An analytical solution to heat conduction with a moving heat source

This paper deals with an analytical solution to heat conduction in the medium subjected to a moving heat source. It evaluates the temperature distribution around a rectangular shape source moving at a constant speed along the axis of a bar. The transient temperature field from a moving heat source was analyzed using a Fourier series procedure. The most interesting result of the theory, is the derivation of a single formula capable of predicting the cooling time and cooling rate with a fairly good accuracy for ranges of temperature. Because of the passage of the heat source, the rise of temperature produced at a given near the source, tends to rapidly become constant. Several sample problems are discussed and illustrated, and comparisons with numerical approaches where these can also be used made. The results show that these solutions are in good agreement with the numerical results.     

Detailed modelling of a moving heat source using multigrid methods

The actual contact between solid surfaces is generally rough and time dependent. This roughness can only be correctly described using a 3D model. Moreover, severe thermo-mechanical loading induces high stress and temperature gradients within a thin subsurface layer. Even higher gradients are obtained in the case of surface coatings. Consequently, the problem is strongly multiscale: from contact to coating to roughness, the characteristic dimensions range from millimeter to nanometer.A straightforward discretization of this multiscale problem would exceed the memory and CPU capabilities of current computers. The aim of this work is to propose a more efficient numerical model able to deal with this multiscale problem: using 10⁹ points and 10³ timesteps.This paper traces the history of the numerical solutions of the heat equation from the pioneering work of Carslaw and Jaeger, to the current era.The proposed model is based on multigrid techniques within a finite difference frame work. Localized refinement is implemented to optimize memory and computing time costs. The numerical performance of the solver is presented through a comparison with analytical results using different types of boundary conditions. A multisource contact is studied as a first approximation to real asperity interaction.     

Adaptive finite element solution for the heat conduction with a moving heat source

We have demonstrated that some of the parabolic type problems encountered in such branches of engineering as heat conductions with a moving source can be analyzed successfully by means of the finite element method. Adapted mesh generation technique is implemented for solving heat transfer involving a moving heat source so that small elements can be used in areas of large time rates of change of temperature. It has been adjusted to steep gradients of the solution with respect to the relatively large time interval. A program has been developed for the case of two-dimensional triangular elements, and algorithm is possessed a number of usual advantages that made solutions very divergent. Numerical results have shown that the adaptive gridding scheme is effective in localizing oscillations due to the sharp gradients or discontinuities in the solution. Furthermore, the numerical results near the region of moving source from the present method are under and over estimated the solution of traditional finite element method by almost 3% respectively. The several examples are given to illustrate the validity and practicality of the method. The results of various sample solutions are evaluated and discussed.     

General temperature rise solution for a moving plane heat source problem in surface grinding

In the present study, the general solutions for a transient state as well as for the temperature rise formed everywhere in the workpiece due to a rectangular-shaped moving plane heat source arising at the grinding zone are derived. The present analysis starts from a point heat source solution by applying the method of separation of variables to a three-dimensional heat conduction problem. Because the workpiece moving velocity is quite small, the convective term related to the workpiece velocity is first excluded from the heat conduction equation. This workpiece velocity effect will be included in the model by slightly modifying the coordinate variable in the sliding direction shown in the solution of the point heat source. Therefore, the general three-dimensional solution of the stationary temperature rise can be expressed in an integral form as a function of the product value of the unknown initial condition and the particular solution of temperature rise. The unknown initial temperature rise in the solution can be replaced by the point heat source due to frictional that multiplying the product of the Dirac delta functions defined for three directions. Using the definition of the Dirac delta function, the temperature rise solution for a point heat source can thus be obtained. This solution is further extended to obtain the moving and uniform heat sources arising in a rectangular grinding zone. A comparison among the experimental result and the theoretical results predicted by the present model and Jaeger’s model [Jaeger JC (1942) Proc Roy Soc, NSW 76:203–224.] show that the present model is quite accurate and is generally superior to Jaeger’s model; it can be applied to predict the three-dimensional temperature rise distributions in the workpiece.     

Residual stresses in a laminated shell during cure

In this paper, a viscoelastic finite element analysis was performed to investigate residual stresses occurred in a laminated shell during cure. A viscoelastic constitutive equation that can describe stress relaxation during the cure was defined as functions of degree of cure and temperature, and derived as a recursive formula used conveniently for numerical analyses. The finite element program was developed on the basis of 3-D degenerated shell element and the first order shear deformation theory, and was verified by comparing with an exact solution of the one dimensional problem. Effects of chemical shrinkage and stacking sequence on the residual stresses in the laminated shell during the cure were investigated. The results showed that there were big differences between viscoelastic stresses and linear elastic stresses calculated by considering thermal deformation and the chemical shrinkage induced by the degree of cure.     

