•T ••−1 •δ• •δ• •

• •• •

asbbwdeffcfctftfykfE Egfhmx,my,mxy

nnE

ni

nN

nx, ny, nxy

ppFrssbfsbmax

tttx, ty, txyuvx, vyww

wcr

z

c

υ

c

AAAs

At

Au

CCT

DDT

DcT

DLT

EE0

Ec

Es

ET

FFE EGGGf

I1JJ2, J3Lc

Le

MNPTVV

T

T

1, 2, 31, 2, 3

e

T

Te

e

n

α, β

ααE , αR

αβββtεεε1, ε2, ε3εctεcuεc1εcu1εIεVφφϕϕϕc

γγE , γRκκp

κdμE Eννθθϑρρs�sσσ1, σ2, σ3σE Eτττbfτbmax

ωξ

εεεpκσσσ′

Φ

Σ

= · υ

υυ n

ux, uy υ

ε = · υ

ευ

nυ

ε σ

σ = f(ε).

t

ε =∂ε

∂t, σ =

∂σ

∂t

ε σ

∫VδεT · σ V =

∫Vδ T · V +

∫At

δ T · A

VA At

A δδε δ T , δεT

σ, ,

σ

σ ,δ

δε

δ Au A

δ = · δυ, δε = · δυ

δυT ·[∫

V

T · σ V

]= δυT ·

[∫V

T · V +

∫At

T · A

]

δυT , T , T δυ , δυ

=

=

∫V

T · σ V

=

∫V

T · V +

∫At

T · A

υ , n

σ = (ε) ε = ·υ nυ

σ = · ε

= · υ, =

∫V

T · · V

· υ = .

υ = −1 · .

σ ευ

= ( x y z )T

I

= ( r s t )T

e

= ( u v w )T

ϕ = ( ϕx ϕy ϕz )T

Ec

fc εc1εcu1

σε

σ =Ec ε

1 +(

EcEc1

− 2)

εεc1

+(

εεc1

)2 , 0 ≥ ε ≥ εcu1

Ec1 = −fc/εc1 fcσ = −fc ε = εc1

Et =∂σ

∂ε=

Ec

(1− ε2

ε2c1

)(1 +

(EcEc1

− 2)

εεc1

+ ε2

ε2c1

)2Et = Ec ε = 0 Et = 0 ε = εc1 ET < 0

ε < εc1, |ε| > |εc1|

σ =

{Ec ε, ε ≤ fct/Ec

0,

fctET = Ec ε ≤ fct/Ec

[ ] [ −1]

[ −3][ −6]

B

A

bw

L = 0.5 Ac = 0.1×0.1

Le = 0.001

Ec = 36 000 /fct = 3.5 /

RR = 0.03

uN = 0.1333 · 10−3

ε = 0.2667 · 10−3

AA

B B

εc(x)x1 ≤ x ≤ x2 bw = x2 − x1

bw

w =

∫ x2

x1

εc(x) x

w = bw εc

εc εc(x) εc

εct fct εcrσc = 0 εc ≥ εcr εc = εct

gf =

∫ εcr

εct

σ(ε′) ε′

σ(ε′)

Gf = bw gf

Gf

σ − εc

σ0 = 3.0 /

ε0 =σ0/E0 = 0.1

t = 100

ε

Es

fyk ftεu ft > fyk

εy =fykEs

, ET =ft − fykεu − εy

fy−fy

σ =

{Es (ε− εp) εp − fy

Es≤ ε ≤ εp +

fyEs

ε fy

fy εp

εp = ε

fy = ET |ε|}

ε ε > 0 |σ| = fy

ET 2

σ = 0 |ε| > εu

ET = 0 T

Te

σ, ε

T

T =Fs

x=

Fc

x

Fs Fc

s

s

T s

T = fT (s)

Uτ = T/U

τ = fτ (s)

τmax

τf

τbmax, τbfsbmax, sbf

s, τ < 0

(s′, τ ′)τ − s

(−s′,−τ ′)

u(x)

ε =1

Le[(Le − bw) εu + bw εc]

= (1− ξ) εu + ξ εc, ξ =

bwLe

Le εubw εc

Le

bw ξ0 < ξ ≤ 1

Le

bw ≤ Le

εu, εc

σ = fc(εc)

εc = f−1c (σ)

σ = fu(εu)

εu = f−1u (σ)

σ

ε = (1− ξ) f−1u (σ) + ξ f−1

c (σ) = d(σ)

d

σ = d−1(ε)

σ = fu(εu) = E εu

E

σ = fc(εc) =

⎧⎨⎩ fct

(1− εc − εct

εcr − εct

), εct < εc ≤ εcr

0, εcr < εc

εct = fct/E fct εcrGf = 1

2bwfct(εcr − εct) εcr = 2Gf/(bwfct)+ εctGf

fctE < ε ≤ εcr

εu, εc

εu =σ

E, εc = (εcr − εct)

(1− σ

fct

)+ εct

σ = fctξ εcr − ε

ξ εcr − εct, εct ≤ ε ≤ ξ εcr

ξ = 1 σ = fct (εcr− ε)/(εcr− εct)

ξ < 1 ξ → εct/εcr σ ξ

Gf = 0 ε > εct

σ = 0 εc > ξ εcr

x u

x

fct

σ =

{Ec ε, ε ≤ fct

Ec

0,

Ec

τbmax τbfsbmax, sbf

Δu sF T Le

,σ

T

ni = 0

L = 1.0 Ac = 0.1 × 0.11 � 16 As = 2.01

Us = 5.02

Le = 0.01 bw =0.01

Gf bw = Le

w = Le ε

bw = Le εcε

Ec /fct /

Es / 200 000fsy /

τbmax /sbmax

τbf /sbf

τmax ≈ 1.8 fct

uN = 2.40 ≤ t ≤ 1

εmean = 2.4

t = 0.5, uN = 1.2 t = 1.0, uN = 2.4

u x

w ≈ 0.6

y [ ]