Two-dimensional Stress Research Articles

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.

Investigation of Two-Dimensional Gravitational Stresses in Anisotropic Media Based on the Sherman-Type Integral Equations

Investigation of Two-Dimensional Gravitational Stresses in Anisotropic Media Based on the Sherman-Type Integral Equations

Electromagnetic Behavior Simulation of REBCO Pancake Coils Considering REBCO Tape Rotation Under High Magnetic Field

Since the screening current induced in rare earth-barium-copper-oxide (REBCO) magnet generates an irregular magnetic field, a few screening current simulation methods have been proposed. For an insert REBCO magnet generating ultra-high magnetic field, a new screening current method has been proposed with consideration of the coil-deformation effect. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">In this paper, we have developed a new screening current method based on a partial element equivalent circuit (PEEC) model coupling with a two-dimensional stress finite element analysis to accurately the time-transient distribution of accurate screening current. In the presented method, the changes of self/mutual inductances are considered due to the coil deformation. As the simulation results, it was found that the coil deformation affected the screening current distribution. When considering the coil deformations, the screening current-induced fields to operating current are different for charging cycles. It is necessary to simulate the coil deformation to accurately estimate the screening current and the screening current-induced field.

Assessment of residual stresses in welded T-joints using contour method

Welded T-joints are commonly used in manufacturing of stiffened panels or crossbeams. However, welding of web-to-flange joint induces residual stresses and distortions resulting in significant challenges during both manufacturing and service. These issues can ultimately impact the structural performance, including fatigue and stability, and lifetime of the weldment. The purpose of current paper is determining the distribution and magnitude of manufacturing-induced longitudinal residual stresses in welded T-joints. Influence of hot rolling, thermal cutting, prestressing, welding, etc. are all included using the so-called contour method. Deformed surfaces are measured using coordinate measuring machine after wire electrical discharge machining of specimens. An inverse linear elastic finite element analysis is carried out for assessing residual stresses after data smoothing. Two-dimensional longitudinal residual stress patterns are evaluated in representative cross-sections clearly indicating high stress peaks near the welds, even exceeding yield strength of the base material. In addition, longitudinal membrane residual stresses are determined based on through-thickness stresses. In addition, a simplified residual stress model, satisfying the requirement for equilibrium of internal forces and bending moments, is developed for welded T-joints with double-sided fillet welds based on contour method results.

An analytical investigation of soil arching induced by tunneling in sandy ground

The soil mass above the tunnel generally yields due to the tunnel volume loss. Depending on the amount of the volume loss, soil unit weight, overburden height and internal friction angle, a stress arching zone and a loosened zone can be formed above the tunnel. In this study, a two-dimensional stress field resulting from the tunnel volume loss inside the soil mass above the tunnel is investigated analytically. By presenting an analytical solution in rectangular coordinates, new equations were developed to evaluate and estimate the magnitude of vertical, horizontal and shear stress components inside the soil mass. The vertical stress changes resulting from the tunnel volume loss were analyzed and the height of the arching zone and the loosened zone above the tunnel were determined. In addition, a parametric study was conducted to evaluate the effect of soil unit weight, soil internal friction angle, surcharge on the ground surface and overburden depth on the stress distribution in the soil above the tunnel. The results showed that depending on the depth of the tunnel and the strength of the surrounding soil, the arching and loosened zones may or may not be created. In order to validate the proposed solution, the vertical stress distribution along the tunnel crest has been compared with some experimental and numerical data. It can be stated that the proposed solution provides relatively good approximations for the vertical stress component. The results of this research show that a safe and economical support system for tunnels in sandy ground can be designed using the proposed solution.

