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 After taking the logarithm (18), the equation of the straight line for finding , g B ω is rewritten as: * ln( ) ln(( 1) ) ln( ) t m B g ω = − + − σ . Modes of deformation under tension of rods Let’s consider the process of damage accumulation for two modes of deformation under tension of rods made of an alloy with a non-monotonic dependence * ( ) c ε σ on the example of steel 12Cr18Ni10Ti. Elastic strains are neglected. In view of the fact that in (12)–(14) ( ) sh( / ) A A f B c σ = σ ⋅ σ and ( ) g c B ω ϕ σ = σ , the expression for the creep strain rate is written as: 1 sh( / ) (1 ) c A m B c d dt α α+ σ ε = ω − ω . (19) In the case of mode 1 ( 0 const i σ = σ = ), it follows from (11) that ( ) ( ) ( ) 1/(1 ) 1/( 1) 0 1 1 1 1 m g m B t +α + ω   ω = − − + α + σ     and 0 0 sh( / ) ( ) ( ) c A g B c t t B ω σ ε = ω σ . (20) In the case of mode 2 ( 0 0 sh( / ) const A B c η = σ = ), it follows from (12), (14) that ( ) 1 0 sh( / ) 1 / m A c B α+ α σ = − ω ω η and 2 ln 1 n n c F F   σ = + +     , (21) where ( ) ( ) 1 1 0 0 ( ) 1 / 1 sh( / ) m m n A F B c α+ α α+ α ω = − ω ω η = − ω ω σ . Substituting the expression for σ in (13), we obtain the equation for finding of ω : 2 1 ln ( ) ( ( )) 1 (1 ) g n n m B d c F F dt ω α α+ ω     = ω + ω +         ω − ω . (22) To numerically solve (21), (22), one can use, for example, the Runge-Kutta method. The number of deformation modes can be considered much more. In [34], in relation to the problems of shaping a hemispherical shell from a flat workpiece, the modes of deformation under the action of constant pressure, linearly increasing pressure, or when the law of variation of the deflection in time is specified is investigated. Results and discussion Determination of the parameters of steel 12Cr18Ni10Ti Experimental data for steel 12Cr18Ni10Ti show that the fracture strain * c ε in the stresses range from 40 MPa to 80 MPa first decreases monotonically with increasing stress, and at 0 60 σ ≈ MPa begins to increase monotonically. In Fig. 1, a points 1–4 show the experimental dependences ( ) c t ε corresponding to