Three-dimensional Elastic Stress Research Articles

3D elasticity solutions for stress field around square hole in functionally graded plate

Since the square holes are widely used in many structures and machines, the purpose of this work is to obtain analytical solutions for the three-dimensional (3D) elastic stress field of transversely isotropic functionally graded (FG) infinite plates containing a single square hole with small round corners subjected to far-field loads or on the edges of the holes. Based on the generalized England-Spencer plate theory, material parameters can vary continuously and arbitrarily along the thickness direction of the plate, the solutions are obtained by using the conformal mapping technology combined with the Cauchy integral method. Concentration factors of internal forces Kij around the square hole are explicitly expressed. The obtained explicit expressions of solutions can be validated against the classical two-dimensional (2D) solutions when the material degenerates into an isotropic and homogeneous material. The detailed numerical examples are given to further investigate the influence of several parameters such as the material gradient factor and load conditions on the stress field around the hole. The present 3D analytical solutions can not only provide an in-depth understanding of the 3D stress field of FG plates with holes, but also be utilized as a benchmark to verify the numerical solutions of the same problems.

Shakedown analysis of ballasted track structure using three-dimensional finite element techniques

Shakedown analysis is an attractive method for determining the capacity of geostructures to sustain repeated loads involving a large number of cycles, e.g. rolling and sliding train wheel loads. Its main advantage is that it comes at a significantly reduced computational cost, compared to standard time-domain analyses. An essential component of shakedown analysis is the derivation of closed-form solutions to compute stresses due to the external repeated loads, a task that is not always feasible for complex problems as the ballasted rail track discussed herein. To tackle this, we present in this paper the use of finite element tools to obtain a quasi-lower-bound shakedown load numerically. The proposed method is based on the computation of the three-dimensional elastic stress field numerically, and the estimation of the shakedown load iteratively via an optimisation subroutine implemented in ABAQUS. Following a short presentation of this concept, we compare the elastic stress fields from models featuring varying degree of complexity, with the aim of identifying an optimal discretisation of the problem. This approach can be used for optimising the design of ballasted track structure, and this concept is briefly presented via a parametric study.

The Possibility of High-Speed Laser Glass Cutting

In recent years, the appearance of high-power fiber lasers, more efficient and easier to operate than other types of lasers, made it necessary to conduct systematic studies and develop a technology for cutting glass by the method of controlled thermal splitting. The paper considers the question of the possibility of high-speed cutting of glass by modern lasers with a power of up to several tens of kilowatts in the near-IR radiation range. In order to achieve the goal set in the work, the problem of determining the field of elastic stresses arising in a glass plate when it is heated by laser radiation has been solved. The method of investigation is based on the theory of thermoelasticity. We have proposed a mathematical model for calculating the three-dimensional temperature distribution and elastic stresses caused by the irradiation of the glass by a moving laser beam and have presented the results of numerical simulation of temperature fields and elastic stress profiles at different irradiation spot velocities. At high radiation powers, the maximum achievable cutting speed of glass is shown to be directly proportional to the power. The data on the speed of Bohr glass at low radiation powers (less than 400 W) available in the literature we have extrapolated to a high power (up to several tens of kilowatts). The investigation shows that the cutting speed can reach 70 cm / s at a laser radiation power of 20 kW. For cutting large-sized glass sheets in mass production, the fiber ytterbium lasers having a wavelength of 1.065 m are recommended. These lasers meet the requirements for power level, irradiation stability and service life. In addition, they have the ability to transport radiation through a fiber optic cable and the ability to automatically control the spatial location of the radiation spot. The results obtained can be used to develop new technologies for cutting glass on industrial lines for the production of float glass, in the process of cutting glass sheets and cutting 3D articles. The method of laser controlled thermal splitting, in comparison with other methods of laser cutting glass, is more efficient and stable, and also has a much wider range of applications, because allows increasing in several times the thickness of the divided glass.

Inclined Hole Under Different Loading Conditions: A Review of Recent Results

Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness effect. The stress results are compared with those provided by a recent theory which reduces the 3D governing equations of elasticity to a differential equation system, which includes a biharmonic equation and a harmonic equation. They provide the solution of the corresponding in-plane and out-of-plane notch problem, respectively, and have to be concurrently satisfied. Comparing numerical results and theoretical stress distributions, a good agreement is found.

