Uniform Stress State Research Articles

Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics

We consider the internal stress field of a three-phase elliptical inclusion bonded to an infinite matrix through an interphase layer when the matrix is subjected to remote uniform stresses. The elastic materials comprising all the three phases belong to a particular class of harmonic materials, and the formed interfaces are two confocal ellipses. A condition leading to internal uniform hydrostatic stresses is derived. This condition relates the two remote principal stresses with the geometric parameters (the thickness of the interphase layer and the aspect ratio of the elliptical inclusion) of the three-phase elliptical inclusion. When this condition is met, the hoop stress in the interphase layer along the entire interphase/inclusion interface is also uniform. Five special situations of practical importance are discussed in considerable detail to demonstrate the unique phenomena inherent in harmonic materials. Our discussions indicate that when this condition is met, it is permissible for the two remote principal stresses to have opposite signs and that for given geometric and material parameters, the remote loading ratio is no longer constant and multiple external loading states exist leading to internal uniform hydrostatic stresses. It is found that this condition can be written into a hyperbola for the two remote principal stresses when the interphase layer is extremely compliant or relatively stiff or when the inclusion is almost rigid. When the magnitudes of the remote stresses are sufficiently large, this condition becomes a very simple one relating the remote loading ratio with the geometric parameters of the composite. Interestingly, it is clearly observed from the simple condition that for given geometric parameters of the three-phase elliptical inclusion, there exist two different values of the remote loading ratio, both of which lead to an internal uniform hydrostatic stress state.

Effect of Heat Treatment on the Microstructure and Residual Stresses in (Ti,Al)N Films

(Ti,Al)N films were fabricated by arc on plating (AIP) and then annealed within a range of temperatures from 200 to 500 degrees C for 30 min in vacuum. The results indicate that the average residual stresses decrease slightly from -5.84 to -4.98 GPa with increasing annealing temperature. The stress depth distribution evolves from a sharp bell shape to a mild bell shape, suggesting a more uniform stress state in the annealed films. The microstructure of the films was also investigated in detail. The as-deposited film consists of fine columnar crystals with an amorphous phase at the interface. During heat treatment, the columnar subgrain growth was observed; meanwhile, the phenomenon of crystallization has been identified at the interface. Further more, the relationship between the residual stresses and the microstructure of the films was explored and highlighted. In addition, there is no hardness degradation of the films during heat treatment.

Buckling of Thin Cylindrical Shells Under Locally Elevated Compressive Stresses

Thin cylindrical shells used in engineering applications are often susceptible to failure by elastic buckling. Most experimental and theoretical research on shell buckling relates only to simple and relatively uniform stress states, but many practical load cases involve stresses that vary significantly throughout the structure. The buckling strength of an imperfect shell under relatively uniform compressive stresses is often much lower than that under locally high stresses, so the lack of information and the need for conservatism have led standards and guides to indicate that the designer should use the buckling stress for a uniform stress state even when the peak stress is rather local. However, this concept leads to the use of much thicker walls than is necessary to resist buckling, so many knowledgeable designers use very simple ideas to produce safe but unverified designs. Unfortunately, very few scientific studies of shell buckling under locally elevated compressive stresses have ever been undertaken. The most critical case is that of the cylinder in which locally high axial compressive stresses develop extending over an area that may be comparable with the characteristic size of a buckle. This paper explores the buckling strength of an elastic cylinder in which a locally high axial membrane stress state is produced far from the boundaries (which can elevate the buckling strength further) and adjacent to a serious geometric imperfection. Care is taken to ensure that the stress state is as simple as possible, with local bending and the effects of internal pressurization eliminated. The study includes explorations of different geometries, different localizations of the loading, and different imperfection amplitudes. The results show an interesting distinction between narrower and wider zones of elevated stresses. The study is a necessary precursor to the development of a complete design rule for shell buckling strength under conditions of locally varying axial compressive stress.

