IV. Engineering

Plasticity

4.1 Engineering plasticity

4.2 Elastoplaticity

References

1. 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 Plasticity

2

1

Tensor quantities and their indicial notation

Coordinate axis

Mechanical quantities

Unit vector

Permutation symbol ijk

Summation

Partial differentiation

Kronecker delta ijδ

First and second order

Examples

Divergence theorem

i

j

k1e

2e

3e

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

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

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

11 12 13

21 22 23

31 32 33

ori

xx xy xz

ij ij yx yy yz

zx zy zz

u u

u

0 if

1 ifij

i j

i j

2

, , , ,, , ,iji

i ij i j ij j

i i j i jx x x x x

3

, ,

1

0 0

Free index: once in a term

Dummy index: twicein a term

ij j i ij j i

j

f f

0

1 , , 1,2,3 2,3,1 3,1,2

1 , , 1,3,2 2,1,3 3,2,1

ijk

if i j or j k or k i

if i j k or or

if i j k or or

,

,

,

2

,

i i

i ijk j k

i

i i

ijk k j

ii

W W a b

c c a b

grad

div

curl

a b

a b

v

v

.. , ..

: Outwardly directed unit normal vector

jk m i jk m iV S

i

Q dV Q n dS

n

Tangential plane

n

S

Outwardly directed

unit normal vector

S

Mechanical quantities and their correlation

0th order

tensor

(scalar)

Temperature,

Effective strain,

Effective strain

rate, Energy,

Power

1th order

tensor

(vector)

Displacement,

Velocity,

Force

2th order

tensor

(Dyadic)

Stress,

Strain,

Strain rate

Displacement

Strain-displacement relation

Velocity

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 i

v v

ij ij

T

,ij j i if v

,0

ij j if

( , , , ), .ij ij ij ij

T ect

( , , , ), .ij ij ij ij

D M ect

ij

T

ij

ij

i

i

dva

dt

i

i

duv

dt

, ,

1( )

2ij i j j i

u u

( , )i i j

u u x t

Tensile test

0 0

0

0

0

0 0

,

ln ln 1

1

e e

f

t e

t

f

e e

P

L A

L

L

AP P

A A A

LP

A L

Predictions of tensile test

Hooke's law in uniaxial loading

,t tE E

0L

0

LLf

P

Current

area A

P

Initial

area A0

,x x xx xxE E

: engineering

: true

e

t

Engineering strain, Engineering stress

(Engineering = Conventional=Nominal)

Before necking occurs

Tensile test

Engineering stress-engineering strain curve

Definition of and ee

: Yield strength

: Tensile strength

: Elongation

Y

U

max 100(%)

lat t

long a

Poisson’s ratio

True strain

0

0

ln ln 1

e

f

t e

L

L

L

Engineering strain

True strain

0L

0fL 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

f

e

L L L n

dL

L i L

L L

L L L

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

0 ( 2)L n

True strain and True stress

True stress

0

0

0 0

(1 )

e

t e e e

P

A

AP P l

A A A l

Engineering stress

True stress

0 0

0

0

0

0 0

,

ln ln 1

1

e e

f

t e

t

f

e e

P

L A

L

L

AP P

A A A

LP

A L

True and engineering stress-strain curves

Specimen Yield Point

Necking Point

Fracture Point

O Y N F

Engineering

True

u

eln(1 )u u

t e

N

F

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

Down

P

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

0

3

2

0

2

2 0

1

2

Internal force, stress vector in tensile test

θ 60θ 45

θ 30θ 90

θθ

θ 0

θ=0

n

nnn

n

0

0A

0

θo 0

sin

AA

cos sini j n

0 0P A i

θ

( )

0 sinP

iA

n

(n)T t

θ

=

o o o oo

o o0 sin

0 cos sin nt

2

0 sin nn

x

y

0

3

4

0

1

4

i

j

0 0P A

nt

Stress vector

( ) 2

0 0sin cos sinn t nt

0 0P A

θ

0

0

3

4n

0

1

2n

0

1

4n

Internal force and stress components

θ 60θ 45

θ 30θ 90

θ

θ

θ

θ = 0

n

nnn

n

0

0A

0

o

o o o oo

θ

0

3

4nt

0

1

2nt

0

3

4nt

= sinN P

= cosT P

0

sin

AA

2 2

0

0

sin sinnn

N P

A A

0

0

cos sin cos sinnt

T P

A A

t

t

t

P P P P

θ

n

o

t

NT

P

0

Normal stress

Shear stress

0 0P A

( ) 2

0 0sin cos sin nn ntn t n t nt

Stress vector and stress

F1

2

F

F3

F

F4

F

F

n

F

F

F

A

n t

t(n) n

(-n)

t(n)

