Distributions Of Stress Components Research Articles

NGHIÊN CỨU PHÂN BỐ ỨNG SUẤT TẠI BIÊN NGÀM CỦA VỎ TRỤ FGM CHỊU TẢI TRỌNG TĨNH SỬ DỤNG LÝ THUYẾT BIẾN DẠNG CẮT BẬC CAO QUASI-3D

This article presents the results of investigating FGM closed cylindrical shells using quasi-3D higher-order shear deformation theory. The law of volume distribution of materials according to the power function. In this article, the results of the analysis of the stress state at the clamped boundary area of the FGM cylindrical shell subjected to radial local loads distributed according to different laws. The mathematical model and the calculation program in Maple are compared and verified with the published results. The article has shown that at the clamped boundary edge, the values of stress components increase sharply, and at the same time, the strong influence of the load parameters (the variation law, the size of the load area), the power-law index on the values and distribution of stress components in the boundary area is demonstrated.

Numerical modeling of edge inhomogeneity in a macro cantilever beam

This study explores the inhomogeneity problem near the edge of a cantilever beam. Common in gears and turbine blades, inhomogeneities under local edge contact, coupled with macroscale bending effects, result in extreme and complex stress concentration. This model employs the Conjugate Gradient Method to combines the macroscopic beam theory and microscopic elasticity. The Eshelby inclusion method for simulating inhomogeneities. The Hetényi image method is used for address edge contacts in the beam. Enhanced by the Discrete Convolution fast Fourier Transform algorithm, the model’s accuracy and efficiency are validated with the Finite element method. Parametric analysis highlights the influence of distance between indenter and edge, stress components distribution and potential failure in cross-scale challenges.

Numerical modelling and analysis of current induced stresses in REBCO coil stacks

This study reports the effect of the radial, axial, and hoop electromagnetic stresses caused by the large current density and the self-field (screening current-induced magnetic field) on REBCO superconducting coil stacks. A two-dimensional numerical model is used to study the stress distribution of a superconducting coil stack under current. The electromagnetically homogenized coil stack is assumed to be composed of four equally spaced coils with rectangular cross-section. Initially, the verification of the model is done by comparing the analytical calculations (which were in good agreement) of stress distributions of a superconducting slab exposed to a variable magnetic field and in a superconducting strip exposed to a variable current. Then, the two-dimensional model is used to obtain the components of stress that occurred in the coil stack under sinusoidal varying currents. The stress component distributions are calculated at different time intervals, currents, frequencies, and coil distances. Besides, one-dimensional stress distributions are plotted to show how the stress distributions change while the current increases. The upper and lower coils have much higher magnetic stress than the inner coils. Results obtained from the one-dimensional stress distribution calculations show that the stress occurring in the radial direction is lower than the stress in the z-direction in the coil stack. The stress distributions do not show a significant change as the frequency of the applied current changes. The stresses in both the radial and axial directions decrease as the distance between coils increases. All results are presented, and the possible reasons are explained.

Thermal stress analysis of variable angle tow composite plates through high-order structural models

This work focuses on thermo-elastic analysis of Variable Angle Tow (VAT) composite plates. Emphasis is given to the through-the-thickness 3D stress component distributions as a consequence of constant temperature profiles. High-order structural models are used, in order to take into account the volumetric and the distortion contributions. The last contribute arise from boundary conditions or asymmetric staking sequences. Governing equations of the uncoupled thermo-elastic problem are derived in the domain of the Carrera Unified Formulation (CUF), which in turn is coupled with a classical finite element method to obtain opportune numerical solutions. The advantage of using CUF lies in the fact that different theories of structures can be implemented automatically and a critical study on the use of standard to high-order plate finite elements can be performed with ease. In this manner we classify models based on their efficiency and depending on the degree of accuracy needed, both in terms of displacements and stresses. Different numerical problems are considered and it is demonstrated that layerwise approximations are needed whenever shear stresses trends are of interest. Boundary Conditions influence more the choice of the adequate expansion theory otherwise.

