PDE Standardization Analysis and Solution of TypicalMechanics Problems

A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed.

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.

Effects of applied potential on the stress corrosion cracking behavior of 7003 aluminum alloy in acid and alkaline chloride solutions

Potentiodynamic polarization tests and slow strain rate test（SSRT） in combination with fracture morphology observations were conducted to investigate the stress corrosion cracking（SCC） behavior of 7003 aluminum alloy（AA7003） in acid and alkaline chloride solutions under various applied potentials（Ea）. The results show that AA7003 is to a certain extent susceptible to SCC via anodic dissolution（AD） at open-circuit potential（OCP） and is highly susceptible to hydrogen embrittlement（HE） at high negative Ea in the solutions with p H levels of 4 and 11. The susceptibility increases with negative shift in the potential when Ea is less than-1000 m V vs. SCE. However, the susceptibility distinctly decreases because of the inhibition of AD when Ea is equal to-1000 m V vs. SCE. In addition, the SCC susceptibility of AA7003 in the acid chloride solution is higher than that in the alkaline solution at each potential. Moreover, the effect of hydrogen on SCC increases with increasing hydrogen ion concentration.

A CLOSED FORM SOLUTION OF STRESS INTENSITY FACTORS FOR THREE DIMENSIONAL FINITE BODIES WITH ECCENTRIC CRACKS

In this paper, a new analytical-engineering method of closed form solution about stress intensity factors for three dimensional finite bodies with eccentric cracks is derived by means of energy release rate method. The results of stress intensity factors can be obtained. The results provided ir this method are in nice agreement with those of the famous alternating method by which only special cases can be solved.

Thermoelastic stress analysis of multilayered films in a micro-thermoelectric cooling device

This paper presents an analytical solution for the thermoelastic stress in a typical in-plane＇s thin-film micro- thermoelectric cooling device under different operating con- ditions. The distributions of the permissible temperature fields in multilayered thin-films are analytically obtained, and the characteristics, including maximum temperature dif- ference and maximum refrigerating output of the thermo- electric device, are discussed for two operating conditions. Analytical expressions of the thermoelastic stresses in the layered thermoelectric thin-films induced by the tempera- ture difference are formulated based on the theory of mul- tilayer system. The results demonstrate that, the geometric dimension is a significant factor which remarkably affects the thermoelastic stresses. The stress distributions in layers of semiconductor thermoelements, insulating and support- ing membrane show distinctly different features. The present work may profitably guide the optimization design of high- efficiency micro-thermoelectric cooling devices.

Stresses induced by post-tensioned anchor in jointed rock mass

A new analytical study on stresses around a post-tensioned anchor in rocks with two perpendicular joint sets is presented. The assumptions of orthotropic elastic rock with plane strain conditions are made in derivation of the formulations. A tri-linear bond-slip constitutive law is used for modeling the tendon-grout interface behavior and debonding of this interface. The bearing plate width is also considered in the analysis. The obtained solutions are in the integral forms and numerical techniques that have been used for evaluation. In the illustrative example given, the major principal stress is compressive in the anchor free zone and compressive stress concentrations of 815 k Pa and 727 k Pa(for the anchor load of 300 k N) are observed under the bearing plate and the bond length proximal end, respectively. However, large values of tensile stresses with the maximum of-434 k Pa are formed at the bond length distal end. The results obtained using the proposed solution are compared very those of numerical method(FEM).

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.

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.

Apparent Stress Changes of the 2010 Luanxian Earthquake Sequence

On March 6,2010,an earthquake of M L4. 5 took place in Luanxian,Hebei Province,with plenty of foreshocks and aftershocks. From December 2009 to March 2010,a series of M L≥ 2. 5 earthquakes were recorded by the digital seismic network of the capital region,which were selected to calculate the apparent stress in this region. The results show that,firstly,a high value anomaly of apparent stress appeared before the M L4. 5 and peak value appeared on the main shock, which then decreased after the ML4. 5 earthquake. The apparent stress of the main shock is much greater than that of most aftershocks,the sequence type is considered as a main shock-aftershock. Secondly,the size of apparent stress perfectly reflects the state of the stress field in the hypocenter region,and we can discuss seismic sequence properties through the changing process of apparent stress,in combination with the traditional methods to identify a sequence more accurately. Finally,in the case of magnitude less than or equal to M L3. 3,correlation between magnitude and apparent stress is positive.

