ENDURANCE LIMIT OF THE AXIAL-PISTON HYDRAULIC MACHINE CYLINDER BLOCK

The fatigue resistance of the cylinder block (CB) of an axial-piston hydraulic machine (APHM) made of tin bronze CuSn12 has been investigated. Fatigue tests of smooth and notched standard samples were performed and fatigue curves were constructed. The stress state of the CB is investigated by the finite element method. It is established that the maximum stresses are localized on a small part of the partition between axial cylinders holes loaded by internal pressure. A method for recalculating the established length of this zone of partition with an uneven distribution of dangerous stresses by an equivalent length with a constant stress equal to the maximum is proposed. The use of the statistical theory of the similarity of the fatigue failure of Serensen-Kogayev, the results of the fatigue tests of the samples and the calculation of the equivalent length made it possible to determine the endurance limit of the CB considered. The acceptability of the proposed method estimating the endurance limit by the known results of resource tests of the CB APHM 210.25 is confirmed.


INTRODUCTION
The main trends in the development of the volumetric hydraulic drive are an increase in nominal pressure from 20...25 MPa to 32...45 MPa and a reduction in overall dimensions while maintaining the requirements for its reliability. The solution of these problems is obtained by improving the designs, increasing the quality of the materials of the parts, and the obligatory observance of maintenance requirements. The most important indicator of reliability is the durability of machines. The increase in pressure in axial piston hydraulic machines (APHM) leads to an increase in volumetric and contact stresses and a magnification in the frequency of failures, of which approximately 30% account for only two parts, such as the cylinder block (CB) and the distributor [1,2]. These failures are of two types: wear of contact surfaces and fatigue destruction. The reduction of wear in APHM is achieved by providing lubrication of contact surfaces with working fluid (mineral oil) and using up materials of details with high antifriction properties [3,4]. It should also be taken into account that when the nominal pressure increases, the clearances in some connection increase. This leads to an increase in leakage and a decrease in the volumetric efficiency of the APHM [5,6]. Because of the large areas of the friction surfaces, to minimize wear, CB from a tin bronzes are produced. At high levels of cyclically changing pressure, the relatively low strength of bronze is the cause of CB fatigue destruction. The complexity of the multiply connected CB design does not allow to determine its stress-strain state and to establish the fatigue resistance characteristics by traditional analytical methods.
This article proposes a numerical-analytical method for calculating of CB APHM for fatigue resistance, which allowed find its endurance limit. On the basis of this method, it is possible to construct a fatigue curve for СВ, and the resource may be evaluated at given loading regimes.

ANALYSIS OF THE LITERATURE DATA AND STATEMENT OF THE PROBLEM
Stresses that arise in unevenly loaded details of complex shape are often localized in small internal volumes and it is only experimentally possible to effectively determine their effect on strength and fatigue resistance. Such works were carried out for local sections of notched samples of material [7], individual structure units [8,9] and vehicles [10,11] and even for multi-storey buildings [12] on special stands. Making stands and performing experiments are requires considerable expenditure of resources and time, therefore, in modern conditions, the analysis of strength and fatigue resistance is carried out numerically more often.
For the details by the finite element method (FEM) in a three-dimensional formulation are DIAGNOSTYKA, Vol. 21, No. 1 (2020) Khomiak Y, Kibakov O, Medvedev S, Nikolenko I, Zheglova V.: Endurance limit of the axial-piston… 72 determine all components of the stress tensor. The characteristics of strength and fatigue resistance are established for samples of the detail material on standard stands experimentally. Then, analysis of strength or fatigue resistance of a given detail is performed using approved calculation algorithms [13][14]. For details of simple shapes (rods, some plates), the methodology for calculating fatigue resistance in the current regulatory documents is proposed [15,16]. These documents cannot cover a wide range of details complex shape, including CB, so special methods to calculate them are used [17][18][19][20]. The CB APHM to a complex details class is related. The main load for the CB is the internal pressure p acting on the parts of the axial cylindrical channels. The length of these parts is determined by the position of the pistons moving in the axial direction z. This type of loading leads to the appearance in the CB of a cyclically varying uneven stresses. The maximum values of these stresses in the short region of the inter-cylinder partition (ICP) are localized.
The aim of this paper is to create a method for calculating the fatigue resistance of details with the working stress localization, which is applied to CB APHM. The basis of the method is the statistical theory of the similarity of fatigue failure (STSFF) of Serensen-Kogayev [22], on which GOST 25.504-82 [16] is founded.
To achieve this purpose in the work of the following tasks: -to build a fatigue curve and establish the endurance limit of the material CB, tin bronze CuSn12; -to determine the distribution and the stresses magnitude arising in the CB at the most unfavorable pressure propagation in its axial cylinders by the FEM; to demonstrate the procedure of calculating the fatigue resistance for details with a significantly uneven distribution of stresses on the example of determining the endurance limit of a CB APHM. Maximum value of first principal stress σ-1

