Random processes in modeling durability of building and structure

Abstract

The problem of buildings and structures durability during their long-term operation in aggressive environments is discussed. It is closely connected with the problem of destruction of composite materials as a random process of birth, development, and death of defects at chemical corrosion. The content of the study is to analyze the influence of non-stationary chemical reactions occurring in real conditions on these processes. The analysis is carried out taking into account the incidental physical phenomena affecting corrosion, the most important of which is the diffusion of initial substances and reaction products occurring under different hydrodynamic conditions, i.e., the nature of fluid motion and its pressure which stimulates corrosion. The synthesis of three processes – chemical kinetics, diffusion, and hydrodynamics – allows us to study one of the possible scenarios of corrosion process development as a Markov process based on the statistical theory of the “weakest” link and the joint distribution of the random material and geometric parameters. Numerical analysis of changes in elastic and electrical properties of composite materials under these conditions allows us to substantiate the technology of non-destructive control of changes in the state of the structure. The research results are based on mathematical and statistical modeling.

1. Introduction

The importance of the durability problem of structures using composite materials, in particular, concrete, is beyond doubt. Durability is defined as the ability of material to retain the specified physical and mechanical properties for a long period of time in unfavorable conditions of external influences. This ability decreases abruptly in the form of a random response of the system to a smooth change in external conditions. The modern approach to solving the problem of durability assumes that by creating an ideal composition at the initial stage of the project it is possible to ensure the prolonged existence of the structure before its destruction. However, the conditions in which the structure is living can differ from those in which the design composition is tested. Moreover, after a period of time, the material properties change to such an extent that they have little or nothing common with those originally designed. The obvious difference is that a specimen subjected to strength testing at the design stage under conditions of constant temperature and unilateral compression is virtually free of macroscopic defects and is not exposed to corrosive fluids. All these effects occur quite slowly and simultaneously, and their role is difficult to assess through physical experiments. On the other hand, under conditions of long-term operation of a structure, its elements are subjected to all-round compression by stresses, including those applied at infinitely remote boundaries. Estimation of elastic energy loss on specimens is performed under unilateral loading and on specimens of finite dimensions in the absence of sharp temperature fluctuations, frequent moistening and drying, the impact of chemically active air and water environments, leading to dissolution and leaching of individual components under the action of running water, sulfate, chlorine, or alkaline corrosion. At each stage of chemical corrosion there is a simultaneous occurrence of mass transfer processes – diffusion of solvent molecules and chemical reactions, leading, in the end, to changes in the physical properties of the used composite material, including loss of its strength and destruction of the whole structure, thus, to a catastrophe, often resulting in human casualties and economic losses.

The objective of the current research is to prevent disasters by continuously monitoring and predicting the state of the object. This goal can be achieved by solving the following problems, partially discussed in (Vilge, 2016):

a)

solving the problem of composite material durability in order to optimize their designed properties under conditions of physical and chemical corrosion requires a set of experimental studies to determine for each type of material the following parameters: the diffusion coefficient of the liquid “giving birth” to a defect in the material, the rate constant of dissolution of a particular composite material, the rate of chemical reactions in it, and the rate and chemical composition of groundwater;

b)

development of the technology of continuous control (monitoring) of the composite material state and determination of the measured parameters.

c)

development of special methods of processing and interpretation of measurement results related to the random nature of the measured parameters since it is practically impossible to carry out direct assessments of the state of materials.

The study of the effect of chemical corrosion on the durability of composite materials requires the development of mathematical and statistical models which take into account the complex physical and chemical processes occurring in the material and the environment. Our research is based on the statistical theory of the “weakest element” (Freudenthal, 1969) and the assumption that the corrosion is a Markov process which describes a sequence of random changes in the state of the system, independent of the previous ones, over time. Each next state of the system depends only on the state at the current point in time and is independent of how the system arrived at the last state. This approach was justified and used in (Landau & Lifshitz, 1987; Liebowitz, 1969). It is important to note that not only corrosion processes are nonstationary, since chemical reactions and hydrodynamics vary in time, but also physical parameters, such as diffusion coefficients and rate constants of chemical reactions, are functions of time and randomly distributed in space. The latter circumstance requires clarification of the concept of the “weakest” link. This is important because in the process of monitoring the system state in real conditions all measured parameters are indirect, and the very definition of critical strength of an element as the “weakest” (Freudenthal, 1969) link needs to be clarified. The analysis of the listed processes is carried out by methods of mathematical and statistical modeling. They have no alternative in the study of long-term processes and are realized on a simplified but relevant model of a random macroscopic element of the medium containing an elementary, macroscopic defect. The adopted model (Freudenthal, 1969) transforms a homogeneous medium with unknown homogeneous properties (strength) into a medium with known inhomogeneous properties by means of assumptions formulated in (Landau & Lipshitz, 1987; Liebowitz, 1969; Polyanin & Zielinski, 2016). This goal is achieved by analytical and numerical analysis of boundary value problems for the model of the medium describing the nature of the mutual influence of diffusion rates and homogeneous chemical reaction under conditions of mass transfer across the interface. The transport of matter in a moving fluid is caused by two quite different mechanisms. Firstly, in the presence of concentration difference in the liquid there is molecular diffusion, and secondly, particles of dissolved substance in the liquid are captured by the latter during its motion and are transported together with it. The combination of both processes is convective diffusion of matter in a liquid.

The paper is organized as follows. After Introduction, Section 2 describes the medium model and the formulation of the physical and chemical kinetics problem for a single cylindrical defect of a finite radius and a finite extent “born” and “developing” in the medium element. Section 3 describes internal and external problems: Subsection 3.1 presents the results of solving the internal problem of the process of “birth” of the defect, development of chemical corrosion by analytical method and analyzes the process of destruction of the solid “film”; Subsection 3.2 considers the results of numerical solution of the external problem in the absence of flow of the solution of aggressive liquid filling the defect; Subsection 3.3 is devoted to the analysis of the influence of hydrodynamic conditions on the process of physical and chemical kinetics; Subsections 3.4–3.6 present the results of the analysis of the numerical solution of the external problem, from which follows the formula of the dimensionless similarity criterion, linking chemical, physical and hydrodynamic parameters of the process of formation and destruction of the solid “film”. Section 4 shows the relationship between the elastic energy and radius of cylindrical defects in a closed macroscopic element under bilateral compression and its critical strength. Section 5 is devoted to the analysis of the problem of random distribution of three types of the set of physical and geometrical parameters of the joint distribution for the generalized characteristic of the model: critical strength, thickness of the solid “film” and chemical reaction rate. Section 6 presents result of numerical experiments related to the substantiation of methods for monitoring changes in the electrical and elastic properties of the medium element, supplementing the results in (Vilge, 2011). Section 7 concludes that continues analysis of changes in the elastic and electrical properties of composite materials under conditions of chemical corrosion would make possible to control changes that lead to a decrease in the strength and reliability of the structure.