A circular inhomogeneity with interface slip and diffusion under in-plane deformation

We consider the in-plane deformation of a circular elastic inhomogeneity embedded in an infinite elastic matrix subjected to remote uniform stresses or uniform heat flow. The inhomogeneity and matrix have different material properties. The rate-dependent slip and mass transport by stress-driven diffusion concurrently occur on the inhomogeneity/matrix interface. For the remote uniform stress case, it is observed that the internal stresses within the inhomogeneity are quadratic functions of the coordinates x and y, and decay with two relaxation times. Interestingly the average mean stress within the circular inhomogeneity is in fact time-independent. As time approaches infinity, the internal stress field within the inhomogeneity becomes uniform and hydrostatic. In addition the change of strain energy due to the introduction of the circular elastic inhomogeneity is derived, containing various existing results as special cases. Furthermore, a simple condition leading to an internal uniform stress state within the inhomogeneity is found. This condition, which is independent of the elastic properties of the inhomogeneity and matrix, gives a simple relationship between the interface drag and diffusion parameters. For the remote heat flow case, the internal thermal stresses are linear functions of the coordinates x and y and decay only with a single relaxation time. Numerical results are presented to demonstrate the obtained solution and the corresponding physics.

Fatigue and inelasticity of metals under nonuniform stressed state

Based on the review of the available publications and the findings of the original studies, the author demonstrates that with cyclic stresses in the maximum-stressed surface layer being equal the inelastic strains in the presence of stress gradients are smaller than those in the material under uniform stressed state. As a result, there is a difference in the cyclic stress–strain diagrams between these cases. An equation for the cyclic stress–strain diagram is put forward, which allows for the stress gradient effect. A model is substantiated, which provides an explanation for the difference between fatigue strength characteristics under uniform and nonuniform stressed states.

Lateral oedometer testing of anisotropic clay as affected by surface roughness

To examine the state of anisotropy of a natural (glaciated) clay sample lateral stress oedometer tests are carried out to determine the magnitude and direction of the principal stresses. Initially, this was done by assuming no slippage along the boundary, thus for a perfectly rough oedometric ring with uniform anisotropic stress state and zero in-plane deformation. Here the other extreme case, the perfectly smooth oedometric ring, is examined by accounting for the analytical solution of a circular cylindrical continuum for isotropic linear elasticity, using appropriate boundary conditions. After determining experimentally the frictional coefficient, this solution is used in a semi-empirical interpolation method for estimating the influence of the roughness of the oedometric ring. For the assumption of a rough oedometric ring for the investigated Potclay the anisotropy ratio decreases in the range 0·58 to 0·30 with increasing axial effective stress in the range 355 ≤ σax ≤ 1860 kPa. For the observed reduced wall roughness the anisotropy ratio is shown to increase to the range 0·70 to 0·40, which is a relative increase of 21% to 34%. The relative error in this estimated anisotropy ratio is estimated in the range ±9% to ±13%, increasing with increasing axial effective stress. The orientation of the in-plane anisotropy is shown to be independent of wall roughness.

Remarks on elastic buckling for sectorial plates

This work investigates the edge-buckling experienced by a sectorial plate in a uniform bi-axial state of stress and subject to in-plane bending. Since the governing differential equations have variable coefficients, it turns out that the neutrally stable eigenfunctions can be qualitatively quite different as the mode number varies. Our interactive boundary-layer analysis succeeds in capturing the most dangerous mode associated with the global minimum of the marginal stability curve, while a complementary WKB route supplies an explanation for the morphological transitions experienced by the eigenmodes. The validity of our analysis is confirmed by direct numerical simulations of the full fourth-order buckling equation, which are in excellent agreement with the theoretical considerations.

Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: Approach to understanding Wolff's law

Ever since Julius Wolff proposed the law of bone transformation in the 19th century, it has been widely known that the trabecular structure of cancellous bone adapts functionally to the loading environment. To understand the mechanism of Wolff's law, a three-dimensional (3D) computer simulation of trabecular structural changes due to surface remodeling was performed for a human proximal femur. A large-scale voxel finite element model was constructed to simulate the structural changes of individual trabeculae over the entire cancellous region. As a simple remodeling model that considers bone cellular activities regulated by the local mechanical environment, nonuniformity of local stress was assumed to drive the trabecular surface remodeling to seek a uniform stress state. Simulation results demonstrated that cell-scale (∼10 μm) remodeling in response to mechanical stimulation created complex 3D trabecular structures of the entire bone-scale (∼10 cm), as illustrated in the reference of Wolff. The bone remodeling reproduced the characteristic anisotropic structure in the coronal cross section and the isotropic structures in other cross sections. The principal values and axes of a structure characterized by fabric ellipsoids corresponded to those of the apparent stress of the structure. The proposed large-scale computer simulation indicates that in a complex mechanical environment of a hierarchical bone structure of over 10 4 length scale (from ∼10 μm to ∼10 cm), a simple remodeling at the cellular/trabecular levels creates a highly complex and functional trabecular structure, as characterized by bone density and orientation.