List of symbols
Endurance limit of the sample −1D Endurance limit of material at symmetrical stress cycle −1e Average value endurance limit of a detail at symmetrical cycle 0D Endurance limit under pulsation σu Ultimate strength , σW Weibull's distribution parameters

THE BASIS DESIGN DEPENDENCE
The median value of the endurance limit of details is determined by the expression The standard contains recommendations for estimating the parameters of equation (1): a median value of the endurance limit of the sample 1 σ − and the characteristics of the detail Kt, νσ, KV, KA, KF [16]. The relative similarity criterion of fatigue failure is a ratio The parameter of the detail L means the length of the perimeter of the dangerous section or the part of it where the maximum stresses act.
The main advantage of STSFF is the assertion that samples and details with different sizes and shapes, but with the same criterion values ) / ( G L have equal endurance limits. Consequently, the limit of endurance of real details can be determined from the results of tests of laboratory samples.

FATIGUE TESTS OF SAMPLES OF CB MATERIAL
To determine the characteristics of the fatigue resistance of bronze CuSn12, from which the CВ is made, fatigue tests of corset (smooth) and notched samples were performed ( Fig. 1).
Here is also presented the theoretical stress concentration factor Kt (CSCF) for a cylindrical surface, which is determined by the Neuber formulas for the notch a hyperbolic profile [16] where the theoretical stress concentration coefficients for shallow and deep notches are respectively equal ρ 2 1 ; a=0.5dthe radius of dangerous section; tthe depth of the notching (Fig. 1). The design radius of curvature is ρ=1.05R because a hyperbolic notch profile is replaced by an equivalent V-notch rounded profile [21]. Similarity criterion of fatigue failure G L / is set according to [16]. To prepare the samples, 20×20 mm blocks were used, which were obtained from the longitudinally cut billet for a CB with a diameter of 120 mm, followed by turning and grinding to a surface finish Ra0.4 μm (Rz1.6 μm). The manufacturing process was kept constant for the whole batch.
The tests were carried out with a pure circular bending with a loading frequency of 50 Hz. The test base was NB=10 8 cycles. To describe the fatigue curve, the power equation where σ and Nthe current value of the test stress and the corresponding durability before failure; m and Cthe parameters of equation (6); 1 σ −the endurance limit of the material for a symmetric stress change cycle; NGthe abscissa of the fracture point of the fatigue curve plotted in double logarithmic coordinates.
The fatigue curves for smooth samples (round symbols) and for notched samples (quadrate symbols) constructed in double logarithmic coordinates according to the results of fatigue tests is presented in Fig. 2.
Dark symbols correspond to broken samples; light ones correspond to samples that did not crumble until the base of test 10 8 cycles. As a result of fatigue testing and correlation and regression analysis, the characteristics of eq. (6) are obtained, which are presented in Table 2. Limits of endurance are defined as stress levels at which three of the two samples did not collapse.  The calculated large values of the modulus of correlation coefficients r indicate that there is a strong correlation between logσ and logN. Therefore, the use of equation (6) for the study bronze CuSn12 is quite justified.
To solve the problem considered in the article, it is necessary to determine the coefficient of sensitivity of the metal to stress concentration and the influence of absolute dimensions νσ. For this we use the formula [22] ) ν 36 . , see Table 1 and Fig. 2

DETERMINATION OF THE STRESS STATE OF THE CB
The object of research in this work is the CB of a seven-piston APHM 210.25 (Fig. 3). The quantity of cylinders in the injection zone is variable, 3 or 4, respectively in the suction zone -4 or 3.
The large volumes of axial cavities being loaded and the variability of pressure along their length cause a complex, cyclically variable nature of the stress fields that in the CB arise. Known models for the analytical calculation of the CB are either too simplistic (the cylinder was considered as a pipe of constant thickness, loaded with internal pressure) or bulky. Therefore, for stress field calculations FEM is used. These calculations at a pressure p=25 MPa for a various quantity of axial cylindrical holes in the injection zone were performed. It is established that the greatest magnitude of stresses is achieved when four axial channels are located in the injection zone. Some results of the FEM-calculation are shown in Fig. 4. It is installed that the maximum values of the first principal stress of σmax=102104 MPa act in the ICP on a section about 3 mm long. The region of action of stresses with a level of 94104 MPa has a length of about 22 mm, Fig. 4a.
It should be borne in mind that it is not possible to obtain exact values of stresses in many cases. Therefore, the recommendation of the standard that as L should be assigned as a "part of the perimeter of the working section, which is adjacent to the zone of increased tension" in this case does not allow unambiguous determination of the value of this parameter. Below we propose a calculation method for determining the length L, based on the positions of the STSFF.
During APHM operation, stresses in the CB are changed by a pulsating cycle, whereas the STSFF methodology establishes procedures for calculating the fatigue resistance characteristics for the symmetrical stress cycle. Therefore, for further calculations, the resulting stress values are reduced to an equivalent symmetrical cycle.
Calculations The stress diagrams (9) and (10) are shown in Fig. 5. The dots represent the stresses recalculated from formula (8).
Stresses, the value of which is below the boundary values u=55 MPa, do not lead to the accumulation of fatigue damages and into calculations are not taken. For the cross-sectional of the ICP the equation of surface of the equivalent stresses is obtained by combining expressions (9) The surface of equivalent stresses (11) is shown in Fig. 6.
The stresses in the longitudinal section of the partition σ-1е(x, z) accept the maximum values at x=0.
When calculating the relative stress gradient G , from these values we should take the largest,  (11); 2the boundary plane u=55 MPa.

