teorija plasticnosti_2010_11
TRANSCRIPT
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Inkrementalna teorija
plastičnosti(rate theory of plasticity)
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Klasične teorije nelinearnog ponašanja materijala
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Mehanika kontinuuma
0
Za izotermale i statičke uvjete vrijedi
Prvi zakon termodinamike : u :
Drugi zakon termodinamike : :
sa :
u specifičnaunutarnjaenergija
tenzor deformacija
tenzornaprezanja
Helmholtz ova slobod
D
naenergija
disipacijeenergijeD
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div
(m) m
T
Jednadžba ravnoteže : g 0
Tenzor deformacija (Boltzman - ov continuum) :
1(U I) ; m 0
m
sa :
U F : F right stretch tensor
F deformation gradient tensor
1Green Lagrange strain tensor (m 2) : (F F I)
2
Biot strain te
nsor (m 1) : U I
Logaritmic strain tensor (m 0) : lnU
Mehanika kontinuuma
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Konstituitivni zakon :
: ( , )
( , ) ( , ): :
D
D
Mehanika kontinuuma
za elasticni materijal :
( )konstitutivni zakon
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• Povijest:
- Coulomb (1773)
- Tresca (1864)
- Saint Venant (1870)
- Levy (1871)
- Haar i von Karman (1909)
- von Mieses (1913)
Teorija plastičnosti
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Osnovne pretpostavke:
• Nelinearnost je posljedica prirasta plastične
deformacije
• Materijal je linearno elastičan
• Kriterij tečenja materijala
• Totalna deformacija se sastoji od elastičnog i
plastičnog dijela
• Pretpostavka maximalne disipacije energije
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e p ε ε εDekompozicija deformacija:
e e e p( ) σ D ε D ε εKonstitutivni zakon (elastičnost):
Pravilo tečenja:
Uvjet plastičnog tečenja (ploha tečenja): 0),(f qσ
),(λp qσg ε
Zakon očvršćenja materijala: )();,(λ κhqκσkκ
f je funkcija tečenja; g definira smjer plastične deformacije;
je tzv. Plastični multiplikator; q & k su parametri vezani uz
svojstva materijala.
λ
Osnovne jednadžbe:
Iz uvjeta maksimalne disipacije energije (ireverzibilna termodinamika):
gdje je: = plastični potencijal( , )
g σ qσ
( , )
k σ κq
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0
1
2
0
0
el
el
el pl
tot pl pl pl pl
pl
el
elel pl
el
pl
pl
: ( , )
( , , ) : D : ( )
Za elastični materijal :
D :
: :t t
:t
: q
D
D
D
D 0 q h( )t
Pravilo tečenja & očvršćenje materijala
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0
0 0
0 0
pl
Y
pl
: q qt
Lagrange ova metoda maximizacije funkcionala :
( ,q) gdje je : ( ,q) ( ,q) (q)
Maximalna disipacija:
; g( ,q); k( , ); g ; kq q
uz uvjete :
D
D
0 0
Pravilo tečenja & očvršćenje materijala
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Pravilo tečenja
pridruženo, (elasto - plastic stiffness matrix symetric)
/ normal to the yield surface (normality rule)
p
g f
f( , )
f
σ q ε
σ
σ
nepridruženo,
(elasto - plastic stiffness matrix non - symetric)
g f f
kombinacija
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elastično stanje : f < 0
funkcija tečenja: f = 0 0p ε
0λ
plastično stanje : f = 0 0λ
Karush-Kuhn-Tucker-ovi uvjeti opterećenja-rasterećenja
(iz uvjeta maximalne disipacije):
Uvjet konzistentnosti plastičnog tečenja:
0fλ,0λ,0f
0fλ
Kriterij tečenja materijala (tzv. ploha tečenja)
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Uvjeti opterećenja / rasterećenja
f 0 elastic loading / unloading
f 0 f 0 0 plastic loading
f 0 f 0 0 elastic unloading
f 0 f 0 0 neutral loading
f = 0
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Drucker-ov postulat stabilnosti
0d)(,0dd p*TTpT εσσεσ
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Očvršćenje materijala
)(F)(F),(f :hardening isotropic 1 qσqσ
stressbackα
F)α(F),(f :hardening kinematic 1
-σqσ
ncombinatio
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Numerički algoritam
( , )p g σ qε
0ff
f
TT
q
qσ
σ gεDσ λe
kκ
hκ
κ
hq
λ
κh/f/f/withε
λ qσTqe
Tσ
eTσ