2. Environment model and problem statement

The environment model should provide a solution to the problem of statistical processing of the results of indirect measurements of physical and geometric parameters using non-destructive testing methods. The model of the medium considers an elastic body in the form of a parallelepiped of limited dimensions, the outer surface of which is surrounded by an aggressive liquid of infinite volume with a given concentration of active solvent. We describe the entire volume of the body under study as divided into macroscopic elements (Freudenthal, 1969; Landau & Lifshitz, 1987; Liebowitz, 1969; Vilge, 2016), large enough for any of them to contain a significant number of defects of various sizes. These defects are small in comparison with the smallest size of the volume element to ensure mutual independence of stress fields and concentrations of dissolution products around each of the defects. Then the elastic energy and its change for any volume element will be predominantly determined by the most dangerous, “weak” defect, i.e., the one at which the joint distribution of physical and geometrical parameters is most likely to lead to the element failure. We assume that inside the element there is a defect in the form of a circular cylinder of finite length and radius, small at the initial stage of dissolution. Suppose the defect axis is perpendicular to the outer surface of the element, and the defect itself has direct contact with the surrounding liquid through the cylinder section on one of the outer surfaces of the element. The defect is subjected to the continuous influence of three unsteady processes: diffusion, macroscopic chemical kinetics, and hydrodynamics. The process of interaction between the aggressive fluid and the environment leads to the separation of the material into the internal region of the defect and the region external to it. The necessity to consider two classes of problems – internal and external – is due to the important circumstance that at the first stage the processes are practically independent of hydrodynamic conditions inside the defect, except for external pressure and temperature. The solution of the internal problem of diffusion kinetics is analytical, and the solution of the external problem of physicochemical kinetics is numerical. A specific feature of diffusion and chemical dissolution is that they occur at the boundary between the inner and outer regions. At the initial stage, a solid “film” a thin surface layer of reaction products is formed (Frank-Kamenetzky, 2015; Sherwood et al., 1975). The emerging new structure of a macroscopic element of the medium leads to the loss of elastic energy and strength of the entire element. A unified model was investigated in the study of changes in the elastic and electromagnetic properties of the medium when the defect geometry changes in the process of chemical corrosion. The obtained results are part of the basics of the technology of nondestructive control of the object’s condition.

3. Internal and external problems

3.1 Internal problem: The first stage of the defect’s origin

Correlation connections of measured parameters with physical and geometric properties of the environment are the main content of the analysis of internal and external problems for subsequent statistical processing of monitoring results.

Suppose in some arbitrary element of the body there is a defect in the form of a circular cylinder of finite length $L$ and radius $R$ . The axis of the cylinder coincides with the direction of the $Y$ axis of the Cartesian coordinate system (XOY), the origin of the coordinates is at the point $O$ belonging to the cylinder axis, and the outer surface of the element is parallel to the $X$ axis. Suppose that the element is immersed in an aggressive medium of infinite volume, in which the solvent concentration is equal to $C_0$ . Inside the defect during its formation and development, dissolution processes occur at the interface of two phases. Simultaneously with the formation of reaction products there is diffusion of these products inside the defect volume and their further diffusion into the external medium surrounding the element. The fastest part of the process at the initial stage is the reaction of dissolution of the outer surface of the defect and the formation of a solid “film” (SF) on it. The slowest stage of the process is the reaction of destruction of the (SF) because of diffusion of solid particles – reaction products from the interface of the two phases due to the difference in their concentration in the internal area of the defect. If the defect was isolated from the external environment, then between volume occupied by the SF and the internal area of the defect after some time will establish the equality of concentrations and diffusion at the interface of the two phases will stop. At violation of equilibrium the process would go in the opposite direction and so on until a stationary state is established. If the defect is associated with an external unbounded medium, the equilibrium is broken all the time as a result of diffusion of dissolution products into the unbounded volume. Once the SF is destroyed, the dissolution process resumes. In other words, the dissolution process is cyclical. It begins from the birth of the SF, followed by the process of its destruction and the resumption of the process of dissolution of the solid with the formation of a new SF. The most important stage in the initiation of the process is the stage of destruction of the (SF), since with its completion the birth of a new defect of larger radius and extent begins (length).

The process of destruction of SF in a medium element is described by the following diffusion equation (Vilge, 2010) in cylindrical coordinates:

${\displaystyle \frac{\partial C(r,y,t)}{\partial t}}=D\ast \left({\displaystyle \frac{{\partial }^{2}C}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial C}{\partial r}}+{\displaystyle \frac{{\partial }^{2}C}{\partial y^2}}\right).$(1)

The initial condition is $C(r<R,y,t)=$ 0 at $t=$ 0; the boundary conditions are:

$C(r,y,t)=g_1(R,y,t)\text{ by }r=R;C(r,y,t)=g_2(r,0,t)\text{ by }y=0;C(r,y,t)=g_3(r,L,t)\text{ by }y=L.$(2)

Suppose that the concentration of dissolution products at the outer boundaries of the defect is:

$g_1(R,y,t)=C_1(\alpha ,t)=C_1^0\ast \exp (-\alpha t);g_2(R,0,t)=0;g_3(r,L,t)=C_3(\beta ,t)=C_3^0\ast \exp (-\beta t),$(3)

where: $t$ – time [sec]; $(r,y)$ – coordinates [cm]; $C(r,y,t)$ – concentration of dissolution products in the defect volume; D [cm²/sec] – diffusion coefficient of solid particles forming a (SF) in an aggressive liquid into the inner space of the defect; ( $\alpha$ , $\beta$ ) [sec ^{- 1}] – rate constants of SF destruction as a result of change of concentration difference dissolution products in the “film” and the internal volume of the defect. These parameters should be determined as a result of experiments.

The following remark is important. To simplify the estimates, we assume that at the boundary $y⩽$ 0 there is a solvent reservoir in which the concentration of dissolution products of the surface thin layer (SF) is much lesser than their concentration inside the defect, 0 $⩽r⩽R$ , and 0 $⩽y⩽l$ . In other words, the diffusion of dissolution products into the open space occurs much faster than they accumulate inside the defect. Indeed, the diffusion coefficient of particles – dissolution products from the SF inside the defect is $D\approx$ 5*10 ^{- 9} [cm²/sec], and the coefficient of their diffusion into the outer space is $D\approx 2.5$ *10 ^{- 5} [cm²/sec]. The solution of the system Eqs (1)–(3) has the following form (Polyanin & Zielinski, 2016).

