Random “Birth and Death” process of defects yielding destruction of composite materials

Abstract

Composite materials destruction as a “Birth and Death” process of defects with chemical corrosion is discussed. The problem is closely connected with durability of materials in case of their long-term usage in corrosive environment. The statistical concept of the “weakest link” helps to consider a model of completely elastic continuous medium with a layer of small thickness extending to the maximum compression, and dissolution product is a Bingham viscous-plastic liquid filling up the layer. The equation for mathematical expectation of destruction process with continuous time and discrete but limited number of events enables us to estimate a number of defects in accordance with the energy parameters and the entire ideal elastic body, and the time to eventual fracture. One of the possible results of the random of “Birth and Death” process of defects is a total loss of medium strength. Non-destructive methods of strength control by monitoring material conductivity and spectrum of random signals upon the process of “Birth and Death” of defects are suggested.

Keywords

Statistical concept of the “weakest link”model of medium elastic modulus defect crack “Birth and Death” process of defects strength nondestructive methods

1. Introduction

Durability of materials is understood as their ability to preserve a set physical and mechanical properties during a long-term usage in conditions of negative exposure, for example, violent temperature fluctuations, frequent wetting and drying, exposure to chemically active air and aqueous media, etc. Such exposure can be divided to physical action (freeze resistance, water resistance, resistance to alternate wetting and drying), and chemical action (dissolution and movement of individual components by streaming water and under action of sulfate or alkaline corrosion). Corrosion and destruction, as well as physical and chemical impacts on composite media are discussed in numerous papers (McCauley, 2004; Nakhi et al., 1979; The Site, 2017; Schweitzer, 2004). However, less attention has been paid to analysis of connection between these and the fracture processes, corrosion monitoring, and forecast of the state of materials affected by corrosion. Solving the problem of durability encounters with multiple obstacles, such as complexity of media structure and diversity of corrosion processes, their duration in time, considerable economic and technical complications in performance of experiments, and difficulty of analytical and numerical analysis even for the simplest physical and mathematical models. Furthermore, each type of corrosion requires a specific approach to formulation of the phenomenon’s model and analysis of the fracture process. Physical properties of material formed as a result of dissolution and the rate of development of each type of corrosion are practically unknown. Such properties could provide experimental basis for theoretical research. Most adverse effects develop as a result of composite material interaction with fluid or gaseous phase of ambient medium where the fracture process starts at the surface of the structural element simultaneously with the start of its use and gradually moves into the depth. Moreover, in practically perfectly prepared chemically active composite materials, structural defects are always present in the form of crevices (see, for instance, Kamran et al., 1968; with a vast list of references). Surface or near-surface defects are first to be affected by corrosion, and under action of compression strain they may coalesce forming heavy main free-form “crack layers” largely orthogonal to external stress vector. Most commonly, the experimental basis of such research is provided by findings negative cyclic effects of wetting and drying processes. Eventually, contractional and shrinking deformations start to form in the near-surface zones, resulting in the emergency of internal stresses and generation of micro cracks which are gradually increasing and coalescing, causing fracture of composite material in surface zones. Such cyclic effects are relatively short and do not provide explanation on which generalized medium physical parameters corrosion rate depends, and with which intensity a fracture process develops depending on properties of liquid and dissolution materials replacing it. In the final analysis, the absence of answers to these questions prevents the substantiation of the basic physics of methods of structure and material state monitoring.