HqfσfHkfgDf
Df
εHkfgDf
DgfDDσ
Tqe
Tσ
eTσe
e
1)
2)
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Tresca kriterij tečenja (1864)
hardening) (noconstant material k
stress yieldσ
σ2
1k
0k)σσ(2
1)(f
y
y
31
σ
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Von Mises-ov kriterij tečenja (1913)
2 0
2 2 2
2 1 2 2 3 3 1
1 2 3
0
0
2
( ) ( ) 0
1( ) ( ) ( )
6
principal stresses
material constant (no hardening)
1
3
Representative stress :
3
y
f J
J
J
σ σ
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Ekvivalentna naprezanja i deformacije
Kako bi se naprezanje u slučaju višeosnog stanje naprezanje direktno moglo usporediti
s jednoosnim stanjem naprezanje koriste se ekvivalentna naprezanja odnosno deformacije
Ekvivalentno naprezanje
Ekvivalentna deformacija
Ekvivalentni inkrement plastične deformacije
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Von Mises sa očvrčćenjem (1913)
0 2 0
0
0 2 0
0 2 0
( , ) ( ) 0
2: ( )
3
: ( , ) ( )
: ( , ) ( )
T
p p
f J
or h
isotropic hardening f J
kinematic hardening f J
σ σ
hardening σ
σ σ
σ σ
Strain-hardening hypothesis
Work-hardening hypothesis
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Plohe tečenja za kvazi krte
materijale(beton, keramika, ..)
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Tipične plohe (krivulje) sloma betona za stanje više-osnog naprezanja
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2
1
3,2
2,3
3,2
fcm1
2
3,2
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Krivulje sloma i tipični razvoj pukotina za slučaj dvo-osnog stanja
naprezanja – relativno prema jedno-osnoj tlačnoj čvrstoći betona fC
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Hofstetter/Mang 1995
Tipične krivulje opterečenja za slučaj dvo-osnog stanje naprezanja
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Mohr-Coulomb kriterij tečenja (1773)
1 3 1 3
1 3
( ) ( ) (2 cos ) ( )sin 0
0
( ) ( ) 2 0 ( 2 )
σ
σ
f c
c kohezija
kut trenja
za
f c or k Tresca
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Mohr-Coulomb-ov kriterij tečenja
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Drucker-Prager-ova ploha tečenja
tcoefficienfriction α
stresses principalσσσ
3/)σσσ(3/Iσ
σσσI
0τ)α(J)(Iα)(f
321
3211V
3211
021
σσσ
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Plohe tečenja materijala
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Realne plohe tečenja beton + nepridruženo pravilo tečenja
1 2 1 1 2 2 3 2
33 1 2 33/ 2
2
2 2
( , , ) ( ) 1 0
3 3cos3 ( )( )( )
2
1cos arccos( cos3 ) cos3 0
3( )
1cos arccos( cos3 ) cos3 0
3 3
4(1 )cos (2( )
V V V
f I J c I c r J c J
JJ
J
K if
r
K if
e er
Ottosen :
Willam & Warnke :
2
2 2 2 2
1)
2(1 )cos (2 1) 4(1 )cos 5 4e e e e e
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3-parameter model
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5-parameter model
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11-parameter model (ELM)
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“Multi-surface” model plastičnosti
1
2
Rankine
Von Mises
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M
M'
N
N'
15025
30
25
30
P'
P
10
01
00
5
65
50
50
S
T
Primjer: DEN uzorak (M.B. Nooru-Mohamed, 1992)
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DEN uzorak (M.B. Nooru-Mohamed, 1992)
X
Y
Z
X
0. 15. 30. 45. 60. 75. 90. 105. 120. 135. 150. 165. 180. 195.
Y
0.
15.
30.
45.
60.
75.
90.
105.
120.
135.
150.
165.
180.
195.
210.
V3
L1
C1
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“anisotropic damage”pridruženo pravilo tečenja
nepridruženo pravilo tečenjaexperiment
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Plastična veza izmedju naprezanja i deformacije
Levi-Mieses jednadžba
Vrijedi za totalni inkrement deformacije uz uvjet da je elastični dio << od plastičnog
Potreban kriterij tečenja !
Pravilo tečenja
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Prandtl-Reuss-ova jednadžba
Pravilo tečenja
Potreban kriterij tečenja !
