Mathematical modeling of friction stir welding considering dry and viscous friction

In this paper, the investigation of a pre-immersion period of Friction Stir Welding process (FSW) from the mechanical point of view using analytical considerations is presented. This initial period is a part of the real technological process, when a FSW instrument reaches the touchdown point with the workpiece, preheats the contact zone, but does not penetrate the material. This approach enables an insight into FSW process on a level, where the system parameters are known and an experimental evaluation of theoretical results is possible. Beside these considerations an analytical estimation of the power input of dry and viscous friction during instrument rotation and translation in parallel is shown. These phenomena are investigated including friction between a FSW instrument and a workpiece. The influences of both dry and viscous friction are studied. Experiments have been done to validate the calculated results.     

Analytical solution for the fully coupled static response of variable stiffness composite beams

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.     

Numerical investigation of residual thermal stresses in welded joints of heterogeneous steels with account of technological features of multi-pass welding

The paper describes a two-dimensional mathematical model to evaluate stresses in welded joints formed in multi-pass welding of multi-layered steels. The model is based on a system of equations that includes the Lagrange's variational equation of the incremental theory of plasticity and the Biot's variational principle for heat transfer simulation. In the constitutive equations, the changes in the volume which occur as a result of phase transitions can be taken into account. Therefore, the prehistory and impact of thermal processing of materials on macroscopic properties of the medium can be considered.The variational-difference method is used to solve both the heat transfer equation for calculation of the non stationary temperature field and the quasi-static problem of thermoplasticity at each time-step. The two-dimensional problems were solved to estimate the residual thermal stresses (for the case of plane stress or plane strain) during cooling of welds and assessing their impact on strain localization in the heat-affected zone under tensile and compressive loading considering differences in mechanical properties of welded materials.It is shown that at initial stages of the plastic flow, the residual stresses significantly affect the processes of stress concentration and localization of strains in welded joints. To estimate the model parameters and to verify the modeling results, the available experimental data from scientific literature obtained on the basis of the Satoh test for different welding alloys was used.     

Cooling of a stretching sheet in non-Newtonian flow

An analytical solution has been developed for the temperature distribution in a stretching sheet undergoing cooling in a non-Newtonian viscoelastic fluid. A solution has also been obtained for the temperature field in the boundary layer. Sufficient conditions for convergence of the solutions have been established.     

Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity,non-uniform heat source and radiation

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a non-Newtonian viscoelastic fluid over a flat sheet with a linear velocity in the presence of thermal radiation and non-uniform heat source. The thermal conductivity is assumed to vary as a linear function of temperature. The basic equations governing the flow and heat transfer are in the form of partial differential equations, the same have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformation. The transformed equations are solved analytically by regular perturbation method. Numerical solution of the problem is also obtained by the efficient shooting method, which agrees well with the analytical solution. The effects of various physical parameters such as viscoelastic parameter, Chandrasekhar number, Prandtl number, variable thermal conductivity parameter, Eckert number, thermal radiation parameter and non-uniform heat source/sink parameters which determine the temperature profiles are shown in several plots and the heat transfer coefficient is tabulated for a range of values of said parameters. Some important findings reported in this work reveals that combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Nonlocal coupled photo-thermoelasticity analysis in a semiconducting micro/nano beam resonator subjected to plasma shock loading: A Green-Naghdi-based analytical solution

In this article, coupled photo-thermoelasticity analysis is carried out using an analytical method in a semiconducting micro/nano beam resonator, considering Green – Naghdi theory (with energy dissipation) and small scale effects. The governing equations for temperature and displacement fields are derived using Eringen nonlocal theory combined with Rayleigh beam theory. One end of the assumed semiconducting MEMS/NEMS is excited by three types of suddenly increasing carrier density and temperature as the plasma and thermal shock loading. The transient behaviours of carrier density field are studied and the effects of disturbances in plasma field on other fields including temperature and deflection are obtained using the proposed analytical solution. The presented analytical solution is based on Laplace transform. To find the dynamic and transient behaviours of fields’ variables in time domain, an inversion Laplace technique is utilized, which is called Talbot method. The effects of small scale parameter and dimensions of the semiconducting micro/nano beam on the dynamic behaviours of fields’ variables are discussed in detail. The axial wave propagation and the distribution of fields’ variables along axial direction are studied at various times.     

Modelling of heat affected zone in cylindrical steel elements surfaced by welding

Computational models of a temperature field in cylindrical steel elements surfaced by the following methods: controlled pitch, spiral welding sequence and spiral welding sequence with swinging motion of the welding head are presented in the paper. The lateral surface of regenerated cylindrical object, subjected to the welding heat source, has been treated as a plane rolled on cylinder and temperature field of repeatedly surfaced plain massive body was solved. Temperature rises, caused by overlaying consecutive welding sequences and self-cooling of areas previously heated, were taken into consideration in the solution. The computations of the temperature field for continuous casting steel machine roll made of 13CrMo4 steel were carried out.     