This paper describes destruction process caused by dissolution and movement of particular composite medium components with the aim of identification of generalized physical parameters that determine the rate of chemical corrosion processes for the suggested model of a medium. In tackling some dissolution problems, for description of chemical reaction and removal of its products, many authors apply the Newtonian liquid (Levich, 1959), but it is a considerable simplification omitting important aspects of a state of defects formed as a result of dissolution. I suggest a model of medium and process that is valuable because of its simplicity and clearness. It may be fairly useful, as it helps to analyze a potential scenario of evolution of ideal elastic composite body from the beginning of its corrosion to potential brittle fracture mechanism, and to identify generalized physical properties which provide a reliable basis for theoretical calculations. In future, different options of corrosion processes and various spacing effects related to viscous-plastic deformations can be examined. This work studies successive processes of dissolution and deformation up to full or partial compression of a layer between two elastic regions and related physical processes resulting in change of composite material state. The presented approach allows to substantiate the use of nondestructive methods of control of medium physical parameters for experimental study of durability of a particular material. The aim of the work is to give a sequential analysis of a possible scenario of composite material fracture caused by chemical corrosion, more exactly: random “birth” process of a new system of defects and their “death” that may result in structural destruction; state of the layer filled up with reaction products in the form of Bingham viscous-plastic liquid; compression and fracture of the originated “crack layer” defect; to derive equation for mathematical expectation of random fracture process with continuous time and discrete but limited number of events $N$ , to estimate their number and time to eventual fracture of ideal elastic body; to discuss destruction prediction methods based on continuous monitoring using nondestructive methods of control of material condition. Such scrutiny will give a clearer idea of further experimental analysis for tackling the problem of durability of materials and structures, with subsequent creation of a mathematical model of their fraction due to different types of corrosion. Although the mathematical modeling is based on simplified assumptions, they allow to present a possible scenario of successive interrelated stages of corrosion and media fracture. The composite materials’ fracture analysis uses models of ideal elastic body brittle fracture. Our research (Vilge B I & Vilge B B, 2016; Vilge, 2010, 2011, 2013) has shown that such an approach is fairly feasible, at least in respect of cement, sand, and concrete compositions. Analytical relations and generalized physical properties of materials form the basis of a reliable experimental base for tackling the problem of durability of materials exposed to chemical corrosion.

The paper is organized as follows. Section 2 describes model of medium and statement of the problem, Section 3 analyzes the random “Berth and Death” process (BDP) of defects in composite materials resulting from chemical corrosion and causing their fracture, Section 4 (with Annex 1) analyzes compression of a single defect – a “crack layer” filled up with Bingham viscous-plastic liquid, Section 5 (with Annex 2) evaluates the liquid backpressure inside a defect as a function of time, and Section 6 summarizes.

2. Model of the medium and statement of the problem

The basic principle in selecting a model of the medium and processes was the statistical principle of the weakest link (Freudenthal, 1968) successfully used in analysis of strength of cement and sand concrete composition in Vilge B I and Vilge B B (2016). Due to this principle, a random fracture of a sample is generally determined by the local strength of the weakest element in the volume, implying identification of a sample fracture with nonstable propagation of the most critical defect (“crack”) from such an element throughout the entire sample, irrespective of local strength of the other elements on the way of the crack. A possible defect propagation due to progressive coalescence of small cracks is discounted. In other words, strength of a sample or an entire volume equals the strength of its weakest link. Hence, the following steps in the model of medium and processes are applied:

1. 1.
Investigation on corrosion problems requires identification of the processes in the defect of a thin single “crack layer”;
2.
Corrosion products within a single “crack layer” are not Newtonian liquid (The Site, 2017);
3.
Resulting from chemical corrosion replacement of strong material by material with other physical and mechanical properties will lead to reduction of backpressure within the defect and, as a consequence, its compression;
4.
Compression process (exhaustive in time) gives rise to a new system of defects with Poisson distribution in volume;
5.
A corrosion process continues or stops depending on the termination or emergence of new cracks with the attenuation of compression energy and/or probability of emergence of defects near the surface of their volume approaches zero, and the BDPs process dies away.

In accordance with these five steps, it is possible to analyze a two-dimensional ideal elastic brittle body of finite size submerged in chemically active medium. Due to the weakest link theory used in the model, the body has a thin horizontal layer filled up with thickness of composite material, namely, compacted cement and sand concrete composition normally loaded by the compression stress and located close to or touching the outer surface of the body. Such thin layer is exposed to chemical corrosion resulting in its solid phase dissolution and transformation into Bingham viscous-plastic liquid as a result of dissolvent diffusion. Corrosion of materials is the process of an aggressive component transfer in the body’s pores accompanied by solid phase dissolution, chemical reactions with the release of new formations, etc. We do not consider transitional corrosion stages assuming that they had ended and dissolution products (Bingham viscous-plastic liquid) freely move along the layer surface. Replacement of solid material by dissolution products results in reduced resistance of the layer to compression pressure under action of overlying material, finally attaining a full or partial closure of “crack layer” in a finite, fairly short period of time. As a consequence of movement of dissolution material and layer volume deformation, potential energy of body deformation is increased and spent on maintenance of the static balance in the layer area, on heat, and on giving rise in the body area to a system of new, mostly horizontal defects with the Poisson distribution. Some of such defects with a certain probability can be located very close to the body surface, thus, increasing the probability of contact with chemically active liquid surrounding the body. Then the process of corrosion of one or more thin horizontal layers, movement of reaction products, compression and new system formation, etc. repeats itself. Therefore, a branching, possibly, continuous BDPs process of gradual stress reduction and brittle fracture of ideal elastic body emerges. Analysis of processes in suggested model empowers to identify physical phenomena accompanying corrosion processes, such as increased fracture porosity and emergence of elastic oscillations upon compression and fracture, provides the basis for development of nondestructive methods of control and evaluation of strength and durability of structures containing composite materials.
3. Random BDPs of defects in composite materials from chemical corrosion leads to their destruction

