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A material in the welding process can be considered by way of the concept of a body. In the course of thermal and mechanical loadings the body changes its geometrical shape. The consequences of welding processes are small deformations of the welded body and large deformations as it is in friction and spot welding. The deformation and the motion of welding material is described as standard continuum.

The linear momentum of the material occupying the volume V in the configuration at time t by a moving material body B is defined by the relation where v is the velocity of the material particle at the point x.

In order to derive the thermal and mechanical equations of welding processes it is necessary to consider welding as a thermodynamical process.

In welding processes a material is analyzed either in solid or fluid state. If under the action of forces the deformation of the body increases continuously and indefinitely with time we say that a material flows. In the case of a purely viscous material, a stress is only generated if the amount of strain is changing, the stress generated by strain being considered to relax instantaneously.

In the weld and base material complex strains occur being the result of welding heat source acting on the welded structure. The deformation of welded structure is thermo-elastic or thermo-elastic-plastic. Thermo-mechanical coupling in thermo-elastic material is described by the following state equations In thermo-elasticity the free energy Ψ depends on temperature θ and strain

Most of the thermo-mechanical problems in welding can only be solved by using numerical procedures. In realistic model the thermal conductivity and specific heat should be considered as a function of temperature which complicates analytical solutions. Because of possible phase change the analytical methods have a limited range of applications. In most types of welding melting will occur and there will also be convective heat transfer in addition to conductive heat transfer. The heat sources in welding in realistic model are not concentrated in point or line.

The solution of heat flow equations for welding conditions is a complicated problem. In a realistic model the thermal conductivity and specific heat should be considered as a functions of temperature. In most types of welding melting occurs and convective heat transfer in addition to conductive heat transfer takes place. In general, the source of heat is not concentrated at a point or line but is spread out over the workpiece with an unknown distribution of heat input. In addition, most realistic welding problems have heat losses at the boundaries caused by convection, radiation and contact with other bodies, so the precise boundary conditions are often unknown. In order to find analytical solutions to the equations, it is therefore necessary to make many simplifying assumptions. The purpose of this chapter is to present analytical solutions to the heat transfer equations under conditions of interest in welding. The restrictive assumptions will however limit the practical utility of the results. Nevertheless the results are useful in that they emphasize the variables involved and approximate the way in which they are related. In addition, such solutions provide the background for understanding more complicated solutions obtained numerically and provide guidance in making judgments.

The effective solutions of complex thermal problems in welding only recently has become possible. In the last two decades one observes the vigorous development of effective numerical methods of analysis. Automobile, aircraft, nuclear and ship industry are experiencing a rapidly-growing need for the numerical tools to handle complex problems of welding. Efficient numerical methods for nonlinear welding problems are needed because experimental testing in such cases is often prohibitively expensive or physically impossible.

Changes of stresses during welding both in the weld and the base material are the result of welding heat source acting on the welded structure. This chapter analyses thermal stresses during welding. The changes of temperature and resulting stresses that occur during welding are presented in Fig. 9–1. It is assumed that the weld is made along the axis x and the moving welding heat source is located at point .

In the course of heating and cooling processes during welding thermal strains occur in the weld metal and base metal near the weld. The stresses resulting from thermomechanical loadings cause bending, buckling and rotation. These displacements are called distortion in weldments. In welding process three fundamental changes of the shape of welded structure are observed. These transverse shrinkage perpendicular to the weld line, longitudinal shrinkage parallel to the weld line and angular distortion (rotation around the weld line). These changes of welded structure are shown in Fig. 10–1 where classification of distortion has been introduced.