( )n

t

N n

( ) ( )( ) ,

n

i i iT T t n (n)

n (n)T = t

( ) ( )

i it t

( n) (n)

n nt = t

Stress vector

( )i

j ijt e

1 1 2 2 3 3i

i i i (e )

t e e e

A point in 3D mechanics

yface

xface

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 area

1

x xyy

(i)F t i j

1F

2F

3F

4F

x

y

y

x

y( )yx yx

x

( )xy xy

x

y

Thickness=1

Upper Die

Lower Die

Material

( ) ( ) 0

( ) ( ) 0

0 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 y

x y

x y

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

0 ; 0yx yy

yFx y

( , )xy

x x y

2

yF2

xF

1

xF

1

yF

4

yF4

xF

3

xF

3

xF

y

x

1F

Infinitesimal

area

2F

3F

4F

x

y

y

x

xy yx

, , .xx x xy xy etc

0 ;AM

( , ) ( , )( , ) ( , )0

yx yxxx xxx y y x yx x y x y

x y

( , )yx

x y y

( , )xx

x x y

( , )yy x y y

x

yy

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)

Stress

Equation of equilibrium

Symmetry 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 i

S V S V

ij j i ij j i

V

t dS f dV n dS f dV

f dV f

n

A point in 3D mechanics

yface

xface

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 zy

y

yzxz zzz

fx y z

fx y z

fx y z

' '

0 0ijk jk jk kj

S 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

n

x x x yxn

x x yx

F t l l l

t

( )

( )

0 ; cos sin 0

cos sin

n

y y xy yn

y xy y

F t l l l

t

x

cos sinn i j

1

length l

width

sinl

yx

cosl

xy

( )( ),

nn

T t

xn ynO

= =

ys

( )

( )

[cos , sin ] n

x x x yx y ix i

n

y xy x y y iy i

n t n n n

t n n n

( )n

i ji j ji jj

t n n ( )

( )

nx xy xx

nyx y yy

nt

nt

Cauchy’s formula

x

length l

sinl yyx

cosl

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

cos sin cos sin

sin cos sin cos

x x y x xy

y x y yx y

Cauchy’s formula

in 2D

Coordinate

transformation

Since two transformation matrixes are used,

stress is a tensor of order two!i j i p j q pqT T

ABC S

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

body forceif

( )( ) ( )( )

1 2 3

10 : 0

3i i i i i iF t S t n S t n S t n S f hS 31 2 ee en

( )

1 11 21 31 1

( )( ) ( )

2 12 22 32 2

( )

3 13 23 33 3

j

i i j ji j

t n

t t n n t n

t n

n

en n

n

0

0

S

h

(n)t

n

Tangential plane on the surface

Outwardly directed unit normal vector

Traction vector

Cauchy's formula in 3D

Cauchy’s formula in 3D

Stress vector and traction

1

2

3

n

n

n

n

-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 = t

Stress vector

cos( , )i in x n

Characteristic equationNormal comp. of stress vector

Principal stress

( ) ( )

i ji jt n n nt Cauchy's formula

( )

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 n

n n

n n

n

Eigenvalue problem

Stress invariants

1 1 2 3

2

2 2 2

1 2 2 3 3 1

2 2 2

3

1 2 3

1

2

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

t(n)

( )n

t

N n

ij

S

N

( )nt

z

x y

2

3

1

=

Principal stresses and stress invariants

yface

xface

zz

zyzxyz

yyyxxy

xx

xz

z

x

y

z face

Direction of principal stress

associated with k

( )

Nnt n

Examples of stress invariants

20

60

80

1 0 3 .9

4 0 .0

a

b

3 .8 8

ο40

1 1 7 .1

1 7 .1

1

2 o31.7

67.1R

2 1c

2 P

y

b

60xy

x

80

a

20y

80x

1

2

80 20 10 110

80 20 20 10 10 80 60 60 1000

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

I

I

I

10, 0z zx zx

1

2

3

117.1 17.1 10 110

117.1 ( 17.1) ( 17.1) 10 10 117.1 1000

117.1 ( 17.1) 10 20000

I

I

I

1

2

3

103.9 3.9 10 110

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

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

10 40 40 20000

I

I

I

1 1 2 3

2

2 2 2

1 2 2 3 3 1

2 2 2

3

1 2 3

1

2

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

80 60 0

60 20 0

0 0 10

1/ 3 / 3m ii I p

2 2 22 2 2 2

2 1

2 2 22

1 1 2 2 3 3 1

2

1 2

2 2 2

2 1 2 2 3 3 1

1 16( )

