Kinetic equations of creep and damage for description of materials with non-monotonic dependence of fracture strain on stress

OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 23 No. 3 2021 For const σ = from (5) we have 1 1 ( , ) 1 [1 ( 1) ( , ) ] , ( , ) A m c c f T m T t A T + σ ω = − − + ϕ σ = ω ϕ σ , (6) * / , ( , ) / ( , ) A c A A A f T T ∗ ω = = σ ϕ σ , (7) * * 1 / , t ( 1) ( ,T) c t t m τ = = + ϕ σ  . (8) Here * t is the time of fracture. If stress const σ = , then damage parameter is * / / c c A A ∗ ω = = ε ε and the verification of equations (3) should be carried out in the normalized values * * * / / , t/t c c A A ω = = ε ε τ =  . If the material, in addition to the stages of steady-state creep and softening, has a hardening stage, then (3) can be rewritten as [13] 1 ( , ) (1 ) A A m f T dA W dt α α+ σ = = ω − ω , 1 ( , ) (1 ) c m T d dt α α+ ϕ σ ω = ω − ω , (9) where α is the hardening parameter. In this case, in normalized values * * c c A A ω = ≡ ε ε , * t t τ =  , equation of a single normalized curve must also be fulfilled for ω in the form ( 1) 1 (1 ) 1 m α+ + − ω = − τ  . (10) Integrating (9) with const σ = instead of (6) we obtain ( ) 1 1 1 1 * ( , ) 1 1 ( 1)( 1) ( , ) , , ( , ) ( , ) ( , ) , . ( , ) ( , ) A m c c c c A A c c f T m T t A T f T f T T T +α +   σ ω = − − + α + ϕ σ = ω     ϕ σ   σ σ ε = ω ε = σ ⋅ ϕ σ σ ⋅ ϕ σ (11) It follows from the analysis of (11) that in the uniaxial case the parameter characterizes the deformability of the material, i.e. * c c ω = ε ε – reduced deformation, and the dependence * c ε on stress σ can be arbitrary. In the case of a complex stress state, equations (9) can be generalized [13, 18]: 1 ( , ) (1 ) A e A m f T dA W dt α α+ σ = = ω − ω , t ij o , c ij A dt = σ ε ∫  (12) * 1 ( , ) (1 ) c e m T d dt α α+ ϕ σ ω = ω − ω , 0 1 ≤ ω ≤ , (13) , c ij e A ij ij e d W dt ε ∂σ η = = λ λ = ∂σ σ . (14) Here * e e σ , σ are the equivalent stresses. The stress e σ .can be taken, for example, the stress intensity according to Mises 1/2 (3 / 2) e i ij ij σ = σ = σ σ , ij σ – the components of the stress deviator. The choice of an equivalent stress (the criterion of long-term strength), as already noted, allows us to take into account the anisotropic nature of damage accumulation for various stress states. The analysis of the criteria for long-term creep strength is given in [19–21].