Manual

manual

User Manual:

Open the PDF directly: View PDF .
Page Count: 21





σ11 σ12 σ13

σ12 σ22 σ23

σ13 σ23 σ33 

→σ11 σ22 σ33 √2σ23 √2σ13 √2σ12 

C

















C1111 C1122 C1133 √2C1123 √2C1113 √2C1112

C1122 C2222 C2233 √2C2223 √2C2213 √2C2212

C1133 C2233 C3333 √2C3323 √2C3313 √2C3312

√2C1123 √2C2223 √2C3323 2C2323 2C2313 2C2312

√2C1113 √2C2213 √2C3313 2C2313 2C1313 2C1312

√2C1112 √2C2212 √2C3312 2C2312 2C1312 2C1212

















.

a b ˆ

a

ˆ

b

a:b=ˆ

a·ˆ

b

Cˆ

C

a=C:b

ˆ

a=ˆ

C·ˆ

b.

F

l

˙

ε

q= [σF]α= [¯εp]

f=J2(σ) + r2

3σF(¯εp, T ) = ksk+r2

3σF

s=σ−1

3tr σI

∂f

∂σ=s

ksk

∂2f

∂σ∂σ=1

ksk1−1

3(I⊗I)−s

ksk⊗s

ksk

∂2f

∂σ∂q=0

∂f

∂q=r2

3

∂2f

∂q∂q= 0

∂2f

∂q∂σ=0

q=σFXα=¯εpεp

f=J2(σ+X) + r2

3σF(¯εp, T ) = ks+Xk+r2

3σF.

∂f

∂σ=s+X

ks+Xk

∂2f

∂σ∂σ=1

ks+Xk1−1

3(I⊗I)−s+X

ks+Xk⊗s+X

ks+Xk

∂2f

∂σ∂q=1

ks+Xk1−s+X

ks+Xk⊗s+X

ks+Xk

∂f

∂q=hq2

3

s+X

ks+Xki

∂2f

∂q∂q="00

01

ks+Xk1−s+X

ks+Xk⊗s+X

ks+Xk#

∂2f

∂q∂σ="0

1

ks+Xk1−1

3(I⊗I)−s+X

ks+Xk⊗s+X

ks+Xk#

σF(¯εp, T ) = −(σ0+K¯εp)

∂σF

∂¯εp

=−K

σF(¯εp, T ) = −[σ0+R(1 −exp (−δ¯εp))]

∂σF

∂¯εp

=−δR exp (−δ¯εp)

X(εp, T ) = −Hεp

dX

dεp

=−HI

X=

n

X

i=1

Xi

˙

Xi=− 2

3Cin+r2

3γi(¯εp)Xi!˙γ−Air3

2kXikai−1Xi

n= 1

γ(¯εp)

γ(¯εp) = γs+ (γ0−γs)e−β¯εp.

y(σ,α) = g(f(σ,q(α)))

η

g

g

g(f) = fn.

f(σ,q)

y(σ,q) = r3

2*f(σ,q)

p2/3η+n

.

˙

εp=n˙p

n=3

2

σ0−X0

J2(σ0−X0)

J2(Y0) = r3

2Y0:Y0

˙p=J2(σ0−X0)−σa

Dn

X=X1+X2

˙

X1=C12

3(a10 −Q)n−X1˙p−γ1J2(X1)m−1X1

˙

X2=C22

3a2n−X2˙p−γ2J2(X2)m−1X2

˙

Q=d(q−Q) ˙p

˙σa=b(σas −σa) ˙p

b=(bhσas −σa≥0

brσas −σa<0

σas =hA+Blog10 ˙pi

473 K < T < 873 K

˙

εcr =f(σ,εcr, t, T )

εcr

n+1 =εcr

n+fn+1∆tn+1

R=εcr

n+1 −εcr

n−fn+1∆tn+1

J=I−∂f

∂εcr

n+1

∆tn+1

dεcr

n+1

dσn+1

.