3 61 1

3 61

3

1( ) ( ) ( )

6

xx yy yy zz zz xx xy yx zxI I

I

I J

J

2 2 2 2 2 2

2

2 2 2

1 2 2 3 3 1

3 13 6 6 6

2 2

1

2

ij ij xx yy yy zz zz xx xy yz zxJ

Second invariant of stress and

deviatoric stress tensors

Effective (equivalent) stress

Mean stress and

hydrostatic pressure

Second invariant of ij

Deviatoric stress ij

xx m xy xz

ij yx yy m yz

zx zy zz m

=

Effective (equivalent) stress

Tensile test

yface

xface

zz

zyzxyz

yyyxxy

xx

xz

z

x

y

z face

60 50 10 30 50 10

50 20 0 , 30, 50 10 0

10 0 10 10 0 20

ij m ijp

Displacement and deformation, velocity and

rate of deformation

A velocity field in cross wedge rolling

A displacement field

2

,

1, 1

2

x

y

u x y xy

u x y x y

2

8

1 6 x

y

576

3

43

7

2

4

85

1'

(1, 1) 1

1

Deformation of die, exaggerated

Deformation = displacement – rigid-body motion

Velocity field Effective strain-rate

lim

lim

22

xxC O

yyC O

xy xy

O C OC

OC

O E OE

OE

E O C

Undeformed

Deformedxy

Strain tensor

0

0

0 0 0

xx xy

ij yx yy

xx xy xz

ij yx yy yz

zx zy zz

1( )

2

jiij

j i

uu

x x

Displacement-strain relation

Deformation = Displacement – Rigid-body motion

, ,yx z

xx yy zz

uu u

x y z

1( ), .

2

yxxy

uuetc

y x

Strain tensor in 3DPlane strain

fixed0

lim |x xx

y

u u

y 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 rate

t t

t t t

L

L L

10.000 ( )t s

10.001 ( )st

1.00.01

100.0xx

0.01 110.0

0.001

xxxx

t s

,

, ,

xxxx

xx xx

yy yy zz zz

xy xy yz yz zx zx

L LL t Lt t t

L L t t

Ld dt dt

L

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 rate ijExample

t t

0

t

ij ij

ij

ij

dt

d

dt

Quantification of rate of deformation-strain rate

1( )

2

jiij

j j

vv

x x

xx xx t

Strain invariants

Effective strain

xx xy xz x x

yx yy yz y y

zx zy zz z z

n n

n n

n n

iL

1

2 2 2

2

2 2 2

3

1

2

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

1

2 2 2 22 2 2

2

3

26

3

ij ij

xx yy yy zz zz xx xy yz xy

3 2

1 2 3

1 2 3

0

0

, , Principal strain

ij ij

L L L

( )

( ) ( )

( ) ( )

; Direction of principal strain

Directions of principal strains

are orthogonal

k

k ij j k i i i

i

k k

ij j k i

k l

i i kl

For n n n n

n

n n

n n

:

1 03 3

iiv

L

Eigenvalue problem

Characteristic equation

Eigenvector

Incompressibility Deviatoric strain ij

xx m xy xz

ij yx yy m yz

zx zy zz m

Principal strain and effective strain

Volumetric strain

1

2 2 2 22 2 2

2

3

26

3

ij ij

xx yy yy zz zz xx xy yz xy

Eigenvalue problem

Characteristic eq’n

Eigen vector

Strainrate invariants

Incompressibility

Effective strain rate

iL

xx xy xz x x

yx yy yz y y

zx zy zz z z

n n

n n

n n

1

2 2 2

2

2 2 2

3

1

2

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

3 2

1 2 3

1 2 3

0

0

, ,

ij ij

L L L

Principal strain rate

1 03 3

iiv m

L

Effective strain ( ) and effective strain rate ( )

Deviatoric strain rate ij

xx m xy xz

ij yx yy m yz

zx zy zz m

1

0 0

t

ij ijdt dt

Principal strain rate and effective strain rate

( )

( ) ( )

( ) ( )

;

k

k ij j k i i i

i

k k

ij j k i

k l

i i kl

For n n n n

n

n n

n n

Direction of principal strain rate

Directions of principal

strain rates are orthogonal

:

03

2 2 2

1 1 2 2 0Y

1 2max

2 2

Y

1 20

von Mises yield function in 3D problem

Huber-von Mises yield criterion

Yield function of plane stress problem

2 2

2

2 2 2

1 2 2 3 3 1

2 2 2 2 2 2

10,

2 3

1

2

16

2

, , 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

2 2 2 2

1 2 3 1 2 3

1, , , , 0

2f k

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

Tresca yield criterion

1 2 3 0

max 0

2

f k

Yk

2 2 2 2

1 2 3 1 2 2 3 3 1

1, , 0

6f k

<Yield locus> <Yield surface>

Y

Y

von Mises

Tresca

Torsion test

Tensile test

Yield criterion for isotropic hardening material

2 2 2 2 2

1 2 2 1

2 2 2

1 1 2 2

1

2Y

Y

ⅰ)

ⅱ)

ⅲ) ~~ omitted.

