COMPUTATIONAL MODEL FOR DURABILITY ANALYSIS OF STRUCTURE ELEMENTS WITH DEFECTS

A methodology for the determination of number of cycles before the destruction of structure elements exposed to cyclic loading (tension-compression) has been developed. The analysis of the structure element static and dynamic stress-strain state with a usage of numerical methods of finite and boundary elements in order to determine the stress concentration zones is carried out. Model cracks that are placed in the highest stress concentration zones are selected. A database of model cracks is proposed. The initial length at which crack development begins is determined with a usage of the stress intensity factor threshold value. For each crack from the database, a critical number of cycles during which the crack grows to unacceptable sizes, is found based on the Paris criterion. A method for determining stress intensity factors for a structure element with cracks is proposed. The problem is reduced to solving singular integral equations. To obtain a numerical solution of these equations, the boundary element method is used. Densities, which appear as unknown functions in the considered integral equations, are used to calculate stress intensity factors. The analytical and numerical solutions of singular equations are compared. The critical number of cycles for plates with isolated cracks and cracks chains, cracks located at the elements holes and boundaries is determined. It was established that at the same loading level, a smaller critical number of cycles corresponds to a structure element with cracks that are in close proximity to the technological hole. An analysis of the fatigue crack development at holes in an elastic-plastic statement in order to determine the number of cycles before destruction is made, estimated number of cycles before the fatigue crack appearance is given. increase in displacements and stresses after 100 hours of creep compared with a straight cylinder. For bodies of revolution made of a functionally graded material, the cut-out effect on the SSS is less pronounced.


Introduction
Due to the production of the hydro turbine, petrochemical, and power equipment resource at many industrial enterprises in Ukraine, the problem of the possibility of the individual elements service time extension and the need to replace morally and physically obsolete components and parts arises. The solution to this problem will ensure the units' operational reliability during further operation upon fulfilling guarantees in terms of power and efficiency. Since the said equipment works for a long time, usually its elements and components can be weakened by various kinds of microdefects. The development of these defects under operational loadings can lead to the failure of individual parts or to complete structure disruption. There are two main approaches to the structure elements durability estimation. The first approach lies in the defects development modeling (pores, cracks, etc.) in the framework of the fracture mechanics. The second approach lies in the structure elements durability estimation when explicit defects are not considered, and the time between the fatigue cracks formation can be predicted. In this regard, the problem of determining the time for the structure elements with defects destruction under the action of cyclic loadings is relevant for the equipment residual life estimation. A similar problem arises while studying the possibility of safe long-term transportation of structure elements.
The objective of the structure with cracks durability estimation is to determine the time (number of cycles) after which the crack grows to a critical size and disruption occurs. Usually it is not known in advance what exactly defects are present in the studied structure element, since microdefects cannot always be detected even by ultrasound scanning methods. Therefore, it is advisable to study the defects of various configurations and sizes effect on the structure durability in order to find out the most dangerous cracks and de-termine the shortest time to destruction. This time can serve both to estimate the residual life and to determine the timing of the equipment overhaul period.
It is known that small cracks, which are not possible to detect during the structure visual inspection upon repair work, lead to partial or complete destruction of the structure element due to gradual developing under the influence of alternating loadings. Fatigue cracks are usually located near bearing surfaces. They occur due to corrosion, intense temperature and power loadings. The formed crack begins to grow slowly, even if the applied loadings do not exceed the rated values. Then, if a critical crack length is reached, the bearing element suddenly receives a huge amount of kinetic energy, which leads to a catastrophic destruction of the entire structure.
The problem is relevant due to the great interest in the fatigue cracks study in the last decade. By now, a significant amount of experimental and theoretical researches of the patterns of crack growth under the influence of variable cyclic loadings has been accumulated. We note the fundamental monographs of Andreikiv A., Darchuk A., [1] Makhutov N. [2], Panasyuk V., Andreikiv A., Kovchik S. [3], articles [4][5][6], review [7], in which this problem was reviewed in detail. A regulatory document [8] on the hydraulic turbines elements flowing part residual life calculation, including the presence of cracks, has also been developed. The crack resistance of oil and gas pipelines is the subject of [9]. In [10], a database and classification of cracks in pipelines are presented; work [11] is devoted to the aircraft equipment residual life estimation. The latest studies on the theory of cracks and inclusions that form clusters and chains serve as the foundation for the development of modern technologies for the residual resource determination. In [12], interacting surface cracks are studied; the effect of a pore chain on the weld strength is studied in [13]; in [14], an analysis of the strength of structure element with a cracks chain at a temperature loading was made; work [15] was devoted to studying the durability of a hydraulic turbine shaft with surface cracks.
This work's objective is to develop a methodology for the structure elements with defects of various kinds durability estimation with the usage of modern methods of boundary and finite elements.