Theorems (Freudenthal, 1968) and our research (Vilge B I & Vilge B B, 2016; Vilge, 2010, 2011, 2013) suggest that in the zone surrounding the layer exposed to corrosion with chemical dissolution products movement (Annex 1) and compression (Annex 2), there is a free elastic deformation energy, with some of it giving rise to emergence of a new system of defects – lengthy “crack layers”. Let us consider the plane $X Y$ which is fairly stretched in $Z$ -direction, and emerging defects do not appear simultaneously but occur independently one by one, in increments $\Delta t_{i}$ starting and going in time of compression $t_{\textit{compr}}$ . Therefore, the following conditions regarding the defects are met: they occur in non-overlapping domains independently; they emerge one at a time; and they are statistically evenly distributed in the domain with average density $q$ . Then the number of defects $m$ in a domain $D$ is distributed by the Poisson’s law (Ugarte et al., 2016):

$\displaystyle{P}_{m}=\frac{{b}^{m}}{m!}\exp(-b),∼{}m=0,1,2,\ldots$ (1)

For a flat case, the average number of defects in domain is $b=S_{D}q$ , where $S_{D}$ is the domain area. However, we are primarily concerned with defects in the neighborhood of the outer surface of a structure containing composite material and exposed to corrosion with the highest probability.

Let us denote a half-height of domain as $H$ , and the average number of defects on interval $H$ is denoted as $c=\lambda H$ , where $\lambda$ is the average density of relevant defects per unit of length $H$ . It is evident that $\lambda\leqslant q$ . Then probability of appearance of exactly $m$ defects in the vicinity of $H$ equals:

$\displaystyle{P}_{m}=\frac{c^{m}}{m!}\exp(-c),∼{}m=0,1,2,\ldots$ (2)

In this case, the probability of at least one defect appears within the interval $H$ is

$\displaystyle P_{1}=1-\exp(-c).$ (3)

To evaluate a number of the defects within domain $D$ and interval $H$ , we can use tables from Stewart (2009). Let us put $S_{D}/2=$ 2.5 m $*$ 4 m $=$ 10 m ${}^{2}$ ; $q=$ 0.4; 0.5; 0.6; then $b=$ 4; 5; 6, respectively. Mathematical expectation for number of defects for Poisson distribution equals the parameter of this distribution (Stewart, 2009): $M_{b}=b$ , so for our example it is 4, 5, or 6, respectively. Probability of occurrence of at least one defect in the neighborhood of vertical boundary of half of area $D$ when $H=4$ , $c=0.8$ , $\lambda=0.2$ , due to Eq. (3) equals to $P_{1}=$ 0.550671. The results depend on the size of area $S_{D.}$ , primarily, and on the length of initial defect. It should also be noted that calculation was made for the half of area, therefore, by symmetry, the mathematical expectation for the whole area with two outer surfaces submersed into aggressive medium is $2M_{b}=2b$ , and, respectively, the number of points near these surfaces will double with the same probability. In our example, mathematical expectation for total number of defects in the whole area is 8, 10, 12, and the number of defects near outer surfaces of the structure is not less than 2. Therefore, compression of body with one “crack layer” results in increased total fracture porosity, which reduces strength of material, and emergence near the body and aggressive medium surfaces of new defects become a good trigger for additional corrosion processes. Two potential process development scenarios are: attenuation without destruction and destruction before attenuation.