11 2 max 1

00;

2 2

YY

1 21 2 max 1 20 ;

2 2

YY

von Mises

Tresca

Stress at the initial yielding point

in tensile test

Direction of increase

in principal stresses

during torsional test

1

2

Y

1

u

Y

E

•

•

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

A: Elastic

B: Tresca impossible

Mises elastic

C: Mises plastic

D: Both impossible

E: Tresca plastic

Mises elastic

①: Tensile test, uniaxial loading

②: Tortional test, torsion

1

Y

2

1

Isotropic hardening

Y = Y(

1

2

Y

Y

Y

Y

A

BD

C

E

11

0

2 2

YY

2 1

2 2

Y

10

2 2

Y

2 2 2

1 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

E

v

0

y

y

x z y

xy yz zx

E

v

0

zz

x y z

xy yz zx

E

v

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 deformation

Principle of superposition

Coefficient of

thermal

expansion

x

y

z

xy

z x

y

z

isotropic

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

x2

3xx

isotropic

3

xm

1

1

1 xx

11

2xx

3

x

20 0 0 0

3 30 0

0 0 0 0 0 0 03 3

0 0 0

0 0 0 03 3

xxxx

x

xx xxij

xx xx

0 0

0 02

0 02

xx

xxij

xx

: : : :xx yy zz xx yy zz

: :yyxx zz

xx yy zz

constant

3

x

3

x

3

x

3

x

3

x

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

21

2

1,

pq ij ij

ij ij ij ij ij

ij ij

f k

f f

<Normality>

ij ij

ij ij

f f

또는

3

2

21 1 1( ) ( )

2 2 2pq pq ip jq pq pq ip iq ij ij ij

ij ij

fk

2

2 3, 3

3 2ij ij ij ijJ

:

( ) 0ijf

Yield surface

ij

33 23

22 13

12

11

2 2

3 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 deformation

: Plastic strain- rate

: Difference strain- rate

: Elastic strain rate

0

p d

ij ij ij

p

ij

d

ij

d e

ij ij

e

ij

d

ij

Elastoplastic

Rigid plastic

:

:

Idealization of deformation and flow stress

Idealization of flow stress

Perfectly plastic constant

Elastoplastic ,

Rigid plastic

Rigid- viscoplastic ,

Rigid- thermoviscoplstic , ,

e p

p

p p

p p T

: =

: =

: =

: =

: =

Example of flow stress model at room temperature

Example of flow stress model at elevated temper.

1 , ,

: Initial yield stress

: Strength coefficient

: Strain hardening exponent

n

n n

o o

o

Y K Y Kb

Y

K

n

,

: Strain hardening exponent

: Strain rate dependency

: Strength constant of hot material

n m mC C

n

m

C

0

T/T

Flo

w s

tress

m

0.5

1 2,.

,2 1

.

1, 1

.

2 > 1

2 > 1

..

Special boundary conditions: Friction

t

t n

n

:

:

:

:

Low of Coulomb friction

Coefficient of Coulomb friction

Frictional stress

Normal stress

3

t

t n t

mk

m

k Y

mk

:

:

:

: ,

The Tangential component of displacement or velocity

sho

Friction factor

Shear yield stress

Independent 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 conservation

,

,

,

0

0

( ) 0

ii i i

ii i i

i i

u

v

vt

Incompressibility condition

Law of mass conservation

Direction of friction and sticking condition

1

( ) on

( ) on

( )2( ) tan

t n t C

t t C

t tt

g v S

mkg v S

v vg v

a

Strong for sticking

Very weak to sticking

Laws of friction and incompressibility

Hybrid frictional law

( ) on '

