A New Model for the Mechanochemical Corrosion of a Thin Spherical Shell

EPJ Web of Conferences 108, 0 2 0 4 0 (2016 ) DOI: 10.1051/epjconf/ 2016 10 8 0 2 0 40  C Owned by the authors, published by EDP Sciences, 2016 A New Model for the Mechanochemical Corrosion of a Thin Spherical Shell O. Sedova1 , a and Y. Pronina1 , b 1 Saint Petersburg State University, 7-9, Universitetskaya nab., St.Petersburg, 199034, Russia Abstract. A pressurized thin-walled spherical shell under the conditions of double-sided uniform corrosion is considered. The rates of corrosion are supposed to dependent linearly on the effective stress. Previous solutions for thin shells obtained by other authors reflect only the effect of differential pressure (i.e., the difference between internal and external pressure). The model proposed here allows to include also the effect of hydrostatic pressure on the durability of the shell under the corrosion conditions. 1 Introduction Corrosion is a process of material destruction that occurs as a result of chemical interaction between a material and its environment. The most common type of corrosion damages the entire surface of the material at about the same rate, and, as a consequence, results in a failure. If a structural element, immersed in an aggressive environment, is subject to mechanical stresses, then the so-called mechanochemical effect may occur (i.e. corrosion accelerated by stress) [1, 2]. Many scholars have studied the stress state of structural members under mechanochemical corrosion conditions. In the case when the corrosion is understood as a uniform dissolution, there are a few works reporting closed-formed expressions, see, e.g., [3–9]. Stress-assist chemical reactions sustained by diffusion in deformable solids were modeled in [10–13]. Initial boundary value problems with evolving boundaries have to be solved for other types of wear [14–16] and in other areas [17]. Because of the complexity of such problems, most of the research was done using numerical methods. In this paper an analytical solution is presented for a thin-walled spherical shell subject to internal and external pressure under the conditions of double-sided uniform corrosion. The rates of corrosion are supposed to dependent linearly on the effective stresses. Previous solutions for thin shells [4] do not depend on the internal and external pressures themselves, but only on their difference (see [18]). However, the hydrostatic pressure can affect the corrosion rate [1]. Accurate solutions for the doublesided corrosion of shells under internal and external pressure based on the solutions of Lamé problems for thick cylinder and sphere are rather cumbersome [19–25]. The model proposed here is simpler but, it allows to take into account the effect of the hydrostatic pressure on the durability of the thin shells under the mechanochemical corrosion. a e-mail: b e-mail:                4                   Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/201610802040 EPJ Web of Conferences 2 Problem formulation Consider a linearly elastic thin-walled sphere with the inner r and outer R radii subjected to dead internal pr and external pR pressure (figure 1). The sphere is uniformly corroding from the inside and the outside with the dissolution rates vr and vR , respectively. The corrosion process affects the shell sizes: the inner radius increases while the outer radius decreases from the initial time t0 values denoted respectively by r0 and R0 . According to [1, 3], the rates of corrosion on the inner and outer surfaces are: vr = dr = ar + mr σ(r, R, pr , pR ), dt vR = − dR = aR + mR σ(r, R, pr , pR ), dt (1) where the constants ar , aR , mr , and mR are determined experimentally [1]. In general, they are different under tension and compression; σ(r, R, pr , pR ) is the maximum principal stress assumed to be identical through the entire thickness. We suppose that at any time t the maximum principal stress σ (which is a circumferential stress) depends on the instantaneous values of the inner r(t) and outer R(t) radii of the shell and on the given constant pressures pr and pR . The problem is to find the time dependence of the maximum stress at any t > t0 . Consider a spherical coordinate system (ρ, θ, φ) with the origin identical to the center of the hollow sphere. In previous solutions obtained by other authors (for example, [4]), σ was chosen as σ(r, R, pr , pR ) = σφφ = σθθ = (pr − pR )rc , 2h (2) where rc = (R + r)/2 is the radius of the middle surface of the spherical shell, h = R − r is the thickness of the shell. Naturally, the solution derived using (2) does not depend on the pressure values pr and pR , but only on the difference pr − pR . However, it is known that the hydrostatic pressure p = min{pr , pR } could affect the corrosion kinetics [1]. To take into account the effect of the hydrostatic pressure p we suppose that the effective stress is defined by the more accurate formula [26]: σ = σ(r, R, pr , pR ) = σφφ = σθθ = (pr − pR )rc − q, 2h (3) where q = (pr + pR )/2. In both equations (2) and (3), the radius rc is assumed to remain constant, while the thickness h is changing with the time. 3 Problem solution Adding Eqs. (1)1 and (1)2 (with negative signs), we obtain d(R − r) dh = = − (a + mσ) , dt dt (4) where a = ar + aR , m = mr + mR . Differentiating (3) with respect to t and using (4) yields the ordinary differential equation: dσ 2 (a + mσ) (σ + q)2 . = dt (pr − pR )rc 02040-p.2 (5) Mathematical Modeling and Computational Physics 2015 pR pr r R Figure 1. The model of a pressurised hollow sphere The initial condition to be satisfied at t = t0 is σ|t=t0 = σ0 = (pr − pR )rc − q, 2h0 where h0 = R0 − r0 . The solution of (5), with the above initial condition is   m (pr − pR )rc (a + mσ)(σ0 + q) σ − σ0 . t = t0 + ln + 2(a − mq) a − qm (a + mσ0 )(σ + q) (σ + q)(σ0 + q) It is known that a higher hydrostatic pressure may lead to an increase in the corrosion pitting growth probability. Note that for spherically and cylindrically symmetric problems, analytical solutions can often be obtained [27–29, etc.]. In the case of a more complex geometry of the solid (especially, made of nonlinear materials), it is necessary to use numerical methods [30–32]. Some results concerning the stress distribution near corrosion pits in spherical shells are presented in [33, 34]. 4 Comparison with other solutions Let us compare calculation results obtained by the proposed model (using Eq. (3) as the effective stress) with the results based on both the classical thin shell theory formula (Eq. (2) is used as the effective stress) and the Lamé’s formula for a thick pressurised spherical shell [23, 24]. In figure 2 the dependencies of |σ| on t are demonstrated for different corrosion models of the spherical shell with the initial radi (...truncated)