# PSI - Issue 2_A

Petteri Kauppila et al. / Procedia Structural Integrity 2 (2016) 887–894 P. Kauppila et al. / Structural Integrity Procedia 00 (2016) 000–000

890

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3. Specific model

3.1. Potential functions

The reversible prosesses are described by the specific Helmholtz free energy ψ = ψ ( T , ε te , ω ) , which depends on absolute temperature T , thermoelastic strains ε te , and integrity ω . In this study, the following form of the specific Helmholtz free energy is chosen ρψ = ρ c ε T − T ln T T r + 1 2 ( ε te − ε th ) : ω C e : ( ε te − ε th ) , (17) where ε th = α ( T − T r ), is the thermal strain, C e the elasticity tensor, α a second order tensor containing the thermal expansion coe ffi cients and T r is an arbitrary reference temperature. For an isotropic solid the elasticity and the thermal expansion tensor have the forms where I and I are the second- and fourth order identity tensors, respectively. All the material coe ffi cients, the Young’s modulus E , the Poisson’s ratio ν and the linear coe ffi cient of thermal expansion α can depend on temperature. The complementary dissipation potential can be additively decomposed into thermal, damaging and viscoplastic parts as ϕ ( Y , q , σ ; T , ω ) = ϕ th ( q ; T ) + ϕ d ( Y ; T , ω ) + ϕ c ( σ ; T , ω ) , (19) where the thermal part is C e = ν E (1 + ν )(1 − 2 ν ) I ⊗ I + E 1 + ν I , and α = α I , (18) For an isotropic solid the thermal conductivity tensor λ is simply λ = λ I , where the thermal conductivity λ can depend on temperature. Damage a ff ects also to the thermal conductivity, and thus the coe ffi cient of thermal conductivity can also depend on the integrity. Here this e ff ect is neglected. For creep the following Norton type potential function is adopted (21) where ¯ σ = √ 3 J 2 is the von Mises e ff ective stress ( J 2 is the second invariant of the deviatoric stress), t c is a charac teristic time for creep and it is directly related to the relaxation time, h c Arrhenius-type thermal activation function h c ( T ) = exp( − Q c / RT ), where Q c is the creep activation energy and R is the universal gas constant. Choice of the reference stress σ rc (also known as a drag stress) is discussed later. From the creep tests of many metals and alloys it is concluded that the product of minimum creep strain-rate and rupture time is a constant (Riedel, 1987; Nabarro and Villers, 1995), i.e. ˙ ε c min t rup = constant . (22) This constant is almost independent of temperature and stress and it is known as the Monkman-Grant parameter. It is often expressend in a modified form C MG = ( ˙ ε c min ) m t rup , (23) where m is in the range 0 . 8 − 1. The value of the Monkman-Grant parameter equals roughly to the rupture strain C MG ε rup . (24) ϕ c ( σ ; T , ω ) = h c ( T ) p + 1 ωσ rc t c ¯ σ ωσ rc p + 1 , 3.2. Monkman-Grant hypothesis ϕ th ( q ; T ) = 1 2 T − 1 q · λ − 1 q . (20)

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