$C(r,y,t)=2\pi {\displaystyle {\int }_0^l}\xi {\displaystyle {\int }_0^R}f(\xi ,\eta )G(r,y,\xi ,\eta ,t)𝑑\xi 𝑑\eta$ $-\left\{2\pi DR{\displaystyle {\int }_0^t}{\displaystyle {\int }_0^l}{g}_1(\eta ,\tau )\left[{\displaystyle \frac{\partial }{\partial \xi }}G(r,y,\xi ,\eta ,t-\tau )\right]d\eta d\tau [\text{by }\xi =1]\right\}$(4) $+2\pi D{\displaystyle {\int }_0^t}{\displaystyle {\int }_0^l}\xi g_3(\xi ,\tau )\left[{\displaystyle \frac{\partial }{\partial \eta }}G(r,y,\xi ,\eta ,t-\tau )\right]d\xi d\tau [\text{by }\eta =1].$

Green’s function of this problem is:

$G(r,y,\xi ,\eta ,t)=G_1(r,\xi ,t)\ast G_2(y,\eta ,t),$(5)

where:

$G_1(r,\xi ,t)={\displaystyle \frac{1}{\pi R^2}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{J_1^2({\mu }_n)}}{J}_o\left({\displaystyle \frac{{\mu }_n}{R}}r\right)J_o\left({\displaystyle \frac{{\mu }_n}{R}}\xi \right)\exp \left(-{\displaystyle \frac{D{\mu }_n^2}{K^2}}(t-\tau )\right),$(6) $G_2(y,\eta ,t)={\displaystyle \frac{2}{l}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}\sin \left({\displaystyle \frac{n\pi y}{l}}\right)\sin \left({\displaystyle \frac{n\pi \eta }{l}}\right)\exp \left(-{\displaystyle \frac{D{n}^2{\pi }^2(t-\tau )}{l^2}}\right).$(7)

Bulk transformations lead to the following final expressions for the solids’ concentration inside the confined defect volume:

$C_1(r,y,t)=4{\displaystyle \frac{D}{\pi R^2}}{C}_1^0\left[{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{J_o\left(\frac{{\mu }_n}{R}r\right)}{J_1({\mu }_n)}}\right]$ $\left[{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{\mu }_n}{n^2}}\sin \left({\displaystyle \frac{n\pi y}{l}}\right)(1-\cos (n\pi )){\displaystyle \frac{1-\exp \left(-\left(\frac{D{\mu }_n^2}{R^2}+\frac{D{n}^2{\pi }^2}{L^2}+\alpha \right)t\right)}{\left(\frac{D{\mu }_n^2}{R^2}+\frac{D{n}^2{\pi }^2}{L^2}+\alpha \right)}}\right],$(8) $C_3(r,y,t)=4{\displaystyle \frac{D}{R{L}^2}}{C}_3^0\left[{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}n\ast \sin \left({\displaystyle \frac{n\pi y}{L}}\right)\cos (n\pi )\right]$ $\left[{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{\mu }_n}{J_1({\mu }_n)}}{J}_o\left({\displaystyle \frac{{\mu }_n}{R}}r\right){\displaystyle \frac{1-\exp \left(-\left(\frac{D{\mu }_n^2}{R^2}+\frac{D{n}^2{\pi }^2}{L^2}+\beta \right)t\right)}{\left(\frac{D{\mu }_n^2}{R^2}+\frac{D{n}^2{\pi }^2}{L^2}+\beta \right)}}\right].$(9)

The analysis of the obtained Eqs (3.1) and (3.1) allowed us to make the conclusion: the development of a defect having the form of a circular cylinder of small length L, and radius R, occurs under the action of diffusion, dissolution, and external pressure. Indeed, it follows from the obtained relations that the time of the SF destruction is the same at $\alpha =\beta$ . That is, the rate of development of the dissolution process along the radius and at the base is the same. However, this is not true if to take into account the phenomenon of “wedging” in which the growth of the defect along its axis has the form (Landau & Lifshitz, 1987): $L=\frac{E^2{R}^2}{{(1-{\kappa }^2)}^2{K}^2}$ , where $E$ and $K$ are Young’s and shear modulus, respectively; $\kappa$ is Poisson’s ratio. As a result, the length of the defect grows faster than the growth of its radius, reaching the opposite outer boundary of the region. From this point on, the defect development occurs only along the radius. In this case, the defect development is significantly affected by the diffusion of dissolution products inside the defect and hydrodynamic conditions, i.e., the character of fluid flow. The criterion for the cessation of SF destruction is the cessation of flow:

$F(y,t)=D{\displaystyle \frac{\partial }{\partial t}}({\displaystyle \frac{\partial }{\partial r}}{C}_1(r,y,t)+{\displaystyle \frac{\partial }{\partial y}}{C}_3(r,y,t)=0.$(10)

This condition is satisfied in two cases. In the first case, there is no flow from the inner volume of the defect to the outer volume $y<$ 0. In this case, as destruction of the SF at some point comes the equality of the concentration of dissolution products inside the defect volume and in the volume of the solid film. When the equilibrium is broken for various reasons, the process of diffusion of particles goes in the opposite direction, etc., until the equilibrium is reached, and dissolution stops. In the second case, flow from the internal volume of the defect to the unrestricted external one ( $y⩽$ 0) does not stop until the complete destruction of the SF. Indeed, the concentration of dissolution products of the surface thin layer – solid “film” in the external region is much less than their concentration inside the defect (0 $⩽r⩽R$ , 0 $⩽y⩽l$ ). In other words, the difference between the concentration of dissolution products in the open space and inside the defect is always greater than between these products in the SF and inside the defect. Consequently, the accumulation of dissolution products inside the defect does not occur, the (SFs) destroyed completely and the dissolution process resumes with increasing intensity, and the defect radius increases by the value of the film thickness. It follows from Eqs (3.1) and (3.1) that the duration of the defect growth process depends on the value of diffusion coefficients $D\approx$ 5*10 ^{- 9} [cm²/sec], of dissolution products, and parameters ( $\alpha$ , $\beta$ ). In the case of sulfate, chloride or alkaline corrosion, the dissolution products consist of solid particles and water. The second element, water saturated with acid or carbon dioxide of the aggressive liquid, continues the dissolution process filling the pores with solid particles. If all the pores are filled with them, the dissolution process ends and a long process of diffusion of solid particles inside the defect and into the open space around it begins. This process can only be accelerated by fluid movement inside the defect, temperature change and defect deformation (Vilge, 2016).

3.2 External problem: the second stage of the birth and development of the defect

Suppose that the macroscopic element of the medium is a rectangular parallelepiped, and in its center, there is a defect in the form of a cylinder, the axis of which $Z$ is perpendicular to the two vertical opposite faces of the element $S_1(x,y,0)$ , $S_2(x,y,z=L)$ . Thus $(x,y,z)$ are the coordinates of the points of the element. Then the set of points with coordinates $(x,y)\in (x,y,z_0)$ is an arbitrary cross-section of the element perpendicular to the defect axis. Suppose furthermore that for $\forall z_0\in (0<R\ll L)$ the flow velocity of the aggressive fluid $V_0(z)=$ const. Then the processes occurring in the outer region (defined as $\{r=\sqrt{(x^2+y^2)}⩾R;z⩽L\}$ in each section $S(x,y,z_0)$ are described by the equation of unsteady field of concentrations moving with constant velocities $V_1(x,y)$ and $V_2(x,y)$ at simultaneous diffusion and chemical reaction (Sherwood et al., 1975):

