ANALYTICAL SOLUTIONS TO STRESS CONCENTRATION PROBLEM IN PLATES CONTAINING RECTANGULAR HOLE UNDER BIAXIAL TENSIONS

The stress concentration problem in structures with a circular or elliptic hole can be investigated by analytical methods. For the problem with a rectangular hole, only approximate results are derived. This paper deduces the analytical solutions to the stress concentration problem in plates with a rectangular hole under biaxial tensions. By using the U-transformation technique and the finite element method, the analytical displacement solutions of the finite element equations are derived in the series form. Therefore, the stress concentration can then be discussed easily and conveniently. For plate problem the bilinear rectangular element with four nodes is taken as an example to demonstrate the applicability of the proposed method. The stress concentration factors for various ratios of height to width of the hole are obtained.

ANALYTICAL SOLUTIONS OF THERMAL STRESS DISTRIBUTION IN PLASTIC ENCAPSULATED INTEGRATED CIRCUIT PACKAGES

Due to the mismatch in the coefficients of thermal expansion of slicon chip and the surrounding plastic encapsulation materials, the induced thermal stress is the main cause for die and encapsulant rupture. The corner geometry is simplified as the semi_infinite wedge. Then the two_dimensional thermal stress distribution around the corner was obtained explicitly. Based on the stress calculation, the strain energy density factor criterion is used to evaluate the strength of the structure, which can not only give the critical condition for the stresses, but also determine the direction of fracture initiation around the corner.

Analytical solutions for characteristic radii of circular roadway surrounding rock plastic zone and their application

In the non-uniform stress field, the surrounding rock plastic zone of the circular roadway shows different shapes under the different confining pressure conditions. Based on the boundary shape characteristics of the plastic zone, the characteristic radii of the plastic zone were proposed, namely the horizontal,longitudinal and medial axis radii, which could reflect the plastic zone shapes characteristics and classify the sizes of the key parts. On the theoretical basis of elastic-plastic mechanics, analytical solutions for the characteristic radii were obtained by theoretical deduction, and the relationships between the characteristic radii and key influencing factors were analyzed. Finally, the evaluation criterion of the circular roadway surrounding rock plastic zone shapes, evaluation criterion of the location of potential hazards caused by the roadway surrounding rock and evaluation critical points of roadway dynamic disasters based on characteristic radii were proposed. This work could provide a theoretical basis for stability analysis of the surrounding rock, support design, and guide the prevention and control of dynamic roadway disasters.

Stress analytical solution for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform distributed pressure

The stress state around circular openings,such as boreholes,shafts,and tunnels,is usually needed to be evaluated.Solutions for stresses,strains and ultimate bearing capacities of pressurized hollow cylinder are common cases.Stress analytical method for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform pressure on the outer surface and uniform radial pressure on the inner surface is given.The power series method of complex function is used.The stress analytical solution is obtained with the assumption that two layers of a cylinder are fully contacted.The distributions of normal and tangential contact stress along the interface,tangential stress on the inner boundary and stresses in the radial direction at θ=0°,45° and 90°,are obtained.An example indicates that,when the elastic modulus of the inner layer of a double-layered thick-walled cylinder is smaller than that of the outer layer,the tangential stress is smaller than that in the corresponding point for a traditional cylinder composed of homogeneous materials.In that way,stress concentration at the inner surface can be alleviated and the stress distribution is more uniform.This is a capable way to enhance the elastic ultimate bearing capacity of thick-walled cylinder.

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magnetomelectro-elastic （FGMEE） cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.

Mathematic Model and Analytic Solution for a Cylinder Subject to Exponential Function

Hollow cylinders are widely used in spacecraft, rockets, weapons, metallurgy, materials, and mechanical manufacturing industries, and so on, hydraulic bulging roll cylinder and hydraulic press work all belong to hollow cylinders. However, up till now, the solution of the cylinder subjected to the pressures in the three-dimensional space is still at the stage of the analytical solution to the normal pressure or the approximate solution to the variable pressure by numerical method. The analytical solution to the variable pressure of the cylinder has not yet made any breakthrough in theory and can not meet accurate theoretical analysis and calculation requirements of the cylindrical in Engineering. In view of their importance, the precision calculation and theoretical analysis are required to investigate on engineering. A stress function which meets both the biharmonic equations and boundary conditions is constructed in the three-dimensional space. Furthermore, the analytic solution of a hollow cylinder subjected to exponential function distributed variable pressure on its inner and outer surfaces is deduced. By controlling the pressure subject to exponential function distributed variable pressure in the hydraulic bulging roller without any rolling load, using a static tester to record the strain supported hydraulic bulging roll, and comparing with the theoretical calculation, the experimental test result has a higher degree of agreement with the theoretical calculation. Simultaneously, the famous Lam6 solution can be deduced when given the unlimited length of cylinder along the axis. The analytic solution paves the way for the mathematic building and solution of hollow cylinder with randomly uneven pressure.