Analytical solution for the three-dimensional stress fields in anisotropic composite bimaterial corners

The paper presents a theoretical treatise for the three-dimensional elastic stress distributions close to anisotropic bimaterial corners according to which the 3D governing equations are successfully reduced to two uncoupled equations in the two-dimensional space. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. With the new theory an analytical solution for the stress fields for bimaterial corners in thick anisotropic composite plates is presented and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE analyses carried out on bimaterial composite plates and joints.

Comparison of stress singularities of kinked carbon and glass fibres weakening compressed unidirectional composites: a three-dimensional trimaterial junction stress singularity analysis

A three-dimensional eigenfunction expansion technique, based in part on separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion in conjunction with the Eshelby–Stroh formalism, is used to compute the local stress singularity, in the vicinity of a kinked fibre/matrix trimaterial junction front, representing a measure of the degree of inherent flaw sensitivity of unidirectional carbon-fibre-reinforced composites under compression. Micro-kinking, more prominent in the misaligned regions of carbon fibres, is caused by crystallite disorientations, as detected by the Raman and X-ray measurements, as well as dislocation glide in crystallites and intra/intercrystallite disorders. The present analysis explains the test results relating to propagation of failure from such discontinuities in a unidirectional composite under compression. Numerical results presented include the effect of fibre wedge aperture angle on the strengths of the mode I and mode II singularities. Of special practical interest is the comparison of the inherent flaw sensitivity of carbon/epoxy and glass/epoxy composites, because improvement of the compressive strength and kink toughness was earlier accomplished through commingling of highly anisotropic (and crystalline) carbon and isotropic (and amorphous) glass fibres at the tow level. Compression fracture of these composites can be fully explained and quantified by the present three-dimensional linear elastic stress singularity analysis-based method. Finally, numerical results, pertaining to the through-thickness variation of ‘stress intensity factor’ for symmetric uniform load and its skew-symmetric counterpart that also satisfies the boundary conditions on the top and bottom surfaces of a compressed composite monolayer, in the vicinity of a trimaterial junction front, are also presented.

Analytical study on elastic transition in short-fiber composites for plane strain case

An analytical approach for short-fiber-reinforced composites is developed for three-dimensional (3D) elastic stress field distribution subjected to an applied axial load. Two sets of exact displacement solutions for matrix and fiber, which are respectively called far-field and transient solutions, are derived based on the theory of elasticity. The superposition state of these solutions are then used to obtain the analytical expressions for the 3D stress field components over the entire composite system, including the fiber end region, through the adding imaginary fiber technique. The fiber/matrix 3D stress field components fully satisfy the equilibrium and compatibility conditions in the theory of elasticity. The stress field components also satisfy the overall boundary, interface continuity, and axial force equilibrium conditions. The analytical results obtained are then validated by finite element method modeling.

Three-Dimensional Finite Element Analysis of Single-Lap Joints: Effect of Adhesive Thickness and Poisson's Ratio

Three-dimensional (3D) elastic stress distributions in the vicinity of overlap corners of single-lap joints are investigated. A detailed 3D finite element (FE) model is carried out to study the intensity of the in-plane and out-of-plane stress distributions along the plate width direction. The effects of adhesive thickness and Poisson's ratio are also studied. The FE results show the presence of coupled modes at the overlap corners of the joint. In particular, sharp increment of out-of-plane fracture mode very near the lateral free surface of the joint is worth noting.

Three-Dimensional Analyses of Spur Gear Bending Stresses by Global-Local Finite Element Technique

Three-dimensional elastic stresses developed in the root fillet of spur gear tooth due to uniform or non-uniform load distribution are analyzed in detail using global-local technique. Results indicate that root stresses distribute always non-uniformly along tooth width, and the maximum critical root stress (CRS) and its position depend on the face width and load distribution form. The maximum CRS is always higher than the corresponding value obtained under plane assumption and the value on the free surface. So, it is dangerous to use the CRSs calculated in practical gear designs. This research provides knowledge of root stresses for engineer to guard against gear failure and to design for increased loading.

Three-dimensional elastic stress fields ahead of notches in thick plates under various loading conditions

The paper discusses the features of three-dimensional elastic stress distributions ahead of notches in finite thick plates under different loading conditions. It is proved that, under certain circumstances, the three-dimensional governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. The analytical frame is applied to a number of notched geometries, and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE models. Practical consequences on the expected crack paths are also documented for one of the considered models.

