Stress and failure probability analysis for a transversely isotropic, brittle, elastic cylinder subjected to internal pressure and axisymmetric thermal loading

Procedures are developed for the determination of the stresses in and thence the probability of failure of a transversely isotropic cylinder made of a brittle material and loaded by an internal pressure and an axisymmetric radial temperature gradient. As examples of the application of the procedures a cylinder is analysed first with isotropic material properties, then with various degrees of anisotropy including both the “fibrous” and “laminar” types. The treatment is non-dimensional; results are presented graphically in the form of failure probability “contours”. For the dimensions and materials considered it is shown that the probability of failure is affected only slightly by the fibrous form of anisotropy but markedly by the laminar form when the thermal loading predominates.     

Prediction of cutting forces and machining error in ball end milling of curved surfaces -II experimental verification

This paper presents the results of a series of experiments performed to examine the validity of a theoretical model for evaluation of cutting forces and machining error in ball end milling of curved surfaces. The experiments are carried out at various cutting conditions, for both contouring and ramping of convex and concave surfaces. A high precision machining center is used in the cutting tests. In contouring, the machining error is measured with an electric micrometer, while in ramping it is measured on a 3-coordinate measuring machine. The results show that in contouring, the cutting force component that influences the machining error decreases with an increase in milling position angle, while in ramping, the two force components that influence the machining error are hardly affected by the milling position angle. Moreover, in contouring, high machining accuracy is achieved in “Up cross-feed, Up cut” and “Down cross-feed, Down cut” modes, while in ramping, high machining accuracy is achieved in “Left cross-feed, Downward cut” and “Right cross-feed, Upward cut” modes. The theoretical and experimental results show reasonably good agreement.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

New results for the fretting-induced stress concentration on Hertzian and flat rounded contacts

Recent work on fretting fatigue has emphasized the role of stress concentration on fretting damage, while previous work had concentrated on empirical parameters to assess influence of fretting on fatigue life. In particular, analogies with fatigue in the presence of a crack or a notch have been noticed, suggesting that the stress field induced by frictional contact per se may explain the reduction of fatigue life due to fretting.In the paper, new analytical and numerical solutions are produced for the stress concentration induced in typical fretting contacts involving the Hertzian geometry or the flat punch with rounded corners in view of application to the dovetail joints. Normal and tangential load (in the Cattaneo–Mindlin sense) is considered with “moderate” or “large” bulk stresses.     

Specific Features of the Behavior of the Coercive Force in Low-Carbon Plastically Deformed Steels

The relationship between the coercive force in low-carbon steels under plastic extension and compression and the values of deformation and actual and residual stresses are studied. This relationship is investigated for both “ slow” loading (when an equilibrium deformation is attained for each load value) and “fast” loading (when such equilibrium is not attained). It is shown that (i) a comparatively small increase in the coercive force in a loaded condition is due only to an increase in the density of dislocations in the process of plastic extension; (ii) a significant steep increase in the coercive force accompanying removal of the load from a plastically stretched specimen is fully due to residual compression stresses; (iii) the values of the coercive force under “slow” and “fast” loading are significantly different in the region of small deformations less than 2.5%; (iv) these values are close to each other in the loaded state for all deformations up to 10%; (v) a relief of the compression stress that creates plastic deformations causes a steep decrease in the coercive force that is as large as its increase following relief of plastic extension; this is explained by the emergence of a significant residual tension stress. The obtained results are of importance for the use of the method based on measuring the coercive force to test steel structures under the conditions when plastic deformations develop.__________Translated from Defektoskopiya, Vol. 41, No. 5, 2005, pp. 24–38.Original Russian Text Copyright © 2005 by Kuleev, Tsar’kova, Nichipuruk.     