Influence of water on rockburst of surrounding rock in deep circular tunnels under triaxial internal unloading conditions

Rockburst is one of the most serious engineering geological hazards encountered in deep rock mass. In this study, we investigate the influence of water on the rockburst mechanism in circular tunnels considering the whole stress path of high in situ stress + internal unloading + stress adjustment, based on external loading-internal unloading (ELIU) tests on intact air-dried and saturated sandstone cubic specimens (100 × 100 × 100 mm3). The proposed ELIU method uses true triaxial equipment to first load each intact cubic specimen with two-dimensional high stresses, followed by an internal drilling device for internal unloading to form circular holes. For comparison, cubic specimens with predrilled holes 25 mm in diameter were subjected to internal drilling – external loading (IDEL) tests, i.e., drilling first and then loading. The results show that water reduces the uniaxial compressive strength, Young's and initial moduli, and elastic energy storage capacity of sandstone materials, and increases the significance of shear failure. Furthermore, saturated tunnel structures induce static spalling failure, while dry tunnels lead to dynamic rockburst failure. The influence of water on the stability of the rock mass surrounding a tunnel is analyzed, including the induced weakening of surrounding rock strength and the amplification of the unloading effect. In summary, water prevents rockburst through the following mechanism: (i) water reduces the brittleness of the surrounding rock and enhances its plasticity; (ii) water reduces the elastic energy storage capacity and increases the energy dissipation capacity of the plastic zone; and (iii) water induces the redistribution of stress throughout the surrounding rock, which reduces the stress concentration near the free surface and transfers the maximum stress to the deeper surrounding rock.

Approach for the simulation of the load-deformation behaviour of hyperelastic bonded joints using strain and strain rate dependent tangent moduli

In structural glass, Structural Sealant Glazing (SSG) systems are often used to realise facades with maximum transparency. The used silicone adhesives are currently designed in accordance to the European Technical Approval Guideline 002 (ETAG 002) by a simplified two-dimensional stress calculation concept. For a more precise calculation of the adhesive stresses and deformations, finite element analyses are used. The quality of the results, however, strongly depends on the calculation concept and material model used. In this paper, a practice-oriented viscoelastic approach using strain and strain rate dependent tangent moduli within FEA is presented. Required material parameters can be determined with a simple uniaxial tension test only. The proposed model is validated with numerous FE calculations of reality-based adhesively bonded component tests with different geometric ratios that include complex multiaxial stress states. The proposed approach allows a precise prediction of the strain state- and time-dependent stress behaviour of hyperelastic silicone bonds. The prediction quality is significantly more accurate than often used linear-elastic material models or commonly available hyperelastic models.

Predicting the yield surface in a two-dimensional stress system: A student's laboratory experiment

This paper is concerned with the determination of the yield surface in a two-dimensional stress field within an undergraduate Mechanics of Materials student laboratory experiment. In this experiment, the two-dimensional stress field is obtained by subjecting a steel specimen to a combined bending and torsion load. The motivation for the paper was the lack of correlation between the von Mises and Tresca yield criteria and the principal stresses calculated at the load at which students were predicting material yielding. The latter yield loci were based on the uniaxial 0.2% proof stress. The lack of correlation was creating a lot of frustration amongst students. This is undesirable, especially within an undergraduate student experiment. Two hypotheses for the lack of agreement were considered. The first hypothesis was the uncertainty involved in the method that the students were using to predict the onset of material yielding. The second hypothesis was that the specimens being used for the experiment had anisotropic properties, given that little information on their manufacture was provided. The need for determining accurate yield stress values and on methodologies found in literature that are used to determine the material proof stress or lower yield stress in a uniaxial tensile test are discussed. Four methods taken from literature are used and adapted to detect the first yield of the specimen under a combined bending and torsional load. The resulting experimentally yield loci are compared with the theoretical von Mises and Tresca isotropic yield loci over half of the second quadrant of the two-dimensional principal stress field plot. Correlation between the four methods was quite good but not so when compared with the von Mises and Tresca’ loci. A lack of correlation occurred for an increasing torsional load indicating a possible anisotropy in the material properties. These results hinted towards the second hypothesis. A number of tensile test specimens were hence heat treated so as to induce isotropy in the material properties. The combined loading experiments were repeated using the heat-treated samples. A very good correlation was obtained between the experimental yield points for the two-dimensional stress field and the von Mises and Tresca yield loci. This good correlation for the heat-treated specimens confirmed the authors’ second hypothesis on the anisotropic properties in the as-received state and, therefore, the requirement to heat treat the specimens for a meaningful student undergraduate laboratory activity.