Rock mass response ahead of an advancing face in faulted shale

In this study, the rock mass response ahead of an advancing test tunnel in the Opalinus Clay at the Mont Terri Rock Laboratory (Switzerland) was investigated. Characterisation of the excavation-induced damage zone at Mont Terri is a challenging task due to the anisotropic and heterogeneous nature of the shale: pronounced bedding leads to intact rock anisotropy and prevalent small-scale tectonic shears lead to rock mass heterogeneity. Rock mass damage ahead of an experimental tunnel or niche was characterised through single-hole seismic wave velocity logging, borehole digital optical televiewer imaging, and geological drillcore mapping. Three-dimensional elastic stress analyses were completed and showed that rock mass degradation can be correlated to changes in the maximum to minimum principal stress ratio (i.e., spalling limit). Numerical results showed that close to the niche boundary, unloading lowers stress ratios, which correspond with decreasing seismic wave amplitudes and velocities; thus, indicating that strength degradation resulted from increasing crack-induced damage. Considerations of tectonic shears and distance from a previously stressed volume of rock were necessary in understanding both the damage state and extent ahead of the face. By integrating field and numerical data, the investigation showed that geological structures (i.e., bedding and bedding-parallel tectonic shears) were most influential near the entrance but played a lesser role as the niche deepened. Additionally, a portion of the niche is located in the perturbed zone of the intersecting Gallery04.

Effects of Dual-Pore Interaction on Three-Dimensional Elastic Stress Concentration Factor for Die Castings

Effects of Dual-Pore Interaction on Three-Dimensional Elastic Stress Concentration Factor for Die Castings

The effect of rubber contact on the fretting fatigue strength of railway wheel tire

The cause of the ICE train derailment, which occurred in 1998 at Eschede, was fatigue failure originating on the inside of the wheel tire. Rubber-sprung resilient wheels were used for the trailer cars. The wheel tire is mounted on the wheel disc. Thirty-four rubber pads were arranged between the wheel disc and the wheel tire. It was postulated that fretting fatigue between the rubber block and the inner side of the tire might have an influence on the initiation of the incipient crack. In order to clarify the influence of the rubber contact on the fatigue strength of the tire, fretting fatigue experiments under rubber contact conditions were performed. During the fundamental fretting fatigue test using bridge pads and small size carbon steel specimens, no typical fretting damage such as fretting wear and minute cracks were observed due to contact of the rubber. Stress conditions of the rubber-sprung wheel under vertical and lateral wheel loads were evaluated by a three-dimensional elastic stress analysis. Since the rubber is a super-elastic material, the Mooney–Rivlin model was used in the FEM calculation. It was found that the wheel tire is subjected to a cyclic stress during one revolution of the wheel and the maximum stress occurred at the center of the inner surface of the tire where the fatigue crack initiated. Fatigue strength of the wheel tire was determined by the rotating bending fatigue testing of specimens taken from the tire. It was found that the tire with an 862 mm diameter at a wheel load of 80 kN had a safety factor more than 3.5 from a fatigue limit diagram with a failure probability of 0.01. To confirm the fretting damage under the rubber contact and the result of the fatigue strength evaluation, fatigue tests of a full size wheel were made. After 20 million cycles at the wheel load of 280 kN, which was just below the endurance limit estimated by the endurance limit diagram, no fretting damage and no fatigue cracks were observed. The wheel was, however, fractured at 1.56 million cycles under the maximum load of 308 kN, which was just above the endurance limit. The estimation of the safety factor of 3.5 estimated from the endurance diagram was confirmed by the full size fatigue testing. It was concluded that there was no effect of fretting due to the rubber contact on the fatigue strength of the rubber-sprung single-ring railway wheel.