${\displaystyle \frac{\partial C}{\partial t}}=D\left({\displaystyle \frac{{\partial }^{2}C}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}C}{\partial y^2}}\right)+V_1(x,y){\displaystyle \frac{\partial C}{\partial x}}+V_2(x,y){\displaystyle \frac{\partial C}{\partial y}}+KC+Q,$(11)

where $C=C(x,y,t)$ is the concentration of the substance at the point $(x,y,t)$ ; $C_0(x,y,t)$ is the initial concentration at $t=$ 0; $D$ [cm²/sec] is the diffusiocoefficient; $K$ [1/sec] is the dissolution rate constant; $KC$ [gr/cm³] is the amount of substance formed due to chemical reaction per unit time; $Q$ [gr/cm³] is the concentrationsolvent located inside the defect ( $r<R$ ). Substitution (Polyanin & Zielinski, 2016) as follows:

$C(x,y,t)=\exp (A_{1}x+A_{2}y+Bt)\ast U(x,y,t),$(12)

with the notations:

$A_1=-{\displaystyle \frac{V_1(x,y)}{2D}};A_2=-{\displaystyle \frac{V_2(x,y)}{2D}};B={\displaystyle \frac{V_1^2+V_2^2}{4D}}.$(13)

The function $U(x,y,t)$ is the solution to the equation (Sherwood et al., 1975):

${\displaystyle \frac{\partial U(x,y,t)}{\partial t}}=D\left({\displaystyle \frac{{\partial }^{2}U(x,y,t)}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}U(x,y,t)}{\partial y^2}}\right)+KU+Q.$(14)

In Eq. (12) the exponential multiplier describes the change in the solute concentration in the external medium, ( $r⩾R$ , $z>0$ ), due to the flow, in particular, of a viscous Bingham fluid (Vilge, 2016) inside the defect with velocity $V_0(z)$ . These velocities depend on the Reynolds number, where L is a characteristic size, for example, the length of the defect. Suppose that the flow velocity of the Bingham fluid is small, and the viscosity is large, then the Reynolds number is not large: $R⩽$ 10. As shown in (Landau & Lifshitz, 1987), at the boundary between a solid and a liquid solvent at laminar flow inside the defect, there is a thin boundary layer. The flow in it has two velocity components: parallel and perpendicular $V_0(z)$ for both the planar and cylindrical surfaces of the interface. If the axis of the cylindrical surface $Z$ coincides with the direction of the laminar flow having velocity $V_0(z)$ , then a laminar flow with velocity $V_1(x,y)$ and perpendicular to it with velocity $V_2(x,y)\ll V_1(x,y)$ . It changes very rapidly and decreases at a distance of the order of the boundary layer thickness. The flow with velocity $V_1(x,y)$ changes slowly in the direction appendicular to the interface, from zero at the interface to a maximum value and then to zero. Therefore, the mass transfer of solid particles – dissolution products – by the flow which destroys the SF starts inside the boundary layer, gradually approaching its boundaries. Average velocity $V_1(x,y)\approx V_0(z)$ , while $V_2\approx \frac{V_1}{\sqrt{R}}$ (Landau & Lifshitz, 1987). Let us denote, that $V_n=\sqrt{V_1^2+V_2^2}$ – the velocity component normal to the inner surface of the cylindrical defect. The thickness of the (SF) may be greater than the thickness of the boundary layer. Consequently, there is a gradual displacement of the boundary layer itself until the (SF) is destroyed.

Let us supplement the Eq. (12) with the initial and boundary conditions in the domain of $G(x,y,z):\{-h<x<h;-h<y<h;0⩽z⩽L=H\}$ . The $(x,y)$ axes are located in the cross section of the cylindrical defect, and its axis coincides with the $z$ axis. In accordance with the conditions (Vilge & Vilge, 2016) in all points surrounding the defect, the concentration of chemical reaction products satisfies the conditions: $C(x,y,z,t)=$ 0, $t=$ 0. For $\forall r=\sqrt{x^2+y^2}<R,\forall z$ , and $\forall t⩾0$ , the solvent concentration inside the defect: $C_0=$ Const. Numerical solution of the Eq. (12) in the region $S(x,y,z_0):\{x_s\gg h,y_s\gg h$ , provided that at the outer boundary $r=H$ , $z=H$ , the concentration of dissolution products $C_s=$ 0 for the entire observation time $t_s<$ 10¹⁰ [sec]. Suppose $z_0$ are far enough away from the boundaries of the outer volume such that boundary effects no longer affect the concentration distribution inside $G(x,y,z)$ over the observation time 10¹⁰ sec.

An important feature of the processes related to the influence of liquid flows inside the defect region on the concentration of dissolution products in a solid body is that the processes of formation of SF (as shown below) occur in a very short period. Their destruction by liquid flows in the boundary layer (in real conditions of low velocities) is a significantly longer process. Therefore, there are two processes going on: the birth of the SF, and its destruction. They can be considered separately, as the case A) $V_0(z)=$ 0; and the case B) $V_0(z)\ne$ 0.

3.3 Results of analysis of the numerical solution of the external problem of diffusion kinetics

Let us consider the case A) $V_0(z)=$ 0; $V_1(x,y)=V_2(x,y)=$ 0. The solution of the boundary value Eq. (14) is sought by the finite element method, where instead of the constants $D$ and $k$ we consider durability problem with the time dependence of the parameters $D(t)$ and $K(t)$ . At the beginning of the diffusion stage of the process both these quantities (Frank-Kamenetzky, 2015) can vary approximately in the ranges: $D(t)$ from 10³ to 10 ^{- 4} [cm²/sec]; $K(t)$ from 10 ^{- 1} to 5*10 ^{- 2} [1/sec]; $t$ from 0 to 10¹⁰ [sec]. Numerical analysis of the diffusion kinetics of the external problem in these ranges of parameter changes and process times in seconds not only allows us to study the phenomenon in detail, but, no less importantly, to determine the measurable parameters for evaluating such important characteristics of chemical corrosion as the depth of penetration of the chemical reaction, its speed, and the thickness of the solid film. They are the necessary basis for the technology of continuous monitoring of the structure state and its durability. The penration depth (PD) of chemical reaction into the medium at a given concentration of the solvent inside the defects is determined by the value of macroscopic substance reaction rate (SR) – the total quantity of substance reacting per unit of time per unit of surface layer (Frank-Kamenetzky, 2015). The SR depends on the relative defect radius R/H and/or length (R/L) (in this case H $=$ L). If the order of the chemical reaction is $n=$ 1 (Frankamenetzky, 2015), then:

$SR=\frac{dm}{dt}=C\sqrt{D\ast K}$ is the change in flux over time; $D=\frac{dC}{dr})=\text{𝐹𝐿𝑂𝑊}[\frac{\text{𝑔𝑟𝑎𝑚𝑚𝑖𝑜𝑛𝑠}}{\text{𝑠𝑒𝑐}\ast {\text{𝑐𝑚}}^2}]=KC$ is the change of flux in space. Hence,

$C=\text{𝐹𝐿𝑂𝑊}/K,$(15) $SR=\text{𝐹𝐿𝑂𝑊}*\sqrt{D/K},$(16)

which is macroscopic rate of chemical reaction. These formulas allow us to determine numerically and by measurements the concentration of matter between two surfaces in the process of mass transfer during the “development” of defects. L us introduce some definitions for SF: TB – time of birth; TS – time of stabilization; TD – time of death; FT – thickness of SF; $FL(t_i)$ – the value of aggressive liquid flow at consecutive moments of time $(t_i)$ .