Cutting speed dependence of material removal mechanism for monocrystal silicon

In advanced semiconductor packaging, silicon wafers are typically thinned to tens of microns in thickness using ultra-precision grinding techniques, and the cutting speed greatly determines their surface integrity. However, an in-depth understanding of fundamental material removal mechanism considering the strain rate effect has not been revealed yet. This paper systematically explores the impact of cutting speed on the penetration depth, subsurface damage characteristics and scratch-induced stress field for monocrystal silicon material. First, nanoscratch tests in varied-load and constant-load modes are performed under different scratch speeds on the silicon specimen surface. The surface morphology of generated groove is observed to identify its brittle and plastic features. Next, the geometric relationship and contact pressure between the indenter and workpiece during the scratching process are analyzed. A theoretical penetration depth model is developed considering scratch speed, indenter tip radius and elastic recovery. The corresponding parameters are solved via the particle swarm optimization algorithm and agree very well with the experimental data. Then, the subsurface damage under different scratch speeds is measured by a transmission electron microscope. The atomic scale lattice defects are presented, involving the amorphous layer, stacking faults and phase transformation. Finally, a scratch-induced stress field model is established to interpret the distortion degree variation in damaged regions with scratch speed. The distribution of stress components in the scratch region is presented.

Mathematical significance of strain rate and temperature rate on heat conduction in thermoelastic material due to line heat source

The primary purpose of the present article is to investigate the effect of strain rate and temperature rate factors on a homogeneous, unbounded isotropic elastic medium originating due to continuous line heat source. This study is based on the modified Green-Lindsay Model (MGL) theory proposed by Yu et al. (2018). This model eliminates the discontinuous nature of the displacement field reported under the temperature-rate-dependent thermoelasticity theory (GL) established by Green and Lindsay. The present work obtains the analytical solution for the distributions of stress components, temperature and displacement through the potential function approach accompanied by the Laplace transform method. The inverse Laplace transformation is performed by using short-time approximation method to find the approximated analytical solution of the problem in space-time domain. A detailed analysis of solution is discussed for MGL model and compared to the results predicted by other existing generalized thermoelastic models. The effect of strain-rate and temperature-rate-terms is acknowledged explicitly in mathematical formulation and other significant effects are notified. However, this new model predicts the infinite speed of disturbance analogous to classical theory.

Undrained semi-analytical solution for cylindrical cavity expansion in anisotropic soils under biaxial stress conditions

This paper presents an undrained semi-analytical elastoplastic solution for cylindrical cavity expansion in anisotropic soil under the biaxial stress conditions. The advanced simplified SANICLAY model is used to simulate the elastoplastic behavior of soil. The cavity expansion is treated as an initial value problem and solved as a system of eight first-order ordinary differential equations including four stress components and four anisotropic parameters. The results are validated by comparing the new solutions with existing ones. The distributions of stress components and anisotropic parameters around the cavity wall, the expansion process, the stress yield trajectory of a soil element and the shape and size of elastoplastic boundary are further investigated to explore the cavity expansion response of soils under biaxial in situ stresses. The results of extensive parameters analysis demonstrate that the circumferential position of the soil element and the anisotropy of the soils have noticeable impacts on the expansion response under biaxial in situ stresses. Since the present solution not only considers the anisotropy and anisotropy evolution of natural soil, but also eliminates the conventional assumption of uniform radial pressure, the solution is better than other theoretical solutions to explain the pressure test and pile installation effect of shallow saturated soil.

Three-dimensional thermoelastic analysis of a functionally graded truncated conical shell with piezoelectric layers

In this study, three-dimensional thermoelastic behavior of a functionally graded (FG) truncated conical shell with piezoelectric layers perfectly bonded to its inner and outer surfaces under electro–thermo–mechanical loads is investigated. The material properties of the FG truncated conical shell are assumed to be graded along the thickness according to a power-law function while the Poisson’s ratio is considered constant. The differential quadrature method (DQM) is utilized to obtain numerical results. The findings of this method are compared with the results published in the literature and excellent agreement with significantly fast convergence is shown. Parametric studies are performed to demonstrate the effects of the FG grading index, edge boundary conditions, semi-vertex angle of the shell, and thickness of the piezoelectric layers on the distribution of stress components, displacements, and electric potential.

Drained expansion responses of a cylindrical cavity under biaxial in situ stresses: numerical investigation with implementation of anisotropic S-CLAY1 model

Cavity expansion theory has been well developed in the past few decades, but little progress has been made to cavity expansion theory regarding biaxial in situ stresses. Owing to the two-dimensional nature of cavity expansion under anisotropic in situ stresses, a rigorous analytical or semi-analytical solution is no longer available for such cavity expansion problems. In this paper, a numerical study is performed to investigate the drained expansion responses of a cylindrical cavity under biaxial in situ stresses. The advanced anisotropic S-CLAY1 model integrated by the fully implicit backward Euler algorithm is implemented as a user-defined materials (UMAT) subroutine in the finite element model (FEM) to represent the elastoplastic behaviors of the anisotropic soils during cavity expansion. The FEM is validated by comparing with a benchmark semi-analytical solution under the uniform in situ stress. The expansion responses under biaxial in situ stresses, including the distributions of stress components, the evolutions of expansion pressures, stress paths, and yield surfaces, are subsequently investigated on leverage of the FEM and compared with those from approximate solutions. The present FEM in conjunction with the UMAT subroutine provides a benchmark for the validation of approximate solutions and the outcomes give significant insights into practical geotechnical problems pertaining to cavity expansion under biaxial in situ stresses.