A Comparative Numerical Study of the Stress States in Flat and Cylindrical Lap Joints

The stress states along the bondline in flat and cylindrical lap joints are analysed with the finite element method. The cross section of both specimen configurations is the same and is modelled with the same two-dimensional finite element mesh. The simulations are performed for the plane strain and axisymmetric cases which correspond to the flat and cylindrical specimens, respectively. The elastic properties of the adhesive are varied in a wide range and the influence on the stress distribution is analysed and compared for the two configurations. The most uniform stress state is obtained for the cylindrical specimen with the smallest diameter and the derived diagrams can be used for error prediction if average stress values are used for the estimation of elastic properties or failure loads.

Stress concentration factors of a general triaxial ellipsoidal cavity

ABSTRACTThe equivalent inclusion method is applied to solve the stress concentration problem concerning the disturbing effect of a general triaxial ellipsoidal cavity on an otherwise uniform normal stress state. Several useful solutions in simple form for limiting cases are derived and numerical results for general cases are obtained. These findings show the general features of the stress concentration factors around the base equator of the cavity. It is found that (1) when Poisson's ratio of the material is zero, the stress concentration factor does not vary along the equator of the cavity; (2) When the aspect ratio c/b of the cavity is very small, the stress concentration factor is also constant along the equator; (3) In general, the variation of stress concentration factors around the equator is less than 0.17 regardless of Poisson's ratio of the material. Thus, it is concluded that the stress concentration factor may be treated as constant around the equator of an ellipsoidal cavity with only a slight error.

Development of an X-ray stress measurement technique for a round corner inside a deep groove

A new X-ray stress measuring technique, namely the pseudo- ψ angle-changing method, has been developed to measure the circumferential stress on a round corner located at the bottom of a deep groove such as the tooth root of a gear or the cold-rolled fillet of a crankshaft. This is a multiple-point measurement technique applicable to a curved surface having a uniform stress state. An X-ray path is taken in the plane that does not intersect the sidewalls near the corner. Instead of ψ rotation, the irradiation position is changed along the circumference of the corner with the specimen's attitude against the X-ray path unchanged; diffraction angles at ψ = 0° are measured for different positions of the corner. The position change causes a shift in the diffraction peak since the irradiated area forming a part of the round corner has a different angle of inclination to the incident X-rays. The circumferential stress on the corner is obtained from the relation between the irradiation position and the diffraction angle at ψ = 0°. The validity of this method was verified by applying this technique and the conventional method to model specimens of a rack and a gear having an almost uniform residual stress state introduced by sand blasting, and comparing the stresses measured by the two different methods.

Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics

Anti-plane and plane-strain, time-harmonic, small-amplitude vibrations of an elastic layer on an elastic half space are considered, superimposed upon a state of finite, uniform stress and strain. A (compressible) elastic material is considered, orthotropic with orthotropy axes aligned parallel and orthogonal both to the layer and the prestress principal directions. A non-uniform mass density is assumed in the layer. A formal long-wave asymptotic solution is derived under the assumptions of high contrast between the stiffnesses of the layer and the half space and between certain prestress components and the current elastic shear modulus. It is shown that (i) the layer asymptotically behaves as a beam subject to transversal and axial vibrations; (ii) the response of the half space can be found in a closed-form, under the assumption of plane wave motion (which becomes consistent when the density of the layer is uniform), otherwise it is represented by a hypersingular integral equation; (iii) if the nonlocality introduced by the hypersingular integral equation is restricted to an influence area of finite extent, the integral can be analytically approximated, so that a Winkler-type spring model representing the half space is rigorously derived. For uniform density of the layer, the constants defining the spring model are given as functions of the prestress and anisotropy parameters of the half space; and, finally, (iv) the asymptotic solution provides new analytical expressions for incremental displacement of the layer, which, compared to the exact numerical solution (also included), are shown to perform quite well, even for values of parameters much beyond the limits imposed by the asymptotic analysis. The asymptotic analysis allows us to explore, for the first time, dynamic properties of a periodic layer bonded to an elastic half space and subject to a uniform prestress state. We find that the system exhibits band gaps (ranges of forbidden frequencies) and that the prestress can be used as a parameter tuning the filtering properties of the structure, an effect which may have important consequences in the design of resonant devices.