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.     

Analytical transient-time solution for temperature in non perfused tissue during radiofrequency ablation

Radiofrequency ablation (RFA) with internally cooled needle-like electrodes is a technique widely used to destroy cancer cells. In a previous study we obtained the analytical solution of the biological heat equation associated with the RFA problem in perfused tissue, i.e. when the governing equation which models the temperature distribution in tissue includes the blood perfusion therm. We also found that under these circumstances the temperature profiles always reach a steady state (limit temperature). However, the analytical solution of the RFA thermal problem without perfusion (e.g. conducted on an organ in which atraumatic vascular clamping is performed to temporally interrupt blood perfusion), cannot be directly obtained by setting the blood perfusion term to zero in the previously obtained solution. In fact, it is necessary to address the mathematical resolution in a totally different way. Our goal was to obtain the analytical expression of the temperature distribution in an RFA process with internally cooled needle-like electrodes when the biological tissue is not perfused. We consider two spatial domains: A finite domain which represents the real situation, and an infinite domain, which only makes sense from a mathematical point of view and which has been traditionally employed in analytical studies. Even though considering infinite time is not realistic, these approaches are surely worth considering in order to understand what happens “far from the electrode” or for “very long periods of time.” The results indicate that the temperature value is finite both when the spatial domain is finite (which implies that a steady state is reached), and when time is finite for any spatial domain. From this it can be concluded that a steady state is never reached if the spatial domain is infinite.     

Elastic analytical solution of shallow tunnel owing to twin tunnelling based on a unified displacement function

A unified displacement function is used as the displacement boundary condition of the cross-section of each tunnel, which can be used to capture the asymmetrical deformation behaviors about the horizontal and vertical center line. Complex variable method is adopted to obtain the analytical solution of a single tunnel. Based on the Schwartz alternating method, the analytical solution of twin tunnels is obtained and the correctness of the analytical solution of twin tunnels is verified. A series of analyses are carried out to investigate the effects of the displacement boundary condition of the tunnel cross-section and the distance of twin tunnels on the ground surface settlement trough curve and the ground displacement field. Based on the proposed method, the parameters of displacement mode of the cross-section of twin tunnels are calibrated through field monitoring data of the Bangkok Subway Tunnel project, and the displacement field of the whole ground are analyzed. The proposed method can provides guidance in the conceptual stage of the design process of twin tunnels.     

Thermal analysis of moving induction heating of a hollow cylinder with subsequent spray cooling: Effect of velocity,initial position of coil,and geometry

In this paper, temperature analysis of the complete process of moving induction heat treatment is performed using numerical methods. A non-linear and transient magneto-thermal coupled problem with a moving coil which is considered as moving heat source, is investigated by an efficient finite-element procedure. A vertical hollow circular cylinder is heated by the moving coil at a given velocity along it, and the heated parts then quenched by a moving water–air spray. The effects of natural convection with air on the both inner and outer surfaces of cylinder, and also radiation of outer surface of cylinder with ambient are taken into account. For quenching of work-piece, a specific kind of atomized spray cooling which utilizes a mixture of water and air with different mass fractions is used. This procedure includes moving boundary conditions, temperature-dependent properties, and change in magnetic permeability of specified alloy at the Curie temperature. Obtained numerical results have been verified by comparison with analytical solutions using Green’s function methods. Also, the effect of velocity, initial position of inductor and inner to outer radius ratio on temperature distribution are investigated.     

A thermomechanical analysis of interfaces between the mating parts in ultrasonic wire bonding

Ultrasonic welding (USW) is an alternative solution for the bonding process especially in automotive industry. Ultrasonic welding of metals is a joining technique as a combination of applying pressure and frictional vibrations within the range of ultrasonic frequencies. In automotive industry, ultrasonic welding is often used for wired connections. As an alternative for crimping technology of multi-strand aluminum cables in wire bonding, ultrasonic welding is used. This work presents a thermomechanical analysis of the interface between two mating parts in USW. For this reason, the temperature distribution at bonding locations inside a wire bundle due to frictional vibrations and pressure is investigated using the finite element method (FEM). The obvious difference in microsections from different welding samples, which originates from different local temperature rises, was the motivation for this study to further investigate the thermomechanical aspects of the USW by use of finite element simulations. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The scattering of acoustic waves by a cylinder with a non-uniform elastic coating

The problem of the scattering of a plane acoustic wave by a solid cylinder with a radially non-uniform elastic coating is considered. An analytical expression describing the scattered acoustic field is obtained. The equations of motion of the non-uniform elastic cylindrical layer are reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the power-series method. The results of calculations of the directional pattern of the scattered field are presented.     