Table 1 presents some results of calculations of PD values for different (R/L).

Table 1

Dependence of the penetration depth of the chemical reaction (PD) and stabilization time (ST) on the thickness of the solid “film” (TP) with varying (R/L)

Def	R/L	0.03	0.05	0.12	0.21	0.312
1.00E-04	PD [cm]	1	0.312	0.312	0.502	0.75
	TS [sec]	1.00E+07	3.00E+00	3.00E+01	3.00E+02	3.00E+02
1.00E-02	PD [cm]	1	0.12	0.21	0.312	0.9
	TS [sec]	1.00E+04	6.00E+01	1.00E+01	6.00E+00	1.00E+01
1.00E+00	PD [cm]	1	0.312	0.312	0.312	0.5
	TS [sec]	6.00E+01	1.26E+00	3.00E+00	6.00E+00	1.00E+01
1.00E+02	PD [cm]	1	0.12	0.21	0.312	0.4
	TS [sec]	1.00E+00	1.04E+00	3.00E+00	6.00E+00	1.00E+01
1.00E+03	PD [cm]	1	0.12	0.21	0.312	0.4
	TS [sec]	1.00E+00	1.00E+00	3.00E+00	6.00E+00	1.00E+01

The coresponding graphs in Fig. 1 illustrate e process of dissolution and penetration of the chemical reaction.

Figure 1.

Variation of solvent flow rate in time and space. D $=$ 1E+02, K $=$ 5E-01.

The analysis of the presented materials allows us to study the changes in the above characteristics for the specified range of changes in the physical properties of the material. Here TS – stabilization time of SF at K $=$ 1E-01 a chemical reaction rate constant for various of the diffusion coefficients D and (R/L).

The technology of continuous monitoring of the state of the material corroded as a result of chemical reaction allows to estimate the rate of birth and development of SF, as a result of chemical reaction.

Figure 2.

The rate of a chemical reaction. D $=$ 1E+03, K $=$ 5E-01 R/L $=$ 0.312. From the bottom to the top series: 1 – 1. 5.62 sec.; 2 – 1E+01 sec.; 3 – 1E+02 sec; 4 – 1E+03 sec; 5 – 1E+04 sec.

Figure 3 illustrates the results of numerical experiments to estimate the rate of film nucleation for two relative defect radii: on the upper curve (R/L) $=$ 0.312; lower curve (R/L) $=$ 0.21.

Figure 3.

Relative concentration as a function of time.

This illustrates the possibility of implementing reaction rate estimation. Here CRL $=\frac{C}{C_0}$ is the ratio of solvent concentrations in the pores of the solid and in the volume of the defect. The graph also illustrates the dependence of the rate of change in the concentration of chemical reaction products in the (SF).

Figure 4.

Relative concentration as a function of time in the initial part.

It should be noted that at the initial stage of diffusion kinetics with dissolution, the reaction rate is approximately constant, the concentration of reaction products increases linearly. It is the greater for the greater (R/L).

If D $=$ 1E+02, K $=$ 5E-01, and (R/L) $=$ 0.21, then regression equations of concentration is CRL $=-4.8612\ast t^2+13.183\ast t-8.3238$ .

If D $=$ 1E+02, K $=$ 5E-01, and (R/L) $=$ 0.312, then regression equations of concentration is CRL $=-1.8006\ast t^2+5.346\ast t-3.546$ .

Indeed, the rate of change in the concentration of dissolution products, i.e., the rate of a chemical reaction satisfies the regression equation:

${\displaystyle \frac{\partial \text{𝐶𝑅𝐿}}{\partial t}}=f(t,D,\left({\displaystyle \frac{R}{L}}\right))\ast t+b(t,D,\left({\displaystyle \frac{R}{L}}\right)).$

The resulting relationship characterizes the high level of linear correlation between the rate of change in the concentration of dissolution products and the process time. However, with the onset of the second stage of dissolution (Fig. 4), its rate sharply decreases, dissolution acquires a stationary character with the reaction rate tending to zero.

Frank-Kamenetzky (Frank-Kamenetzky, 2015) proposed for the first-order dissolution reaction a formula for determining the thickness of SF. It is valid only for the model of the medium in which a flat boundary between the aggressive medium and the solid body:

$FT\approx D/K$(17)

Another formula (ibid.) obtained theoretically is

$FT\approx \sqrt{2A{t}_0},$(18)

where $A$ – an unknown constant proportional to the diffusion coefficient; $t_0$ – time of birth and development of SF. This formula is valid for a more complex model, including the axisymmetric one, in which the solvent flow is perpendicular to the surface. Numerical analysis allowed us to determine the total birth time (TB), which depends significantly on (R/L) and to determine the coefficient A $=$ D. Thus, the formula for determining the thickness of SF is as follows:

$FT(R/L)=\sqrt{2D{t}_0}.$(19)

Numerical analysis of Eq. (19) made it pble to obtain the explicit dependence. For each diffusion coefficient FT, it has the form of a linear function of (R/L) 10³ with a correlation coficient equals 1.

Figure 5.

Dependence of the film thickness on the relative defect radius.

For those examples, from the top to the bottom for $D=$ 10³, it has the form $FT=$ 2.98*(R/L) $+$ 0.33. For $D=$ 10², it has the form $FT=$ 2.3113*(R/L) $+$ 0.15. For $D=$ 1, it is $FT=$ 1.68*(R/ L). Hence,

$FT(R/L)=A(D)*(R/L)+B(R/L).$(20)

Equation (20) determines the correlation between the measured parameter – FT (film thickness) and the geometric – (R/L), defect radius. The correlation coefficient $(R^2)$ in all cases is close to one.

The general solution to the problem of distribution of dissolution products in the region of SF, is based on a characteristic feature of the problem under consideration: it is that at the initial moment $t=$ 0 the reaction proceeds in the kinetic region, when the concentration gradient and diffusion rate at the interface between the regions at the initial moment are infinite (Frank-Kamenetzky, 2015). When studying the diffusion regime by numerical methods, infinite values are not assumed. As a result, there is a gradual increase of concentration in a fixed region by diffusion in the part of space not yet occupied by dissolution products until a stationary regime is established.