' ( ) on 't n t C n

t t C n

g v S when m k

m 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 conduction

Law of heat convection

( )c qq h T T

Heat transfer coefficient

4 4( )r qq T T

Law of radiation

Stefan-Boltzmann constant

Boundary 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 etc

x

Equation 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 z

t

g

T T T Tk k k q c

x x y y z z t

Thermal capacity

xx x x

q Tq q Δx k

x x x

Theory and law of heat conduction

(0.9 1.0)gq

Equation of heat conduction

Rigid-plasticity

Heat transfer

Elasticity

S

P

S

V

iuS

itS

V

S

SDie

ti

Sit

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 zy

y

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

, ,

2 ,

x y z

xy xy

u v w

x y z

u v

y x

Detailed description of an elastic problem

,

,

0

0

0

ij

i

i i

ij i i

i

ij i i

fx

f

f

Indicial notation

2ij ij kk ij

, ,

1

2ij i j j iu u

Equation of

equilibrium

Stress-strain

relation

Displacement-

strain relation

,yz zx

v w w u

z 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

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 dx

x x x x t

2

2

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

y x t y x tT x p x t x dx

x x t

(0, ) 0,y t ( , )

( ) 0x L

y x tT x

x

⊙ Boundary condition

Torsional or bending vibration of bar or beam

( , )( , ) ( )T

x tM x t GJ x

x

2

2

( , ) ( , )( ) ( )

x t x tGJ x I x

x x t

⊙ Boundary condition

⊙ Differential equation

(0, ) 0,t ( , )

( ) 0x L

y x tGJ x

x

⊙ Differential equation

2 2 2

2 2 2

( , ) ( , )( ) ( )

y x t y x tEI x m x

x x t

2 22

2 2

( )( ) ( ) ( ) 0

Y xEI x m x Y x

x x

⊙ Boundary condition

(a) Clamped End at x=0 or x=L

0

( )0

x

dY x

x

( )0

x L

dY x

x

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

2

0

( )( ) 0

x

d Y xEI x

x

2

2

( )( ) 0

x L

d Y xEI x

x

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

2

2

0

( )( ) 0

x

d d Y xEI x

dx x

2

2

( )( ) 0

x L

d d Y xEI x

dx x

Torsional vibration Bending vibration

Vibration of membrane and plate

Differential equation

22

2

wT w p

t

22

2

wT w

t

Boundary condition

0w 1 1( , )a bat

2 2( , )a bat0w

Tn

⊙ Vibration of plate

Differential equation

24

2E

wD w p

t

3

212(1 )E

EhD

v

2

4

2E

wD w

t

Boundary condition

(a) Clamped edge

0w

0w

0w

n

0nM

0nM 0nsn n

MV Q

s

(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 continuity

Incompressible 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 iv

t

1

2 , 1i j i j k k i j i j i j k k i jE

2

3i j i j

,i iq k

, ,

1

2i j i j j iu u

, ,

1

2i 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- strain

relationX

Velocity-strain rate

relation

Constitutive law

IncompressibilityUnnecessary in general.

For incompressible materialX

Boundary conditions

V

S

SDie

ti

Sit

Si

c

V

Sti

Sit

P

s

iu ,iv p

, 0 in Vi j j if

, ,

1

2i j i j j iu u

, ,

1

2i j i j j iv v

2i j i j k k i j

1

1i j i j k k i jE

2

3i j i j

, 0i iu , 0i iv

on

on

i

i

i i u

i j j i t

u u S

n t S

on

on

or on

on

i

i

i i v

i j j i t

t n c

n n c

v v S

n t S

mk S

u u S

, ,

, ,

E

or

,i i

i

n

u v

t

m

u

or

Continuum mechanics – Displacement method

Displacement ,i i ju u x t

Velocity ii

duv

dt

Strain-displacement relation , ,

1

2ij 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 principle

Virtual work-rate principle

Minimum total potential theorem

Hamilton’s principle

etc.

Strain ij

Strainrate ij

Temperature T

Damage

Microstructure

D

M

,ij ij

T

, ,

1

2ij i j j iu u

Strainrate-velocity relation

,i i

i i j j

dv va v v

dt t

Acceleration

,

1 1ij ij i i g

duq q

dt

Thermodynamics

First law (local energy equation)

Second law (Clausius-Duhem inequality)

,

10i

i

ds qe

dt 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 unloading

0 for plastic loading

p

p

If Y

, 0Y

, 0 0,pIf Y

46

One-dimensional elastoplastic constitutive model

4. Plastic flow rule

p sign multiplierplastic:

01

01

aif

aifasign

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 laweE

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 multiplier

During 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 E

signH E H E

Elastoplastic tangent modulus

p

p

dYH H

d

Hardening

modulus:

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

epE

From the above equations, we haveHE

EHE ep

49

Generalization of elastoplastic constitutive model

1. 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

),,( ppN

),,( pH

: A set of internal variables associated

with hardening

Elastoplasticity

Heat transfer

Elasticity

S

P

S

V

iuS

itS

V

S

SDie

ti

Sit

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

0p

ii

Additive decomposition of strain

pe

epE tnn 1

stress integration