Implications for identification of principal stress directions from acoustic emission characteristics of granite under biaxial compression experiments

The rock fracture characteristics and principal stress directions are crucial for prevention of geological disasters. In this study, we carried out biaxial compression tests on cubic granite samples of 100 mm in side length with different intermediate principal stress gradients in combination with acoustic emission (AE) technique. Results show that the fracture characteristics of granite samples change from ‘sudden and aggregated’ to ‘continuous and dispersed’ with the increase of the intermediate principal stress. The effect of increasing intermediate principal stress on AE amplitude is not significant, but it increases the proportions of high-frequency AE signals and shear cracks, which in turn increases the possibility of unstable rock failure. The difference of stress in different directions causes the anisotropy of rock fracture and thus leads to the obvious anisotropic characteristics of wave velocity variations. The anisotropy of wave velocity variations with stress difference is probable to identify the principal stress directions. The AE characteristics and the anisotropy of wave velocity variations of granite under two-dimensional stress are not only beneficial complements for rock fracture characteristic and principal stress direction identification, but also can provide a new analysis method for stability monitoring in practical rock engineering.

Investigation of load-bearing capacity of rigid-plastic reinforced ellipsoidal shells of rotation

For ellipsoidal fiber-reinforced shells of rotation, the upper (kinematic) limit of load-bearing capacity is determined. The structures are clamped along the lower support contour and have rigid polar plate. The material of the components is rigid-plastic with asymmetry in tension and compression. The matrix is orthotropic. The principal axes of matrix anisotropy coincide with meridional, circumferential and transverse directions. Two-layer shell model and refined relations of structural model with two-dimensional stress state in components are used. Extreme problem is formulated and numerically solved for spherical and two different ellipsoidal shells. Influence of reinforcement parameters on limit load is investigated.

Modelling of the two-dimensional stresses in a three-layered adhesively bonded tungsten disulfide/poly(methyl methacrylate) nanocomposite structure under axial load

For the first time a two-dimensional stress-function method describing the stress transfer in a three-layered adhesively bonded tungsten disulfide/poly(methyl methacrylate) nanocomposite structure, subjected to axial load has been developed and applied. The governing fourth-order ordinary differential equation of with constant coefficients for the axial stress in the first layer has been obtained minimizing the strain energy in the whole structure and solved analytically. Depending of the sign of the discriminant of the respective characteristic equation of the 4th order differential equation for the considered nanocomposite structure, two different analytical solutions have been obtained for axial stress in the first layer. The type of solutions depends strongly from the geometry (length and thickness of the layers), material properties and magnitude of applied load. The two-dimensional stresses (axial, shear and peel) in the structure's layers are expressed and calculated as functions of this axial one and its derivatives and illustrated with three-dimensional graphics, for both solutions. The model results for axial stress in tungsten disulfide (WS2) were compared with shear–lag results and show good agreement in elastic region of applied tension load.

Mathematical modeling of a three-component system to investigate the formation and characteristics of glass

The article investigates glass formation and its properties in a three-component system using mathematical modeling. The authors have applied the method of mathematical planning to increase the efficiency of research and to establish the dependences of glass properties on the composition. The authors propose a mathematical model of structure relaxation and two-dimensional stress field in grade glass products, considering the temperature dependencies of glass viscosity of a given chemical composition.

Analytical and numerical study of one-dimensional and two-dimensional stress distribution within an elastic semi-infinite material under the action of an arbitrary rectangular uniform loading

Purpose. The study of stress distribution within an elastic semi-infinite material under the action of an external loading is of great importance in the theory of elasticity. In most cases, the lack of knowledge about the stress distribution within a material can result in incomplete and inappropriate engineering designs, leading to unsatisfactory consequences. The latter include cracks and fractures, created inside the concrete segmental lining in TBM tunneling, as well as indentations that occur in floors due to the lack of pillar design not only in underground mining, but even in civil projects. This study focuses on the one-dimensional and two-dimensional internal stress distribution induced by arbitrary rectangular–square loading, in other words, that applied to an elastic semi-infinite material. Methods. Firstly, this paper uses an analytical method and, subsequently, a numerical method. In the analytical study, the point load equations of Boussinesq and Westergaard are used. Extending these equations to the rectangular loading area, four new equations are introduced. Using the Abaqus finite element software, the numerical study is performed in 3D space. Findings. The results show that the answers from the introduced equations are in high consistency with numerical ones. However, the result of the extended Boussinesq point load equation is closer to the answer obtained by the numerical method. Originality. Four new equations, presented in this paper, describe one-dimensional and two-dimensional stress distribution. Practical implications. The presented equations can provide a simple and convenient way to solve rectangular load problems in many cases such as foundation, civil and mining projects. This study uses initial information on specific segments in the Tabriz Metro line-2 Project.

