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 Entering normalized values * * * / / , t/t c c A A ω = = ε ε τ =  in the analysis of experimental creep curves makes it possible to determine the parameter ω through the values * * , , , c c t t ε ε measured in the experiment, while it remains in no way related to the microstructure of the material (with changes in the density of pores, dislocations, vacancies). The fracture in the experiment is understood as the separation of the sample into parts. The geometric similarity of the curves (10) at constant stresses in the normalized values was obtained for a number of alloys [12, 13, 22–24]. Publications [25, 26] demonstrate the possibility of using equations (12)–(14) to describe materials with a monotonic dependence * ( ) c ε σ in creep diagrams ( ) c t ε . In [25] this is shown by the example of torsion of rods made of an alloy without the first creep stage ( 0 α = ) AK4-1 (Al-Cu-Mg-Fe-Ni) at 250 T = ° С, while the value of the ultimate strain intensity * ( ) c i i ε σ increases mono- tonically. Publication [26] studied a titanium alloy 3B (Ti–Al–V) at 20 T = ° C, which has all three pro- nounced creep stages. Equations (12)–(14) in the variant * const A = describe it, while the ultimate strain intensity, on the contrary, monotonically decreases with increasing of i σ . The experimental data of both alloys are densely located near the “single curve”. The possibility of such grouping into a normalized curve of test data with a non-monotonic dependence * ( ) c i i ε σ is discussed in [22, 23], but the procedure for obtain- ing the parameters of equations (12)–(14) is not given. Method for determining the parameters of kinetic equations Publications [12, 24, 27] discuss the methods for finding the coefficients of kinetic equations (12)– (14). As a rule, the investigated alloys have a monotonic dependence * ( ) c ε σ on the experimental diagrams. If the dependence is non-monotonic, then in these works it is usually averaged and assumed to be monotonic. The exponent m describes softening and is determined by the third section of a single normalized creep curve: if the material has a hardening stage, then after the inflection point located on steady-state section; if the first stage is absent, then only along the last section. If 0 α = , then after taking the logarithm (4) we have ( ) ( ) ( 1) ln 1 ln 1 m + − ω = − τ  . This relation is the equation of a straight line in logarithmic coordinates ( ) ( ) ln 1 ln 1 − ω − − τ  . Its slope determines the exponent m . Here * * / ; / ; c c k k k k t t ω = ε ε τ =  index k means the number of the creep curve const k σ = σ = ; c k ε , k t are the creep strain and time at the point of transition to the third stage; * c k ε , * k t are the ultimate strain and fracture time of the sample. The parameter m is found by averaging its values obtained at different values k σ = σ . If 0 α = , then the parameter m in (4) can be found by the least squares method applied to experimental points in normalized coordinates. Publication [27] demonstrates the obtaining of the exponent α on the basis of experimental data on ten- sion and compression using the dependence of the type of the strain hardening ( ( , ) c c f T −α ε = ε σ  ). When found α in (9), it is accepted ( , ) / c d T dt α ω = ϕ σ ω . Integrating this equation from 0 ω = to the current values ω и t , after subsequent logarithm, we come to the equation of the straight line ( ) ( ) ( 1) ln ln[( 1) ( , )] ln 1 c T α + ω = α + ϕ σ + − τ  .