Unified analytical solution for deep circular tunnel with consideration of seepage pressure,grouting and lining

A new unified analytical solution is presented for predicting the range of plastic zone and stress distributions around a deep circular tunnel in a homogeneous isotropic continuous medium. The rock mass, grouting zone and lining are assumed as elastic-perfectly plastic and governed by the unified strength theory(UST). This new solution has made it possible to consider the interaction between seepage pressure, lining, grouting and rock mass, and the intermediate principal stress effect together. Moreover, parametric analysis is carried out to identify the influence of the related parameters on the plastic zone radius. Under the given conditions, the results show that the plastic zone radius decreases with an increasing cohesion, internal friction angle and hydraulic conductivity of lining and unified failure criterion parameter, respectively; whereas the plastic zone radius increases with the growth of elasticity modulus of lining. Comparison of results from the new solution and the other published one shows well agreement and provides confidence in the new solution proposed.

Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to thermal and uniformly dis-tributed load is investigated,with material parameters being arbitrary functions of the thickness coordinate. The heat conduction problem is treated as a 1D problem through the thickness. Based on the elementary formulations for plane stress problem,the stress function is assumed to be in the form of polynomial of the longitudinal coordinate variable,from which the stresses can be derived. The stress function is then determined completely with the compatibility equation and boundary conditions. A practical example is presented to show the application of the method.

Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated. The analysis is based on the exact elasticity equations for the plane stress problem. The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate. The expressions for stress components are then educed from the stress function by simple differentiation. The stress function is determined from the compatibility equation as well as the boundary conditions by a skilful deduction. The analytical solution is compared with FEM calculation, indicating a good agreement.

Analytic Study on a Plane Stress Tensile Crack in a Power-Law Hardening Material

Aim To construct an analytic solution for the asymptotic field near a tensile cracktip of power-law hardening material under Plane stress condition. Methods Constructing funtion method was used. Results The exact asymptotic field was found. Conclusion The exact analytic solution for the problem is available.

Science Letters:Analytical solution for fixed-end beam subjected to uniform load

A bi-harmonic stress function is constructed in this work. Ariy stress function methodology is used to obtain a set of analytical solutions for both ends fixed beams subjected to uniform load. The treatment for fixed-end boundary conditions is the same as that presented by Timoshenko and Goodier (1970). The solutions for propped cantilever beams and cantilever beams are also presented. All of the analytical plane-stress solutions can be obtained for a uniformly loaded isotropic beam with rectangular cross section under different types of classical boundary conditions.

MATHEMATIC MODEL AND ANALYTIC SOLUTION FOR CYLINDER SUBJECT TO UNEVEN PRESSURES

According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant＇s theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab. When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the famous Lame solution can be induced from this limit. The above work paves the way for mathematic model building of hollow cylinder and for the analytic solution of hollow cvlinder with randomly uneven pressure.

Analytic decomposition and numerical procedure for solving the singular boundary value problem arising in viscous flows

An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.

The singularity in the state-based peridynamic solution of uniaxial tension

We solve the local uniaxial tension of an infinite rod in the framework of non-ordinary state-based peridynamics.The singular solutions of stress and displacement are acquired.When the influencing range of the window function approaches zero,these two solutions will return to the solutions of the classical elasticity.The analysis shows that the singularities of the solutions stem from such a feature of the window function that must be represented by a rapidly decreasing function in physics.Contrary to the classical elasticity,the stress solution of peridynamics is smoother than the displacement solution.In addition,a criterion used to select the window function is proposed in this paper.

