IV. Engineering Plasticity - AFDEX · IV. Engineering Plasticity 4.1 Engineering plasticity 4.2...
Transcript of IV. Engineering Plasticity - AFDEX · IV. Engineering Plasticity 4.1 Engineering plasticity 4.2...
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