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# On the Evolution Theory of Identification of Mathematical Models of Corrosion Destruction at the Optimum Design of Structures

George Filatov

► To cite this version:

George Filatov. On the Evolution Theory of Identification of Mathematical Models of Corrosion Destruction at the Optimum Design of Structures. Mechanics, Materials Science & Engineering Journal, 2015, 10.13140/RG.2.1.1192.8088. hal-01305946

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Submitted on 22 Apr 2016

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# On the Evolution Theory of Identification of Mathematical Models of Corrosion Destruction at the Optimum Design of Structures

George Filatov¹a

1 – Dneproprovsk State Agrarian-Economic University, Ukraine

a – filatovgv@mail.ru

**Keywords:** Optimal design of structures, mathematical modeling of corrosion damage, identification of mathematical models

**ABSTRACT.** The process of optimal design of structures, interacting with aggressive environments, can be viewed as a process of design evolution from suboptimal to optimal state. In most cases, the control variables are taken as the geometric design parameters. When designing the structure during the search of the optimal solutions of these parameters change during the transition from one structure to another intermediate state. And changing the geometrical characteristics of cross-sectional structure characterizing their stiffness, such as the area and inertia moment of the cross sections. Changing the geometrical characteristics of the cross-sections results in a change in stress and strain in the construction. Thus, it can be argued that in the process of design evolution at its optimal design of the stress-strain state (SSS) of the structure varies depending on its stiffness. Natural to assume that a change in the SSS design values of the coefficients that characterize the impact of SSS on the rate of corrosion process, and are subject to change and become functions of SSS. To test this hypothesis, we studied the theoretical aspects of the behavior of mathematical models of corrosion damage at the optimal design of structures and performed extensive numerical experiment on a computer. The experiment was conducted using four objects: membrane cylindrical shell loaded by internal pressure, smooth cylindrical shell compressed in the axial direction, statically determinate beams with rectangular cross-section, statically determinate beams welded I-section.

**The basic pre-conditions and hypotheses.** Over the past few decades in structural mechanics appeared a new direction of research: development of the methods of calculation of structural elements and machine parts interacting with aggressive environments. The emergence of this trend is due to a significant reduction in the carrying capacity of structures, their lifetime, reliability and durability of machines and equipment as a result of chemical or physic-chemical exposure to corrosive environments.

Direct losses from corrosion are enormous. Due to the deterioration of physical-mechanical and physic-chemical properties of materials, the cost of protection against corrosion, the cost of repair affected by corrosion products in the most developed countries of the world consumes about 40% of annual production of the metal. Indirect losses associated with the deterioration of the technological and operational characteristics of the equipment and machinery subject to corrosion, their downtime, disaster recovery, leakage of valuable or hazardous products into the environment, metal inflated costs due to increased tolerance and etc. account for about 10% of national income in many countries. That is why the study of the mechanism of corrosion and to find effective ways to protect metals and other materials is becoming one of the most important and urgent problems of modern science and, in particular, structural mechanics.

In an emerging economic crisis, the increase of the value of construction materials, such as steel, there is another problem – the rational use of available resources, optimal design of structures and equipment. To date, developed powerful techniques to optimize designs, working in a neutral environment: methods for linear and nonlinear mathematical programming, and stochastic gradient search high and effective methods of zero order, multi-criteria optimization, etc. These methods are well established in the design of many objects of construction and engineering industries. However, the simple transfer of the developed techniques for the design and optimization of structures especially interacting with an aggressive environment, is impossible, since the character of the corrosion process, its kinetics are often not known in advance. To clarify the nature of the influence of aggressive environment on the behavior of the material usually put experiment under physical modeling. The results of such simulations allow us to construct a mathematical model of the corrosion process. Physical modeling of corrosion damage structures, being mandatory and very important step in the implementation of direct payments or optimal, due to its high cost and the
duration of the test may not always meet the designer, as the reasons given above can not have a
complete picture of the corrosion process. Using mathematical models it is possible to extrapolate
the corrosion process and more fully explore its kinetics. The second possibility, represented by a
mathematical model is to integrate the corrosion process in the design scheme optimized object in
the form of a system of algebraic, transcendental or differential equations. However, such a model
should be adequate to the real process of corrosion. The adequacy of the mathematical model is
achieved by identifying the model to experimental data. The proximity of the experimental and
simulated data of the corrosion process provides an introduction to the mathematical model of
special factors and adjusting their values in the process of identification. And only when the in
accordance with the selected criterion of similarity of the simulated and real corrosion process can
be administered in such a model calculation scheme and proceed to the calculation of the design,
including its optimal design. Otherwise, the designer runs the risk of a project that does not meet
reality.