The Äspö pillar stability experiment: Part I—Experiment design

The Äspö pillar stability experiment was carried out to examine the failure process in a heterogeneous and fractured rock mass when subjected to coupled excavation-induced and thermal-induced stresses. The rock pillar was created by the excavation of two adjacent large-diameter boreholes. The pillar was loaded by a combination of excavation-induced stresses and heating of the surrounding rock by a rectangular heater pattern. The experiment was designed using the observation design method. The characterzation of the experiment volume showed that the experiment was located in a fractured water-bearing rock mass that was considered typical for the Äspö Hard Rock Laboratory (Äspö HRL) and for the Fenno-Scandinavian shield. Scoping calculations using two- and three-dimensional elastic stress analyses were carried out to reduce the uncertainty for the far-field in-situ stresses, establish the geometry for the access tunnel that would provide a elevated uniform stresses in the floor of the tunnel, and determine the optimum width of the pillar. It was concluded, based on the rock mass characteristics, that a 1 m-wide pillar formed by two 1.75 m diameter boreholes would meet the design objectives. Thermal modelling showed that thermally induced stresses in the pillar were adequate to elevate the pillar stresses above the stress magnitude required to initiate failure. Acoustic emission, displacement, and thermal monitoring systems were installed according to the experiment design without problems. No sensors were lost over the three month duration. After the experiment was completed, a laser scan of the pillar revealed the extent of the damaged pillar. The experiment commenced in January 2002 and was successfully completed in 2006.

Tensile testing of thin films supported on compliant substrates

Elastic properties of thin films are routinely measured by straining them on a flexible carrier substrate. Stress–strain characteristics for the film/substrate composite strip are obtained and a simple mixture theory is used to extract the elastic properties of the film by assuming a uniform one-dimensional tensile stress distribution within the film thickness. In this paper, we examine the validity of using this simple theory by determining the three-dimensional elastic stress field in a composite strip. Three sample geometries and loading configurations are considered. In case A, the film is coated on one side of the carrier substrate while in case B, the carrier substrate is coated on both sides. For both case, the external load is applied only on the substrate. In case C, the geometry was the same as case A, but the external load was applied to both the film and the substrate. For film to substrate moduli ratios less than 5, the mixture theory provided adequate results with errors less than 10% for all cases considered. However, for case A, the rule of mixtures theory tends to significantly overestimate the tensile stress in the film for moduli ratio exceeding 5. An error analysis is also presented which provides guidance regarding the minimum film thickness necessary to attain results with a given accuracy. Overall the results presented in this paper provide researchers with rules to design the sample geometry and film thickness to accurately determine the elastic properties of thin films using the composite film/strip experiment. These rules are exemplified by measuring the elastic properties of a thin sputtered Al film and a hexagonal honeycomb structured Si/polymer composite film produced through an evaporation induced self-assembly process. In both cases, films are deposited on a flexible Kapton substrate and strained using a home-built film tester.

An application of the implicit gradient method to welded structures under multiaxial fatigue loadings

This paper deals with the problem of fatigue strength assessment of welded joints subjected to multiaxial loading. Three-dimensional solid modelling and linear elastic stress analysis, by means of numerical methods, are used to investigate the local stress field at weld toes and roots, geometrically regarded as sharp notches. Starting from the stress field obtained from a linear elastic analysis and taking advantage of the so-called implicit gradient approximation, an effective stress index connected with the material strength is calculated. In particular, there will be an investigation into the possibility of applying the implicit gradient approach to welded structures, under both uniaxial and multiaxial loading conditions, by introducing a multiaxial criterion into the implicit gradient framework. The multiaxial criterion consists of an improvement of the well-established Crossland-like criteria. It will deal with multiaxiality caused by external loadings as well as multiaxial stress fields caused by severe stress raisers. In order to validate the devised approach, theoretical fatigue damage estimations are compared with experimental data. In particular, the proposed approach is applied to a series of applicative examples taken from scientific literature and related to welded joints subjected to uniaxial or in-phase multiaxial fatigue loading.

Three-Dimensional Elastic Stress Fields of Finite Thickness Plates with Elliptical Hole

Based on detailed three-dimensional finite element analyses, elastic stress and strain field of ellipse major axis end in plates with different thickness and ellipse configurations subjected to uniaxial tension have been investigated. The plate thickness and ellipse configuration have obvious effects on the stress concentration factor, which is higher in finite thickness plates than in plane stress and plane strain cases. The out-of-plane stress constraint factor tends the maximum on the mid-plane and approaches to zero on the free plane. Stress concentration factors distribute ununiformly through the plate thickness, the value and location of maximum stress concentration factor depend on the plate thickness and the ellipse configurations. Both stress concentration factor in the middle plane and the maximum stress concentration factor are greater than that under plane stress or plane strain states, so it is unsafe to suppose a tensioned plate with finite thickness as one undergone plane stress or plane strain. For the sharper notch, the influence of three-dimensional stress state on the SCF must be considered.