Let us consider Markovian BDP of defects with a limited number of discrete states and continuous time. By each subsequent step, the body elastic energy grows smaller, but purely statistical distribution of defects does not depend on decreased elastic energy. There are two parallel processes: formation of a total set of defects (increased fracture porosity, and reduced critical strength (see Annex 2, the Eqs (2.1)–(2.5)), and emergence of defects near the surface which are first to be exposed to corrosion. Increased fracture porosity results in diminishing body elastic energy, which, in turn, gives rise to a new set of defects. Coalescence of defects under each compression also lowers elastic energy. Hence, in each event of compression of main “crack layer” defect, a flow of events takes place during the compression time – an array of emerging defects of different size, i.e., defects’ birth process. Assuming the one-dimensional law of distribution of random process $X(t)$ , a flow of events is characterized by the fact that events (defects) occur one by one, thus, flow of events is ordinary; event flow intensity $\lambda(t)$ does not depend on the event of compression number, and compression time interval is the same; and a flow of events for each subsequent case of compression does not depend on the previous one. These conditions define a heterogeneous Poisson process (Harris, 1963, 1965; Hellan, 1988; Stewart, 2009), and for a fixed intensity of time t numerical characteristics of random variable $X(t)$ are:

$\displaystyle M\left[X(t)\right]=D\left[X(t)\right]=b(t)=\int^{t}_{0}{\lambda(% t)dt}.$

At least two scenarios of further development of the state of ideal elastic body exposed to corrosion are possible:

1. 1.

Body destruction upon gradual decrease of body elastic energy and strength (Annex 2, the Eqs (2.1)–(2.5)) with increasing or constant value of $\lambda(t)$ – intensity of “birth” of new defects, defects coalescence, and integration into the main “crack layer” affected by corrosion.

Attenuation of “birth” process upon $\lambda(t)$ minimized to zero related to consistent reduction of body elastic energy and to the fact that potential plastic deformation results in slow compression and “healing” of main crack, i.e., “birth” preventing of new cracks.

If random process $X(t)$ is one-dimensional, stable ( $\lambda(t)=\lambda=$ const, and death process rate $\mu((t)=\mu=\text{const})$ , with continuous time and discrete finite number of states, then $0\leqslant X(t)\leqslant N$ . Mathematical expectation $M(X(t))$ when $M(X(0))=0$ , by Stewart (2009) equals:

$\displaystyle M(X(t))=N\lambda(1-\exp(-(\lambda+\mu)t))/(\lambda+\mu)(t>0),$ (4)

and dispersion $D(X(t))$ equals:

$\displaystyle D(X(t))=N\lambda\mu(1-\exp(-(\lambda+\mu)t))\left(1+\frac{% \lambda}{\mu}\exp(-(\lambda+\mu)t)/{(\lambda+\mu)}^{2}\right).$ (5)

We find the value $N$ from the theorem given in Liebowitz (1968) and also in Annex 2 (bottom part): the total integral compression energy of defects and generation of new defects emerging in the corrosion process is less than or equal to the total elastic energy of the body. Let $W_{0}$ be a total elastic energy of body, then $2W_{h}$ is the total energy of deformation Eq. (2.2), and for each $i$ the flow of events is a random process $X(t)$ . Then condition of fracture as a result of all corrosion processes is as follows: $W_{0}-2W_{h}\int^{N}_{i=0}{i}\geqslant 0$ .

Let $N$ denote the maximum number of events of the process $X(t)$ in which the following condition is satisfied: $W_{0}-2W_{h}\int^{N}_{i=0}{i}=0$ , thus,

$\displaystyle W_{0}-2W_{h}\sum^{N}_{i=0}{i}=W_{0}\left(1-\frac{W_{h}}{W_{0}}% \left(N(N+1)\right)\right)=0.$ (6)

Then we can give the estimate:

$\displaystyle N=\sqrt{{W_{0}}/{W_{h}}}$ (7)

Mathematical expectation for a finite number of destruction moments of defects of ideal elastic body is:

$\displaystyle M(X(t_{d}))=\frac{\lambda\sqrt{{W_{0}}/{W_{h}}}}{(\lambda+\mu)}% \left[1-e^{-(\lambda+\mu)t_{d}}\right]∼{}∼{}(t_{d}>0),$ (8)

where $t_{d}$ is a limit moment of fracture time.

From Eqs (7) and (8) we obtain evaluation of time $t_{d}$ to eventual fracture.

