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Simulations for Design and Manufacturing pp 103–146Cite as

Thermomechanical Simulation of Friction Stir Welding Process Using Lagrangian Method

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Part of the book series: Lecture Notes on Multidisciplinary Industrial Engineering ((LNMUINEN))

Abstract

Friction stir welding (FSW) is a solid-state joining process. Modeling of FSW process provides insight into the mechanism of the process in terms of heat generation, material flow, etc. In the present chapter, modeling of FSW process in DEFORM-3D using Lagrangian method is discussed. Various governing equations involved in the finite element modeling of FSW are explained. Three different solvers, viz., conjugate gradient, sparse, and combination of the two, are compared based on computation time and accuracy of the solution. The developed method is validated with the experimental force and torque. The model is further used to study the temperature distribution and material flow. The latter is studied using particle tracking method.

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Abbreviations

\( A_{iN} \) :

Area contribution of ith element to the node N

\( B_{\text{s}} \) :

Strain rate matrix

\( C \) :

Heat capacity matrix

\( D \) :

Effective strain rate coefficient matrix

\( {\text{E}}\left( {{\dot{\varepsilon }}_{ij} } \right) \) :

Work function

\( F_{i} \) :

Traction

\( J_{\text{b}} \) :

Jacobian matrix

\( K \) :

Thermal conductivity

\( K_{\text{s}} \) :

Global stiffness matrix

\( K_{\text{wt}} \) :

Thermal conductance between the workpiece and tool

\( K_{\text{c}} \) :

Heat conduction matrix

\( K_{\text{wb}} \) :

Thermal conductance between the workpiece and backing plate

\( N^{\text{T}} \) :

Transpose of shape function matrix

\( Q \) :

Heat flux vector

\( Q_{\text{wa}} \) :

Heat transfer between the workpiece and ambient

\( Q_{\text{wt}} \) :

Heat transfer between the workpiece and tool

\( Q_{\text{wb}} \) :

Heat transfer between the workpiece and backing plate

\( S_{F} \) :

Traction surface

\( T_{\text{t}} \) :

Tool temperature

\( T_{\text{b}} \) :

Backing plate temperature

\( T_{\text{w}} \) :

Workpiece temperature

\( T_{\text{a}} \) :

Ambient temperature

c :

Specific heat capacity

\( f_{\text{R}} \) :

Residual nodal point force

\( f \) :

Yield function

\( g \) :

Plastic potential

\( h_{\text{a}} \) :

Convective heat transfer between the workpiece/tool and ambient

\( h_{\text{wt}} \) :

Convective heat transfer between the workpiece and tool

\( h_{\text{wb}} \) :

Convective heat transfer between the workpiece and backing plate

\( J_{2} ,J_{3} \) :

Second and third variant of stress tensor

\( k \) :

Material constant

\( k_{\text{s}} \) :

Static yield stress in shear

\( m \) :

Shear factor

\( q_{N} \) :

Shape function of element i at node N

\( q_{\alpha } \) :

Element shape function

\( \dot{q} \) :

Heat generation rate

\( q_{f} \) :

Surface heat flux due to friction

\( u_{i}^{\alpha } \) :

Velocity component at the αth node

\( u_{i} \) :

Velocity component

\( v \) :

Nodal velocity

\( v_{\text{s}} \) :

Sliding velocity at the interface of tool workpiece

\( x, y, z \) :

Cartesian coordinate system

\( V^{{\prime \prime }} \) :

Viscosity constant

\( \sigma_{\text{b}} \) :

Stefan–Boltzmann constant

\( \varepsilon_{\text{b}} \) :

Emissivity

\( \sigma_{ij}^{{\prime }} \) :

Deviatoric stress

\( \sigma_{ij} \) :

Cauchy stress

\( \delta_{ij} \) :

Kronecker delta

\( \sigma_{m} \) :

Hydrostatic stress

\( \sigma_{Y} \) :

Yield stress

\( \varepsilon_{ij}^{p} \) :

Plastic strain

\( \lambda \) :

Positive proportionality constant

\( \dot{\gamma }_{xy}^{{}} \) :

Shear strain rate

\( \tau_{xy} \) :

Shear stress

\( \dot{\varepsilon }_{x} \) :

Strain rate in x-direction

\( \dot{\bar{\varepsilon }} \) :

Effective strain rate

\( \bar{\sigma } \) :

Effective stress or flow stress

\( \dot{\varepsilon }_{ij} \) :

Strain rate component

\( \bar{\tau } \) :

Contact stress

\( \tau_{ \hbox{max} } \) :

Shear yield strength

\( \lambda_{\text{P}} \) :

Penalty constant

\( \dot{\varepsilon }_{\text{v}} \) :

Volumetric strain rate

\( \dot{\bar{\varepsilon }}_{0} \) :

Limiting strain rate

\( \bar{\varepsilon }_{N} \) :

Effective strain rate at node N

\( \xi ,\eta ,\zeta \) :

Natural coordinate system

\( \varphi \) :

Inelastic heat fraction

\( \psi \) :

Constant

\( \rho \) :

Density

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Authors and Affiliations

  1. Friction Stir Welding Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, West Bengal, India

    Rahul Jain & Surjya K. Pal

  2. Department of Metallurgical and Materials Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Shiv B. Singh

Authors
  1. Rahul Jain
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  2. Surjya K. Pal
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  3. Shiv B. Singh
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Corresponding author

Correspondence to Surjya K. Pal .

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Editors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, India

    Uday S. Dixit

  2. Indian Institute of Technology Ropar, Rupnagar, Punjab, India

    Ravi Kant

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Jain, R., Pal, S.K., Singh, S.B. (2018). Thermomechanical Simulation of Friction Stir Welding Process Using Lagrangian Method. In: Dixit, U., Kant, R. (eds) Simulations for Design and Manufacturing. Lecture Notes on Multidisciplinary Industrial Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-10-8518-5_4

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  • DOI: https://doi.org/10.1007/978-981-10-8518-5_4

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