Elastic Stress Field Research Articles

Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory

ABSTRACTIn this article, an analytical model for the wave propagation analysis of inhomogeneous functionally graded (FG) nanobeam in thermal environment is developed based on nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. The nanobeam is modeled through a higher order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field supposed to have a nonlinear distribution across the nanobeam thickness. Temperature-dependent material properties of nanobeams are spatially graded based on Mori–Tanaka model. The governing equations of the temperature-dependent functionally graded (FG) nanobeam are derived using the Hamilton’s principle. Numerical examples show that the characteristics of the wave propagation of FG nanobeam are influenced by various parameters such as nonlocality parameter, length scale parameter, gradient index, and temperature changes.

Mechanics and energetics modeling of ball-milled metal foil and particle structures

The reported research establishes a semi-analytical computational predictive model of fractal microstructure in ball-milled metal foils and powder particulates, with emphasis on its transformation mechanics via an energy-based approach. The evolving structure is composed of reconfigurable warped ellipsoid material domains, subjected to collisions with the ball milling impactors following Brownian motion energetics. In the first step of the model, impacts are assumed to generate ideal Hertzian elastic stress fields, with associated bulk deformations quantified as per Castigliano's strain energy methods. In the second stage of the model, elastic energies are recast to produce frictional slip and plastic yield, thus resulting in surface micro-joints. Only two parameters of the model necessitate experimental calibration, performed by comparison of joint energy with laboratory tensile measurements on ball-milled multilayer Al-Ni foils. Model predictions of evolving internal microstructure are validated against SEM micrographs of Al-Ni powder particulate samples for different ball milling durations. Results demonstrate the capability of the model to accurately capture relevant fractal measures of the microstructure of ball-milled powders.

The general form of the elastic stress and displacement fields of the finite cracked plate

In this paper, the general forms of in-plane fields of a finite cracked plate have been achieved in the elastic range by expanding the potential functions of the infinite cracked plate about the crack tip. Subsequently, the stress intensity at the crack tip has been obtained. In addition, the numerical models have been provided by the finite element method.The effect of finite sizes on the stress and displacement fields has been detected and to validate the obtained analytical relations, the least-squares curve fitting has been used. Moreover, the calculated stress intensity correction factors have been compared with the existing results in the literature.

Simple estimation of effective elastic constants for thin plates with regular perforations and an application to the vibration of distillation column trays

The horizontal perforated sheet metal plates are commonly used in the process industries as trays in distillation columns, important internal parts for fractionating the input liquid mixture. Normally, the operating performance of such trays is satisfactory. However, cases have been reported of abnormally high levels of tray vibration during operation at particular conditions. The trays then experienced fatigue cracking accompanied by the loosening of bolts and fixings, which led to expensive failures. The excitation of structural resonance was suspected as a component in flow-induced vibration. Using linear stress superposition, a simple but robust analytical method is developed to provide high-quality predictions for the stress and strain distributions for in-plane loaded thin perforated plates with periodic hole arrangements. This approach is built on the classical solution for the elastic stress field around a single circular hole in a large plate. The perforated plates with square penetration patterns are investigated in this article, although the same approach is applicable to any regular penetration pattern. Stress concentration factors as well as the effective elastic constants, which can be used to describe the bending properties of the perforated plates, are then verified against both the established theoretical solutions and the results from finite element simulations. Excellent agreement to both previously published physical experiments and complex modelling is observed in all cases, with small-to-medium (up to 40%) hole-area fraction. The proposed analytical method is much simpler and computationally efficient than finite element analysis. The computed effective elastic constants are used in a finite element modal analysis to estimate the free vibration frequencies of a stiffened distillation column tray example; the first 30 vibration modes are found to be almost uniformly distributed between 25 and 70 Hz, which matches the vibration frequency range reported from plant operations.

Experimental study on stress intensity factor for an axial crack in a PMMA cylindrical shell

In the paper, the stress intensity factor (SIF) for an axial crack in a polymethyl methacrylate (PMMA) cylindrical shell is studied using the optical caustic method. First, the optical analysis of transmitted caustics is proposed based on the geometry conditions of PMMA cylindrical shell. Then, the governing equations that are related to the caustic measurements and the elastic stress fields at an axial crack tip in the cylindrical shell are derived in terms of the SIF, material constant, shell radius and shell thickness. Finally, the characteristics of stress singularity for mode I edge crack in the PMMA cylindrical shell specimen with different crack length, shell thickness and shell radius are experimentally investigated using the caustic method under three-point-bend loading, also the SIF obtained from the caustic measurement shows a good agreement with finite element results.