Modelado numérico del proceso de soldadura FSW incorporando una técnica de estimación de parámetros

Numerical models of heat transfer and fluid flow used in the simulation of the friction-stir welding (FSW) process have contributed to the understanding of the process. However, there are some input model parameters that cannot be easily determined from fundamental principles or the welding conditions. As a result, the model predictions are not always in agreement with experimental results. In this work, the Levenberg-Marquardt (LM) method is used in order to perform a non-linear estimation of the unknown parameters present in the heat transfer and fluid flow models, by adjusting the temperatures results obtained with the models to temperature experimental measurements. These models are implemented in a general-purpose software that uses a numerical formulation developed from the finite element method (FEM). The unknown parameters are: the friction coefficient and the amount of adhesion of material to the surface of the tool, the heat transfer coefficient on the bottom surface and the amount of viscous dissipation converted into heat. The obtained results show an improvement in the numerical model predictions from the incorporation of parameter estimation techniques.     

大气运动基本方程组的解析解

在已知大气运动基本方程于光滑函数类中具有最好的稳定性前提下,讨论了它的局部解的解空间构造．根据它的解空间构造,分析了这个方程具有代表性和应用性的第三初值问题,在解析函数类中给出了适定的第三初值问题的解析解的计算方法以及具体的关系表达式,在局部解意义下完整的解决了这一点初值问题的解析解所涉及的理论与计算问题．指出其它类型定解问题都可以仿照文中的计算方法和步骤,求出所需要的稳定的解析解．     

Approximate solution for two stage open networks with Markov-modulated queues minimizing the state space explosion problem

Analytical solutions for two-dimensional Markov processes suffer from the state space explosion problem. Two stage tandem networks are effectively used for analytical modelling of various communication and computer systems which have tandem system behaviour. Performance evaluation of tandem systems with feedbacks can be handled with these models. However, because of the numerical difficulties caused by large state spaces, considering server failures and repairs at the second stage employing multiple servers has not been possible. The solution proposed in this paper is approximate with a high degree of accuracy. Using this approach, two stage open networks with multiple servers, break downs, and repairs at the second stage as well as feedback can be modelled as three-dimensional Markov processes and solved for performability measures. Results show that, unlike other approaches such as spectral expansion, the steady state solution is possible regardless of the number of servers employed.     

The Lambert Function on the Solution of a Delay Differential Equation

Recently, a new method for computing the analytical solution of a delay differential equation was developed considering a constant initial function. It is based on the existence of a specific class of polynomials in the delay. In this article, we extend this new method to the case of a continuous initial function. We also show the relationship between the new solution's method and the solution expressed in terms of the Lambert function.     

Modeling of thermal phenomena in single laser beam and laser-arc hybrid welding processes using projection method

This paper concerns mathematical and numerical modeling of thermal phenomena accompanying single laser and laser-arc hybrid butt welding of steel sheets. Coupled heat transfer and fluid flow in the fusion zone were described respectively by transient heat transfer equation and Navier–Stokes equation. Laser beam and electric arc heat sources were modeled using different heat source power distributions. Latent heat associated with the material’s state changes, buoyancy forces and liquid material flow through a porous medium were taken into account in considerations. Differential governing equations were numerically solved using projection method combined with finite volume method. Elaborated solution algorithm was implemented into computer solver used for simulation of heat transfer and fluid flow during welding. The geometry of the weld and heat affected zone as well as cooling rates were estimated on the basis of numerically obtained temperature field.     

Electro-thermo-mechanical behaviors of FGPM spheres using analytical method and ANSYS software

A hollow sphere made from functionally graded piezoelectric material (FGPM) such as PZT_4 has been considered. One-dimensional analytical method for electro-thermo-mechanical response of symmetrical spheres is used. For asymmetric three-dimensional analysis, ANSYS finite element software is employed in this study. Loading is combination of internal and external pressures, a distributed temperature field due to steady state heat conduction and a constant electric potential difference between its inner and outer surfaces for analytical solution. In three-dimensional solutions closed and open spheres with different boundary conditions subjected to an internal pressure and a uniform temperature field are studied. All mechanical, thermal and piezoelectric properties except the Poisson’s ratio are assumed to be power functions of radius. It has been found from analytical solution that the induced radial and circumferential stresses of an imposed electric potential is similar to the residual stresses locked in the homogeneous sphere during the autofrettage process of these vessels. It has been concluded from the three-dimensional analysis that the magnitudes of effective stresses at all node points are higher for the clamped-clamped boundary condition and are lower for the simply-simply supported condition.     

Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.