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 slope of this straight line similarly to the exponent m determines the exponent α : after processing and averaging the experimental data of the first sections of the creep curves at const k σ = σ = or according to the data of the normalized curve ( ) ω τ  to the transition point to the steady-state stage by the least squares method. Dependences of a power-law or exponential form can be selected as functions ( , ) c f T σ and ( , ) c T ϕ σ [2, 13]: n B σ ; 1 exp( ) B βσ ; 2 3 (exp( ) 1) B βσ − and etc. If ( , ) n c f T B ε σ = σ , then the coefficients , B n ε are found from the experimental data at the steady-state stage of the curve ( ) c t ε at const σ = (in this case, in (12)–(14) 1 ( , ) , n A A A f T B B B + ε σ = σ = ). After taking the logarithm of the ratio c n A B ε = σ  , we get ln( ) ln( ) ln c A B n ε = + σ  . Averaging the n obtained for different const k σ = σ = , we calculate the values of the coefficients n and A B . If ( , ) g c T B ω ϕ σ = σ , then from (11) it is follows ( ) * 1 / ( 1)( 1) g t m B ω = + α + σ . Taking the logarithm of the last expression, we obtain the equation of the straight line * ln( ) ln(( 1)( 1) ) ln( ) t m B g ω = − + α + − σ for finding the coefficients , g B ω . The functions ( , ) c f T σ and ( , ) c T ϕ σ taken in a power-law form allow us to describe the deformation of materials with a monotone dependence of the ultimate dissipation work * A (strain * c ε ) on stress. Publications [25, 26] shown this using the example of alloys AK4-1 (Al–Cu–Mg–Fe–Ni) at 250 T = °C and 3V (Ti– Al–V) at 20 T = ° C, which were satisfactorily described by power functions within the approach of a single normalized damage curve. Analysis of the creep tests results shows that the function * ( ) c ε σ for some alloys may be non-monotonic, namely, in a certain stress range have a minimum (12Cr18Ni10Ti, 850°C; 15Cr1Mo1V, 565 °C) or a maximum (Ti-Al-Mn, 500°C; Al-Zn-Mg-Cu, 165ºC; Al-Mg-Mn, 165ºC; Al-Mg-Sc, 500 °C) [22, 23, 28– 31]. The energy model of the kinetic equations of creep and damage in the initial version * const A = is applicable only in a narrow range of temperatures and stresses. The non-monotonic form of the function * ( ) c ε σ also complicates the description of deformation processes using models that take into account the microstructure. Authors of [2, 22, 28, 30] discuss the possibility of a mathematical description of such materials using a phenomenological approach. Let consider a technique for finding the coefficients of kinetic creep equations with a scalar damage parameter for materials with a non-monotonic function * ( ) c ε σ . Publication [28] studied steel 12Cr18Ni10Ti at 850 T = °C. Uniaxial tensile tests were carried out on the equipment of the Institute of Mechanics of Moscow State University. The monograph by A. M. Lokoshchenko [32] gives a description of a typical IMekh-5 device used for carrying out experiments on tension and torsion at creep. The experiments were carried out on tubular samples with an outer diameter of 12 mm, a wall thickness of 0.5 mm and a working length of 70–100 mm at a constant temperature of 850 ° C. In the tests, a constant acting load was applied to the sample. The change in the cross-section during creep was assumed to be insignificant, and therefore it was assumed that at a constant load, the stress in the cross-section is constant until fracture. The original curves at 39,2; 49; 58,8; 78,4 σ = MPa are published in [33]. From 2 to 8 tests were performed for each stress. The averaged results [28] showed that at a stress of 60 σ ≈ MPa, the dependence * ( ) c ε σ has a minimum in diagrams with creep curves ( ) c t ε . Similarly to [28], we will use the functions ( ) sh( / ) A A f B c σ = σ ⋅ σ and ( ) g c B ω ϕ σ = σ to determine the coefficients in equations (12)–(14). Note that the authors [28] used a dependence of type (2) with different exponents in the denominators to approximate