R=0=⇒0 = dR=∂R

∂σn+1

:dσn+1 +∂R

∂εcr

n+1

:dεcr

n+1 +∂R

∂tn+1

:dtn+1

R=0=⇒0 = dR=∂R

∂σn+1

:dσn+1+∂R

∂εcr

n+1

:dεcr

n+1+∂R

∂tn+1

:dtn+1+∂R

∂tn+1

:dTn+1

0=∂R

∂σn+1

+∂R

∂εcr

n+1

:dεcr

n+1

dσn+1

+∂R

∂tn+1

:dtn+1

dσn+1

+∂R

∂tn+1

:dTn+1

dσn+1

dεcr

n+1

dσn+1

=−∂R

∂εcr

n+1 −1∂R

∂σn+1

dεcr

n+1

dσn+1

=J−1∂f

∂σn+1

∆tn+1.

f∂f

∂εcr

∂f

∂σ

∂f

∂t

∂f

∂T

J2

˙

εcr =g(σeq, εeq, t, T )3

2

s

σeq

s=σ−1

3tr (σ)

σeq =r3

2s:s

εeq =r2

3εcr :εcr.

g

f=g3

2

s

σeq

∂f

∂σ=3

2∂g

∂σeq −g

σeq  s

σeq ⊗s

σeq +g

σeq

I3

2I−1

31⊗1

∂f

∂εcr =s

σeq ⊗∂g

∂εeq

εcr

εeq 

∂f

∂t =∂g

∂t

3

2

s

σeq

∂f

∂T =∂g

∂T

3

2

s

σeq

g=Aσn

eq

∂g

∂σeq

=Anσn−1

eq

∂g

∂εeq

= 0

∂g

∂t = 0

g=mA1/mσn/m

eq (εeq)(m−1)/m

∂g

∂σeq

=nA1/mσn/m−1

eq (εeq)(m−1)/m

∂g

∂εeq

= (m−1) A1/mσn/m

eq (εeq)−1/m

∂g

∂t = 0

σn+1 =Cn+1 :εn+1

An+1 =Cn+1

C

•εe

n+1 =εe

nσn+1 =Cn+1 :εn+1 −εn+εe

n+1

•f(σn+1)

εp

n+1 =εp

n+∂fn+1

∂σn+1

∆γn+1

f(σn+1,αn+1)=0

σn+1 =Cn+1 :εn+1 −εp

n+1

x=σn+1 ∆γn+1 

R=−εn+1 +C−1

n+1 :σn+1 +εn+1 −εe

n+∂fn+1

∂σ∆γn+1

fn+1 =R1

R2.

J="C−1

n+1 +∂2fn+1

∂σ2∆γn+1 ∂fn+1

∂σ

∂fn+1

∂σn+1 0#=J11 J12

J21 J22 

Φ=C−1

n+1 +∂2fn+1

∂σ2∆γn+1−1

An+1 =Φ−hΦ·∂fn+1

∂σi⊗hΦ·∂fn+1

∂σi

h∂fn+1

∂σ·Φ·∂fn+1

∂σi2

˙

εp=γg(σ,α)

˙

α=γh(σ,α)

γ≥0

f(σ,α)≤0

γf (σ,α)=0

•αn+1 =αnεp

n+1 =εp

nσn+1 =Cn+1 :εn+1 −εp

n+1

•f(sn+1,αn+1)

εp

n+1 =εp

n+gn+1∆γn+1

αn+1 =αn+hn+1∆γn+1

f(σn+1,αn+1)=0

σn+1 =Cn+1 :εn+1 −εp

n+1

R=



−εp

n+1 +εp

n+gn+1∆γn+1

−αn+1 +αn+hn+1∆γn+1

fn+1 

=



R1

R2

R3

.

