PSI - Issue 2_A

Petteri Kauppila et al. / Procedia Structural Integrity 2 (2016) 887–894

891

P. Kauppila et al. / Structural Integrity Procedia 00 (2016) 000–000

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3.3. Damage potentials

In this study two variants for the damage potential are compared. Both of them are of Kachanov-Rabotnov type (Kachanov, 1958, 1986):

k

Y Y r

r + 1

Y r t d ω

h d ( T ) r + 1

(model 1)

(25)

ϕ d ( Y ; T , ω ) =

,

k

Y Y r

1 2 p + 1

h c ( T )

Y r t d ω

ϕ d ( Y ; T , ω ) =

(model 2)

(26)

,

( 1 2 p + 1)(1 + k + p )

where t d is a characteristic time for damage evolution, h d is an Arrhenius-type thermal activation function for damage processes h d ( T ) = exp( − Q d / RT ), where Q d is the damage activation energy and R is the universal gas constant as in the creep activation function. The reference value of the thermodynamic force, Y r , is chosen by Y r = σ rd 2 / (2 E ), where σ rd is a reference stress for the damage process. The first version (25) results in a more general model, whereas the second one (26) is restricted to satisfy the Monkman-Grant hypothesis (with m = 1) exactly and therefore its parameters are coupled to the creep model. For these two models the Monkman-Grant parameter have the values

t d h c t c h d

σ σ r

p − 2 r

1 1 + k + 2 r

t d t c

C MG = ˙ ε c

min t rup =

(model 1) and C MG =

(model 2) .

(27)

To account for di ff erent damage evolution in tensile and compressive regions the damage potentials could be changed to the form ϕ d ( Y ; T , ω, ε e ) ∝ ( Y / Y r + ξ tr ε e ) r + 1 , (28) where ξ is an extra material parameter. Due to the lack of material data available to the authors this form has not been used.

3.4. Constitutive equations

From the general constitutive equations (14) and with the specific choices (17), (20) and (21) the following consti tutive equations are obtained

h c t c

¯ σ ωσ rc

p ∂ ¯ σ ∂ σ

σ = ω C e : ε e , ˙ ε c =

q = − λ grad T .

(29)

,

The integrity rates resulting from the considered two models (25) and (26) are

k

Y Y r

k

Y Y r

1 2 p

r

h d t d ω

h c (1 + k + p ) τ d ω

˙ ω = −

(model 1) and ˙ ω = −

(model 2) .

(30)

,

4. Response in uniaxial creep test

The material parameters can be determined from the uniaxial creep tests at di ff erent temperatures and under several constant stress values. In this simplified loading case, the constitutive equations can be integrated in a closed form, the resulting integrity development and the creep rupture time for the model (25) are ω =   1 − (1 + k + 2 r ) h d σ σ r 2 r t t d   1 / (1 + k + 2 r ) , t rup = 1 (1 + k + 2 r ) h d σ σ r − 2 r t d . (31) The yield stress is selected as the reference value in the creep and damage evolution equations, i.e. σ rd = σ rc = σ y0 ( T ) ≡ σ r . For the total strain, the following formula is obtained

p − 2 r 

(1 + k + 2 r − p ) / (1 + k + 2 r )   ,

 1 − 1 − (1 + k + 2 r ) h d σ σ r

h c h d

σ σ r

t d

p t

1 1 + k + 2 r − p ·

σ E +

(32)

ε =