Static analysis and boundary effect of FG-CNTRC cylindrical shells with various boundary conditions using quasi-3D shear and normal deformations theory

In this paper, static analysis and the stress concentration phenomenon of functionally graded carbon nanotubes reinforced composite cylindrical shells with various boundary conditions are explored by using higher-order shear-normal deformation theory and an analytical approach. Five types of distributions of reinforced carbon nanotubes are considered, that is, uniform and four kinds of functionally gradient distribution of carbon nanotubes along with the thickness of the shell. Effective material properties of carbon nanotubes reinforced composite are estimated to the rule of mixture. The governing equations for FG-CNTRC cylindrical shells with various boundary conditions are solved analytically using simple trigonometric series and the Laplace transform. Moreover, the transverse shear and normal stress are reconstructed using 3D linear elasticity equilibrium equations. The present model is validated by comparing its numerical results with those of published works, including the 3D exact model. Using higher-order shear-normal deformation theory, this article has focused on investigating the stress concentration phenomenon, the distribution of stress components in the near-boundary region, and also assessing the effects of various parameters on the stress state of the FG-CNTRC cylindrical shells.

Flat problem to determine the forces of destruction of pieces n disintegrators while being grabbed in thick layer

Purpose. Research on analytical dependences of destructive stresses, acting on a piece of non-isometric shape at quasi-static deformation of a relatively thick layer of rock mass in disintegrators, on parameters of the piece shape, the pieces spatial orientation, also on the relative piece size in layer, taking into consideration the discrete nature of contact force application. Methodology. The flat scheme of a non-isometric shaped piece contacts in a thick layer of rock mass is obtained by composition of the central rectangular piece and the round pieces of average size for the given layer. The distribution of stress components in the layer of loose rock mass is accepted on the basis of the classical theory of elasticity and the theory of loose medium. The geomechanics criterion showing relationship of equivalent destructive stress and ultimate compressive strength of rock is used as a criterion of piece destruction in complex stress state. All force schemes of the piece loading are reduced to three-point bending schemes and two-point shear schemes, both across the long and the short sides of the piece. The most dangerous loading scheme is determined from the analysis of the mentioned schemes for each particular case. Dimensionless parameterization is applied both to specify the geometric parameters of pieces and to analyze the resulting destructive stresses. Findings. Analytical dependences of equivalent destructive stresses for an oblong piece are obtained depending on the piece relative length, the relative piece size in the rock layer, the angle of piece orientation relative to the direction of the maximum main stress and the side thrust coefficient in the layer. It has been set that lamellar pieces, especially those smaller than the average size for the layer, are destroyed mainly from the implementation of bending schemes across the long side, the shape of their fragments is improved by reducing the pieces relative length. Increasing the uniformity of the force field in the working zone of disintegrator also leads to improvement in the shape of fragments. On the other hand, as the shape of the piece approaches the isometric one, as well as when the piece relative size in layer raises, the probability of implementing shear schemes increases and the probability of implementing bending schemes decreases, including with a deterioration in the fragments shape compared to the original piece. At the same time, larger values of destructive stresses for the lamellar smaller pieces are proved analytically compared to the isometric bigger ones, all other things being equal. Originality. The versatility of application of the three-point bending scheme of a non-isometric shaped piece in combination with the two-point scheme of its shear for analysis of its destruction in the thick layer of rock mass is substantiated. For the first time, the dependences of equivalent destructive stresses for the non-isometric piece on its relative length, its relative size in layer, the angle of deviation of the pieces main axis from the main stress direction and on the side thrust coefficient in the layer have been obtained. Practical value. The results obtained allow making reasonable choice of parameters of disintegrators operational parts for destruction of materials in the thick layer, as well as predicting the change in lamellar pieces fraction during disintegration process. They give the possibility to determine key parameters of operational parts for new designs of disintegrators. This creates the basis for the development of calculation techniques for operational parts of modern samples of crushing and grinding equipment.