Most of mathematical models of corrosion damage are nonlinear functions of the parameters to be
optimized. Therefore, in this study to identify the mathematical models of corrosion damage are
encouraged to use one of the methods of nonlinear stochastic programming - the method of random
search. This method is a method of zero-order, does not require the calculation of derivatives, well
take into account the functional and geometric constraints, trained in the search process and has
good convergence.

All mathematical models using an external parameter of damage can be divided into two types: the
model does not take into account the effect of the stress-strain state (SSS) to the design speed of the
corrosion process, and models that take into account the impact of SSS. Most mathematical models
of corrosion damage, both the first and second groups include empirical coefficients determined by
identifying models from experimental data. It is believed that the results thus the model coefficients
are constant. Such a claim can be considered valid only for the first group of mathematical models
whose coefficients depend on the state of the corrosive environment: temperature, concentration,
pressure, etc. The coefficients of the second group of models depend on the structure of SSS and, if
in the process of changing its structure calculation or geometric parameters of the physical constants
(Poisson's ratio, modulus of elasticity) is changed, the SSS structure. The process of optimal design
of structures can be viewed as a process of design evolution from suboptimal to optimal state. In
most cases, the control variables are taken as the geometric design parameters. When designing the
structure during the search of the optimal solutions of these parameters change during the transition
from one structure to another intermediate state. And changing the geometrical characteristics of
cross-sectional structure characterizing their stiffness, such as the area and moment of inertia of the
cross sections. Changing the geometrical characteristics of the cross-sections results in a change in
stress and strain in the construction. Thus, it can be argued that in the process of design evolution at
its optimal design of the stress-strain state of the structure varies depending on its stiffness. Natural
to assume that a change in the SSS design values of the coefficients that characterize the impact of
SSS on the rate of corrosion process, and are subject to change and become functions of SSS. To
test this hypothesis, we studied the theoretical aspects of the behavior of mathematical models of
corrosion damage at the optimal design of structures and performed extensive numerical experiment
on a computer.

**2. The theorem on the influence function of the stress-strain state (SSS) on the speed of corrosion process.** Consider some of the theoretical aspects of the problem of identification of mathematical models of corrosion damage. To this end, we formulate a criterion of quality received at the identification of mathematical models of corrosion damage. With regard to the mathematical model taking into account the impact of SSS on the rate of corrosion process, this criterion has the form:
$$J = \sum_{j=1}^{n} [\delta(t_j) - \delta_j^e]^2 = \sum_{j=1}^{n} \left[ \alpha t_j + \beta \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1}) - \delta_j^e \right]^2 \quad (1)$$

Take the partial derivative of the functional (1) on the parameters $\alpha$ and $\beta$, equate the resulting expressions to zero – we obtain:

$$\frac{\partial J}{\partial \alpha} = \alpha \sum_{j=1}^{n} t_j^2 + \beta \sum_{j=1}^{n} \left[ t_j \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1}) \right] - \sum_{j=1}^{n} \delta_j^e t_j = 0; \quad (2)$$

$$\frac{\partial J}{\partial \beta} = \alpha \sum_{j=1}^{n} \left[ t_j \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1}) \right] + \beta \sum_{j=1}^{n} \left[ \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1}) \right]^2 - \sum_{j=1}^{n} \left[ \delta_j^e \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1}) \right] = 0. \quad (3)$$

Solving equation (3) with respect to the rate of the corrosion process unstressed material, we get:

$$\alpha = \frac{\sum_{j=1}^{n} \delta_j^e}{\sum_{j=1}^{n} t_j}. \quad (4)$$

Equation (4) shows that the corrosion rate depends only on time and experimental date of depths of corrosion damage and does not depend on the level of stress and strain in the construction. In other words, the value of corrosion rate unstressed material depends on the condition of the corrosive medium, for example, the concentration of aggressive substances, temperature, etc. Equating the corrosion rate to zero and solving the equation (3) with respect to the coefficient of influence of SSS on the corrosion rate, we obtain:

$$\beta = \frac{\sum_{j=1}^{n} \delta_j^{\text{exp}}}{\sum_{j=1}^{n} \sum_{k=1}^{j} f_{k-1}(t_k - t_{k-1})} \quad (5)$$

From (5) it follows that the coefficient depends not only on the state of the environment, but also the function of SSS. SSS function always inversely proportional to stiffness and directly proportional to the stress. From this it follows that the dependence of the influence of SSS on the stiffness of the structure is directly proportional to stiffness: a reduction in stiffness and decreases the coefficient of influence of SSS on the speed of the corrosion process, which is confirmed by the results of numerous experiments. At the same time, the value of the coefficient of influence of SSS on the rate of corrosion process is inversely proportional to the level of stress in the structure. Consequently, the coefficient is not the number and is the function of the stiffness and simultaneously the function of SSS of structure. The theoretical findings allow us to formulate the following theorem.

**Theorem.** The influence function of SSS on the speed of the corrosion process is directly proportional to rigidity and inversely proportional to the stress. The proof given above. From this theorem there is a consequence.