Incorporation of Lagrangian Multipliers into an algorithm for finding exact natural frequencies or critical buckling loads

An existing algorithm enables natural frequencies or critical load factors to be found with certainty when “exact” stiffness matrices are used. This algorithm is extended to permit Lagrangian Multipliers to be used to couple the “exact” stiffness matrices of component structures to represent connections between the structures. The new algorithm also permits coupling of the stiffness matrices for different assumed wavelengths of sinusoidal response of a given structure with the stiffness matrices of other structures to satisfy required constraint conditions. The algorithm applies to problems formulated using real or complex arithmetic.     

Crack propagation in non-proportionally scaled elastic structures

Scaling laws for load, stress, displacement and crack velocity are given for crack propagation in non-proportionally scaled elastic structures where the scaling factors for height, width, thickness and crack length can all be different. Problems are addressed where only one or two of the scaling factors are important, in contrast to geometrically similar scaling where all factors are equal. Examples discussed include pipelines and plate-like structures of ships or bridges. The conditions under which strength of materials design criteria, based on yielding, prove inadequate for such large structures are investigated. In addition, the validity of determining “brittle” fracture toughness by “equivalent energy” scaling procedures from small laboratory specimens is discussed.     

Influence of end-conditions during tube hydroforming of aluminum extrusions

Tube hydroforming experiments were conducted to develop the forming limit diagram of AA6082-T4 by utilizing three types of end-conditions: (i) “free-end”, (ii) “pinched-end” or “fixed-end” and (iii) “forced-end”. It was found that “free-end” hydroforming gives the lowest forming limits followed by “pinched-end” and “forced-end” hydroforming. It was noticed that the tube failure occurs within 5° to the extrusion weld in the “free-end” experiments, within 7° in the “pinched-end” condition and extended up to 10° in the “forced-end” hydroforming experiments. Finite element simulations were carried out to capture the effects of the weld geometry, weld mechanical properties and the end-conditions of the extruded tube on the maximum induced stress and location of the maximum von Mises stress. It was found that the anisotropy of the weld material and the end-condition used during hydroforming experiments have the largest influence on the failure location with respect to the weld center.     

Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure

The dynamic plastic collapse of energy-absorbing structures is more difficult to understand than the corresponding quasi-static collapse, on account of two effects which may be described as the “strain-rate factor” and the “inertia factor” respectively. The first of these is a material property whereby the yield stress is raised, while the second can affect the collapse mode, etc. It has recently been discovered [6,7]that structures whose load-deflection curve falls sharply after an initial “peak” are much more “velocity sensitive” than structures whose load-deflection curve is “flat-topped” (Fig. 1a); that is, when a given amount of energy is delivered by a moving mass, the final deflection depends more strongly on the impact velocity. In this paper we investigate strain-rate and inertia effects in these two types of structure by means of some simple experiments performed in a “drop hammer” testing machine, together with some simple analysis which enables us to give a satisfactory account of the experimental observations. The work is motivated partly by difficulties which occur in small-scale model testing of energy-absorbing structures, on account of the fact that the “strain-rate” and “inertia” factors not only scale differently in general, but also affect the two distinct types of structure differently.     

A numerical/experimental study of contact damage in non-planar porcelain/metal bilayers

In this report, we investigate and visualize the effect of shape irregularity on contact damage in a brittle coating on a stiff metal substrate. Hertzian contact damage in a dental porcelain layer of thickness between 0.25 and 0.75 mm, fused onto a Ni–Cr alloy substrate in both curved and planar geometries was studied with the aid of the finite element method and experimental investigation. Three failure modes were examined with varying porcelain layer thickness: cone cracking at the upper surface of the porcelain, median or interface cracking at the layer/substrate interface and plastic deformation below the contact area in the substrate. It is shown that curvature has very little effect on the initiation of surface cone cracks in this system, but substantial effect on the initiation of interface radial cracks. In particular, curvature reduces the critical load for the onset of interface cracks.     

