PSI - Issue 48

Shanyavskiy A. et al. / Procedia Structural Integrity 48 (2023) 119–126 Shanyavskiy et al/ Structural Integrity Procedia 00 (2023) 000–000

124

6



    / 1 2 ~ 1 1        V

~



8



10

V B B

 





for σ̃ –1 < σ eq < σ –1 +Δ σ –1

1

1  

eq





 1  

5





where is the width of the bifurcation region. The expressions in triangular brackets are “markers” of the fatigue curve branch and defined as follows: ‹ f › = f H ( f ), where H ( f ) is the Heaviside function. The exponent γ varies in the range 0 < γ < 1. The development of damage in a material particle leads to an effective decrease in the elastic moduli, in the general case according to a nonlinear law, and in the proposed version of the model, according to a piecewise linear law of the following form. Material degradation at ψ < ψ * , λ ( ψ ) = λ 0 (1– κψ ), μ ( ψ ) = μ 0 (1 – κψ ). Complete fracture at ψ * ≤ ψ ≤1, λ =0, μ =0. Here, ψ * ≤ 1 is the critical value of damage function at which a state of complete fracture occurs. Narrow extended zones of complete fracture in the indicated sense will be called “quasi-cracks”. The numerical method for calculating damage zones consists in step-by-step (by loading cycles) calculation of the elastic stress-strain state of a structural element, in parallel with the numerical solution of the nonlinear equation for damage (2) and the correction of the elastic moduli of the medium in areas where the damage function is different from zero. The difference approximation of equation (2) is performed by direct integration over the interval of two discrete values of cycles N n and N n +1 , which allows obtaining an expression for damage on the upper layer in terms of the number of cycles (Nikitin et al. (2022)):           n n k k n n n n k n k B N B N                                   21 1 0.5min , 21 1 1 1 2 1 1 1 2 1 , where ψ k n +1 is the value of the damage function at the k -th spatial node on n +1 step, Δ N n = N n +1 – N n is the step by loading cycles. The relationship between the values of elastic characteristics and damage functions is taken in the following form: E k n +1 = E 0 (1– κψ k n +1 ) ( H ( ψ * – ψ k n +1 ) +0.001), where E k n +1 is the value of the elastic modulus at the new step, E 0 is the Young's modulus of the undamaged material, κ is the degradation coefficient of the modulus, which is established in the course of computational experiments. Based on the proposed method, mathematical modeling of the process of initiation and development of a fatigue quasi-crack in a rapidly rotating gear transmission was carried out at the contact interaction of the teeth of a gear Z 4 . Gears material is steel grade 20Kh3MVF. The steel density is equal to ρ = 7800 kg/m 3 , Young's modulus is equal to E = 207 GPa. As already mentioned, the surface layer is nitrocarburized in depth of 1.2 mm and has a Rockwell hardness 59HRC. The hardness of the bulk material lies in the range of (34-41)HRC (Shanyavskiy et al. (2022)), for definiteness we choose the average value of 37. The reference ultimate tensile strength and fatigue limit of gear bulk material are σ U = 1010MPa, σ –1 = 430 MPa. The characteristic value of the fatigue limit of the VHCF is, as a rule, 20-30% lower than the classical fatigue limit; therefore, we take the value σ̃ –1 = 310 MPa. Since the strength characteristics of materials are strongly correlated with their Rockwell hardness, to determine the strength characteristics of a nitrocarburized surface layer, we multiply the standard values for this steel by a factor of 59/37~1.6. Then, for the characteristics of the nitrocarburized surface layer to a depth of 1.2 mm, the following values can be taken: σ U = 1610MPa, σ –1 = 690MPa, σ̃ –1 = 490 MPa. In the calculation scheme, one gear is rigidly fixed on the shaft, a torque of 420 N∙m is applied to the second gear from the side of the shaft. Cyclic loading with a rotational speed of 14,150 rpm or frequency of 236 Hz is considered. Elastic calculations of the loading cycle were performed using the ANSYS software package, supplemented with a code for calculating the kinetics of fatigue damage in accordance with the algorithm described above.      U L 4.2. Numerical method for solving the equation for damage         4.3. Development of a fatigue quasi-crack in a gear 1 10 