3.4 Results of analyzing the numerical solution of the outer problem

Let us consider the case B) $V_0(z)\ne 0;V_1(x,y),\ne 0,V_2(x,y)\ne 0$ . It follows from Eq. (12) that simultaneously with the fast process of SF formation, there are alterative slow processes preventing its formation or leading to its destruction. The reason for this is that at laminar flow of liquid inside the defect, the laminar and vortex flows are formed in the boundary layer at the contact of solid body with liquid. Laminar flow carries out the transfer of particles – dissolution products along the interface, and vortex flow carries out the transfer of particles in the direction perpendicular to the interface. The SF degradation time depends on the flow velocity $V_0(z)\ne 0$ , and the diffusion coefficient in the range is adopted as 10 ^{- 4} $⩽D⩽$ 10³. Dependence of SF formation time ( $t_0$ ) and destruction time (T0) of (SF) as a function of flow rate and diffusion coefficient at various (R/L) are presented in Table 2.

Table 2

Dependence of solid “film” formation time (t0 [sec]) and destruction time (T0 [sec]) of solid “film” as a function of flow rate and diffusion

R/L $=$ 0.15				R/L $=$ 1.045
Vy	Vx	$t_0$	T0	Vy	Vx	$t_0$	T0
D $=$ 1E-04
0.00E+00	0.00E+00	1.00E+02	1.00E+04	0.00E+00	0.00E+00	1.00E+02	1.00E+04
1.00E-04	3.16E-05	1.00E+02	3.16E+00	1.00E-04	3.16E-05	1.00E+02	3.16E+00
1.00E-03	3.16E-04	1.00E+02	1.00E+00	1.00E-03	3.16E-04	1.00E+02	1.00E+00
1.00E-02	3.16E-03	1.00E+02	1.00E+00	1.00E-02	3.16E-03	1.00E+02	1.00E+00
1.00E-01	3.16E-02	1.00E+02	1.00E+00	1.00E-01	3.16E-02	1.00E+02	1.00E+00
1.00E+00	3.16E-01	1.00E+02	1.00E+00	1.00E+00	3.16E-01	1.00E+02	1.00E+00
D $=$ 1E+03
0.00E+00	0.00E+00	1.00E+00	1.00E+04	0.00E+00	0.00E+00	1.00E+00	1.00E+04
1.00E-04	3.16E-05	1.00E+00	1.00E+04	1.00E-04	3.16E-05	1.00E+00	1.00E+04
1.00E-03	3.16E-04	1.00E+00	1.00E+04	1.00E-03	3.16E-04	1.00E+00	1.00E+04
1.00E-02	3.16E-03	1.00E+00	1.00E+04	1.00E-02	3.16E-03	1.00E+00	1.00E+04
1.00E-01	3.16E-02	1.00E+00	1.00E+07	1.00E-01	3.16E-02	1.00E+00	1.00E+07
1.00E+00	3.16E-01	1.00E+00	1.00E+05	1.00E+00	3.16E-01	1.00E+00	1.00E+05
1.00E+01	3.16E-01	1.00E+00	1.00E+02	1.00E+01	3.16E-01	1.00E+00	1.00E+02

Table 2 shows examples of the influence of the laminar flow velocity of the liquid inside the defect on the time T0 – destruction of SF in comparison with the time $t_0$ of its formation at different diffusion coefficients D and defect radius (R/L) $=$ 1.045. Table 2 shows the birth time $t_0$ and development of a SF for R/L $⩾$ 0.03 of several tens of seconds, while the time of its destruction T0 can exceed hundreds of thousands of seconds, after which the process of dissolution with the formation of a SF begins again, and then goes the next stage of its destruction. In this case, each next stage begins with a defect, the radius of which is larger by the thickness of the previous SF. The corrosion process is cyclic, i.e., it is a “step” function of time. As follows from the previous analysis, as (R/L) increases, the FT thickness of the (SF) increases. In (Vilge, 2016) it was shown that the corrosion process has the most important properties of Markov processes. It is also shown there under which conditions the defect can die.

3.5 The criterion of analogy of birth and development of defect

The main conclusion from the realized analysis is that the rate of destruction of a solid containing a defect with an aggressive liquid is determined by the rate of birth and death of SF in Eqs (12) and (13). Let us represent this regularity in the form of equation:

$BV=K_{\text{𝐵𝑖𝑟𝑡ℎ}}{t}_0+K_{\text{𝐷𝑒𝑎𝑡ℎ}}T0,$(21)

or

$BV=K{t}_0+{\displaystyle \frac{(V_1^2+V_2^2)}{4\ast D_{ef}}}T0,$(22)

in which the rate constant of the birth of SF is

$K_{\text{𝐵𝑖𝑟𝑡ℎ}}=K,$(23)

and the rate constant of the destruction of SF is, in [1/sec]:

$K_{\text{𝐷𝑒𝑎𝑡ℎ}}={\displaystyle \frac{(V_1^2+V_2^2)}{4\ast D_{ef}}}.$(24)

Thus, each stage of corrosion is a sequence of two opposite processes – birth and death of SF with a common dimensionless term “BV”. As noted above, $K_{\text{𝐵𝑖𝑟𝑡ℎ}}{t}_0\ll K_{\text{𝐷𝑒𝑎𝑡ℎ}}{T}_0$ . Then approximately,

$B{V}_{\text{𝑎𝑝𝑝𝑟}}\approx {\displaystyle \frac{(V_1^2+V_2^2)}{4D}}{T}_0.$(25)

3.6 Relationship of similarity criterion (BV) with the thickness of SF and relative radius (R/L)

From the formulas $FT=\sqrt{2D{t}_0}$ and $D=\frac{F{T}^2}{2t_0}$ the expression follows:

$B{V}_{\text{𝑎𝑝𝑝𝑟}}\approx {\displaystyle \frac{V_1^2+V_2^2}{4D}}{T}_0,$(26)

and from Eq. (20) we get:

$B{V}_{\text{𝑎𝑝𝑝𝑟}}={\displaystyle \frac{V_1^2+V_2^2}{2{\left[\left(\frac{R}{L}\right)+B\right]}^2}}{T}_0{t}_0.$(27)

The similarity criterion should be determined for monitoring the corrosion process. Having established by means of experimental data processing, the form of the obtained dependences for certain physical and geometrical conditions can be used further the found dependence to calculate any processes occurring in similar geometrical and physical conditions, but at other velocities and physical properties of the substance. For this purpose, the monitoring technology should provide measurements of groundwater flow velocity $(V_1^2+V_2^2)$ , the values $\frac{\partial C}{\partial r}$ . and $\frac{\partial C}{\partial t}$ , the FLOW value of the concentration, and the start and stop time of the birth of the reduced film and its thickness FT. The measurements will allow to determine such properties of the substance as $D$ , $K$ , and the value of concentration $C=(\text{𝐹𝐿𝑂𝑊})/\sqrt{D\ast K}$ .

3.7 Velocity of chemical corrosion process

Consider the time $t_0(i)+T0(i)$ of one i-the step of corrosion. If $S(i)$ is the leading front of the reaction flow passes the path $FT(i)$ equal to the variable thickness of SF, then the velocity of one step equal to $V_i(s_i)\approx FT(i)/[t_0(i)+T_0(i)]$ . According to Eq. (21), the velocity will be $V_i(s_i)\approx [A(D)(R/L)+B(D)(R/L)]/[t_0(i)+T_0(i)]$ , for each diffusion coefficient $D$ .