Stress field at V-notch tip in polymer materials using digital gradient sensing

ABSTRACTThe digital gradient sensing (DGS) technique was used to study the stress field at the V-notch tip of polymer materials. First, DGS governing equations at the V-notch tip were deduced using the elastic singular stress fields. Then, theoretical angular deflections of light rays at the V-notch tip were simulated, and the effect of notch angle on the angular deflections was analyzed. Finally, the DGS experiments were conducted, and the angular deflection contours were obtained. The results show that the stress intensity factors at the V-notches extracted from the angular deflections agree well with the results calculated from the finite element method.

Effect of Mg2Si particles on low-temperature fracture behavior of A356 alloy

The variation of tensile properties of A356alloy at 20°C~−60°C has been investigated. the distribution of dislocation slip bands near the fracture surface was studied by scanning electron microscopy SEM. Dislocation distribution was observed by transmission electron microscope TEM. The elastic stress field near the Mg2Si phase at low temperature was analyzed by ANSYS. When the temperature is decreased from 20°C to −60°C, the tensile strength and yield strength of A356 alloy increase and the elongation decreases. Under the effect of cold shrink, the elastic stress produced by the alloy matrix hinders the movement of the dislocation. In low temperature environment, the edge dislocation whose excess half atom facing toward Mg2Si experiences a repulsing force. On the contrary it experiences a gravitational force.

Prediction of the equivalent elastic modulus of mush zone during solidification process coupled with phase field simulations

A phase field model with the consideration of applied and thermal stress is developed in this paper. The elastic stress and strain fields inside the solid skeleton can be calculated out by phase field model together with the phase morphologies. By adding an external constant stress boundary condition, the total strain can be obtained by solving the mechanical equilibrium equations. We can get the equivalent elastic modulus under different phase morphologies from these calculations, which is useful for FEM or FDM simulations on a larger scale. This model is applied to as-cast and semi-solid casting process, respectively. The equivalent elastic modulus under dendritic and spherical solid skeleton are both obtained.

Influence of microstructure and mechanical properties on the tribological behavior of reactive arc deposited Zr-Si-N coatings at room and high temperature

Abstract Varying the Si-content in Zr-Si-N coatings from 0.2 to 6.3 at.% causes microstructural changes from columnar to nanocomposite structure and a hardness drop from 37 to 26 GPa. The softer nanocomposite also displays lower fracture resistance. The tribological response of these coatings is investigated under different contact conditions, both at room and elevated temperatures. At room temperature tribooxidation is found to be the dominant wear mechanism, where the nanocomposite coatings display the lowest wear rate of 0.64 × 10 − 5 mm 3 /Nm, by forming an oxide diffusion barrier layer consisting of Zr, W, and Si. A transition in the dominant wear mechanism from tribooxidation to microploughing is observed upon increasing the test temperature and contact stress. Here, all coatings exhibit significantly higher coefficient of friction of 1.4 and the hardest coatings with columnar structure display the lowest wear rate of 10.5 × 10 − 5 mm 3 /Nm. In a microscopic wear test under the influence of contact-induced dominant elastic stress field, the coatings display wedge formation and pileup due to accumulation of the dislocation-induced plastic deformation. In these tests, the nanocomposite coatings display the lowest wear rate of 0.56 × 10 − 10 mm 3 /Nm, by constraining the dislocation motion.

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

The longitudinal vibration analysis of small-scaled rods is studied in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions for the vibration analysis of small-scaled rods are derived by employing the Hamilton principle. The model contains a nonlocal parameter considering the significance of nonlocal elastic stress field and a material length scale parameter considering the significance of strain gradient stress field. The analytical solutions of predicting the natural frequencies and mode shapes of the rods with some specified boundary conditions are derived. A finite element method is developed and can be used to calculate the vibration problem by arbitrarily applying classical and non-classical boundary conditions. It is shown that the nonlocal strain gradient rod model exerts a stiffness-softening effect when the nonlocal parameter is larger than the material length scale parameter, and exerts a stiffness-hardening effect when the nonlocal parameter is smaller than the material length scale parameter. The higher-order frequencies are more sensitive to the non-classical boundary conditions in comparison with the lower-order frequencies, and the type of non-classical boundary conditions has a little effect on mode shapes.

