Numerical and experimental study of mitigation of welding distortion

Welding stresses and deformations are closely related phenomena. During the heating and cooling cycles thermal strains may occur in the weld and adjacent area. The strains produced during the heating stage of welding are always accompanied by plastic deformation of the metal. The stresses resulting from these strains combine and react to produce internal forces that cause a variety of welding distortions. Welding deformation needs to be minimized and also the designer should know before hand the extent of deformation so that it can be accounted for in the design as well as in the construction stages.In this paper, heat sinking as a method of distortion mitigation has been studied. Heat sinking has been affected by circulating water through channel clamped at the bottom surface of the plates undergoing welding. The pseudolinear equivalent constant rigidity concept has been used in this investigation for thermo-mechanical analysis of plates undergoing welding with simultaneous heat sinking. The initial nonlinear problem with varying modulus dependent on temperature is transformed into a pseudolinear equivalent system of constant rigidity that is solved by linear analysis.The numerical results compared very well with those of the experimental ones. The proposed concept is found to be computationally more efficient and simpler to model compared to FEM for solving similar thermo-elasto-plastic nonlinear problems. The procedure presented in this work and the results thus obtained, holds a great promise for determining the heat sinking parameters for effectively controlling welding distortion.     

Finite element modeling of welding processes

This work deals with the simulation of fusion welding by the Finite Element Method. The implemented models include a moving heat source, temperature dependence of thermo-physical properties, elasto-plasticity, non-steady state heat transfer, and mechanical analysis. The thermal problem is assumed to be uncoupled from the mechanical one, so the thermal analysis is performed separately and previously to the mechanical analysis at each time step. The mechanical problem is based on the thermal history. A special treatment is performed on mechanical elements during the liquid/solid and solid/liquid phase changes to account for stress states. The three-dimensional stress state of a butt-welded joint is obtained as an example of an application.     

双圆柱体结构的热耦合微分方程及其有限元解

针对物体的热耦合问题,给出了内外形式的双圆柱体结构的热耦合分析的基本微分方程,提出了采用有限元进行求解的方法,并给出了具体算例.热耦合是一个复杂的多元微分方程的求解问题.求解得知,热应力分布情况不仅与几何形状有关,也与热载荷与材料物理特性有关,在一定温度变化范围内,物体由于温度变化所产生的等效应力会随温度呈近似线性变化.此外,结构几何尺寸对应力分布变化的影响大于对温度分布变化的影响.     

杂交元假设应力模式的变形刚度分析

通过本征变形模式提出识别杂交元零能变形模式和假设应力场中零能应力模式的新方法,同时给出了在假设应力场中增加应力模式时杂交元变形刚度的计算公式．从而从理论上阐明了在假设应力场中增加零能应力模式不仅不能抑制单元零能变形模式而且可能增加非零能变形模式的刚度,因此不宜用来假设应力场;同时进一步指出寄生应力模式将使单元产生虚假应变能而使单元显得过刚,因此即使它能够抑制单元零能变形模式也不宜用来假设应力场,从而为假设应力场提供了合理的建议．数值算例说明了包含零能应力模式和寄生应力模式时单元的性能．     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity

Steady mixed convection micropolar fluid flow towards stagnation point formed on horizontal linearly stretchable melting surface is studied. The vortex viscosity of micropolar fluid along a melting surface is proposed as a constant function of temperature while dynamic viscosity and thermal conductivity are temperature dependent due to the influence of internal heat source on the fluid. Similarity transformations were used to convert the governing equation into non-linear ODE and solved numerically. A parametric study is conducted. An analysis of the results obtained shows that the flow-field is influenced appreciably by heat source, melting, velocity ratio, variable viscosity and thermal conductivity.     

Nonlinear dynamic response for functionally graded shallow spherical shell under low velocity impact in thermal environment

Based on Giannakopoulos’s 2-D functionally graded material (FGM) contact model, a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment. The FGM shallow spherical shell, having temperature dependent material property, is subjected to a temperature field uniform over the shell surface but varying along the thickness direction due to steady-state heat conduction. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko–Midlin theory. And the nonlinear motion equations of the FGM shallow spherical shell under low velocity impact in thermal environment are founded in terms of displacement variable functions. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated and effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.     

Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers

The main aim of this article is to obtain characterizations of the solution set of two non-linear programs in terms of Lagrange multipliers. Both the programs have pseudolinear constraints but the objective function is convex for the first program and pseudolinear for the second program, where all the functions are defined in terms of bifunctions.     