Analysis of undrained cylindrical cavity contraction in anisotropic soils under constant total vertical stress condition

A semi-analytical solution to cylindrical cavity contraction in anisotropic soils under constant total vertical stress and undrained condition is presented in this paper. The elastoplastic constitutive relationship for the problem considered is obtained via Hooke’s law in the elastic stage and the anisotropic modified Cam-Clay model in the plastic stage. By employing an auxiliary parameter to transform the equilibrium equation from Eulerian scheme into Lagrangian scheme, the problem reduces to an initial value problem with the vertical strain and the effective radial, circumferential, vertical stresses as four basic unknowns. The governing equations of this problem in the plastic region are solved as an initial value problem with the mechanical states of soil at the elastic-plastic boundary as the initial values. Parametric analyses show that, owing to the large deformation in the vertical direction, the material particle at cavity surface reaches the critical state much more quickly under constant total vertical stress condition than that under plane strain condition. The distribution of stress components does not manifest a critical-state region as a traditional plane strain solution does. Also, the overconsolidation ratio has an obvious influence on contraction responses. The proposed solution is expected to provide a more accurate prediction of stress and displacement around a drilled wellbore and to other similar geotechnical problems.

Undrained cylindrical cavity expansion/contraction in stiff clays using a two‐surface plasticity model

AbstractThis paper presents a semi‐analytical solution of undrained cylindrical cavity expansion/contraction problem using a two‐surface plasticity model for natural stiff clays from the view point of poromechanics. The present solution reduces the classical boundary problem into a group of seven first‐order ordinary differential equations via an auxiliary independent variable ξ under undrained conditions. It strictly follows the premise of constant soil mass (CSM). Meanwhile, the solution for the constant soil volume (CSV) problem is used for comparison. It shows that CSM assumption is of paramount importance in developing solutions under high stress condition, while CSV induces non‐negligible errors in effective stress and pore pressure distributions. Based on the premise of CSM, extensive parametric studies on cavity expansion/contraction are undertaken to investigate the effects of the model key parameters on the stress components distribution, excess pore pressure distribution, and cavity expansion curves. The present solution could be useful while dealing with geotechnical and petroleum engineering problems involving stiff clays or soft claystones. The application in Praclay Gallery excavation in natural Boom clay well demonstrates the validity of the present solution.

Consistent Euler – Bernoulli beam theories in statics for classical and explicit gradient elasticities

The Euler – Bernoulli beam bending theory in elementary (engineering) mechanics uses two fundamental assumptions, first that the material behaviour is isotropic elastic and secondly that plane cross sections remain plane, rigid and perpendicular to the beam axis. It is well – known, that this theory suffers from the inconsistency that, e.g., the shear strain is always vanishing, whereas the shear stress is supposed to be not vanishing. In the present paper we remove this inconsistency in engineering mechanics by postulating the material response to be a limiting case of anisotropic elasticity. We show that all results within the “inconsistent” isotropic elasticity remain valid. Then, the anisotropic elasticity approach is extended to model bending of Euler – Bernoulli beam in explicit gradient elasticity. Especially, attention is focused on deriving distributions of stress components and discussing limiting responses in dependence of a material length parameter inherent in the elasticity law.

Thermoelastic Analysis of Pressurized Hollow Spherical Vessels with Arbitrary Radial Non-Homogeneity

In this study, a general analysis of one dimensional steady-state thermal stresses of a functionally graded hollow spherical vessel with spherical isotropy and spherically transversely isotropy is presented with material properties of arbitrary radial non-homogeneity. The material properties may arbitrarily vary as continuous or piecewise functions. The boundary value problem associated with a thermo-elastic problem is converted to an integral equation. Radial and tangential thermal stress components distribution can be determined numerically by solving the resulting equation. The influence of the gradient variation of the material properties on the thermal stresses is investigated and the numerical results are presented graphically.

Size dependent nano-spherical pressure vessels based on strain gradient theory

This study investigates the effect of size scale material parameters on stress distribution and radial displacement of nanosphere based on strain gradient theory. This model is more capable of studying mechanical behavior than classical elasticity theory as the size scale effect of the nanosphere is also considered. Minimum total potential energy is used to derive governing differential equation of nanosphere under internal hydrostatic pressure. Using the efficient numerical generalized differential quadrature (GDQ) method, the governing equation and corresponding boundary conditions are solved. The classical elasticity equation is obtained by setting the value of size scale material parameters to zero. With the comparison of these theories, the importance of the size scale material parameters is achieved. It is found that the radial displacement of nanosphere predicted by strain gradient theory is less than those predicted by classical elasticity theory but comparing the distribution of stress components along radius is more complex. The effect of the size of the nanosphere on the radial stress components is also studied. With an increasing outer radius of the nanosphere, the mechanical behavior predicted by strain gradient theory tends toward those in classical elasticity theory.