ANALYTICAL SOLUTION FOR FIXED-FIXED ANISOTROPIC BEAM SUBJECTED TO UNIFORM LOAD

The analytical solutions of the stresses and displacements were obtained for fixed-fixed anisotropic beams subjected to uniform load. A stress function involving unknown coefficients was constructed, and the general expressions of stress and displacement were obtained by means of Airy stress function method. Two types of the description for the fixed end boundary condition were considered. The introduced unknown coefficients in stress function were determined by using the boundary conditions. The analytical solutions for stresses and displacements were finally obtained. Numerical tests show that the analytical solutions agree with the FEM results. The analytical solution supplies a classical example for the elasticity theory.

Analytical Research of Temperature Distribution and Thermal Stresses within Circular Micro-hotplates

Micro-hotplate （MHP） technology is one key part in the manufacturing of gas sensors. The pursuit of analytical solutions for the temperature distribution and also thermal stresses within the MHP is of intrinsic scientific interest. In this study, analytical solutions for the temperature field, and both radial and tangential stresses and van Mises stress for circular MHP were obtained. Two geometries were considered： one had a circular heater at the center and the other had a circular heater at the center and an annular heater within the membrane part. Internal heat generation was incorporated in the energy equation for the MHP and different values of convection heat transfer coefficient were used at the upper and lower surfaces of the MHP. It has been shown that the MHP with two heaters can provide more uniform temperature field compared with the MHP with one heater. The main objective of this work is to provide an exact analytical solution for thermal stresses within the circular micro-hcater with a simple geometry as a benchmark, from mathematical point of view, against which the accuracy of new numerical schemes can be checked. To make sure that the analytical procedure is correct, the analytical results are checked against numerical solutions derived from finite element simulation. Since the analytical models for the temperature field and especially for the thermal stresses of MHP ace seldom investigated in the literature, the obtained results are believed to facilitate the design and performance evaluation of MHPs as well.

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.

Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of residual surface stress and surface elasticity are included.Airy stress function method and Fourier integral transform technique are used.The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all.Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane.Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane,while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded.In the remaining situations,combined effects of surface elasticity and residual surface stress should be considered.The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects.The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.

Stress-deformed state of cylindrical specimens during indirect tensile strength testing

In this study, the interaction between cylindrical specimen made ofhomogeneous, isotropic, and linearlyelastic material and loading jaws of any curvature is considered in the Brazilian test. It is assumed thatthe specimen is diametrically compressed by elliptic normal contact stresses. The frictional contactstresses between the specimen and platens are neglected. The analytical solution starts from the contactproblem of the loading jaws of any curvature and cylindrical specimen. The contact width, correspondingloading angle （2 ^0）, and elliptical stresses obtained through solution of the contact problems are used asboundary conditions for a cylindrical specimen. The problem of the theory of elasticity for a cylinder issolved using Muskhelishvili＇s method. In this method, the displacements and stresses are represented interms of two analytical functions of a complex variable. In the main approaches, the nonlinear interactionbetween the loading bearing blocks and the specimen as well as the curvature of their surfacesand the elastic parameters of their materials are taken into account. Numerical examples are solved usingMATLAB to demonstrate the influence of deformability, curvature of the specimen and platens on thedistribution of the normal contact stresses as well as on the tensile and compressive stresses actingacross the loaded diameter. Derived equations also allow calculating the modulus of elasticity, totaldeformation modulus and creep parameters of the specimen material based on the experimental data ofradial contraction of the specimen.

Investigation of the short-term stress distribution in stopes and drifts backfilled with cemented paste backfill

Cemented paste backfill(CPB) is largely used in underground mines worldwide.A key issue associated with application of CPB is to estimate the stresses in backfilled stopes and on barricades.Recent numerical and experimental results show that arching effect is absent shortly after the placement of CPB in stopes.However,stress decreases in barricade drift with increasing distance between the measurement points and drawpoint have also been observed,demonstrating arching effect shortly after the pouring of CPB.To explain these paradoxes,CPB is considered as Bingham fluid having a yield shear stress.Three dimensional analytical solutions are proposed to evaluate the short-term total stresses in backfilled stopes and on barricades,accounting for the CPB's yield shear stress-induced arching effect.Stress diminution due to such arching effect in the backfilled stopes and on barricades is indeed obtained.But the reduction becomes insignificant using typical yield shear stress and stope geometry.More analyses indicate that the typical yield shear stress values do not fully correspond to field conditions where the yield shear stress would increase exponentially due to apparent consolidation(loss of water by drainage,a phenomenon similar to the desiccation of overly saturated fine-grained materials).