Comparison of experimental and theoretical residual stresses in welds: The issue of gauge volume

Welding residual stresses have an effect on many aspects of the integrity of structures but are normally one of the largest unknown stresses. Residual stresses are difficult to measure and to estimate theoretically but are often significant when compared with the service stresses on which they superimpose. High tensile residual stresses can lead to loss of performance in corrosion, fatigue and fracture.In this research, measurement of residual stresses by the neutron diffraction technique is compared to an analysis of a sample geometry by theoretical finite-element procedures. The results indicate good qualitative agreement. One of the key issues in this comparison relates to what is termed “gauge volume” in the measurement technologies and what might be described as a “calculation volume” in theoretical approaches.     

A simple approach to estimate failure surface of polymer and aluminum foams under multiaxial loads

From mechanical point of view, it is required to have a criterion for evaluating the failure of cellular solids (foams) under multiaxial loads. Well-documented experimental results in the literature show foams could fail by several mechanisms, e.g., elastic buckling, plastic yielding, brittle crushing or brittle fracture. In the previous years, both theoretical and phenomenological approaches have been applied to obtain the failure surface of various foams. The purpose of this paper is to present a simple approach to estimate the complete failure surface of “non-textured” foams. The predicted results of polymer and aluminum foams are compared with the experimental results reported in the literature. It is found that three selected tests will be sufficient to estimate the complete failure surface of a foam. The recommended testing stress states are σ₁=σ₂=σ₃>0, σ₁=σ₂=σ₃<0, and σ₁=−σ₂=−σ₃ (or σ₁=−σ₂, σ₃=0).     

Geometry of “developable cones”

When a thin disc is supported on the rim of a bowl, and its centre is pushed down by a finger, it adopts a characteristic conformation, known as a “developable cone”, and sketched in Fig. 1(a): the main, broadly conical, shape can only form if about one-quarter of the disc buckles upwards. There is a curved intersection between the two parts, which takes the form of a crescent-shaped “crease” near its apex, but with the flanking regions less tightly deformed. The “developable cone” is a recurring motif in a wide range of physical situations—crumpling, buckling, draping—and its mechanics provides a key to understand the phenomena, whether the disc deforms in the elastic or the plastic range. The task of this paper is to study only geometrical features of the “developable cone”. The first step is to replace the actual crease (Fig. 1(a)) by an idealised “sharp” crease (Fig. 1(b)). The second step is to study the apparently “large-rotation” problem of kinematics by means of an adaptation of the classical “yield-line” pattern of folding, but with a crucial added constraint that springs from Gauss's analysis of inextensional deformation. We illustrate the method via a graded sequence of examples, and we close with a discussion.     

Coating thickness effects on initial wear of nitrogen-doped amorphous carbon in nano-scale sliding contact: Part I—in situ examination

This paper, the first of a two-part series, presents the empirical data obtained from in situ examination on the generation of wear particles on carbon nitride coatings by a spherical diamond counter-face during repeated sliding contacts. In particular, the effect of coating thickness, varying from 1 to 500 nm, on the generation of wear particles was examined.Based on the in situ examination, the shape transition maps for generated wear particles were obtained for carbon nitride coatings of various thickness. The results show that the critical number of friction cycles, Nc, for the transition from “no observable wear particles” to “wear particle generation” generally increased with increasing coating thickness. It was noted that up to 20 friction cycles, the maximum Hertzian contact pressure, P_max, for “no observable wear particles” regime can be increased from 1.39Y to 1.53Y if silicon was coated with carbon nitride coating thicker than 10 nm, where Y is defined as the yield strength of silicon.     

Nano-abrasion machining of brittle materials and its application to corrective figuring

A method of ultraprecision abrasion machining named “Nano-abrasion machining” is proposed for optical finishing of brittle materials. The fundamental characteristics and its applicability for corrective figuring to improve form accuracy of optics of brittle materials are investigated. It is experimentally ascertained that the material removal rate and surface roughness are suitable for optical finishing. However, the cross-sectional profile of the machined spot that is dependent on the collision angle is a combination of V- and W-shape, which is unsuitable for the corrective figuring. Therefore, circular motion machining is introduced and a preferable profile with an axis-symmetric V-shape is realized. The machining method is applied to corrective figuring of optical glass of BK7. The NC program is generated with a computer program developed by modifying the scanning motion and the form accuracy is predicted. According to the simulation results, corrective figuring is performed. The flatness is improved from PV = 151 to 29 nm. From the experimental results, it is clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials.     