Soil Layer Disturbance Caused by Pipe Jacking: Measurement and Simulation of a Case Study

In this paper, areas of soil disturbed by pipe jack construction are categorized and analyzed in detail. Mechanisms of soil disturbance are discussed and patterns of soil deformation are studied using random medium theory. Lateral deformations in deep soil, pore water pressures, stratified settlement, and earth pressures are investigated using measurements from an electrical transmission pipeline project in China. The measurements show that soil layer movement can be predicted by monitoring changes in pore water pressure, and the amplitude of soil disturbance transverse to the pipeline is larger than that above the pipeline in this project. The distributions of stress components in the direction of, and vertical to, the pipeline axis are studied by numerical simulation. It shows that the closer to the pipeline axis, the greater the additional stress, the faster the attenuation speed, and the smaller the range of influence. Relationships of positive propulsion, friction, and stratum loss with longitudinal and transverse surface deformations are also discussed. The ground displacement is a coupling of the three factors to the action of the soil. The surface transverse subsidence with a main influence range of about ±4 D (D is outer diameter of pipe jacking). The findings from this study can be used as technical foundation and reference for the construction of similar projects.

Finite Element Modeling of Stress Behavior of FGM Nanoplates

The mechanical response investigation of nanoplates especially the stress distribution plays a very important role in engineering practice, which is a condition to help test the durability as well as design and use the nanoplate structures most effectively. This pioneering paper uses the finite element method to simulate the stress field of FGM nanoplates based on the first‐order shear deformation theory of Mindlin. The finite element formulations are derived by taking into account the effect of the nonlocal coefficient to analyze the mechanical response of nanometer‐scale plates. This work presents the distribution of stress components in the xy‐plane of plates with different boundary conditions. The numerical results also show clearly that the nonlocal coefficient has a significant influence on the deflection and stress of FGM nanoplates. These numerical results are very new and stunning which clearly show the position of the stress reaching the maximum value. This work is also the basis for scientists in testing the durability of FGM nanoplates.

Expansion responses of a cylindrical cavity in overconsolidated unsaturated soils: A semi-analytical elastoplastic solution

This paper presents a semi-analytical elastoplastic solution for cylindrical cavity expansion in overconsolidated (OC) unsaturated soils under constant water content condition, in which both the OC features and the coupled hydro-mechanical behaviour of soils are properly addressed. The unified hardening model for OC unsaturated soil is further coupled with a void ratio-dependent soil-water characteristic curve to model the coupled hydro-mechanical soil responses during expansion. With the aid of an auxiliary variable, the coupled hydro-mechanical constitutive matrix for the investigated problem is finally formulated as a set of first-order ordinary differential equations in terms of Lagrangian description, with the three net stress components, the suction, and the void ratio as the basic unknowns. The governing differential equations are solved as an initial value problem with the code developed based on the Runge-Kutta algorithm. Parametric studies show that both the overconsolidation ratio and the suction have remarkable impacts on the expansion responses, including the distribution of stress components, the change of degree of saturation as well as the expansion pressures. The proposed solution could provide a reasonable tool for interpretation of the pressuremeter tests and prediction of pile installation effects in overconsolidated unsaturated soils.

A micro plasticity model for pure bending analysis of curved beam-like MEMS devices

Computational models are investigated to analyze plane strain pure bending of rectangular cross sectional curved microbeams in elastic and partially plastic states. A generalization of classical von-mises deformation theory of plasticity is adopted to include size effect, by considering intrinsic material length scale parameter, in the modeling. Assuming linear and power-law hardening model, the governing nonlinear equation in elastic-plastic zones are discretized by generalized differential quadrature (GDQ) method and are solved by direct iteration approach. The numerical calculations of proposed model are validated with published literature. Semi-analytical solution is obtained from the special case of curved microbeam with elastic-rigid plastic constitutive model and linear hardening law. The influences of initial curvature and size effect on yielding initiations at the inner and outer surfaces of microstructure are evaluated. Also, distributions of stress components along the microbeam thickness are studied, for different values of the couple moments, at elastic limits and beyond.