Effect of geometrical stress concentrators on the current-induced suppression of the serrated deformation in an aluminum–magnesium AlMg5 alloy

The effect of an electric current on the band formation and the serrated deformation of planar specimens made of an aluminum–magnesium AlMg5 alloy and weakened by holes is experimentally studied. It is found that the concentration of elastic stress fields and the self-localized unstable plastic deformation field near a hole decreases the critical strain of appearance of the first stress drop and hinders the currentinduced suppression of band formation and the serrated Portevin–Le Chatelier deformation. These results are shown not to be related to the concentration of Joule heat near a hole.

A comparison of split sleeve cold expansion in thick and thin plates

Split sleeve cold expansion is a widely used process in the aerospace industry to enhance the fatigue life of rivet holes in the aircraft structures. In the experimental investigation presented in this article, the full-field in-plane residual strains and the out-of-plane surface deformations around open cold-expanded holes were measured using stereoscopic digital image correlation in aluminium specimens of two different thicknesses giving thickness-to-diameter ratios of 0.25 and 1. The results demonstrate that the mechanics of hole deformation is significantly different for the thick and thin specimens. The specimens of 1.6 mm thickness underwent a combination of global bending and significant local warping during the cold expansion process. This localised warping caused a decrease in the minimum principal residual strains close to the edge of the hole, which cannot be predicted by the existing theoretical models as they do not account for the complex out-of-plane deformations that have a significant influence on the shape of the resulting residual strain profiles. In contrast, 6.35-mm-thick specimens did not bend globally mainly because of the higher second moment of area of their cross sections. The material close to the hole edge bulges out from both the faces of the specimen as a result of plastic deformation during the cold expansion process and the out-of-plane deformations are much more localised and lower in magnitude in comparison to the thin specimens. The plastic zone developed around the expanded hole is more axisymmetric and larger in size for the thick specimens. These results imply that the existing split sleeve cold expansion process is not as effective in creating a uniform compressive residual elastic stress field around the fastener holes in thin as it is in the thick specimens.

Evaluation of effective stress along the border of lateral notches

AbstractThis paper analyses the issue of defect nucleation due to fatigue loading in notched components where cracks occur in the proximity of the zone where the major stress gradient occurs. On the basis of recent experimental results where the defects do not nucleate at the notch tip, the effective stress evaluated by means of the implicit gradient approach, can explain the reasons for this apparent discrepancy.Furthermore, in order to find a kind of notch that underlines the nucleation outside the maximum stress zone, rectangular lateral notches have been investigated. Rectangular notches could provide an interesting theoretical and experimental benchmark or reference case in order to validate effective stress definitions. The linear elastic stress field at edges, corners and in the surrounding material of rectangular, sharp or rounded lateral notches has been analysed by means of the implicit gradient approach. The consequent effective values of these notches are evaluated in relation to brittle fractures or their predicted fatigue strength values.

Free vibration analysis of nonlocal strain gradient beams made of functionally graded material

A size-dependent Timoshenko beam model, which accounts for through-thickness power-law variation of a two-constituent functionally graded (FG) material, is derived in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. The model contains a material length scale parameter introduced to consider the significance of strain gradient stress field and a nonlocal parameter introduced to consider the significance of nonlocal elastic stress field. The influence of through-thickness power-law variation and size-dependent parameters on vibration is investigated. It is found that through-thickness grading of the FG material in the beam has a great effect on the natural frequencies and therefore can be used to control the natural frequencies. The vibration frequencies can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam exerts a stiffness-softening effect. When the material characteristic parameter is larger than the nonlocal parameter, the FG beam exerts a stiffness-hardening effect.

Fatigue assessment of weld toe and weld root failures in steel welded joints according to the peak stress method

The peak stress method is an engineering, finite element (FE)-oriented application of the local approach based on the notch-stress intensity factors (NSIFs), which quantify the intensity of the local, linear elastic stress field at the potential crack initiation points (either the weld toe or the weld root). All effects due to plate thickness, joint shape and loading mode (axial or bending) are fully included in the NSIFs, such that a single design curve is sufficient to assess the fatigue strength of arc-welded joints in structural steels, tested in the as-welded conditions. The design stresses to analyse the fatigue strength are the peak stresses evaluated by FEM at the weld toe or the weld root, which are modelled as sharp notches with a tip radius equal to zero. The mean value of the FE size adopted to calculate the singular peak stresses can be chosen within a range of applicability, while the FE pattern of elements to use is the free mesh generated automatically by the Ansys™ numerical code. Due to the rather coarse meshes required to apply the PSM and its suitability to be used directly with results of 3D FE analyses, the method is rather useful in everyday design practice.

