PSI - Issue 6

Irina Stareva et al. / Procedia Structural Integrity 6 (2017) 48–55 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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One of the widely used structural elements is cylindrical tube (Isotov et al. (2001) and (2004), Kolpak and Ivanov (2015), Pavlovsky et al. (2015)). V.M. Dolinskii (1967) was the first to consider the mechanochemical corrosion of a steel tube under longitudinal tension using the linear corrosion kinetics equation. Stability of a thin-walled tube subjected to longitudinal compressive force and external corrosion with the rate exponentially dependent on stress was investigated by Bergman et al. (2006). The comparison of solutions for a stretched pipe using the linear, quadratic, and cubic relationships between stress and corrosion was given by Elishakoff et al. (2012). The optimal design of a compresses column with a circular cross-section exposed to corrosive environment was discussed by Fridman (2014), when the exponential corrosion model introduced by E.M. Gutman and R.S. Zaynullin was adopted for the analysis. The analysis revealed that in some situations the use of Dolinskii’s linear stress corrosion model allowed one to get a su ffi ciently accurate solution. The papers by Fridman and Elishakoff (2013) and (2015) were devoted to the optimal design of a compressed bar and a tube utilizing Dolinskii’s corrosion model. The same model was used to the assessment of the lifetime of a tube under the combined action of longitudinal force and pressure by Pronina (2010). In all the mentioned studies a longitudinal force was assumed to be constant over time. However, the longitudinal stress was changing due to the gradual thinning of the cross-section in the process of dissolution. Herein, the situation is considered when the longitudinal force is changing along the tube axis and in time. To be more precise, a vertically standing or hanging long initially cylindrical tube is considered subjected to mechanochemical corrosion under its own weight using Dolinskii’s linear corrosion model. Of course, at the initial moment of time when the cross-section of the tube is constant along the tube axis, the corrosion rate is linearly dependent on the vertical coordinate of the tube. Since different cross-sections of the tube are dissolved with different rates, the stresses at more strained cross-sections are also intensified by the reduction in their area. Therefore, the corrosion rate becomes strictly nonlinearly depending on the vertical coordinate and accelerating with time. The highest acceleration of the corrosion rate is at the most strained cross-section. It is clear that the own weight of the tube gives a rather small increase in the corrosion rate for relatively short tubes. The following questions arise. At what length of the tube do we need to take into account its own weight for the life assessment? Is there any simple approach to this consideration? These questions are investigated in the present paper. A linearly elastic vertically standing or hanging steel tube loaded with its own weight is considered. The tube is subjected to inside and outside mechanochemical corrosion (i.e. general dissolution) with the rates r v and R v , respectively, so the inner radius r of the tube increases with time t , while the outer radius R decreases (non-uniformly along the tube). Let the inner and outer radii of the tube at the initial time 0 0  t be denoted by 0 r and 0 R . The length of the tube is denoted by l . The rates of corrosion on the inner and outer surfaces are supposed to be linearly dependent on mechanical stress (see Dolinskii (1967)): , r r r r a m dt dr v     (1) , R R R R a m dt v dR      (2) Here, r m , R m , r a , and R a are experimentally determined constants, which in general are di ff erent for tension and compression; r  and R  are the maximum (in absolute value) principal stresses on the relevant surface of the tube, and, according to Pavlov et al (1987), r r m  sign sign  and R R m  sign sign  . It is required to determine the stress in the tube, its thickness for 0  t (both changing along the tube axis and with time), and to assess the lifetime of the tube. The vertically standing tube is supposed to be supported to avoid a buckling due to own weight; therefore the loss of stability is not taken into account. 2. Problem formulation