**Consequence of Theorem.** "Optimal" value of the coefficient of mathematical model of corrosion damage, taking into account the influence of SSS on the speed of the corrosion process, ensures the
convergence of the search for the optimal solution without additional procedures of multiple identification of mathematical model. In other words, the design achieves the optimum state under "optimal" value function of influence of SSS on the rate of corrosion process. This corollary allows avoiding the procedure of multiple identification mathematical model corrosion damage at the optimal design of structures. This fact is quite important, as it gives an opportunity to significantly reduce the loss on search. The consequence asserts that if there is an "optimal" set of coefficients of a mathematical model and optimized function has an extremum, the optimal solution can be found from any point in the range of the permitted parameters.

**3. Briefly on Numerical Experimentation on a computer.** The experiment was conducted using four objects: membrane cylindrical shell loaded by internal pressure, smooth cylindrical shell compressed in the axial direction, statically determinate beams with rectangular cross-section, statically determinate beams welded I-section (Fig. 1).

<figure><img src="image_3.png" /><figcaption>Figure 1. The objects</figcaption></figure>

The object of investigation was considered factor $\beta$ of influence of SSS on the corrosion rate for two models:

$$ \text{I.G. Ovchinnikov's model [1]:} \quad \frac{d\delta}{dt} = \beta\varepsilon_i(\sigma_i - \sigma_{\text{thr}}) + \alpha; $$

$$ \text{V.G. Karpunin's model [2]:} \quad \frac{d\delta}{dt} = \beta\sigma_i + \alpha, $$

where $\beta$ – coefficient taking into account the influence of SSS on the rate of corrosion process;

$\delta$ – current depth of corrosion damage;

$t$ – time corrosion;

$\varepsilon_i$ – intensity of deformation;
$\sigma_i$ – stress intensity; stress threshold stress below which the influence of SSS on the rate of corrosion process is missing: if $\sigma_i \le \sigma_{thr}$ the taking $(\sigma_i - \sigma_{thr}) = 0$;

$\alpha$ – the corrosion rate of the unstressed material.

The order of the experiment was as follows:

1. Formulate the problem of optimal design. The target function takes a cross sectional area of the structure.

2. In the allowed parameter selects the starting point for which is realized an identification of a mathematical model based on experimental data.

3. From the starting point of the selected design optimization was performed.

4. In the search path chosen 30 starting points, each of which performs the identification.

5. The graph of dependence of coefficient $\beta$ on the current stiffness of the optimized construction is built (Fig. 2).

<figure><img src="image_4.png" /><figcaption>Figure 2. The graph of the function influence of SSS on the rate corrosion process on the stiffness of the membrane shell</figcaption></figure>

From the graph clearly shows that the influence of the SSS on the corrosion rate is not constant and decreases significantly when optimizing the design. If now with the optimal factor attempt to perform optimization of any arbitrarily selected point of the field parameters, it is possible to obtain an optimum result of the project. The saving of the weight of design is significant. However, the procedure of multiple identification is rather cumbersome. Therefore, in the work have been proposed several empirical methods for determining the parameter of the SSS on the stage that precede to optimization.

**Summary.** 1. As a result of multiple identification of mathematical models of corrosion damage, the dependence of the parameter influence of SSS on the speed of the corrosion process from the stiffness of cross-sectional design with its optimization is established. Thus, the parameter of influence of the SSS is not constant, but is a function of the stiffness of the structure at its optimal design.

2. The presence of the "optimal" value of the parameter influence of SSS on the corrosion speed is established. The main feature of the "optimal" value of the parameter influence of SSS on the corrosion rate of SSS is to comply with the optimal state structure.

3. The design, which is in optimum state, has the lowest speed of corrosion caused by the influence of SSS.

4. The theoretical aspects of the problem under investigation: formulate and prove a theorem on depending of corrosion process rate function from the rigidity of the optimized design, the
technique of determining the parameters of the mathematical model of corrosion damage, providing
an evolutionary transition structure in an optimal state.

5. The computer program to determine the optimal set of coefficients of a mathematical model of corrosion damage, allow you to create an effective project, is developed.

The entire contents of experimental and theoretical investigations is given in the monograph [3].

References

[1] Petrov, VV Calculation of structural elements, interacting with aggressive media [Text]: monograph / I.G.Ovchinnikov, Yu.M.Shihov. - Saratov: Saratov State University. 1987. – 288 P.

[2] Karpunin, VG Study bending and stability of plates and shells based on a solid corrosion [Text]: Author. Dis.cand. tehn. Science / VG Karpunin. - Sverdlovsk. 1977. - 24 P.

[3] Filatov, G. Theoretical Foundations of evolution matmodeley corrosion damage [Text]: monograph / GV Filatov. - Saarbrucken: LAP LAMBERT Academic Publishing.