Dislocations and crowdions in two-dimensional crystals. Part III: Plastic deformation of the crystal as a result of defect movement and defect interaction with the field of elastic stresses

A continuation of the theoretical study of the intrinsic properties of dislocation and crowdion structural defects in 2D crystals [V. D. Natsik and S. N. Smirnov, Fiz. Nizk. Temp. 40, 1366 (2014) and V. D. Natsik and S. N. Smirnov, Fiz. Nizk. Temp. 41, 271 (2015)]. The atomic lattice model of conservative (glide) and non-conservative (climb) defect movement is discussed in detail. It is shown that given a continuum description of the 2D crystal, an individual defect can be examined as a point carrier of plastic deformation, its value being determined by the topological charge, which is compliant with the crystal geometry defect parameters. It is found that the strain rate depends on the rate at which the defect center moves, as well as its topological charge. The elastic forces acting on the dislocation and crowdion centers in the field of applied mechanical stresses, and the forces of elastic interaction between defects, are calculated in terms of the linear theory of elasticity of a 2D crystal. The non-linear effect pertaining to the interaction between defects and bending deformation of the crystalline membrane, which is specific to 2D crystals, is also discussed.

Multiple crack problems in nonhomogeneous orthotropic planes under mixed mode loading conditions

Analytical method based on continuous dislocation technique is developed to examine the mixed mode behavior of a functionally graded orthotropic plane. First, the elastic stress fields in a functionally graded orthotropic infinite plane containing single Volterra types climb and glide edge dislocations are obtained. Using the dislocation solutions, the problem of interacting cracks is reduced to a system of singular integral equations for dislocation density functions on the surfaces of smooth cracks which are solved numerically for the dislocation density. Then, the dislocation density function is applied to calculate the stress intensity factors for several different cases of crack configurations and arrangements.

Determination of dissipated Energy in Fatigue Crack Propagation Experiments with Lock-In Thermography and Heat Flow Measurements

Lock-in thermography and heat flow measurements with a new developed peltier sensor have been performed during fatigue crack propagation experiments. Lock-in thermography allows space-resolved measurements. Moreover elastic stress fields (E-Amplitude) as well as dissipated energies (D-Amplitude) can be determined. In case of thermographic measurements the specimens have to be painted to enhance the emissivity, but the thickness of the coating influences the results and therefore quantitative measurements are problematic. The heat flow measurements are easy to perform and provide quantitative results, but only integral values in an area given by the size of the peltier element can be achieved. In order to get comparable results the evaluation of the thermographic measurements were performed in the same area as the peltier measurements. In case of the mean temperature measured by thermography and the heat flow determined with the peltier sensor a good agreement was found. The measurement of elastic stresses with the peltier sensor is restricted to low loading frequencies due to the response characteristic of the sensor. For the measurement of dissipated energies the cooling of the specimen by heat conduction into the clamps and heat radiation has to be taken into account.

Interfacial fracture strength property of micro-scale SiN/Cu components

The strength against fracture nucleation from an interface free-edge of silicon-nitride (SiN)/copper (Cu) micro-components is evaluated. A special technique that combines a nano-indenter specimen holder and an environmental transmission electron microscope (E-TEM) is employed. The critical load at the onset of fracture nucleation from a wedge-shaped free-edge (opening angle: 90°) is measured both in a vacuum and in a hydrogen (H2) environment, and the critical stress distribution is evaluated by the finite element method (FEM). It is found that the fracture nucleation is dominated by the near-edge elastic singular stress field that extends about a few tens of nanometers from the edge. The fracture nucleation strength expressed in terms of the stress intensity factor (K) is found to be eminently reduced in a H2 environment.

V-notched components under non-localized creeping condition: numerical evaluation of stresses and strains

Geometrical discontinues such as notches need to be carefully analysed by engineers because of the stress concentration generated by them. Notches become even more important when the component is subjected, in service, to very severe conditions, such as the high temperature fatigue and imposed visco-plastic behaviour such as creep.The aim of the paper is to present an improvement and extension of the existing notch tip creep stress-strain analysis method developed by Nuñez and Glinka, validated for U-notches only, to a wide variety of blunt V-notches.The key in getting the extension to blunt V-notches is the assumption of the generalized Lazzarin-Tovo solution that allows a unified approach to the evaluation of linear elastic stress fields in the neighbourhood of both cracks and notches.Numerous examples have been analysed up to date, and the stress fields obtained according to the proposed method were compared with appropriate finite element data, showing a very good agreement.In view of the promising results, authors are considering possible further extension of the method to sharp V-notches and cracks introducing the concept of the Strain Energy Density (SED).