Numerical investigations on the damage tolerance of adhesively bonded pultruded GFRP joints with adhesion defects

Adhesive bonding has gained increasing acceptance in joining pultruded glass fiber reinforced polymer (GFRP) sections. However, one of the major issues hindering the widespread application of adhesive bonding in building and construction industries is the difficulty in predicting the mechanical behavior of the joints in the presence of adhesion defects. This paper investigates the damage tolerance of the adhesively bonded pultruded GFRP double-strap joints under quasi-static tensile loading using finite element analysis. Defects of various locations (along the bond length), sizes, and numbers were embedded within the bondline. A two-dimensional (2D) stress analysis was conducted to assess the effects of the bondline defects on the out-of-plane displacement, stress distribution, and peak stresses of the joints. Three-dimensional (3D) progressive damage models were further developed to predict residual joint capacity and provide insights into the effects of the adhesion defects on the damage mechanism of the joints. The progressive damage model uses the cohesive zone model (CZM) to simulate progressive interlaminar damage within the adherends as well as adhesive-FRP interface debonding. The proposed FE model was validated with published experimental data, and an excellent correlation was found between experimental and numerical results. The validated model was then used in a parametric study to assess the effects of design parameters such as adhesive types (from highly brittle to highly ductile) and joint geometry (adhesive chamfering, adherend chamfering, and recessing) on the damage tolerance of adhesively bonded joints. This paper also reports a comparative study on the damage tolerance between GFRP double-strap, single-strap, and single-step joints. The stress analysis results revealed that the presence of the bondline defects could increase the out-of-plane displacement of the adherends and the peak shear and peel stresses at the overlap ends. High-stress concentrations were also observed in regions adjacent to the defect extremities. The double-strap joints exhibited better damage tolerance than the single-strap and single-step lap joints. The damage tolerance of the joints can be improved by using stress-reduction methods such as adhesive chamfering, adherend chamfering, and recessing.

Spatial Constitutive Modeling of AA7050-T7451 with Anisotropic Stress Transformation.

The mechanical properties of anisotropic materials are generally characterized based on the orthotropy or transverse isotropy. However, the two-dimensional plane stress problems cannot comprehensively characterize the anisotropy of materials. In this study, based on the theory of elasticity and the transformation of the three-dimensional space coordinate system, combined with the projection relationship of the Cauchy stress tensor of an arbitrary section, the transformation relationship of the elastic modulus, shear modulus, and stress–strain between the orthogonal and load coordinate systems are obtained. The orthotropic Johnson-Cook (JC) constitutive model of AA7050-T7451 aluminum alloy is modified by fitting, and the constitutive relationship at any spatial angle is theoretically calculated by combining the obtained spatial coordinate transformation matrix. The generated spatial constitutive model is verified and modified through experiments. The results demonstrate that the theoretical mechanical properties and the modified spatial constitutive model can accurately reflect the effect of the spatial angle on the material stress distribution.

Determination of bi-dimensional normal residual stress distributions in metallic laser-based powder bed fusion parts

Disadvantageous complex residual stress distributions are common in parts manufactured by laser-based powder bed fusion metals (PBF-LB/M). Thus, the residual stress state determination is essential for understanding the part’s limitations. One residual stress measurement technique applicable to PBF-LB/M parts is the slitting method. This destructive technique allows through-thickness measurements and copes well with discontinuities in the material, yet is limited to one-dimensional stress profiles. Conversely, other stress measurement techniques, e.g. X-ray diffraction, are restricted to the part’s surface. This article presents an inexpensive and straightforward stress determination approach that combines an implementation of the slitting method with X-ray diffraction measurements to create a continuous two-dimensional residual stress map along a part’s cross-section. The approach was numerically validated using finite element models that simulate, on the one hand, the PBF-LB/M process and, on the other hand, the measuring process; further, it was experimentally tested in PBF-LB/M AlSi10Mg samples. The use of linear elastic fracture mechanics allowed a straightforward formulation of the approach, which enabled a high degree of automation. The accurate stress distribution results and the correlation with the simulations and previous studies demonstrate the approach’s robustness and effectiveness for complex residual stress states determination in two dimensions.