悬臂梁大挠度问题的双参数摄动解

利用拟线性分析方法中的一阶导数代换对基本方程进行了简化处理,将基本的微分积分方程转化为非线性代数方程组,并结合双参数摄动研究了悬臂梁的大挠度问题．与已有的研究结果比较表明: 将拟线性方法用于研究悬臂梁的大挠度问题,计算较为简便, 同时又具有良好的精度．     

Remarks on a Mean Field Equation on $\mathbb{S}^2$

In this note, we study symmetry of solutions of the elliptic equation\begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2},\end{equation*} that arises in the consideration of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.     

Estimations on Thermo-mechanical Dynamics of Vibration Elastomeric Isolators

This analysis deals with one of the basic problem category of vibratory systems, means the complete and complex characterization of elastic and viscous isolators behaviour under dynamic loads such as vibrations, seismic waves, shocks, etc. Usually, the dynamic characteristics of vibration isolators made by elastomeric materials are considered to have a constant shape for a certain practical case. It is ignored the thermal phenomenon inside the isolator block during the exploitation cycles and its influences on the proper characteristic parameters. This usual approximation leads to more or less significant differences between simulation and practical evolution of a vibration isolator subjected to the same dynamic load. Continuous changes of rigidity modulus and/or dissipative characteristics due to internal thermal effects imply aleatory evolution of the isolated system, unstable movements and resonance imminence danger. The partial results of this analysis dignify the linkage between thermal effects into the elastomeric isolator and its essential dynamic parameters. Using of these correlations frames the seismic shock and vibration protective devices designing and deployment areas. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类退化椭圆型方程边值问题的适定性

研究一类特殊退化椭圆型方程边值问题的适定性,该类问题与双曲空间中的极小图的Dirichlet问题,曲面的无穷小等距形变刚性问题等等的研究密切相关,而这类方程的特征形式在区域上是变号的,其适定性是值得深入讨论的.最后,得到这类边值问题的H~1弱解的存在性和唯一性.     

Equal-Stress Reinforcement of Elastoplastic Momentless Shells with a Protective Coating under Thermal and Force Loadings

The problem of equal-stress reinforcement of three-layered momentless shells with fibers of constant cross section under thermoelastic and thermoplastic deformation of phase materials is formulated. A qualitative analysis of systems of resolving equations is carried out. Analytical solutions to the axisymmetric problems of equal-stress reinforcement of shells of revolution under elastic and inelastic deformation are constructed. It is shown that the problem can have several alternative solutions, which can be additionally controlled by varying the reinforcement intensities on the shell contour.     

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.     

Lagrange Multiplier Characterizations of Solution Sets of Constrained Nonsmooth Pseudolinear Optimization Problems

This paper deals with the minimization of a class of nonsmooth pseudolinear functions over a closed and convex set subject to linear inequality constraints. We establish several Lagrange multiplier characterizations of the solution set of the minimization problem by using the properties of locally Lipschitz pseudolinear functions. We also consider a constrained nonsmooth vector pseudolinear optimization problem and derive certain conditions, under which an efficient solution becomes a properly efficient solution. The results presented in this paper are more general than those existing in the literature.     

The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α.For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α. We also show injectivity and stability for g and α in a generic class G including real analytic ones.For the nonlinear problem, we show rigidity for real analytic simple (g,α), rigidity for metrics in a given conformal class, and locally, near any (g,α)∈G. We also show that simple magnetic systems on two-dimensional manifolds are always rigid.     

The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet

An analysis has been carried out to study the momentum and heat transfer characteristics in an incompressible electrically conducting non-Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly non-linear coupled ordinary differential equations by similarity transformations. The effect of variable fluid viscosity, Magnetic parameter, Prandtl number, variable thermal conductivity, heat source/sink parameter and thermal radiation parameter are analyzed for velocity, temperature fields, and wall temperature gradient. The resultant coupled highly non-linear ordinary differential equations are solved numerically by employing a shooting technique with fourth order Runge–Kutta integration scheme. The fluid viscosity and thermal conductivity, respectively, assumed to vary as an inverse and linear function of temperature. The analysis reveals that the wall temperature profile decreases significantly due to increase in magnetic field parameter. Further, it is noticed that the skin friction of the sheet decreases due to increase in the Magnetic parameter of the flow characteristics.     

Quenching rate for a nonlocal problem arising in the micro-electro mechanical system

In this paper, we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro-electro mechanical system. This question is equivalent to the stabilization of the solution to the transformed problem in self-similar variables. First, some a priori estimates are provided. In order to construct a Lyapunov function, due to the lack of time monotonicity property, we then derive some very useful and challenging estimates by a delicate analysis. Finally, with this Lyapunov function, we prove that the quenching rate is self-similar which is the same as the problem without the nonlocal term, except the constant limit depends on the solution itself.