$\displaystyle e^{-(\lambda+\mu)t_{d}}=1-\frac{(\lambda+\mu)}{\lambda N}M(X(t_{% d})),$ (9)

therefore,

$\displaystyle t_{d}=-\frac{1}{(\lambda+\mu)}\ln\left[1-\frac{(\lambda+\mu)}{% \lambda\sqrt{{W_{0}}/{W_{h}}}}M(X(t_{d}))\right].$ (10)

It follows from Eq. (9) that when $t_{d}\to\infty$ , then $\frac{(\lambda+\mu)}{\lambda}\frac{M(X(t_{d}))}{N}\to 1$ , and $M(X(t_{d}))\to\frac{\lambda N}{(\lambda+\mu)}$ . In other words, $M(X(t_{d}))\leqslant N$ , so destruction of ideal elastic body may take place earlier than condition Eq. (6) is fulfilled, as invariably $\frac{\lambda}{(\lambda+\mu)}\leqslant 1$ . Equation (10) helps to evaluate time to destruction $t_{d}$ by the known energy parameters ${W_{0}}/{W_{h}}$ .

The simple special case of ideal elastic body with constant parameters $\lambda(t)=\lambda=$ const, and $\mu(t)=\mu=$ const discussed above, is a rude approximation to an actual process of fracture of composite material. Numerical evaluation of possible process of development requires knowledge of physical parameters of actual materials, which may be obtained following a considerable range of purposeful experimental research. Without this information the durability prediction becomes a virtually insoluble problem.

Physical processes taking place in elastic medium as a result of chemical corrosion, dissolution and products movement, lead to material compression and, on the other hand, to increased fracture porosity. Both phenomena can provide physical basis of nondestructive methods of control of a structure state. Also, the fundamental changes take place in defects of structure that according to statistical concept of the weakest link (Freudenthal, 1968; Vilge B I & Vilge B B, 2016), determine the composite materials’ strength. Among all of nondestructive methods used for the state control and monitoring in the process of their hardening and strengthening, such methods with measurable parameters that correlate with physical parameters of defects (i.e., viscosity, elastic moduli, activation energy, etc. of the medium that fills up its defects) are promising. Physical techniques of “Resistivity/conductivity measurement method” (MMR/MMC) and “Acoustic method” (AMM) meet the above requirements. They may prove to be useful for monitoring and forecasting of structure state both at the stage of formation of strengthening and hardening structure (see, for instance, Vilge B I & B B, 2016; Vilge, 2010, 2011, 2013; and references therein), and at the stage of control of the state changes as a result of random BDPs of defects, with a Bingham liquid within the defect.

Let us point out some key features of these methods based on the fact that prior to destruction, corrosion causes change of structure, medium porosity, and, as a consequence, change its electrical conductivity due to saturation with ions of liquid mixed with products of dissolution, and partially replaces them. Sensors for measuring electrical conductivity placed in particular sections of the investigated structure will enable to get signals related to emerging changes, i.e., BDPs of new defects. The discussed model of composite medium is one of the important class of models which may display instability of the state if parameters describing the medium state reach extreme values. Such an unstable state may cause destruction of solid deformable body when mechanical stresses exceeding tensile strength. As shown in Annex 2, in case of defects filled up with a Bingham liquid, unsteadiness may be followed by a not instant compression of solid body, which in turn causes wave processes of compression and rarefaction throughout the body – waves of compression and shearing stresses. Duration of compression of single defect is given in the Eq. (2.3). The so-called spectral method for studying the wave propagation effects and oscillatory occurrences is quite widely used in physics and engineering. It permits to perform analysis of behavior of waves of arbitrary shape and intensity where the principle of superposition is fulfilled. Detection, registration, and analysis of such processes by placing acoustic sensors inside the structure will allows to record kinematic and dynamic characteristics of a particular spectrum of compression waves which is the purpose of AMM.

4. Conclusion

The results obtained in this paper can be summarized in the following list.

1. 1.

Random “Birth and Death” process of defects emerging in chemically active composite materials as a result of chemical corrosion is a key factor determining their durability. Equation for mathematical expectation for random fracture process $M(X(t))$ with continuous time and discrete finite number of events $N$ permits to estimate their number and time $t_{d}$ to eventual destruction of an ideal elastic body. Value $N$ is determined from condition of equality of all elastic energy of relevant body, all deformation energy caused by material compression (“death” of defects), and new (“birth”) defects at each subsequent stage of random process in the regions containing defects due to corrosion.