A Three-dimensional Stress Analysis of a Penny-shaped Crack Interacting with a Spherical Inclusion

A three-dimensional elastic stress analysis is performed on an infinite solid to study the interaction between a penny-shaped crack and a spherical inclusion. In our derivation, a two-step superposition scheme is utilized to obtain the stress field over the imaginary crack site. The Duhamel—Neuman analogy is employed to transform an elasticity problem of a heterogeneous solid into an equivalent problem of a homogeneous solid in which the inclusion is replaced by the void and the boundary conditions modified accordingly. The effect of the inclusion and the crack—inclusion interaction on crack propagation is interpreted in terms of the stress intensity factor for a penny-shaped crack. Finally, the proposed analytical approximations are compared with the noninteracting solution, the exact solution, and other theoretical approximations to validate the current framework.

Investigation of elastic stresses in an adhesively bonded single lap joint with functionally graded adherends in tension

In this study three-dimensional elastic stress state of an adhesively bonded single lap joint with functionally graded adherends in tension was investigated. The adherends compose of a functionally gradient layer between a pure ceramic (Al 2O 3) layer and a pure metal (Ni) layer. Stress concentrations are observed along the free edges of the adhesive layer and through the corresponding zones in the upper and lower adherends. The adhesive layer experiences stress concentrations along the left and right free edges in the horizontal plane, and the normal stresses and the shear stress σ xy are critical. Whereas the middle overlap region has a uniform low stress distribution the zones in the upper adherend corresponding to the left free edge of the adhesive layer and the zones in the lower adherend corresponding to the right free edge of the adhesive layer are subjected to higher stresses. The normal stress σ xx among the normal stresses and the shear stress σ xy among the shear stresses are dominant in both upper and lower adherends. The normal stress σ xx changes uniformly from compression in the ceramic layer to tension in the metal layer through the upper plate-thickness and from tension in the ceramic layer to compression in the metal layer through the lower plate-thickness. In the adhesive layer, the normal stress σ yy becomes peak at the left free edge of the upper adherend–adhesive interface and at the right free edge of the lower adherend–adhesive interface and then decreases uniformly across the adhesive layer towards the other adherend–adhesive interface. The functionally gradient region across the adherend thickness was modelled using the layers with the mechanical properties calculated based on the power law. However, a layer number larger than 20 has a minor effect on the through-thickness profiles and magnitudes of von Mises and normal stresses in both the adherends and the adhesive. In addition, increasing the ceramic phase in the material composition (compositional gradient exponent n) of the functionally gradient region does not affect the through-thickness profiles of von Mises and normal stresses in the adherends and adhesive whereas their magnitudes in the ceramic rich layer of both adherends and along the adherend–adhesive interfaces increase considerably. On the contrary, the layer number and compositional gradient exponent have an evident effect on the through-thickness profiles and magnitudes of the critical stress components in the adherends and adhesive layer of the functionally graded adhesively bonded joints.

A new analytical model for three-dimensional elastic stress field distribution in short fibre composite

A new analytical model has been developed for three-dimensional elastic stress field distribution in short fibre composite subjected to an applied axial load and thermal residual stresses. Two sets of the matrix displacement solutions, which were called respectively as the far-field solution and the transient solution, were exactly derived based on the theory of elasticity. These two sets of the displacement solutions were then superposed to obtain simplified analytical expressions for the matrix three-dimensional stress field components and the fibre axial stress field components in the entire composite system including the fibre end region with the use of the technique of adding imaginary fibre. The matrix three-dimensional stress field components satisfy exactly the equilibrium and compatibility conditions in the theory of elasticity, the fibre axial stress field components satisfy the equilibrium requirements within the fibre and between the fibre and the matrix. The above stress field components also satisfy well the overall boundary conditions including the surface conditions, the interface continuity conditions and the axial force equilibrium conditions. The finite element numerical calculations were conducted to examine the validity of the analytical model. Considerable good agreements have been obtained between the analytical and numerical predictions for all the stress, strain and displacement components.