Numerical study of vortex-induced vibrations of a circular cylinder at different incidence angles

The effects of four different incidence angles (α) on the vortex-induced vibration (VIV) response of a two degrees-of-freedom cylinder were studied using the two-dimensional unsteady Reynolds-averaged Navier-Stokes equation and shear stress transport k-ω model, combined with the fourth-order Runge-Kutta method. The VIV characteristics, including the structural vibration amplitude, vibration frequency, lock-in regime, the pattern of vortex shedding, Strouhal number, hydrodynamic coefficient, and power conversion efficiency were systematically researched. The numerical results showed that a change of the incidence angle made the VIV response of the cylinder display multifrequency characteristics. As the incidence angle increased, the amplitude in the y direction gradually decreased; comparatively, the amplitude in the x direction gradually increased. The VIV responses for different incidence angles all presented an obvious lock-in phenomenon; however, the width of lock-in region changed slightly with variation of the incidence angle. As the incidence angle increased, the root-mean-square value of the hydrodynamic force coefficient in the y direction and the mean value of the hydrodynamic force coefficient in the x direction both had a decreasing tendency.

One- and Two-Dimensional Analytical Solutions of Thermal Stress for Bimodular Functionally Graded Beams under Arbitrary Temperature Rise Modes

In this study, we analytically solved the thermal stress problem of a bimodular functionally graded bending beam under arbitrary temperature rise modes. First, based on the strain suppression method in a one-dimensional case, we obtained the thermal stress of a bimodular functionally graded beam subjected to bending moment under arbitrary temperature rise modes. Using the stress function method based on compatibility conditions, we also derived two-dimensional thermoelasticity solutions for the same problem under pure bending and lateral-force bending, respectively. During the solving, the number of unknown integration constants is doubled due to the introduction of bimodular effect; thus, the determination for these constants depends not only on the boundary conditions, but also on the continuity conditions at the neutral layer. The comparisons indicate that the one- and two-dimensional thermal stress solutions are consistent in essence, with a slight difference in the axial stress, which exactly reflects the distinctions of one- and two-dimensional problems. In addition, the temperature rise modes in this study are not explicitly indicated, which further expands the applicability of the solutions obtained. The originality of this work is that the one- and two-dimensional thermal stress solutions for bimodular functionally graded beams are derived for the first time. The results obtained in this study may serve as a theoretical reference for the analysis and design of beam-like structures with obvious bimodular functionally graded properties in a thermal environment.

Measurement of two-dimensional residual stress in nanocrystalline superelastic NiTi fabricated with pre-strain laser shock peening

The measurement of residual stress is a challenging issue in industrial fields. In this work, focused ion beam (FIB) and digital image correlation (DIC) are combined together to measure the two-dimensional residual stress in nanocrystalline NiTi plates processed with pre-strain laser shock peening (LSP). A four-point bending experiment verifies the accuracy of this measurement method. The pre-strain LSP-treated surfaces are found to have significant compressive residual stress along the pre-strain direction and tensile residual stress along the vertical to pre-strain direction, which verifies pre-strain LSP as an efficient approach to create gradient residual stress layers in nanocrystalline NiTi plates. The FIB-DIC method has also proven to be an attractive tool to measure the two-dimensional residual stress in phase transition nanocrystalline materials.

A rigid line inclusion in a nonlocal elastic medium – Mode I deformation

In this paper, the problem of a rigid line inclusion in a nonlocal elastic medium is solved. Two special inclusion orientations are considered. These are (1) inclusion is parallel to the border of the medium, and (2) inclusion is perpendicular to the border of the medium. The two-dimensional stress and strain field and the effective mechanical property of the medium with inclusion are obtained. Because of the nonlocal effect, the stresses near the tips of the inclusion are significantly smaller than those based on the classic continuum mechanics framework. Essentially, under the nonlocal theory, there is no stress singularity at the tips of the inclusion. When the nonlocal effect vanishes, the results for the classic continuum theory of elasticity are recovered. In addition to the single inclusion, results for collinear inclusions are also presented.