Method for the evaluation of the service life under random loading based on the energy criterion of fatigue fracture

We propose a method aimed at the assessment of the service life of structural elements under random loads based on the energy criterion of fatigue fracture and the model of cyclic deformation of materials after overloading. The possibility of determination of the maximum amplitude of stresses according to the number of loading cycles prior to fracture is substantiated. The results of evaluation of the service life under conditions of uniform stressed state are compared with the experimental data obtained by Swanson, Raikher spectral hypothesis of summation of defects, and linear hypothesis of summation. The dependence of the right-hand side of the linear hypothesis equation on the dispersion of the random loading process is established.

Stresses in a Multilayer Thin Film/Substrate System Subjected to Nonuniform Temperature

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to uniform film stress and system curvature states over the entire system of a single thin film on a substrate. By considering a circular multilayer thin film/substrate system subjected to nonuniform temperature distributions, we derive relations between the stresses in each film and temperature, and between the system curvatures and temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part, which reflects the effect of temperature of other points on the location of scrutiny. We also derive relations between the film stresses in each film and the system curvatures, which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. The interfacial shear tractions between the films and between the film and substrate are proportional to the gradient of the first curvature invariant, and can also be inferred experimentally.

Assessement of tensile strength of graphites by the Iosipescu coupon test

Polycrystalline graphites are widely used in the metallurgical, nuclear and aerospace industries. Graphites are particulated composites manufactured with a mixture of coke with pitch, and changes in relative proportions of these materials cause modifications in their mechanical properties. Uniaxial tension tests must be avoided for mechanical characterization in this kind of brittle material, due to difficulties in making the relatively long specimens and premature damages caused during testing set-up. On other types of tests, e.g. bending tests, the specimens are submitted to combined stress states (normal and transverse shear stresses). The Iosipescu shear test, is performed in a beam with two 90° opposite notches machined at the mid-length of the specimens, by applying two forces couples, so that a pure and uniform shear stress state is generated at the cross section between the two notches. When a material is isotropic and brittle, a failure at 45° in relation to the beam long axis can take place, i.e., the tensile normal stress acts parallel to the lateral surface of the notches, controls the failure and the result of the shear test is numerically equivalent to the tensile strength. This work has evaluated a graphite of the type used in rocket nozzles by the Iosipescu test and the resulted stress, ~11 MPa, was found to be equal to the tensile strength. Thus, the tensile strength can be evaluated just by a single and simple experiment, thus avoiding complicated machining of specimen and testing set-up.

Theta-like specimens for measuring mechanical properties at the small-scale: effects of non-ideal loading

Abstract As material structures, and devices fabricated from them, shrink to the nano-scale, it is ever more difficult to measure accurately the mechanical properties needed for designing small-scale devices and for assessing reliability. A promising test configuration for localized, small-scale uniaxial tensile measurements is the theta specimen. Stress distributions in this test configuration are examined via finite element analysis. When loaded uniformly with a double-articulating loading platen, a uniform uniaxial stress state is realized in the test gauge. When alternative, more easily realized experimentally, loading configurations are used (i. e., line or point contact), large secondary stresses develop. For each loading configuration, influences of non-ideal load application are elucidated. Two design modifications are considered to mitigate undesirable stresses: a compliant layer and an extended upper loading tab. The latter yields the most favorable stress state and is the most tolerant to non-ideal loading.

