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N. Bhardwaj et al.
Physical boundary value problems can be categorized as (i) steady state or equilibrium, (ii) eigenvalue and (iii) transient problems. The steady-state problems are timeindependent and involve evaluation of displacement or stress distribution, temperature and heat flux distribution, or velocity and pressure distribution in spatial directions. Eigenvalue problems involve determination of critical values of certain parameters in addition to equilibrium configurations, for example, determination of buckling
loads or natural frequencies in structural problems and stability of laminar flows in
fluid mechanics. Transient problems include analyzing a body under time-dependent
force or heating/cooling of a particular point with respect to time [64].
Different commercial software applications are available for solving FEM problems in engineering like ABAQUS, ANSYS, DEFORM-3D, etc. The software applications perform the FE analysis in three steps: (1) In pre-processing, the geometry of
the parts, loads applicable, material properties and boundary conditions are specified.
Finite element mesh is generated by an in-built automatic mesh generation module
where the type, size and remeshing criteria are specified. (2) In numerical analysis,
the stiffness matrices (element characteristics) and load vectors are generated and
assembled to generate the system of equations. The specified boundary conditions
are implemented, and the nodal field variables (e.g., displacement) are obtained by
solving the equations to compute the element resultants (e.g., stress and strains). (3)
In post-processing, the solutions are displayed either in tabular form or graphically.
It is extremely important to verify the results obtained from FE analysis from the
software applications by comparing them with known solutions.
3.5.2 Thermal Modeling
Thermal model governs the mechanical and microstructure models. Understanding
of the heat generation is vital to understanding the process and its related phenomena.
Material flow, which depends on flow stress, is dependent on the thermal cycle during
welding. One of the first thermal models is given by Chao and Qi [19]. It was a
three-dimensional heat transfer model that assumed sliding friction only to generate
a constant heat flux with Coulomb’s law for friction. Pressure was assumed to be
constant. However, the study only used heat generated by the tool shoulder. Frigaard
et al. [32] improved the model by incorporating heat generated at the interfaces of
FSW tool shoulder-workpiece and the pin-workpiece. Smith et al. [78] and Bendzsak
et al. [10] used finite difference method for heat transfer modeling as well as material
flow in FSW assuming workpiece material to be a non-Newtonian fluid. Gould and
Feng [34] developed a model using the Rosenthal equation for heat transfer during
FSW. Russell and Shercliff [67] also used the Rosenthal equation to incorporate
heat input as a point source or line source. Khandkar and Khan [46] and Khandkar
et al. [47] proposed a torque-based heat input model. The heat generated is from
contact friction and plastic deformation [13]. Bhardwaj et al. [13] used different
lubricants during FSW to reduce the friction at the tool-workpiece interface and
analyzed the heat contribution from friction and plastic deformation using results
N. Bhardwaj et al.
Physical boundary value problems can be categorized as (i) steady state or equilibrium, (ii) eigenvalue and (iii) transient problems. The steady-state problems are timeindependent and involve evaluation of displacement or stress distribution, temperature and heat flux distribution, or velocity and pressure distribution in spatial directions. Eigenvalue problems involve determination of critical values of certain parameters in addition to equilibrium configurations, for example, determination of buckling
loads or natural frequencies in structural problems and stability of laminar flows in
fluid mechanics. Transient problems include analyzing a body under time-dependent
force or heating/cooling of a particular point with respect to time [64].
Different commercial software applications are available for solving FEM problems in engineering like ABAQUS, ANSYS, DEFORM-3D, etc. The software applications perform the FE analysis in three steps: (1) In pre-processing, the geometry of
the parts, loads applicable, material properties and boundary conditions are specified.
Finite element mesh is generated by an in-built automatic mesh generation module
where the type, size and remeshing criteria are specified. (2) In numerical analysis,
the stiffness matrices (element characteristics) and load vectors are generated and
assembled to generate the system of equations. The specified boundary conditions
are implemented, and the nodal field variables (e.g., displacement) are obtained by
solving the equations to compute the element resultants (e.g., stress and strains). (3)
In post-processing, the solutions are displayed either in tabular form or graphically.
It is extremely important to verify the results obtained from FE analysis from the
software applications by comparing them with known solutions.
3.5.2 Thermal Modeling
Thermal model governs the mechanical and microstructure models. Understanding
of the heat generation is vital to understanding the process and its related phenomena.
Material flow, which depends on flow stress, is dependent on the thermal cycle during
welding. One of the first thermal models is given by Chao and Qi [19]. It was a
three-dimensional heat transfer model that assumed sliding friction only to generate
a constant heat flux with Coulomb’s law for friction. Pressure was assumed to be
constant. However, the study only used heat generated by the tool shoulder. Frigaard
et al. [32] improved the model by incorporating heat generated at the interfaces of
FSW tool shoulder-workpiece and the pin-workpiece. Smith et al. [78] and Bendzsak
et al. [10] used finite difference method for heat transfer modeling as well as material
flow in FSW assuming workpiece material to be a non-Newtonian fluid. Gould and
Feng [34] developed a model using the Rosenthal equation for heat transfer during
FSW. Russell and Shercliff [67] also used the Rosenthal equation to incorporate
heat input as a point source or line source. Khandkar and Khan [46] and Khandkar
et al. [47] proposed a torque-based heat input model. The heat generated is from
contact friction and plastic deformation [13]. Bhardwaj et al. [13] used different
lubricants during FSW to reduce the friction at the tool-workpiece interface and
analyzed the heat contribution from friction and plastic deformation using results