Study of the tribologically transformed structure created during fretting tests

In fretting, without a lubricant, structures fail through wear involving debris. Various metallographic assessments show that this debris creation derives from a special structure, hereafter called the “tribologically transformed structure” (TTS). The latter is nucleated at the very beginning of a contact's life, and eventually masks the initial material's tribological behaviour. This TTS is very hard and brittle, and is quickly destroyed, which explains why TTS is rather difficult to observe. This study focuses on initiation mechanisms which, up to now, are still highly controversial. Large amounts of TTS are observed after fretting tests, which is the reason why they were chosen as the means of investigation. In this paper, the various possible creation mechanisms for TTS will be analysed. From this overview, two nucleation models are chosen: one is based on “mechanical alloying” considerations, and the other is linked to high-strain-recrystallisation phenomena. Transmission electron microscopy (TEM) investigations are used to check these assumptions.     

Analysis of plane and rough contacts, subject to a shearing force

We analyse the surface traction conditions induced in plane contact between two bodies whose surfaces are rough. It is assumed that the roughness may be idealised by a surface of regularly spaced cylindrical “bumps”, and the overall geometry may be in the form of a cylinder, flat ended punch or wedge. The stick-slip regime experienced by each individual asperity contact is found, and hence it is shown how the applied shearing force produces concentrated regions of surface damage. Conditions for crack initiation are then discussed, and compared with equivalent results found for nominally smooth contacts.     

An artefact for traceable freeform measurements on coordinate measuring machines

The present paper describes an artefact based approach to obtain traceability of freeform measurements on coordinate measuring machines. First, the requirements for the traceability of freeform measurements and a strategy for the development of a feasible solution are presented. A new concept of artefact, called the “Modular Freeform Gauge” (MFG) has been developed. It is based on physical modeling of a given freeform surface by a combination of items with regular geometry, well calibrated on their dimensions and form. The relative position is accounted for during the procedure; this information is used to generate a “calibrated” CAD model as reference for freeform measurements. The architecture of the artefact, its collocation in the traceability chain, and the calibration procedure are described.Finally, a procedure for the uncertainty assessment of actual freeform measurements is presented. The work here described has been focused on implementation of the uncertainty assessment procedure for freeform measurements on turbine blades. A task-specific Modular Freeform Gauge was developed for this application.     

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t^1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t^1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t^2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t^2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t^1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.     

Some observations on the CLNA model in fretting fatigue

Using the Atzori–Lazzarin criterion, the author has recently proposed a unified model for Fretting Fatigue denominated Crack-Like Notch Analogue—CLNA model, considering only two possible behaviours: either “crack-like” or “large blunt notch”. In a general FF condition, the former condition is treated with a single contact problem corresponding to the MIT Crack Analogue (CA) improved in some details also by the author. The latter, with a simple peak stress condition, i.e. a simple Notch Analogue model, simply stating that below the fatigue limit, infinite life is predicted for any size of contact. In the typical condition of constant normal load and in phase oscillating tangential and bulk loads, both limiting conditions are immediately written, and the CLNA model permits to collapse the effect of the contact loads on a single closed form equation (differently from many other models which do not permit this flexibility). For not too large contact areas (“crack-like” contact) no dependence at all on geometry is predicted, but only on 3 load factors (bulk stress, tangential load and average pressure) and size of the contact. Only in the “large blunt notch” region occurring typically only at very large sizes of contact does size-effect disappear, but the dependence on all other factors including geometry remains. The model compares favourably with some experimental results in the literature. In this paper, some aspects of the CLNA model are further elucidated.