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 the experimental data. We assume that there is no hardening, i.e. 0 α = . It follows from (11) that the fracture strain is * sh( / ) ( ) c A g B c B ω σ ε σ = ⋅ σ . (15) Differentiating twice * ( ) c ε σ , we get 2 * 2 1 2 ( 1) 2 sh ch c A g d B g g g c c c d B c + ω   ε  σ +  σ σ     = + −         σ     σ σ     . (16) Let transform the expression in the square brackets of the right side of (16). We decompose the hyperbolic sine and cosine into a Taylor series and group the coefficients at the same degrees, omitting the multiplier before the bracket: 1 : ( 1) / g g c − ; 2 σ : 3 1 ( 1) 2 1 3 ! 2 ! g g g c +   + −     4 σ : 5 1 1 ( 1) 2 3 ! 5 ! 4 ! g g g c +   + −     ……. 2 k σ : 2 1 1 1 ( 1) 2 (2 1) ! (2 1) ! (2 ) ! k g g g k k k c +   + + −   − +   ( N k ∈ ) The obtained coefficients at 2 k σ can be generalized in a following form 2 2 2 1 1 ( 2 ) 2 (2 1) ! k k a g k g k c k +   = − − +   + , 0,1, ... , k N = . (17) As a rule, the coefficient 2 g > . The coefficients 2 k a are always greater than zero, except for the case when the parameter g is in the range 2 2 1 k g k < < + . In other words, all coefficients 2 k a are positive, except for one. However, the contribution of this negative term to the total sum in the required range of stresses and values of parameter c is small compared to the rest of the terms, and we can assume that 2 * 2 0 c d d ε ≥ σ . Note that if g turns out to be in the range 2 1 2 k g k − < < , then the expression in square brackets (17) is always greater than zero and all the terms of the series are positive. From the condition * 0 c d d ε = σ we get the equation cth( / ) c g c σ ⋅ σ = ⋅ , solving which we find the minimum. Taking the logarithm of the expression for the creep strain rate at the steady-state stage sh( / ) A B c η = σ , we obtain the equation of the straight line for finding the parameters A B and c : ln( ) / ln( /2) A c B η ≈ σ + . From (8) (or (11) at 0 α = ) follows ( ) * 1 / ( 1) g t m B ω = + σ . (18)