One of possible models of destruction of elastic two-dimensional body exposed to chemical corrosion is a serial process of chemical dissolution and movement of material that fills up the defect in the form of a lengthy “crack layer”, replacement of such material by viscous-plastic liquid, body compression under action of stress orthogonal to defect surface, “BDPs” of the system of new defects and, in the final analysis, structure fracture with nonzero probability.

Concept of the “weakest link” enables us to consider scenarios of development for processes of composite materials fracture due to chemical corrosion via the processes in a model of ideal elastic body with single “crack layer” defect at the initial stage as the weakest element.

Change of composite material state due to corrosion can be detected and predicted by using nondestructive methods of control, including the method of measurement of effective electrical resistance (conductivity) and acoustic method using spectral measurements of dynamic parameters of elastic “compression-rarefaction” waves. Measurable parameters of such methods are directly connected with activation energy, elastic modules, and viscosity of liquid inside structure defects and with spectrum of “compression-rarefaction” waves and, ultimately, with its strength.

Viable solution of the problem of durability of composite material exposed to physicochemical corrosion requires a complex of experimental studies to determine for each type of materials the following parameters: coefficient of molecular transport velocity, diffusion coefficient of liquid that corrodes defect material, compression energy, and intensities of generation and death of the defects.

Footnotes

Annex 1. Analysis of compression of a single defect of “crack layer” filled up with Bingham viscous-plastic liquid

In the course of dissolution, the “crack defect” space is filled up, as may be supposed, with “Bingham” liquid, which in our model is a layer of viscose-plastic material in the central area of the crack. The space between inner surface of the crack and plastic material is filled up with liquid with shearing properties. This area is primarily susceptible to material movement out of it. During the movement of the solution, the crack becomes unstable, which leads to the closure of shear, and in some time plastic the area.

Consider the case in which the maximum unsteadiness occurred after full movement of dissolution products, for time $t>T_{dis}$ . Hence, compression stresses ${\sigma}_{\infty}$ and tension stress back-pressure $p$ force on the newly emerged surface of unsteady crack. (Note that dissolution and movement of material with its possible replacement by liquid other than “Bingham” liquid, results in the appearance of a new free surface of the crack and change of inner backpressure). When ${\sigma}_{\infty}>p$ , compression and forcing away of viscous-plastic liquid from layer volume resulting in its flowing, intensity of shear stresses $\tau$ in some points to be equal or more than critical shear stress $\tau_{0}$ , then the condition $\tau\geqslant$ ${\tau}_{0}$ is fulfilled. The flowing medium area at points $\tau\geqslant{\tau}_{0}$ is the shearing flow area which demonstrates both viscous and plastic medium properties (Gnoevoi et al., 1996; Reiner, 1960). It was shown (Goldstein & Osipenko, 2010) that for medium defects in the form of cut cracks, the compression/tension strength ratio is 2.66. Experimental information does not conform constancy of such a ratio with the material mix of substantial importance.

To evaluate development of a “crack layer”, it is important to estimate the compression speed in the shear area. Let us consider isothermal two-dimensional flow of isotropic, incompressible “Bingham” medium in a thin layer under action of stress directed perpendicularly to its surface. It was shown (Gnoevoi et al., 1996) that for “Bingham” liquid flow in a thin layer, the layer structure model has two flow areas: a shearing flow area: $\{h_{1}\leqslant z_{1},z_{2}<h_{2}\}$ , and a plastic flow area $\{z_{1}\leqslant z_{2}\}$ , with $2z_{0}$ – the cross section (width) of the shearing flow area. Equation for determination of $z_{0}$ as an interface between plastic and shearing medium flow areas (Gnoevoi et al., 1996) is as fllows:

(1.1) $\displaystyle\frac{\partial}{\partial x}\left(\frac{{hz}^{2}_{0}-\frac{1}{3}z^% {3}_{0}}{h-z_{0}}\right)=-\frac{{2Sh}_{1}}{S}\frac{\partial h}{\partial t},$

where $h$ – half of width of the whole medium flow area: $h=(h_{2}-h_{1})/2$ ; $z_{0}$ – distance from the lower or upper layer surface to interface between shearing and plastic flows; ${Sh}_{1}=\frac{l}{\overline{t}\overline{U}}$ – Strouhal number; $\overline{t}$ – characteristic time; $\overline{U}$ – characteristic velocity; $S$ – $\frac{{l\tau}_{0}}{\mu U}$ – Saint-Venant number; $l$ – length of Bingham liquid flow area; $U$ – flow velocity in the shear area; $\tau_{0}$ – critical shear stress; $\mu$ – liquid viscosity in the shear area.