Residual stresses and sliding wear

The residual stresses that develop during the wear of AISI-SAE 1018 and 4340 steels have been examined. The entire three-dimensional stress tensor was obtained. A normal stress perpendicular to the surface, predicted by theory, has been found, but its magnitude is too small to affect the wear rate. There are also significant shear stresses. The wear process rapidly alters any initial stress distribution produced by heat treatment or peening to such a degree that the wear rate is not affected by these stresses, unless they are initially larger than those that can be produced by the wear parameters.     

Simulation of thermal stresses due to grinding

An efficient finite element procedure has been developed to calculate the temperatures and stresses arising due to a moving source of heat. The procedure is applied to calculate the thermal stresses produced in hardened steels during grinding. The thermal load during grinding is modeled as a uniformly or triangularly distributed, 2D heat source moving across the surface of a half-space, which is insulated or subjected to convective cooling. The grinding of elastic and elastic–plastic workpiece materials has been simulated. The calculated transient stresses and temperatures in an elastic solid are found to be in good agreement with prior analytical and numerical results. In an elastic–plastic workpiece material, for which no analytical solution is available for the residual stress distributions, the finite element calculations show that the near surface residual stress is predominantly tensile and that the magnitude of this stress increases with increasing heat flux values. Based on an analysis of the effects of workpiece velocity, heat flux magnitude and convective cooling, on the residual stress distributions in an elastic–plastic solid, it is seen that the calculated thermal stress distributions are consistent with experimentally measured residual stresses on ground surfaces. Furthermore, the results explain often cited observations pertaining to thermally induced grinding stresses in metals.     

Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a functionally graded rectangular plate due to nonuniform heat supply. The thermal and thermoelastic constants of the rectangular plate are assumed to vary exponentially in the thickness direction. The transient three-dimensional temperature is analyzed by the methods of Laplace and finite cosine transformations. We obtain the three-dimensional solution for the simple supported rectangular plate. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the nonhomogeneity of the material is investigated.     

Mathematical modeling of moving heat source shape for submerged arc welding process

An attempt is made in this paper to find out the analytical solution of the thermal field induced in a semi-infinite body by a moving heat source with Gaussian distribution by selecting appropriate inside volume for submerged arc welding process. Three different types of heat source shapes in the form of oval, double ellipsoidal, and conical forms were considered and compared with the experimental result. The study shows that for heat input of submerged arc welding process, the best suitable heat source shape is in the form of an oval. The study also shows two alternate ways of predicting the size of the heat-affected zone.     

A study on large-area laser processing analysis in consideration of the moving heat source

This study proposed a way of reducing the time and memory required for analysis by adjusting the heat input area, to overcome the difficulties in analyzing large-area laser processing. An analytical model was manufactured using existing research results, and based on these results and the volume of the analysis, the efficiency of the new analysis method was verified. The new analysis method proposed in this study can be used as a simple analysis method utilizing a commercial analysis program for diverse areas, including the effect of temperature distribution on the material, the effect of temperature transfer on the mechanical structure, and thermal displacement prediction, in the thermal treatment after processing.     

Residual stresses in aluminium alloy friction stir welds

Residual stresses are detrimental to the fatigue, fracture and corrosion resistance of welds. The literature on residual stress measurements in aluminium alloy friction stir welds is reviewed. The results of a large number of longitudinal residual stress measurements performed by the slitting method on friction stir welds in 2024-T3, 6082-T6 and 5754-H111 aluminium alloys are compared and their origin discussed. From the current investigation, it can be derived that the type of machine used for welding has only little influence on the residual stress profile. The influence of alloy type and welding parameters on the magnitude of the residual stresses and the shape of their distribution across the weld is investigated. Their magnitude is far below the room temperature yield strength of the base material. A distribution with an ??M-shape?? is always found on age hardenable structural alloys (albeit more pronounced in 6082-T6 alloy than in 2024-T3 alloy), while a ??plateau?? is found in the case of the strain hardenable 5754 H111 alloy. The low magnitude and the differences in distribution of the longitudinal residual stress are attributed mainly to the microstructural changes in the weld centre and are discussed based on the hardness profiles performed across the welds. The paper also discusses the reasons why those results are in disagreement with a number of numerical simulations from the literature that do not account for the influence of the welding thermomechanical history on the material microstructure and properties.     

Transverse and longitudinal vibrations of a frame structure due to a moving trolley and the hoisted object using moving finite element

This paper aims at presenting a technique to replace the moving load by an equivalent moving finite element so that both the transverse and the longitudinal inertial effects due to the moving mass may easily be taken into account simultaneously. Where the mass, damping and stiffness matrices of the moving finite element are determined by the transverse () inertia force, Coriolis force and centrifugal force of the moving mass, respectively. From the numerical examples illustrated, it has been found that, in addition to the conventional transverse () responses, the inertial effects of the moving load also affect the longitudinal () responses of the portal-frame structure significantly.     