Methodology for the structure elements durability determination
Based on the available experimental and theoretical data, the following methodology for the structure elements durability estimation is proposed. At the first stage, an analysis of the static and dynamic stress-strain state is carried out, the frequencies of the structures free vibrations are determined with a usage of experimental or numerical methods [16][17][18]. These studies make it possible to find out unwanted vibration frequencies during transportation, to determine the highest stress concentration zones in the structure element. Such zones are usually located near the holes, elements boundaries and welds.
After the analysis, model cracks, which are placed in the highest stress concentration zones, are selected, Fig. 1.
The well-known approach [19] described further is used next. First, the stress-strain state of a structure element without cracks under the action of the given loading is considered. In this case, stresses σ arise in the vicinity of the model crack. Further, it is assumed that predetermined loadings act on the crack edges, р=-σ, and the element contour is free of loadings, and the stress-strain state of the element with a crack is determined. Then, the stress-strain state of the structure element with a crack under the action of given loading is determined as the sum of the solutions of the two problems described above.
We consider flat structure elements that are under the action of an alternating tensile extensioncompression in the perpendicular to the crack line direction. It is assumed that the model crack is located along the contour L 0 . Let ( ) be the contours bounding the considered element. We use the integral equations method [20]. Denote the unknown densities on the contour L 0 as 2 1 ,α α , and on the contours be the components of the internal unit normals to the considered contours. We set that the structure element is under the action of predetermined external loadings S 11 , S 12 , S 22 . In [20,21], it was indicated that unknown functions The integrals with kernels ( ) ξ in the equations of system (1) are hypersingular [21], and in the there are logarithmic singularities or singularities of Cauchy type if the points i ξ and j x coincide. The numerical solution of the integral equations system (1) is carried out by the boundary element method [18], [22].
To determine the critical number of cycles, the Paris dependence [23] is used, and it is expressed by the formula ( Here l is the characteristic defect size; N − critical number of cycles; where max K , min K − maximum and minimum SIF for one loading cycle; m − fatigue curve exponent; C − fatigue curve characteristic constant.
It is assumed that the loading cycle is symmetrical, i.e.
Then max min max . The number of cycles to destruction is determined by relation (2) We note that cracks do not develop at th K K ∆ < ∆ , and at K ∆ >K 1C an avalanche-like crack development that leads to the structure element destruction occurs. Thus, we have the following formula for the cycles critical number calculation: where 1 0 ,l l − initial and final crack lengths. In order to apply the formula (3), it is necessary to know the initial and final crack dimensions and have an analytical or algorithmic expression for SIF depending on the crack length. With a usage of the integral equations system solutions (1), we determine SIF by the formulas [19], [21] ( ) ( ) where r − the distance between the crack tip and the observation point, G − shear modulus, ν − Poisson's ratio. The development of rectilinear cracks located perpendicular to the loading is researched. Then only the coefficient 1 k will be nonzero. To determine the length 0 l with which crack development begins, the following relation is used Cracks with an initial size less than the calculated value 0 l , according to the Paris criterion, do not develop.
We note that cracks with dimensions more than 0.005 m are usually considered unacceptable, i.e., after scheduled inspections, structure elements in the presence of such cracks are subject to repair or modernization [8]. Thus, it is of interest to find the number of cycles during which the crack grows from the initial size calculated by the formula (5) to the final size 1 l =0.005 m. We also note that the value 1 l must satisfy the condition otherwise, an avalanche-like crack growth occurs.
The following model cracks were considered: defect 1 -an isolated crack in the elastic plane ( Fig. 1, a), defect 2 -a crack that extends to the half-plane boundary (Fig. 1, b), defect 3 -a cracks chain in the weld vicinity (Fig. 1, c) defect 4 -two symmetrical cracks located in the hole area, (Fig. 1,d).
We give singular integral equations for the unknown densities 2 1 ,α α determination. For an isolated crack with a length l 2 ( Fig. 1, a), we have the following system of hypersingular equations [20]: For a crack that is perpendicular to the half-plane boundary (Fig. 1, b), a hypersingular integral equation is obtained in the form For a cracks chain under the action of loading perpendicular to the cracks location line (Fig. 1, c), the corresponding hypersingular equation takes the form where l − the crack half length; d − the distance between the cracks centers in the chain.
The problem of determining the stress concentration in a structure element with a technological hole of radius R and two symmetric cracks with a length l (Fig. 1, d) that extend to the hole boundary is reduced to the singular equation [19] , Analytical solutions for equations (7) and (9), [19], [20] are known. Numerical solutions of the singular equations (7)-(10) were obtained with a usage of 80 boundary elements with a constant approximation of the density inside the elements [21], [22]. Figure 2 shows the numerical and analytical solutions of hypersingular equations (7), Fig. 2, a, and (9), Fig. 2 The solid lines in Fig. 2 correspond to analytical solutions, dots indicate numerical solutions, accuracy ε=10 -3 is achieved in the calculations.
Having determined the unknown functions 2 1 ,α α , we find the stress intensity factors by the formulas (4). a b Fig. 2