εp

n+1 =εn+1 −C−1

n+1 :σn+1

∂σn+1

∂εp

n+1

=−Cn+1

J=





−I−∂gn+1

∂σn+1 :Cn+1∆γn+1 ∂gn+1

∂αn+1 ∆γn+1 gn+1

−∂hn+1

∂σn+1 :Cn+1∆γn+1 −I+∂hn+1

∂αn+1 ∆γn+1 hn+1

−∂fn+1

∂σn+1 :Cn+1 ∂fn+1

∂αn+1 0





=



J11 J12 J13

J21 J22 J23

J31 J32 J33 



Rεn+1,εp

n+1,αn+1,∆γn+1= 0

dεn+1

dεp

n+1

dεn+1

dαn+1

dεn+1

d∆γn+1

dR1=∂R1

∂εn+1

dεn+1 +∂R1

∂εp

n+1

dεp

n+1 +∂R1

∂αn+1

dαn+1 +∂R1

∂∆γn+1

d∆γn+1 = 0

dR2=∂R2

∂εn+1

dεn+1 +∂R2

∂εp

n+1

dεp

n+1 +∂R2

∂αn+1

dαn+1 +∂R2

∂∆γn+1

d∆γn+1 = 0

dR3=∂R3

∂εn+1

dεn+1 +∂R3

∂εp

n+1

dεp

n+1 +∂R3

∂αn+1

dαn+1 +∂R3

∂∆γn+1

d∆γn+1 = 0

0=∂R1

∂εn+1

+∂R1

∂εp

n+1

dεp

n+1

dεn+1

+∂R1

∂αn+1

dαn+1

dεn+1

+∂R1

∂∆γn+1

d∆γn+1

dεn+1

0=∂R2

∂εn+1

+∂R2

∂εp

n+1

dεp

n+1

dεn+1

+∂R2

∂αn+1

dαn+1

dεn+1

+∂R2

∂∆γn+1

d∆γn+1

dεn+1

0 = ∂R3

∂εn+1

+∂R3

∂εp

n+1

dεp

n+1

dεn+1

+∂R3

∂αn+1

dαn+1

dεn+1

+∂R3

∂∆γn+1

d∆γn+1

dεn+1

0=A

B+JKK JKE

JEK JEE  K

E

K=dεp

n+1

dεn+1 K

K=JKK −JKE J−1

EE JEK −1JKE J−1

EE B−A

A=∂R1

∂εn+1

=∂gn+1

∂σn+1

:Cn+1∆γn+1

B="∂R2

∂εn+1

∂R3

∂εn+1 #="∂hn+1

∂σn+1 :Cn+1∆γn+1

∂fn+1

∂σn+1 :Cn+1 #.

dεp

n+1

dεn+1

dσn+1

dεn+1

=C:I−dεp

n+1

dεn+1 

˙

σ=˙

σσ,q,˙

ε, T, ˙

T , t

˙

q=˙

qσ,q,˙

ε, T, ˙

T , t

R1=−σn+1 +σn+˙

σσn+1,qn+1,˙

εn+1, Tn+1 ˙

Tn+1, , tn+1∆tn+1 =0

R2=−qn+1 +qn+˙

qσn+1,qn+1,˙

εn+1, Tn+1,˙

Tn+1, tn+1∆tn+1 =0

J=Jσσ Jσq

Jqσ Jqq ="−I+∂˙

σn+1

∂σn+1 ∆tn+1 ∂˙

σn+1

∂qn+1 ∆tn+1

∂˙

qn+1

∂σn+1 ∆tn+1 −I+∂˙

qn+1

∂qn+1 ∆tn+1 #.