We can require the width of shear area $z_{0}$ be proportional to the half-width of flow areas $h$ , or $z_{0}=k(x)h$ , when $k(x)<1$ . In contrast to Gnoevoi et al. (1996) where $z_{0}$ $\to 0$ , let us consider a general case of a random width $z_{0}$ . We can obtain an expression for $h=h(t)$ from this equation:

(1.2) $\displaystyle\frac{\partial}{\partial x}\left(\frac{{h(kh)}^{2}-\frac{1}{3}({% kh)}^{3}}{h-kh}\right)=-\frac{{2Sh}_{1}}{S}\frac{\partial h}{\partial t}.$

Let us denote $A=\frac{S}{2Sh_{1}}$ , then the transformations Eq. (1.2) produces:

(1.3) $\displaystyle\frac{\partial h(x,t)}{\partial t}=2Ak^{2}\left(\frac{1-k/3}{1-k}% \right)h\frac{\partial h}{\partial x}+Ah^{2}\frac{\partial}{\partial x}\left[k% ^{2}\left(\frac{1-k/3}{1-k}\right)\right].$

(1.4) $\displaystyle\frac{\partial h(x,t)}{\partial t}=2Ak^{2}\left(\frac{1-k/3}{1-k}% \right)h\frac{\partial h}{\partial x}+2Ah^{2}k\left[\left(\frac{1-k/3}{1-k}% \right)+\frac{1}{3}\frac{k}{{(1-k)}^{2}}\right]\frac{\partial k}{\partial x}.$

Differential Eq. (1.4) allows separation of variables if the function $h(x,t)$ is presented as the product of two functions, one of them depending only on time, and the other only on coordinate $x$ :

(1.5) $\displaystyle h(x,t)=f(x)*\varphi(t).$

Following the substitution of $h(x,t)$ into Eq. (1.4) and dividing its left and right sides by the product $f{\varphi}^{2}$ , yields:

(1.6) $\displaystyle\frac{\varphi^{\prime}}{{\varphi}^{2}}=2Ak^{2}\left(\frac{1-k/3}{% 1-k}\right)f^{\prime}+Af\left[\left(\frac{1-k/3}{1-k}\right)+\frac{1}{3}\frac{% k}{{(1-k)}^{2}}\right]\frac{\partial k^{2}}{\partial x}=\lambda,$

The variables become separated, resulting in two equations for functions $f(x)$ and $\varphi(t)$ :

(1.7) $\displaystyle\varphi(t)=\frac{1}{\lambda t-C_{1}};f(x)=\frac{1}{\left(2+\frac{% 1}{3}\frac{k}{(1-k)\left(1-\frac{k}{3}\right)}\right)k}\left[\frac{\lambda}{2A% }\int\frac{dx}{k(x)\left(\frac{1-k(x)/3}{1-k(x)}\right)}+{C}_{2}\right],$

where $\lambda$ – a separation parameter.

To simplify analysis, let us consider $k(x)=k=$ Const, $h_{0}=h\left(x,0\right)=$ Const. We find these coefficients ${{C}}_{1},{{C}}_{2}$ based on $z_{0}\to 0$ : ${{C}}_{1}=-\frac{\lambda}{2aA}$ ; ${{C}}_{2}=\frac{b\lambda}{2aA}$ .

General solution in this case is as follows:

(1.8) $\displaystyle h(x,t)=\frac{a^{2}2aA}{t+1/2aA}*\left(\frac{x}{b}+\frac{b}{a}% \right).$

The obtained Eq. (1.8) is an important result of the assumption that chemical corrosion produces a material filling up a thin layer which is a Bingham viscous-plastic fluid, not a Newtonian fluid. Otherwise, the layer would be compressed throughout its length immediately, rather than during a finite time, as it is described and used in Section 3.

Annex 2. Evaluation of liquid backpressure inside a defect as function of time.