. Comparison of numerical and analytical solutions:
a -numerical and analytical solutions to the problem of an isolated crack; b -numerical and analytical solutions to the problem of a cracks chain

Structure elements crack resistance analysis
We calculate the cycles number for which cracks of different types grow to unacceptable sizes. In this case, we will consider the same loading conditions. According to the regulatory documents [8], the permissible stresses of the base metal are σ 0 =147 MPa, and for the weld they are σ w =95 MPa. We assume that the applied loading is σ=ασ 0 if the crack is located in the base metal, and σ=ασ w for cracks located in the weld zone. A material with the following characteristics is considered: С=3.2⋅10 -11 [24]. c − for a cracks chain; d − for two symmetrical cracks that emerge from a circular hole Figure 3, a shows SIF for an isolated crack in the elastic plane, defect 1; Figure 3, b -SIF for a crack extending to the half-plane boundary, defect 2; Figure 3, c -SIF for a cracks chain, d=0.025 m, defect 3; Figure 3, d shows the change in SIF for two symmetrical cracks that emerge from a circular hole of radius R=0.1 m, defect 4.
Analyzing the diagrams shown in Fig. 3, we establish that at σ=0.1σ 0 =14.7 MPa for the base metal and σ=0.1σ w =9.5 MPa for the welds, none of the considered defects develops.
Next, we calculate the initial crack sizes for loadings σ=0.75σ 0 for the base metal, and σ=0.75σ w for cracks located in the weld zone. We obtain the following values: 0 l =0.0005 m for an isolated crack and a crack extending to the boundary in the base metal zone, 0 l =0.0012 m for a cracks chain located in the weld zone, The presented results show that if the loading amplitude is 75% of the permissible, there is a rapid destruction of the structure element with cracks. Defects located near the technological hole turned out to be the most dangerous; in this case, the number of cycles until the crack grows to unacceptable sizes is essentially smaller than for isolated cracks and for cracks chains.
We note that the obtained data do not include cyclic operating time indicators before the appearance of initial fatigue cracks. Therefore, it is of interest to estimate the number of cycles before the appearance of fatigue cracks for the most dangerous defect (symmetrical cracks extending to the technological hole boundary, defect 4, Fig. 1, a). For this purpose, the calculation of the cyclic strength in the system ANSYS, [25] was conducted.

Calculation of a structure element with a technological hole for cyclic strength
We considered a steel plate with a central hole with the following mechanical and geometric characteristics: density ρ=7850 kg/m 3 , Young's modulus E=200 GPa, Poisson's ratio ν=0.3, plate length L=1 m, width b=0.4 m, thickness h=0.01 m, hole radius R=0.1 m, Fig. 4.
The main material characteristic used to describe its ability to resist the process of multi-cycle fatigue is the Weller's curve (fatigue curve). For this material, this curve in the logarithmic scale is shown in Fig. 5.
We considered plate extension in the horizontal direction by forces of 10 6 N. The distribution of the von Mises equivalent stresses in the plate material is shown in Fig. 6. The maximum stress was 108.7 MPa. The highest stresses were observed at the circular hole boundary The calculation for multi-cycle fatigue showed that the smallest number of cycles until destruction is observed at the circular hole boundary, in the same areas where the highest stresses are reached, as shown by the static calculation (Fig. 7).
When calculating cyclic fatigue, the structure was loaded according to a sinusoidal law with amplitudes of ±108.7 MPa and a frequency of 1 Hz. The calculation showed that the minimum number of cycles until destruction is N=271300. It was shown above that for the considered structure element, the development of a fatigue crack from the initial crack size 0 l =0.0002 m to the maximum allowable size 1 l =0.0047 m occurs in 1102 cycles, which is significantly less than the number of loading cycles during the multi-cycle fatigue calculation. Thus, the main contribution to the resource estimation is made by the cyclic operating time indicator, i.e., the number of loading cycles until the appearance of fatigue cracks. Knowing the total number of cycles to destruction (in the considered case 272402) and the frequency of loading, we determine the time to the structure element destruction. The obtained data make it possible to estimate the structure element transportation safety if there are real transportation cyclograms.

Conclusion
A method for the calculation of critical number of structure elements loading cycles under conditions of alternating cyclic loading has been developed. It is assumed that in the highest stress concentration zones such defects as cracks may occur. The number of stress cycles, which leads to the growth of cracks to unacceptable sizes, is determined according to the Paris criterion. It is proposed to use the model crack base, which contains the isolated crack, the cracks chain, the crack extending to the structure element boundary, and symmetrical cracks extending to the structure hole boundary. Formulas based on the application of numerical solutions of singular and hypersingular integral equations are used to calculate stress intensity factors. It is established that cracks that are in the vicinity of the holes are the most dangerous. An estimation of the number of loading cycles until the appearance of fatigue cracks is given for this case precisely.