dR1=∂R1

∂σn+1

:dσn+1 +∂R1

∂qn+1

:dqn+1 +∂R1

∂εn+1

:d˙

εn+1 +∂R1

∂Tn+1

:dTn+1 +∂R1

∂tn+1

:dtn+1 =0

dR2=∂R2

∂σn+1

:dσn+1 +∂R2

∂qn+1

:dqn+1 +∂R2

∂εn+1

:d˙

εn+1 +∂R2

∂Tn+1

:dTn+1 +∂R2

∂tn+1

:dtn+1 =0

Jσσ :An+1 +Jσq :dqn+1

d˙

εn+1

+∂R1

∂˙

εn+1

+∂R1

∂Tn+1

:dTn+1

d˙

εn+1

+∂R1

∂˙

Tn+1

:d˙

Tn+1

d˙

εn+1

+∂R1

∂tn+1

:dtn+1

d˙

εn+1

=0

Jqσ :An+1 +Jqq :dqn+1

d˙

εn+1

+∂R2

∂˙

εn+1

+∂R2

∂Tn+1

:dTn+1

d˙

εn+1

+∂R2

∂˙

Tn+1

:d˙

Tn+1

d˙

εn+1

+∂R2

∂tn+1

:dtn+1

d˙

εn+1

=0

Jσσ :An+1 +Jσq :dqn+1

d˙

εn+1

+∂R1

∂˙

εn+1

=0

Jqσ :An+1 +Jqq :dqn+1

d˙

εn+1

+∂R2

∂˙

εn+1

=0

X=∂˙

σ

∂˙

εn+1

∆tn+1

Y=∂˙

q

∂˙

εn+1

∆tn+1

Jσσ :An+1 +Jσq :dqn+1

d˙

εn+1

+X=0

Jqσ :An+1 +Jqq :dqn+1

d˙

εn+1

+Y=0

An+1 =Jσσ −Jσq :J−1

qq :Jqσ−1:Jσq :J−1

qq :Y−X

Tn+1 =An+1

1

∆tn+1

˙

σ(σ,q,ε, T, t) = C:˙

ε−gγ˙γ−gT˙

T−gt

˙

q(σ,q,ε, T, t) = hγ˙γ+hT˙

T+ht

∂˙

σ

∂σ=C:−∂gγ

∂σ˙γ−gγ⊗∂˙γ

∂σ−∂gT

∂σ˙

T−∂gt

∂σ

∂˙

σ

∂q=C:−∂gγ

∂q˙γ−gγ⊗∂˙γ

∂q−∂gT

∂q˙

T−∂gt

∂q

∂˙

q

∂σ=∂hγ

∂σ˙γ+hγ⊗∂˙γ

∂σ+∂hT

∂σ˙

T+∂ht

∂σ

∂˙

q

∂q=∂hγ

∂q˙γ+hγ⊗∂˙γ

∂σ+∂hT

∂q˙

T+∂ht

∂q

∂˙

σ

∂˙

εn+1

=C

∂˙

q

∂˙

εn+1

=0

σep

n+1 εep

n+1,εep

n,hn,∆tn+1, Tn+1

εcr

n+1 εcr

n,σcr

n+1,∆tn+1, Tn+1

εn+1 =εep

n+1 +εcr

n+1

σcr

n+1 =σep

n+1.

R=εep

n+1+εcr

n+1 εn−εep

n,σn+1 εep

n+1,εep

n,hn,∆tn+1, Tn+1,∆tn+1, Tn+1−εn+1

εep εep

n+1

dR

dεep

n+1

=I+dεcr

n+1

dσn+1

dσn+1

dεep

n+1

dσn+1

dεep

n+1

dεcr

n+1

dσn+1

dσn+1

dεn+1

= dεcr

n+1

dσn+1

+dσn+1

dεep

n+1 −1!−1

g=kT

µb3ln ˙ε0

˙ε.

[(σ1,A1,h1),(σ2,A2,h2),...,(σn−1,An−1,hn−1),(σn,An,hn)]

[g1, g2,...gn−1].

(σ,A,h) = 









(σ1,A1,h1)g≤g1

(σi,Ai,hi)gi−1< g ≤gi

(σn,An,hn)g > gn−1.

σ0

n+1 =σ0(σn+1,hn+1,ωn+1)

ωn+1 =w(σn+1,hn+1,ωn+1)

σn+1 hn+1

ωn+1

σ0

n+1

h0

n+1 =hn+1 ωn+1 

A0

n+1 =dσ0

n+1

dεn+1

.

σ0

n+1 = (1 −ωn+1)σn+1

ωn+1 =w(σn+1,hn+1, ωn+1)

ωn+1 ∈[0,1]

A0

n+1 = (1 −ωn+1)An+1 −dωn+1

dεn+1 ⊗σn+1

σ0

n+1 = [1 −ωn+1]σn+1

˙ωn+1 = ˙ωωn+1,σ0

n+1,˙

¯εp

˙

¯εp=r2

3˙

εp:˙

εp

R1

R2=σ0

n+1 −[1 −ωn+1]σn+1 (∆εn+1)

−ωn+1 +ωn+ ˙ωn+1∆t=0

0

x=σ0

n+1 ωn+1 

˙ωωn+1,σ0

n+1,˙

¯εp= ˙ωpωn+1,σ0

n+1˙

¯εp+ ˙ωoωn+1,σ0

n+1

R1

R2=σ0

n+1 −[1 −ωn+1]σn+1 (∆εn+1)

−ωn+1 +ωn+ ˙ωp∆¯εp+ ˙ωo∆t=0

0

J11 J12

J21 J22 ="I σn+1

∂˙ωp

∂σ0

n+1 ∆¯εp+∂˙ωo

∂σ0

n+1 ∆t+ ˙ωp∂∆¯εp

∂σ0

n+1

∂˙ωp

∂ωn+1 ∆¯εp+∂˙ωo

∂ωn+1 ∆t−1#

∆¯εp

˙

εp=˙

ε−˙

εe=˙

ε−C−1:σ

˙

¯εp=r2

3˙

εp:˙

εp=r2

3˙

ε:˙

ε+C−1:σ:C−1:σ−2˙

ε:C−1:σ

σ0

n+1 = (1 −ωn+1)σn+1

˙ωn+1 =∂φn+1

∂yn+1

φn+1 =φ(ωn+1, yn+1, sn+1,˙

¯εp)

yn+1 =−σ2

eq

2E(1 −D)2"2

3(1 + ν) + 3 (1 −2ν)σm

σeq 2#

sn+1 =σeps2

3(1 + ν) + 3 (1 −2ν)σm

σeq 2

˙

¯εp=r2

3˙

εp:˙

εp