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Plastično tečenje anizotropnih materijala (Hill, 1948)
Pravilo tečenja: Prandtl-Reuss-ova jednadžba
Kriterij tečenja
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Uvod u mehaniku oštećenja materijala
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E0
Ed
mikropukotine
E0
Ed
naprezanje
deformacija
Mehanika oštećenja– mehanika loma
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Mehanika oštećenja – osnovne jedn. za 1D
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Mehanika oštećenja (naprezanje –deformacija)
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Mehanika oštećenja (naprezanje –deformacija)
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Mehanika oštećenja (naprezanje –deformacija)
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Mehanika oštećenja (naprezanje –deformacija)
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Mehanika oštećenja (naprezanje –deformacija)
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Mehanika oštećenja (naprezanje –deformacija)
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Jednostavni izotropni model
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Jednostavni izotropni model
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Jednostavni izotropni model
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Evolucija oštećenja
Ekvivalentna deformacija
Zakon oštećenja
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max( )
loadingunloading
Evolucija oštećenja (matematički opis)
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32
1
1,2,3
( ) ( );
( ) : ( , ) ( )
.
/
Rankine - ov kriterij :
1 1 1max
i
i
T
e
e e ei ii
g ili samo za monotono opterecenje g
ploha krivulja ostecenja f
McAuley eva zagrada
or E
orE E E
Beton :
D
D D D
32
1i
Ekvivalentna deformacija
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( ) 0
( )
f
g
Evolucija oštećenja (matematički opis)
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Evolucija oštećenja (matematički opis)
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Evolucija oštećenja (matematički opis)
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Evolucija oštećenja (matematički opis)
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Evolucija oštećenja (matematički opis)
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1n
n
Evolucija oštećenja (inkrementalni postupak)
Poznato:
Odrediti:
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sec tan
secsec
secsec
secsec
secsec
sec e
sec e tan sec e
: d : d
dd d: :
dt dt dt
dd d d: : :
dt d dt dt
dd : d : : d
d
d dd : : d : : d
d d
dd : d : : : d
d
d dd : : : d : :
d d
D D
DD
DD
DD
DD
D D
D D D D D
Tangentna matrica krutosti
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0
0
00
0
omeksanje : exp ; model parameter
0
ostecenje : ( )1 exp
t f
f
t
f
f
if
g fif
E
Izotropni model oštećenja betona (vlak)
0
tf
0E
![Page 66: Teorija plasticnosti_2010_11](https://reader034.vdocuments.mx/reader034/viewer/2022051314/5537b97a550346722e8b4630/html5/thumbnails/66.jpg)
Anizotropni model ostećenja
e
pqrsijklpqrs
e
ijklijkl DDD
Pojednostavljenja
• Princip ekvivalentnih deformacija
• Princip ekvivalentnih naprezanja
• Princip ekvivalentne energije
![Page 67: Teorija plasticnosti_2010_11](https://reader034.vdocuments.mx/reader034/viewer/2022051314/5537b97a550346722e8b4630/html5/thumbnails/67.jpg)
M
M'
N
N'
15025
30
25
30
P'
P
10
01
00
5
65
50
50
S
T
Primjer: DEN uzorak (M.B. Nooru-Mohamed, 1992)
![Page 68: Teorija plasticnosti_2010_11](https://reader034.vdocuments.mx/reader034/viewer/2022051314/5537b97a550346722e8b4630/html5/thumbnails/68.jpg)
![Page 69: Teorija plasticnosti_2010_11](https://reader034.vdocuments.mx/reader034/viewer/2022051314/5537b97a550346722e8b4630/html5/thumbnails/69.jpg)
DEN uzorak (M.B. Nooru-Mohamed, 1992)
X
Y
Z
X
0. 15. 30. 45. 60. 75. 90. 105. 120. 135. 150. 165. 180. 195.
Y
0.
15.
30.
45.
60.
75.
90.
105.
120.
135.
150.
165.
180.
195.
210.
V3
L1
C1
![Page 70: Teorija plasticnosti_2010_11](https://reader034.vdocuments.mx/reader034/viewer/2022051314/5537b97a550346722e8b4630/html5/thumbnails/70.jpg)
X
Y
Z X
Y
Z
PH= 10 kN
PH
PV
Double-Edge-Notched
uzorak
Izotropno oštećenje Anizotropno oštećenje