4. Elastic energy of an arbitrary element of a solid containing a cylindrical defect of variable radius

It should be noted that there is a difference in the magnitude of stress on the surface of a defect which depends on its R/L, in the case when the external compressive stress is set at a limited distance from the defect, from the case when it is applied at an infinitely large distance from it. Let us consider the elastic energy of a random element of a solid body containing a defect of variable radius $(R(i)/L)$ , where $i$ is the number of the defect radius; $T(i)$ – surface tension energy of the defect in the case of application of compressive stress at the boundary of the element; and $T_0$ is the surface tension energy of the defect in the case when a compressive stress is applied to the outer boundary of the body containing the defect.

Figure 6.

Elastic energy of a random element of a solid body, containing a defect of varying radius (R/L); the model of the regression eqation: $(T(i)/T_0)=2.477\ast {(R(i)/L)}^2+0.8036\ast (R(i)/L)+0.021;R^2=0.9984$ .

Let us discuss dependence of the relative change of surface tension energy on $(R\{i\}/L)$ , denoting $Ti/T_0$ the ratio of surface tension energies at $(R(i)/L)$ to the surface tension energy $T_0$ at the boundary of the region under unilateral loading. It illustrates the effect of stress applied at a finite distance from the defect. The surface tension energy increases with increasing defect radius faster than ${(R/L)}^2$ , compared to the case of applying tension energy at infinity. This circumstance should be kept in mind when considering the results of the monitoring of the state of structural elements located at a small distance from the boundary of the region. Such correlation dependence makes it possible to evaluate the strength of a structural element in the presence of a defect of rius $R/L$ when interpreting the results of statistical processing of monitoring data.

Let us consider the general case when t elements contain a defect located at infinity from the boundary of the applied compressive stress.

Before the hole was cut, the volume element had the strain energy of the system under plane stress state and bilateral compression (Liebowitz, 1969):

$W_0={\displaystyle \frac{\pi R^2{\sigma }^2}{16\mu }}(\kappa -1){(1+\epsilon )}^2+2{(1-\epsilon )}^2$(28)

where R – the radius of the defect, $R\ll L$ ; $\epsilon$ – the ratio of stresses applied to the body under compression; $\kappa =(3-4\nu )/(1+\nu )$ – elastic constant for the generalized plane stress state; $\sigma$ – stress applied at the boundary of the region containing the defect; $\nu$ – Poisson’s ratio; $\mu =\frac{E}{2}\ast (1+\nu )$ – shear modulus; $E$ – Young’s modulus. If $\epsilon =$ 0 – one-sided compression, then

$W_0={\displaystyle \frac{3\pi R^2}{16\mu }}{\sigma }^2(\kappa -1)$(29)

The introduction of a cylindrical cavity increases the free energy by the amount $W_1$ when a stress $\sigma$ is applied at an infinitely distant body boundary. For two-sided compression ( $\epsilon =$ 1), we have:

$W_1={\displaystyle \frac{3\pi R^2}{14\mu }}{\sigma }^2(\kappa +1)$(30)

in which $W_1$ – the excess of deformation ener of the body due to the existence of a cylindrical defect in it. Suppose that $T$ – the surface energy (Liebowitz, 1969) of the defect. In the case of a circular cross-section of the defect we obtain: $T=4\gamma RE(k)$ , were $\gamma$ – surface energy per unit area; $E(k)$ is the full elliptic integral of the second kind with modulus $k$ equal to a constant value, $E(k)\approx$ 1.5667 at $k=$ 1 in our case. If the material properties are known, the values of $W_1$ and $T$ will depend only on the size of the defect. In the case of a brittle fracture, the surface energy is identified with the surface tension $\gamma$ . Then the parameters of the critical equilibrium state are obtained from the condition (Liebowitz, 1969) for the Griffiths energy criterion:

${\displaystyle \frac{\partial F}{\partial r}}=0.$(31)

Let $F$ denote the whole potential energy of the system. The change of the free energy of the system is $F=T-W_1$ . From Eq. (31) we have $\partial F/\partial r=\partial T/\partial r-\partial W_1/\partial r=$ 0. Parameters of the critical equilibrium state and, in particular, the value of the critical strength can be obtained as in the form

${\sigma }_{cr}^2{\displaystyle \frac{3\pi R(\kappa +1)}{8\mu \gamma E(1)}}=1,\text{ or }{\sigma }_{cr}^2=\text{𝐶𝑜𝑛𝑠𝑡}{\displaystyle \frac{\mu \gamma }{\pi R(\kappa +1)}}.$(32)

Denoting $\frac{{\sigma }_{cr}^2}{\text{𝐶𝑜𝑛𝑠𝑡}}=\frac{\frac{\mu \gamma }{\pi (\kappa +1)}}{\text{𝐶𝑜𝑛𝑠𝑡}\ast R}$ , $M=\frac{\mu \gamma /\pi (\kappa +1)}{\text{𝐶𝑜𝑛𝑠𝑡}}$ – the material characteristics of an element of the system, and $R$ – the geometric characteristic of the system. Hence, the magnitude of the critical strength squared is

5. Random character of the object state

The random nature of the state of the structure is associated with the random distribution of physical properties of the composite material in its volume. One of the possible scenarios for statistical analysis of such a case is considered in (Vilge, 2016). Below we discuss the random joint distribution of physical and geometric parameters of the medium in a random element of the structure. A comprehensive interpretation of local and spatial distributions allows us to develop the fundamentals of statistical methods for analyzing the results of non-destructive testing methods.

The random nature of a state of the object exposed to chemical corrosion is a consequence of random processes of birth, development, and death of defects (Vilge, 2016). Repeatedly emphasized characteristic feature of monitoring the state of the object is impossibility of the direct measurements of each of the physical and geometric parameters. Their determination by monitoring can only be obtained from the results of continuous or discrete indirect measurements. In this case, the required parameters, such as film thickness, time of its birth, rate of development and death of the defect, depth of penetration of the reaction, critical strength, etc., depend simultaneously on a set of independent random variables and the distribution of each physical and geometric parameter in space. In other words, only the analysis of their joint distribution allows us to assess the state of the entire structure. Consequently, there is a need to develop new methods of processing and interpretation of measurement results in order, to predict the state of the object and its durability. To solve the task at the first stage, it is natural to assume that physical and geometric parameters, being random variables, are distributed in space according to the normal law. In this case, the results of theoretical analysis allow us to formulate the requirements for the technology of monitoring the state of the object, in particular, the requirements for the distribution in space of the corresponding sensors – training sample of parameters characterizing the whole process of birth, development, and death of defects.