Analytical solution for interface stresses due to concentrated surface force

The two-dimensional analytical solution for interface stresses due to concentrated surface force has been deduced, by introducing infinite mirror points which are the images of the load point reflected by the interface and the free surface, and adopting the interchange law of differentiation. The analytical solution can be represented in terms of the summation of the “partial” Goursat's complex stress functions defined in the local coordinate systems with their origins placed at each of the mirror points. It is found that the “partial” stress functions corresponding to a higher order mirror point can be determined from those to the lower one. It is also found that the contribution of the “partial” stress functions to the stress field decreases with the increase of the corresponding mirror point order, therefore, only considering the stress functions corresponding to the first several order mirror points can give the accurate enough solution. Numerical examination by the use of boundary element method has also been carried out to verify the theoretical development.     

Residual stresses in friction stir welded parts of complex geometry

Residual stresses play a key role on the mechanics underlying the fatigue crack growth propagation of welded joints. Indeed, compressive residual stresses may induce a beneficial enhancement of the fatigue life under loading condition whereas tensile residual stresses may act to increase the stress distribution at crack tip, resulting in a life-threatening condition of the welded structure. In-process distortion and final geometry of welded joints are also affected by residual stresses. In this paper, the longitudinal residual stress distributions in friction stir welding (FSW) joints were investigated for butt and skin–stringer geometries, including lap and T configurations. To measure residual stresses, the cut-compliance and the inverse weight-function methodologies were adapted for skin–stringer FSW geometries via finite element analysis. AA2024-T4 and AA7075-T6 aluminum alloys were used to weld dissimilar skin–stringer joints whereas butt joints were made of AA2024. The effect of most relevant process parameters as well as the cooling during welding process was also investigated for a better understanding of welding residual stresses. Our findings suggest that FSW of complex skin–stringer geometries produces higher residual stresses than those of butt joints, and that the cooling water flux further reduces residual stresses. Changes of process parameters did not affect markedly residual stress distribution.     

Residual stresses in cold-deformed thin items and sheet tempered metal

Using solutions of the mathematical theory of plasticity for a thin elastoplastic layer, four theoretically possible types of distributions of residual stresses over the thickness of cold-deformed items and tempered sheet metal are determined. The results are obtained by studying different types of diagrams presenting the dependence of the stress intensity on the deformation intensity. The obtained theoretical dependences are confirmed by experimental studies. The effect of the residual stresses on the mechanical properties and formability of sheet cold-rolled steel is considered.     

On the prediction of the design criteria for modification of contact stresses due to thermal stresses in the gear mesh

The mechanism of surface failure due to temperature rise is a very important problem in gear design. Thermal considerations have received considerable attention from the gear researchers but only for scoring failures when the destruction of lubrication film occurs as a result of temperature rise. In spite of the wealth of literature on this subject, this problem is not fully analyzed.The objectives of this paper are to consider the mechanisms of thermal stresses and the thermal cycling in contact zone, during the gear mesh. This research has been conducted for the first point of contact based on consideration of transient heat transfer, elastohydrodynamic lubrications, and surface roughness and gear material.A procedure presented in this paper evaluating the stresses (thermal and mechanical) and predicting the design criteria for modifying the contact stresses due to thermal stresses. The effect of the material, oil film thickness, surface roughness and geometric operating parameters on modification parameter is illustrated. Also the effects of a load on the temperature rise and the modification parameters are evaluated.     

Thermal stresses due to time exponentially decaying laser pulse: elasto-plastic wave propagations

In the present study, thermal stress developed in the substrate material, which is subjected to a laser heating pulse is formulated. The closed form solutions for the temperature and stress fields due to time exponentially decaying laser pulse are presented. The Laplace transformation method is employed when deriving the governing equations. The elastic and plastic propagation of the stress waves are considered and the depth of the plastic zone is predicted. In order to account for the recoil pressure generated during the evaporation process, stress boundary at the free surface of the workpiece is considered. It is found that the magnitude of stress wave, due to stress boundary at the surface, well exceeds the elastic limit of the substrate material. Once the magnitude of the recoil pressure reduces considerably, elastic wave is generated. This occurs after t^*=0.032. Since the elastic wave propagates faster than the plastic wave, both waves meet at some depth below the surface. This, in turn, defines the depth of the plastic zone. In the present case, the depth of elastic zone extends to about x^*=9.2 below the surface. The magnitude of the stress wave generated due to temperature gradient is less than the yield strength of the substrate material; in which case, its magnitude decreases with increasing depth from the surface.