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This book focuses on surface layers fracture of cyclical contacting bodies (machine parts). Calculation models and calculating procedures of stress-strain states of cyclically contacting solids with cracks, are included. Recommendations for the optimization of operating parameters of joints (contact stresses magnitude, friction/lubrication conditions, materials crack resistance etc) for elements of rolling pairs (wheel–rail systems, backup roll – working roll of rolling mills etc.) and some fretting pairs are formulated.

### 1. Contact Lifetime Estimation Methods for Tribojoint Elements. A Survey

Abstract
In this chapter, we give a survey of the available literature dealing with the methods aimed at the evaluation of the contact lifetime of elements of the rolling and fretting couples. We analyze the results of investigations in the scientific directions that can be regarded as basic for the proposed methods, namely, in the fatigue fracture mechanics and in the theory of contact problems of the mathematical theory of cracks. We also consider the applications of the contact fatigue approaches to the problems of the contact lifetime evaluation of the tribojoint elements.
Oleksandra Datsyshyn, Volodymyr Panasyuk

### 2. Calculation Model for Estimation of Cyclic Contact Lifetime of Body with Cracks

Abstract
This chapter contains the key elements of the theory of fatigue fracture, namely, the main stages and fatigue characteristics of the materials; the criteria and diagrams of fatigue fracture, and the characteristics of fatigue crack growth resistance of the materials. We present a brief description of the specific features of cyclic contact interaction accompanied by the crack initiation in the nearsurface contact zone: rolling, rolling with sliding, fretting fatigue, pulsed contact, friction fatigue, etc. We also describe typical contact fatigue defects formed by cracks on the working surfaces of the tribojoint elements. In the same chapter, we propose a new computational model for the investigation of the fracture processes and evaluate the contact residual service life of the bodies (tribojoint elements) under the conditions of their cyclic interaction.
Oleksandra Datsyshyn, Volodymyr Panasyuk

### 3. Singular Integral Equations for Some Contact Problems of Elasticity Theory for Bodies with Cracks

Abstract
This chapter is devoted to the mathematical method of modeling of the deformation and fracture process for elements of moving and fixed joints (tribojoints). In this chapter, we deduce singular integral equations of some contact problems of the elasticity theory for bodies with cracks. The Kolosov-Muskhelishvili complex potentials for the analyzed problems are represented in the form of integral representations with Cauchy-type kernels with respect to the derivatives of the discontinuities of displacements on the crack contours. In the general case, the problems are reduced to systems of singular integral equations of the first kind. We propose singular integral equations (SIE) for the elastic half-plane weakened by a system of curvilinear cracks under the action of various model contact loads applied to the boundary of the half-plane. We briefly describe the Gauss-Chebyshev method of mechanical quadratures that enables one to efficiently construct the numerical solutions of these SIE. We also deduce the relations for the stress intensity factors at the crack tips can be expressed via the solutions of SIE for inside and edge curvilinear cracks in the half-plane. In this chapter, we present both known results available from the literature and new results.
Oleksandra Datsyshyn, Volodymyr Panasyuk

### 4. Rolling Contact Fatigue

Abstract
This chapter contains the results of investigations of the fracture processes (paths of crack propagation) for bodies under the conditions of rolling (rolling with sliding). These results were obtained by using the model proposed in Chap. 2, the solutions of singular integral equations for the corresponding problems, and also the results of evaluation of lifetime under the conditions of contact fatigue. We study and describe the specific features of formation of the typical defects usually observed in rolling bodies, such as pitting, spalling, squat (“dark spot”), checks depending on the operating parameters of the analyzed rolling couple and the characteristics of cyclic crack growth resistance of the corresponding materials. We present examples of evaluation of the residual contact durability of rail, wheel, and roll steels according to the criteria of pitting and spalling formation. The contact fatigue curves (i.e., the dependences of the number Nf of rolling cycles on the maximum value of contact pressure p0) are plotted by using the criteria of pitting or spalling formation on the rolling surfaces. We also give some recommendations useful for the engineering practice.
Oleksandra Datsyshyn, Volodymyr Panasyuk

### 5. Fretting Fatigue Fracture

Abstract
This chapter deals with the investigation of contact interaction of two bodies under the conditions of fretting fatigue. We study the fracture processes in the material (and construct the paths of crack propagation) in the zone of cyclic contact of two bodies under the conditions of fretting fatigue, in particular, depending on the friction coefficient and stick/slip conditions between the bodies, the form of the base of counterbody (the type of modeling contact loading), etc. We also present some examples of evaluation of the residual service life for turbine blades of the GTE (gas-turbine engine) made of TS-5 (TC-5) titanium alloy.
Oleksandra Datsyshyn, Volodymyr Panasyuk