The relations between the main physical and geometrical parameters obtained in the results of studies of corrosion processes in our model of the medium, in the simplest cases, can be presented by generalized functions $f(u,v)$ of the random parameters $(u,v)$ , each distributed according to the normal law. Density functions of the joint distribution of random variables can be of the following forms.

a) $f(u,v)=u/v$ . This is the form of the Eq. (33) of the relation between the critical strength and the ratio of material parameters M and the defect radius R. Let us denote $w=u/v$ . Then $f(u/v)=f(u/v)$ . It is not difficult to show that the joint distribution density of the random variables ( ${\sigma }_{cr}^2$ ) has the form:

$\mathrm{\Omega }(w)={\displaystyle \frac{1}{2\pi }}{w}^2{\displaystyle \underset{0}{\overset{\text{𝑤𝑚𝑎𝑥}}{\int }}}{u}^{-1}\exp \left(-{\displaystyle \frac{u^2}{2w^2}}\right)\exp \left(-{\displaystyle \frac{u^2}{2}}\right)du.$(33)

The “weakest link” is the element of the medium for which the probability of joint distribution of material and geometric parameters is the highest, i.e. the highest probability of critical strength.

b) $f(u,v)=\sqrt{uv}$ . This is the form of the Eq. (19) If $w=\sqrt{uv}$ ; $v=w^2/u$ , then it is easy to show that the joint distribution density of the random variables (FT) has the form:

$\mathrm{\Omega }(w)={\displaystyle \frac{1}{8\pi w}}{\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}{u}^{-\frac{3}{2}}\exp \left(-{\displaystyle \frac{w^4}{2u^2}}\right)\exp \left(-{\displaystyle \frac{u^2}{2}}\right)du.$(34)

c) $f(u,v)=\sqrt{\frac{u}{v}}$ . This is the form of the Eq. (16). If $w=\sqrt{u/v}$ , then the joint distribution density of the macroscopic chemical reaction rate SR in the region containing defects has the following form:

$\mathrm{\Omega }(w)={\displaystyle \frac{1}{8\pi }}{w}^7{\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}{u}^{-3}\exp \left(-{\displaystyle \frac{u^2}{2}}\right)\exp \left(-{\displaystyle \frac{w^4}{2u^2}}\right)du.$(35)

Equations (33)–(35) of the joint distribution density of the listed random variables give an idea of possible methods of processing and interpreting the results of indirect measurements to obtain the distribution of material and geometric parameters of the controlled objects.

6. Monitoring the processes of birth, development, and death of a defect.

The works (Vilge, 2010; Vilge, 2011; Vilge & Vilge, 2016; Vilge, 2019) considered the physical basis for monitoring the state of a composite material using non-destructive methods in which sensitive sensors inserted in the structure throughout the entire period of its existence, with measured electromagnetic and acoustic parameters. In case of electromagnetic method, it can be the ratio of the electric field strength components $Ex/Ey$ which depends on R/L recorded in various points of the object. When using acoustic methods, the measured parameters depending on R/L may be the time of transverse displacement and the single-sided amplitude spectrum.

Figure 7.

Single – Sided Amplitude Spectrum, Series 1 (top): R/L $=$ 0.15; Series 2 (bottom): (R/L) $=$ 0.3.

Physical parameters of the element: Young’s Modulus E $=$ 200E+02, Poisson’s Ratio $=$ 0.25, Mass Density $=$ 2.7 [gr/cm³].

7. Conclusion

The obtained results can be summarized as follows.

1. 1.

Statistical modeling and regression equations are the basis for creating technology for non-destructive methods for monitoring the condition of Buildings and Structures.

2.

The durability of an object’s existence is the time during which its structure undergoes sequential transformations before destruction. In other words, the longevity of the structure is the time of stable equilibrium of the evolutionary system. The process of transition of a structure under conditions of chemical corrosion from a stable state to an unstable one is a sequence of smooth, continuous changes in the structure and its abrupt transformation as a result of changes in some physical and/or geometric parameters. Each element of such a sequence represents a set of processes of dissolution of the composite material with the formation of a solid film and its destruction.

3.

The random process of “birth”, development, and “death” of defects arising in chemically active composite materials because of chemical corrosion is one of the fundamental reasons determining their durability. Prevention of catastrophic destruction of buildings and structures lies on the foundations of the technology of continuous monitoring of the material state. It is based, to a large extent, on mathematical analysis and the synthesis of three processes: physical and chemical kinetics, diffusion, and hydrodynamics. Various possible scenarios of “birth”, development, and “death” of defects should be considered. The choice of an idealized model of the medium can be based on the statistical concept of a “weak link”.

4.

The difference between the technology of nondestructive control of the object state to prevent catastrophe lies in the indirect nature of measurement results. They represent information about the joint distribution in space of physical (material) and geometric properties of the medium. However, their independent estimations can be of main interest. The random character of physical and geometrical parameters, chemical reaction rates, thickness, time of formation and destruction of solid films determines the necessity of theoretical study of a specific possible scenario of corrosion process development. This task is solved by analytical and numerical analysis of internal and external boundary problems of physical - chemical kinetics for the selected model of the medium.

5.

The appearance of the above-mentioned solid “film” is the result of physical and chemical processes at the boundary between the solid and the aggressive liquid. The solid film is a key factor in the process of birth and development of the defect. This process is a sequence of “birth” and formation of a solid film, its destruction because of diffusion of dissolution products inside the defect filled with aggressive liquid and their further diffusion into the external space, formation of a new defect of greater radius and extent, birth of a film at a new interface. This sequence is a step function with increasing time step, and the time of “film birth” is significantly less than the time of its destruction. The first period of the defect birth ends with the formation of the second external interface between the defect and the environment. The second stage of the defect development process is characterized by reduction of the film destruction time due to groundwater movement along the defect, both because of particle transport and vortex flow at the interface between the body and the fluid.

6.

The evaluation of such important parameters of the chemical corrosion process as the depth of reaction penetration, its speed, thickness of the solid film, etc., as well as the determination of relations between them is a necessary basis for the technology of continuous monitoring of the state of the structure and durability, i.e., the forecast of its existence time. Dimensionless similarity criterion of birth and development of a defect in the process of chemical corrosion is a fundamental link between physical and geometrical parameters:

$BV=K_{\text{𝐵𝑖𝑟𝑡ℎ}}{t}_0+K_{\text{𝑑𝑒𝑎𝑡ℎ}}T0;BV=K_{ef}{t}_0+{\displaystyle \frac{V_1^2+V_2^2}{4D_{ef}}}T0.$

7.

New methods of processing and interpretation of the results of monitoring of the object state to assess its durability should provide information about physical and geometric properties of the medium from the results of their random joint distribution. Relationships between the main physical and geometric parameters should be based on the results of studies of physicochemical kinetics phenomena for specific models of the medium. These relations in the simplest cases can be represented by generalized functions $f(x,y)$ of the following forms: a) $x/y$ ; b) $\sqrt{(xy)}$ ; and c) $\sqrt{(x/y)}$ , where $(x,y)$ are random parameters, each of which is distributed according to the normal law. For example, the value of critical strength can be obtained directly from the joint density distribution of physical and geometrical parameters Eq. (33). The weakest element of the environment is the element for which the probability of joint distribution is the highest.