Cylindrical Fretting And Delamination : Axisymmetric Static And Dynamic Analysis
Axisymmetric analysis of cylindrical contacts is considered in the context of axisymmetric assemblies such as shrink-fits. Fretting fatigue induces sub-critical cracks along the contact interface of press fits especially when they are subjected to vibration. The surface and near surface stresses play a major role in the fretting fatigue crack initiation process. Assuming near surface contact stresses to be largely independent of the actual geometry of components in contact, half-plane analyses and experimental results obtained from a strip configuration are often cited in the literature to predict and understand crack initiation in the actual components (ASTM STP 1425). This thesis starts with half plane and strip models for cylindrical contact such as in a shrink fitted shaft. Different traction profiles underpinning a typical fretting contact constitute a study of different geometrical parameters and friction coefficients. The cylindrical shrink fitted contact is considered using mixed boundary formulation. The different cases of contact such as full slip, partial stick-slip and full stick are considered. A formulation for cyclically varying tractions is attempted using dynamic elasticity. Finally, the problem of cylindrical cracks is highlighted to understand interface delamination in a fiber reinforced composite. Stress functions in conjunction with Fourier transforms are used for analysis. Dynamic potentials based on Helmholtz decomposition are used for dynamic loading.For static loading Love’s stress function is used for axisymmetric problems while Airy’s stress function is used for 2D problems. Solution procedures for solving traction boundary and mixed boundary conditions are described. Preliminary experiments are described to appreciate the contact stresses and crack initiation in cylindrical contact. Photoelastic fringes in a cylinder under a band of pressure illustrate fretting contact stresses concentrated close to the surface with the core of the cylinder relatively unstressed. Further, some material testing experiments using a specially designed cylindrical fretting rig demonstrated typical features of fretting fatigue crack initiation for providing the theoretical motivation. Fretting fatigue induces the initiation of a number of sub critical cracks along the contact interface of components in mechanical assemblies especially under vibration. The dominant crack among the initiated cracks may grow in size to the critical length in the presence of bulk cyclic loading finally resulting in fracture of the entire component. Fretting fatigue leads to unexpected failure of the component well below the expected life. It is therefore, critical to analyse, detect and control fretting. The blade root-disk joint in gas turbines as a critical example of fretting fatigue has spurred extensive research effort. There is relatively little literature available on cylindrical fretting in shrink fitted joint focused in this thesis. Analytical solutions for static fretting tractions are presented using both axisymmetric and plane elastic stress functions for later comparison. While Fourier transforms in conjunction with Airys stress functions are exploited for attacking plane problems, Loves axisymmetric stress functions are explored for cylindrical fretting. Near surface stresses are of great interest in fretting fatigue research. Although two dimensional models provide general understanding of stresses caused during fretting, these models become inadequate to explain the interaction of local stresses with the bulk stresses inevitably present in cylindrical components. Global stress analysis tools are desirable for estimating the fatigue life of components experiencing fretting. While numerical techniques immensely aid fatigue life estimation they have their limitation when it comes to coated components. Stress analysis of coated cylinders unveils the intricate influence of the elastic mismatch as well as the width of the loading for varying friction coefficients. Comparison of results obtained from axisymmetric elasticity with plane elasticity is discussed in detail. The validity and scope of relying on plane fretting results to cylindrical fretting contacts is examined by comparing the results obtained for three different traction profiles. Fretting is generally modeled as a stress boundary value problem wherein the normal and frictional shear stresses are prescribed on the cylindrical surface. In reality fretting generally turns out to be a mixed boundary value problem with unknown regions of stick and slip requiring prescribing traction and displacement simultaneously. This belongs to a formidable class of unsolved contact mechanics problems in cylindrical axisymmetric elasticity. The famous spherical axisymmetric Hertz problem has no cylindrical counterpart except in the limiting case of a cylinder of large radius. These aspects are investigated for studying the hub-shaft interfacial geometry. A conformal contact profile is considered to model a shrink fit; the contact pressure is zero at the ends of contact. The case of full slip condition is analysed assuming a frictionless contact. With friction, partial stick-slip condition is analysed. The unknown contact traction is resolved in terms of Chebyshev expansions whose unknown coefficients are solved using Schmidt method. The unknown contact length and stick zone length are determined through an iterative procedure. A rigid uneven undulating axisymmetric hub in total contact over an elastic shaft under full stick condition is analysed for obtaining the near surface stresses for a given value of hub penetration. Even though the stresses oscillate in fretting, almost all the analyses reported in the literature use static formulation. Understanding this need, a dynamic analysis for modeling fretting of a cylinder subjected to harmonic pressure and shear is attempted. The Pochhammer dispersion relation becomes a prerequisite for a dynamic analysis. The results show that the stresses do not decay away from the contact, in contrast to the static results. This shows the propagation of stresses along the axial direction. Further extension of the dynamic analysis to a layered cylinder is also described. The results obtained on contact stresses and contact tractions under the cylindrical contact represent a significant advance to the literature for modeling fretting fatigue crack initiation and propagation. Formulating cylindrical crack problems is somewhat similar to cylindrical contacts. Such cylindrical cracks arise from the debonding along the fiber-matrix interface of a composite. A unified formulation for the problem of a pressurised cylindrical crack as also a pair of 2D parallel cracks in infinite media is attempted using Love’s stress function in conjunction with Fourier transforms. The results obtained for stress intensity factors, strain energy release rate, mode mixity, crack opening and sliding displacements are compared with that of a 2D pair of parallel cracks obtained using the unified formulation. The asymptotic situation of a large crack length to spacing ratio is examined in detail. In the case of a pair of parallel cracks, this implies a single crack in mode-I as far as the total energy release rate is concerned while at the same time retaining an asymptotic value for the mode mixity. This unique feature of near field mixed mode blending smoothly to mode-I in the far field is also seen for the stress field around a symmetrically branched crack. Thus, this thesis presents a collection of cylindrical elastostatic and elastodynamic axisymmetric solutions to provide better understanding of fretting and delamination problems encountered in press fit assemblies.