The compression of a layer takes place when compression stress “ ${\sigma}_{\infty}$ ” is more than back-pressure stress $p$ . Let us find the value of this $p$ , assuming that energy required for the occurrence of “main” crack equals (Hellan, 1988; Liebowitz, 1968):

(2.1) $\displaystyle{W}_{h}=\Delta\sigma*\pi a^{2}\left(1-{\nu}^{2}\right)E^{-1}=\pi(% mgH-p)*a^{2}\left(1-{\nu}^{2}\right)E^{-1},$

where $\Delta\sigma={\sigma}_{\infty}$ – $p\geqslant 0$ ; $m g H$ – compression stress; $H$ – distance from the body surface to the upper boundary of the relevant “layer-crack”; $2h(x,t)$ – tick; $a$ – half of length of cut crack; $\nu,E$ – Poisson ratio and elastic modulus of the body outside the crack, respectively.

Action of compression stress ${\sigma}_{\infty}$ and movement of dissolution products inside shear area: $z_{0}=k(x)*h$ results in back-pressure ${p}$ , with Young modulus for internal area $E^{fr}$ and elastic deformation – $\varepsilon$ : $E^{fr}=\frac{dp}{d\varepsilon}$ . Then it can be written as $d\varepsilon=\frac{dp}{E^{fr}}$ . So a change in the layer volume due to the material movement inducts a change in backpressure ${p}$ . However, volume change is proportional to $\frac{h(x,t)}{h_{0}}$ as suggested in Goldstein and Osipenko (2009). Consequently, $\frac{h(x,t)}{h_{0}}\sim$ $\frac{p}{E^{fr}}$ or $p\sim\frac{h(x,t)}{h_{0}}E^{fr}$ . Thus, with reduction of $h(x,t)$ , the back-pressure decreases under condition of free movement of corrosion products. Then expression Eq. (2.1) becomes:

(2.2) $\displaystyle W_{h}\sim\pi\left(mgH-\frac{h(x,t)}{h_{0}}E^{fr}\right)*a^{2}% \left(1-{\nu}^{2}\right)E^{-1}.$

It means that compression of the layer results in liberation of elastic deformation energy, with part of deformation energy spent for carrying-out of dissolution products, creation of a new free surface, reduction of backpressure and elastic modulus $E^{fr}$ . The other part of deformation energy is spent on heat, for instance, due to friction of particles upon deformation, and for creation of new defects in the volume of deformed body The obtained result is consistent with the theorem given in Liebowitz (1968): “Where inner free boundary expands due to material removal, the resulting work done by invariable external boundary loads is equal to double deformation energy increment (only remaining material is deemed as being in the final state)… If new surfaces are created by gradual removal of forces (stresses) that acted on them earlier, no kinetic energy will occur, and a new steady state will be reached in a simple way. If such new surfaces are formed abruptly, for instance, by outburst of material, fluctuations and waves will occur. In that case, a new balanced state will be reached due to their attenuation”. We showed that in our case new surfaces are not formed abruptly, but during a short finite time, and they can emerge on each finite time interval to overall attenuation of compression process causing “compression-expansion” waves.

Let us consider an ideal case where one half of energy under compression transforms into body compression, and the other half – into creation of new defects. Total compression energy is defined in Eq. (2.2). Without insignificant part of heat energy, the energy for creation of a new system of “crack defects” with “effective” parameter (Vilge B I & Vilge B B, 2016) of half-length $a_{ef}$ equals to:

(2.3) $\displaystyle W^{fr}_{h}=0.5\pi\left(mgH-\frac{h(x,t)}{h_{0}}E^{fr}\right)*a^{% 2}_{ef}\left(1-{\nu}^{2}\right)E^{-1},$

which yields the following expression

(2.4) $\displaystyle a^{2}_{ef}\sim W^{fr}_{h}E\bigg{/}\left[0.5\pi\left(mgH-\frac{h(% x,t)}{h_{0}}E^{fr}\right)*\left(1-{\nu}^{2}\right)\right].$

In accordance with Freudenthal (1968); Vilge B I & Vilge B B (2016), the critical strength of compressed volume for an “effective” crack takes the form:

(2.5) $\displaystyle q^{2}_{cr}\sim W^{fr}_{h}E\bigg{/}\left[0.5\pi\left(mgH-\frac{h(% x,t)}{h_{0}}E^{fr}\right)*a^{2}_{ef}\left(1-{\nu}^{2}\right)\right].$

Thus, the critical strength of material decreases as a result of movement of shear area, and it depends on the time of compression on a layer along its length.

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