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 K. S. Bormotin theoretically and numerically substantiate the use of kinematic modes with a constant creep strain rate within the framework of the energy approach * const A = [14–16]. However, structural alloys can be described by an energy variant of the creep theory * const A = in a fairly narrow range of rates and temperatures. In [17], in order to assess the residual operational life, two deformation modes under uniaxial tension conditions were studied for alloys having a monotonic dependence of the ultimate strain (fracture strain) on stress (AK4-1 (Al–Cu–Mg–Fe–Ni), 250°C; D16T (Al–Mg–Cu), 250°C; VT9 (Ti–Al–Mo–Zr), 600°C; steel 09G2S-12 (Fe–Si–Cu–Cr–Ni–C), 730°C; 3V (Ti–Al–V), 20°C). The deformation modes at constant stresses and at constant strain rates corresponding to these stresses were compared. It is shown analytically and numerically that if the dependence on creep diagrams * ( ) c ε σ decreases monotonically with increasing σ , then the accumulation of damage is less in kinematic modes n / =const c d dt B ε η = ε = σ . Such materials include alloys that are described by the energy approach of the creep theory * const A = and the condition g n ≥ is satisfied. For alloys, in which dependence * ( ) c ε σ monotonically increases in diagrams with creep curves ( ) c t ε , the accumulation of damage is less in the mode const σ = . The purpose of this work is to carry out a comparative analysis of two deformation modes of tensile rods for an alloy with a non-monotonic dependence of the ultimate strain using the kinetic equations of creep and damage: static const σ = and kinematic n =const B ε η = σ . T heory and methods Constitutive relations of creep and damage Equations (1) in [13] are defined as 1 ( , ) (1 ) A f T dA dt q κ σ = − , 2 ( , ) (1 ) c Ò dq dt q κ Φ σ = − , (0 1) q ≤ ≤ . (2) Here 0 0 c t c A ij ij A W dt d ε = = σ ε ∫ ∫ . Replacing q by ( ) 2 1 1/( 1) 1 (1 ) κ −κ + − − ω , the relations (2) can be reduced to the following [13] ( , ) (1 ) A m f T dA dt σ = − ω , ( , ) (1 ) c m T d dt ϕ σ ω = − ω , (3) thus eliminating the arbitrariness in determining the coefficients of the constitutive relations. Under conditions of a uniaxial state, the parameter (0 1) ω ≤ ω ≤ must satisfy the equation of a single normalized curve 1 (1 ) (1 ) m + − ω = − τ  , (4) where 2 0 ( 1) ( , ) t c T dt τ = κ + Φ σ ∫  or 0 ( 1) ( , ) t c m T dt τ = + ϕ σ ∫  is the normalized time. Integrating (4), for we obtain 1 1 1 (1 ( 1) ( , ) ) t m c o m T dt + ω = − − + ϕ σ ∫ . (5)