DETERMINATION OF THE EQUIVALENT LENGTH Le OF THE DANGEROUS ZONE
For CB and similar details in which the stresses are distributed unevenly the value of L, determined by the recommendations of [16], may be unreasonably low, down to the level zero. To solve such problems, we propose a method of replacing the parameter L by its equivalent of Le. The equivalence criterion is the coincidence of the distribution functions of the limits of endurance of a real detail (CB) and her analogue, which is characterized by the constancy σmax on the length of Le. In accordance with the STSFF, the equivalence condition is determined by the similarity equation in the form [22]: Equation (12) is solved with relation to the length L1e. With the ω=2.51 found above and the calculated value of the double integral in the formula (12), I2=5645 for is ξ= σmax /u=1.33, When calculating the equivalent length according to the rules of STSFF, it is necessary to sum the lengths of all equally hazardous areas. The localization of dangerous tensions after the transition of a rotating CB through an upper dead center arises too on the reverse surface of this ICP, see Figures 4c and 6. This situation will take place for all partitions. Consequently, the value of the calculated equivalent length will be Le=2·7 L1e=175 mm for the CB of seven-piston APHM.

THE ENDURANCE LIMIT CALCULATION BY THE SIMILARITY EQUATION
For the CB was obtained the value of the parameter 24. The values of the remaining parameters of equation (1) are found. The parameter КF, which takes into account the surface roughness, at Rz=1.6 μm is determined by the formula [16] 99 . 0 log 1 20 The coefficient of anisotropy (at σu<600 MPa), KA=0.9. The hardening coefficient KV=1 (no hardening) [16].
According to formula (1), 4 .  [23]. For CB seven-piston APHM the maximum stress in her ICP is defined as a linear function of pressure, σ=5.16p [24]. Consequently, the experimentally established endurance limit of the CB is 0 σ =93.9 MPa. Consequently, the calculated endurance limit is less than the experimentally established value by 7.9%.
The error estimation of the calculated value of the endurance limit D 1 σ − was performed for the probability of non-destruction of P=0.9 (90%), according to the standard method [16], which we used in [25]. The coefficient of variation of this quantity is determined taking into account the errors of individual parameters DIAGNOSTYKA, Vol. 21, No. 1 (2020) Khomiak Y, Kibakov O, Medvedev S, Nikolenko I, Zheglova V.: Endurance limit of the axial-piston… The calculations of stress max σ using the FEM were performed. The finite elements of the optimal form, ten-node tetrahedrons were used. Their number was increased to 10723 units, which for stress The coefficient of variation of the CB material endurance limit has a range of 0.04-0.10 [16]. In the calculation takes the average value 07 . 0 The coefficient of variation for Kt, as a function of the curvature radius 0 ρ of the stress concentration zone, is determined by the formula [16] where t K and 0 ρ are, respectively, the average values of TCSF and the radius of curvature of the stress concentration zone (i.e. in a dangerous cross section); is determined by the geometry of the detail in the stress concentration zone. In the considered problem, this is the zone of the minimum thickness of the jumper, Fig. 4c, which is presented as a plate with bilateral cuts at tension. For the ) ρ ( and by the formula (4) 3 1 = t K obtained. For the second parameter, the formula from [16] is used The calculation of the derivative in equation (14) for axial hole with diameter 25Н7 It follows from the structure of formula (13) that the coefficients of variation for the symmetric and pulsating cycles will be the same, MPa can be recommended for designing a wide class of complex shape details with an inhomogeneous distribution of peak stresses. This technique can be useful in finding the optimal structural forms of new details.

DISCUSSION OF THE RESEARCH RESULT
Existing normative documents do not contain recommendations for calculating the characteristics of fatigue resistance of complex shape parts in which significant local stresses arise. Here is a method for estimating the endurance limit of this type of details, confirmed by testing the CB APHM resource. This method should be regarded as an extension of the STSFF provisions to details with a localized stress distribution. The fatigue tests of smooth and notched samples from the CB material made it possible to construct fatigue curves for bronze CuSn12 and determine the parameters of the similarity equation u, ω and νσ. A necessary additional element of the procedure is the finite element analysis of the stressed state of the detail under investigation, on the basis of which are defined the gradient of the first principal stress, the stress concentration coefficient and the equivalent analog of the parameter L. This made it possible effectively the fatigue fracture similarity