A Bayesian approach to estimating tectonic stress from seismological data

SUMMARY Earthquakes are conspicuous manifestations of tectonic stress, but the non-linear relationships between the stresses acting on a fault plane, its frictional slip, and the ensuing seismic radiation are such that a single earthquake by itself provides little information about the ambient state of stress. Moreover, observational uncertainties and inherent ambiguities in the nodal planes of earthquake focal mechanisms preclude straightforward inferences about stress being drawn on the basis of individual focal mechanism observations. However, by assuming that each earthquake in a small volume of the crust represents a single, uniform state of stress, the combined constraints imposed on that stress by a suite of focal mechanism observations can be estimated. Here, we outline a probabilistic (Bayesian) technique for estimating tectonic stress directions from primary seismological observations. The Bayesian formulation combines a geologically motivated prior model of the state of stress with an observation model that implements the physical relationship between the stresses acting on a fault and the resultant seismological observation. We show our Bayesian formulation to be equivalent to a well-known analytical solution for a single, errorless focal mechanism observation. The new approach has the distinct advantage, however, of including (1) multiple earthquakes, (2) fault plane ambiguities, (3) observational errors and (4) any prior knowledge of the stress field. Our approach, while computationally demanding in some cases, is intended to yield reliable tectonic stress estimates that can be confidently compared with other tectonic parameters, such as seismic anisotropy and geodetic strain rate observations, and used to investigate spatial and temporal variations in stress associated with major faults and coseismic stress perturbations.

Modelling reinforcement of rubber with carbon black filler

Mechanical reinforcement of rubber (soft phase) by random distribution of reinforcing units (hard phase), representing carbon black (CB) filler, has been modelled by finite element (FE) analysis. Using a hybrid numerical experimental method, approximate modulus and effective volume fraction of CB reinforcing units, as well as local stress/strain in two phases of the model, were calculated. Non-linear elastic properties of rubber phase and its corrected cross-link density were employed in the model. Strain amplification factor in the rubber phase, as a measure of rubber reinforcement by CB, was estimated and compared with springs in series model of Mullin and Tobin. It was concluded that mechanical behaviour of particulate filled rubber composites agrees with the springs in series model and follows the state of uniform stress rather than the state of uniform strain.

Use of the Master Curve methodology for real three dimensional cracks

At VTT, development work has been in progress for 15 years to develop and validate testing and analysis methods applicable for fracture resistance determination from small material samples. The VTT approach is a holistic approach by which to determine static, dynamic and crack arrest fracture toughness properties either directly or by correlations from small material samples. The development work has evolved a testing standard for fracture toughness testing in the transition region. The standard, known as the Master Curve standard is in a way “first of a kind”, since it includes guidelines on how to properly treat the test data for use in structural integrity assessment. No standard, so far, has done this. The standard is based on the VTT approach, but presently, the VTT approach goes beyond the standard. Key components in the standard are statistical expressions for describing the data scatter, and for predicting a specimens size (crack front length) effect and an expression (Master Curve) for the fracture toughness temperature dependence. The standard and the approach, it is based upon, can be considered to represent the state of the art of small specimen fracture toughness characterization. Normally, the Master Curve parameters are determined using test specimens with “straight” crack fronts and comparatively uniform stress state along the crack front. This enables the use of a single K I value and single constraint value to describe the whole specimen. For a real crack in a structure, this is usually not the case. Normally, both K I and constraint vary along the crack front and in the case of a thermal shock, even the temperature will vary along the crack front. A proper means of applying the Master Curve methodology for such cases is presented here.

Characterizing rate-dependent material behaviors in self-learning simulation

Structural testing, where inhomogeneous stress distribution is introduced within the test specimen, contains far richer information on the material behavior than the conventional material testing with uniform state of stress. We present a methodology that extracts rate-dependent material behavior using load–displacement measurements from the structural test. Self-learning capability of the rate-dependent neural network material model previously proposed by the authors is used in conjunction with the methodology. Unlike other parameter optimization methods, no prior knowledge of the material is required. The model is also capable of improving its performance as further test data become available. As an illustrative example, the method is applied to capture non-linear creep behavior of superalloy.