of problem

Unknown

Equation of equilibrium X

Displacement- strainrelation

XVelocity-strain rate

relation

Constitutive law

IncompressibilityUnnecessary in general.For incompressible material

X

Boundary conditions

V

S

SDie

ti

S it

Si

c

V

Sti

S it

P

s

iu ,iv p

, 0 in Vi j j if

, ,12i j i j j iu u

, ,12i j i j j iv v &

2i j i j k k i j

1 1i j i j k k i jE

23i j i j

&&

, 0i iu , 0i iv

on

oni

i

i i u

i j j i t

u u S

n t S

on

on

or onon

i

i

i i v

i j j i t

t n c

n n c

v v S

n t S

mk Su u S

, ,

, ,

E

&

or

,i i

i

n

u vtmu

or

Continuum mechanics – Displacement methodDisplacement ,i i ju u x t

Velocity iiduvdt

Strain-displacement relation , ,12ij i j j iu u

Stress ij

Coupled analysis

,ij j i if v &

, 0ij j if

Newton’s law of motion Equation of motion, Equation of equilibrium,

Virtual work principleVirtual work-rate principle Minimum total potential theorem Hamilton’s principle etc.

Strain ij

Strainrate ij&

Temperature T

Damage Microstructure

D

M

,ij ij &

T

, ,12ij i j j iu u

Strainrate-velocity relation

,i i

i i j jdv va v vdt t

Acceleration

,1 1

ij ij i i gdu q qdt

&

Thermodynamics

First law (local energy equation)

Second law (Clausius-Duhem inequality),

1 0i

i

ds qedt T

Constitutive law Isothermal Nonisothermal

, , , , .ij ij ij ij D M etc &

, , , , .ij ij ij ijT etc &

( , ), ( ), .i iq q T T u u T etc

4.2 Elastoplasticity

45

One-dimensional elastoplastic constitutive model

Idealization of uniaxial tension experiment. Mathematical model

1. Strain decomposition into sum of an elastic component and a plastic component.

2. Elastic uniaxial constitutive law

pe

eE

3. The yield function and the yield criterion

-yield function

-yield criterion (for plastic yielding)

,Y Y

, 0

0 for elastic unloading0 for plastic loading

p

p

If Y

&

&

, 0Y

, 0 0,pIf Y &

46

One-dimensional elastoplastic constitutive model4. Plastic flow rule

p sign & &multiplierplastic:

0101

aifaif

asign

0

Complementary condition

0

0 0 0p & &

The above equation implies that

0 0Y &

5. Hardening Law

pY Y 0

tp p dt &In a monotonic tensile test

In a monotonic compression test

pp pp p p & &

In view of the plastic flow rule pOne dimensional model. Hardening curve

Uniaxial model. Elastic domain

47

One-dimensional model - Summary

1. Elastoplastic split of the axial strain pe

2. Uniaxial elastic law eE

3. The yield function ,Y Y

4. Plastic flow rule signp

5. Hardening Law pY Y p

6. Loading/unloading criterion 000

48

Plastic multiplier / elastoplastic tangent modulus

Determination of the plastic multiplierDuring the plastic flow, the value of yield function remains constant

0

Consistent condition: During the plastic flow, current stress always coincides with current yield stress

0Y

By taking the time derivative of the yield function

pHsign psign H &&

From elastic law pE & &&

pFrom hardening law

E EsignH E H E

& & &

Elastoplastic tangent modulus

p p

dYH Hd

Hardeningmodulus:

In FEM, we need an elastoplastic tangent modulus, a relationship between stress and total strain rate:

epE &&

From the above equations, we haveHE

EHEep

49Generalization of elastoplastic constitutive model1. Additive decomposition of the strain tensor

pe

2. General elastic law

3. The yield function

4. Plastic flow rule

5. Hardening law

6. Loading/unloading criterion

000

eCεσ

,, p

),,( pp N

),,( pH

: A set of internal variables associated with hardening

ElastoplasticityElastoplasticity

Heat transferHeat transfer

ElasticityElasticity

S

P

S

V

iuS

itS

V

S

SDie

ti

S it

Si

c

Elastoplasticity and heat transfer problems

Input: Young's modulus, Poisson's ratio, Flow stress, Frictional condition, Die velocity, Thermal conditions

eS

TS

qS

V

ivS

itS

itS

cS

V

V

0pii

Additive decomposition of strainpe

epE &&tnn 1

stress integration