13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- The incremental form for a time step from ݐ to ݐ ∆ ݐ with the temperature change from T to T+∆T in a corotational framework can be obtained as ∆ ߪ ൌ ߣ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻΔ ߝ ߜ 2 ߤ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻΔ ߝ ∆ ߝߣ ሺ௧ሻ ߜ 2∆ ߝߤ ሺ௧ሻ (8) ∆ ߝ ൌ∆ε െ∆ ߝ ௧ െ∆ ߝ ௦ ௪ (9) ∆ ߣ ൌ ߣ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻെ ߣ ሺܶ , ܺ,ݐ ሻ, ∆ ߤ ൌ ߤ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻെ ߤ ሺܶ , ܺ,ݐ ሻ (10) where ∆ ߝ represents the total strain increment and ∆ ߝ represents the elastic one, ߝ ሺ௧ሻ depicts the elastic strain at time t; ∆ ߝ ௧ and ∆ ߝ ௦ ௪ mean the thermal strain increment and swelling one as the following ∆ ߝ ௧ ൌሺlnሺ1 ߙ ሺ் ା∆் ሻሺܶ ∆ܶ െܶ ሻെlnሺ1 ߙ ሺ் ሻሺܶ െܶ ሻሻ ߜ (11) ∆ ߝ ௦ ௪ ൌଵ ଷ ߜ ሺln(1+SWሺT ∆T, t ∆t, X))- ln(1+SWሺT, t,X))) (12) Where ܶ is the reference temperature which is set as 350K. 2.2. Constitutive relationship of the cladding 2.2.1. Irradiation hardening model for the cladding material The strain-hardening curve of unirradiated Zircaloy is described as [18] σൌK ߝ ∙ ሺ ఌሶ ଵషయሻ (13) where σ is the true stress(Pa), ε is the true strain, n is the strain-hardening exponent, K is the strength coefficient and m is the strain rate sensitivity exponent. ߝ ሶ is the true strain rate. If 5 10 / s , set 5 10 / s . K ൌ 1.17628 ൈ 10ଽ ܶ ሺ4.54859 ൈ 10ହ ܶ ሺെ3.28185 ൈ 10ଷ 1.72752ܶ ሻሻ (14) n ൌ െ9.49 ൈ 10ି ଶ ܶ ሺ1.165 ൈ 10ି ଷ ܶ ሺെ1.992 ൈ 10ି 9.588ൈ10ି ଵܶ ሻሻ (15) mൌ0.02 (16) where T is the temperature in Kelvin ranged from 300 K to 730K. Due to the irradiation hardening effect, the strain hardening exponent [18] is described by further multiplying the coefficient given in Eq. (17) ݇ ଶ ൌ 1.369 0.032 ൈ 10ି ଶହ ∙ ሺ߶ ∙ ݐ ሻ (17) The strength coefficient under irradiation [18] is given by adding the value given in Eq. (14) to Eq.(18) ݇ ଷ ൌ5.54ൈ10ି ଵ଼ ∙ ሺ߶ ∙ ݐ ሻ (18) The factors in Eqs. (17-18) are both dimensionless, and ߶ ∙ ݐ denotes the fast neutron fluence (n/m2); ߶is the fast neutron flux, which is location-dependent due to the heterogeneous irradiation conditions. 2.2.2. Three-dimensional constitutive relation for the cladding The incremental constitutive relation for the cladding is similar to Eq. (8) as ∆ ߪ ൌ ߣ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻΔ ߝ ߜ 2 ߤ ሺܶ ∆ܶ , ݐ ∆ ܺ,ݐ ሻΔ ߝ ∆ ߝߣ ሺ௧ሻ ߜ 2∆ ߝߤ ሺ௧ሻ Since the large-deformation thermo-elasto-plastic behaviors are considered, the elastic strain increment can be expressed as: Δ ߝ ൌ∆ ߝ െ∆ ߝ ௧ െ∆ ߝ , (19) Where ∆ ߝ can be obtained according to the backward Euler Integration and plastic constitutive
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