STRESS RATE INTEGRAL EQUATIONS OF ELASTOPLASTICITY

The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modem structural engineering. This paper introduces the boundary element method （BEM） into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic me- dia with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media.

Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.

高强高韧铝合金的研究现状及发展趋势

对高强高韧铝合金的发展情况进行了全面的评述。讨论了高强高韧铝合金微观组织与性能之间的关系 ,以及铝合金的主要强化机制。简要介绍了回归再时效 (RRA) ,固溶强化等热处理新技术 ,并就今后的研究开发提出了建议。

An Arbitrary Polygonal Stress Hybrid Element for Structural Dynamic Response Analysis

This paper constructs a new two-dimensional arbitrary polygonal stress hybrid dynamic(APSHD)element for structural dynamic response analysis.Firstly,the energy function is established based on Hamilton's principle.Then,the finite element time-space discrete format is constructed using the generalized variational principle and the direct integration method.Finally,an explicit polynomial form of the combined stress solution is give,and its derivation process is shown in detail.After completing the theoretical construction,the numerical calculation program of the APSHD element is written in Fortran,and samples are verified.Models show that the APSHD element performs well in accuracy and convergence.Furthermore,it is insensitive to mesh distortion and has low dependence on selecting time steps.

SCC investigation of low alloy ultra-high strength steel 30CrMnSiNi2A in 3.5wt% NaCl solution by slow strain rate technique

To evaluate stress corrosion cracking（SCC） mechanism of low alloy ultra-high strength steel 30CrMnSiNi2 A in environment containing NaCl, SCC behavior of the steel in 3.5wt% NaCl solution is investigated by slow strain rate technique（SSRT） with various strain rates and applied potentials, surface analysis technique, and electrochemical measurements. SCC susceptibility of the steel increases rapidly with strain rate decreasing from 1 · 10 5s 1to 5 · 10 7s 1, and becomes stable when strain rate is lower than 5 · 10 7s 1. SCC propagation of the steel in the solution at open circuit potential（OCP） needs sufficient hydrogen which is supplied at a certain strain rate.Fracture surface at OCP has similar characteristics with that at cathodic polarization 1000 mVSCE, which presents characteristic fractography of hydrogen induced cracking（HIC）.All of these indicate that SCC behavior of the steel in the solution at OCP is mainly controlled by HIC rather than anodic dissolution（AD）.

Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate

When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy＇s integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.

Assumed stress quasi-conforming triangular element for couple stress theory

In this paper, a 3-node triangular element for couple stress theory is proposed based on the assumed stress quasi-conforming method. The formulation starts from polynomial approx- imation of stresses. Then the stress-function matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the string-net functions are introduced to calculate strain integration and the stress smooth technique is adopted to improve the stress accuracy. Numerical results show that the proposed new model can pass the Co- 1 patch test with excellent precision, does not exhibit extra zero energy modes and can cap- ture the scale effects of microstructure.

A study on boundary integral equations for dynamic elastoplastic analysis for the plane problem by TD-BEM

The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane.Subsequently,based on Betti reciprocal theorem,by adopting the time dependent fundamental solutions in terms of displacement,traction and equivalent stress,the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem are established.The establishment procedures for the displacement and the stress boundary integral equations,together with the stress equation at boundary points,are presented in details,while the standard discretization both in time and space under the frame of time domain boundary element method(TD-BEM)and the solution of the algebraic equations are also briefly stated.Two verification examples are presented from different viewpoints,for elastic and elastoplastic analysis,for 1-D and 2-D geometries,and for finite and infinite domains.The TD-BEM formulation for dynamic elastoplastic analysis is presented for the plane strain problem as an example,where the formulation is also applicable for the plane stress problem by properly transforming the elastic constants and adopting the corresponding fundamental solutions.