PSI - Issue 50

I.G. Emel’yanov et al. / Procedia Structural Integrity 50 (2023) 57–64 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Here δ ij – displacement in the main system of the method of structural mechanics forces in the direction i of the connection from a unit forces introduced in the direction of the discarded connection j ; Z cos θ i – displacement in the main system in the direction of the dropped i connection, resulting from a unit displacement in the direction of the introduced connection; D – operator connecting the reactive force of the i point of the surface of the base and its displacement; X i – unknown forces of interaction between the shell and the base; E – modulus of elasticity of the material,  – Poisson's ratio, R – shell radius, h – shell thickness. Equations (1) can be represented by a system of linear algebraic equations   . ~ AX B  (2) Here [ A ] are the shell stiffness matrices, X̃ are the contact pressure vector and shell displacement, B is the vector of external loads. Since the dynamic behavior of the shell is being investigated, system (2) is supplemented with terms that consider the inertial and damping properties of the system. The matrix [ A ] and vector B for the steady dynamic motion of the shell under the action of pulsating pressure are given in Vasilenko and Emelyanov (1995), Vasilenko and Emelyanov (2002), Emelyanov (2009). 3. Method for calculating the fatigue of materials In this work, the calculation of the life of the pipeline shell was performed according to the rule of linear summation of damages and to the original technique, which considers the degradation of the material, namely, the decrease the values of tensile strength during operation, Mironov (2001, 2004, 2012), Volkov (1979), Emelianov and Mironov (2012). The linear summation rule, also known as the Miner's rule, is calculated using the well-known formula where n i is the number of loading cycles at stress σ i , N i is the number of cycles to failure at the corresponding stress level σ i , and C is some constant determined from the experimental conditions. It has been experimentally established that C for metals is in the range between 0.7 and 2.2. For engineering calculations, this coefficient can be taken equal to 1, Callins (1984), Chang (2015). This linear hypothesis is not sensitive to the rearrangement of loading cycles, and also cannot take into account single overloads, which cause much more damage compared to ordinary cycles. Despite these limitations and the wide scatter of experimental parameters, the linear damage summation hypothesis continues to be widely used in engineering calculations due to its simplicity. The method of fatigue calculation proposed in the article is a model of cyclic degradation of materials, which is based on experimental studies of full stress-strain curves. The study of the degradation processes occurring in the material with the help of full stress-strain curves makes it possible to represent the static and fatigue failure of the material as an interrelated process. In the experiments carried out, it was found that with fatigue, a decrease in the plastic properties of the material is observed in the process of increasing the number of operating cycles. V.I. Mironov made a significant contribution to the development of the cyclic degradation model, Mironov (2001, 2004, 2012), Volkov (1979), Emelianov and Mironov (2012). Early works, Mironov (2001, 2004, 2012), Volkov (1979), considered the case of block loading, later a method for calculating cycle-by-cycle fatigue accumulation was proposed, Ogorelkov (2018, 2022), Mironov (2017). This method allows modeling quasi-random processes, Ogorelkov (2022), considering damped oscillations, Mironov (2017), and introducing peak overloads, Ogorelkov (2018). The main provisions of the methodology are because the fatigue process mainly accumulates in the surface layer of the sample, and therefore, in order to find the appropriate parameters for some material, it is necessary to cyclically test a certain number of special small samples with subsequent destruction of the samples in machines with increased rigidity, Mironov (2001, 2019). C n i N i n i    1 , (3)