**THE FINITE ELEMENT MODELING OF RESISTANCE SPOT WELDING**

A.  **V. Gohil ^\*^, S. M. Patel ^\#^**

> **ABSTRACT**

Resistance spot welding process is the most significant joining process
in the automobile industry due to its high speed and suitability for
automation. One of the recent demands in the automobile industry is to
reduce the vehicle weight so as to improve the fuel efficiency and to
meet this requirement, aluminium alloys is being considered as an
important alternative for auto-body material. The vehicle corrosion
problem can also be squarely dealt with. However, unlike resistance spot
welding of steel, joining of aluminium through the same process has met
with serious difficulties, because of faster deterioration of
electrodes. High electrical and thermal conductivity, high shrinkage
during solidification and the presence of natural oxide coating are some
of the features that make spot welding process of aluminium alloys
markedly different. At a very high temperature aluminium chemically
reacts with copper alloy (electrode material). Subsequently random
chipping-off of material from the electrode faces takes place and it
results in electrode wear. Further, the electrode life reduces
drastically when spot welding aluminium alloys. In terms of weldability,
this is an extreme negative point since weldability for automobile use
is greatly dictated by electrode tip life.

Since the process is very fast, important information, such as transient
distribution of current density and temperature, are difficult to obtain
from the experimental methods. Thus, it is aimed at the present work to
develop an integrated computer simulation model for analyzing resistance
spot welding process of aluminum alloys by finite element method.
Several calculations have been carried out for different welding
current, weld time and electrode force and for different surface
conditions of aluminium sheets. Non-linear, temperature-dependent,
thermo-physical material properties have been considered. It is
interestingly observed that in most cases the nugget diameter is formed
well within 0.02\~0.04sec and further flow of welding current simply
increases the electrode face heating. Also, the initial surface
condition influences the nugget formation phenomenon to a great extent.
Various other conclusions have been arrived at as a part of this study.

[^1]

**INTRODUCTION**

Resistance spot welding process is the most significant joining process
in the automobile industry due to its high speed and suitability for
automation and thus, any new development of this welding process is
closely influenced by the demand of this industry. The need to reduce
vehicle weight, improve fuel economy, and reduce exhaust emission has
led to increased use of light weight materials such as aluminium alloys.
However, many technical issues have to be solved before the use of
aluminium becomes commonplace in high-volume production. Unlike
resistance spot welding of steel, joining of aluminium through the same
process has met with serious difficulties. Aluminium is a very good
electrical conductor with a bulk resistivity one third that of steel.
Joule heating is proportional to resistance for a given current. It is
understandable that a significant increase in welding current will be
required to join aluminium, compared to an equivalent gauge of steel
sheet. Further, aluminium has a high thermal conductivity and the
localized heat generated by the welding current will be will be
conducted away rapidly. It is therefore necessary to use short weld
times. Aluminium alloys posses a surface oxide layer that varies
depending on the prior thermal and mechanical processing. The oxide has
a high resistivity band high melting point, around three times greater
than pure aluminium. Therefore, spot welding of aluminium alloys has
become an important research area in the last couple of years both in
the academic as well as industrial research laboratories.

The electrical resistance spot welding process for joining two materials
at their common interface is a complicated interaction of electrical,
thermal, mechanical, metallurgical and surface phenomena. In this
process, electrodes press against two or more steel sheet and high
amperage current is passed through the sheet-electrode system. Because
of the electrical contact resistance, heat will be generated at
electrode / work piece interfaces and faying surface. The heat at the
faying face melts the work pieces to form a nugget. To prevent melting
at the electrode / work piece interface, water is circulated in the
cooling chamber of the electrode.

The current carrying zone in the sheet is determined by the region over
which electrodes touches the sheet and this, in turn depends on the
electrode force and consequent plastic flow at sheet-electrode
interface. The complete phenomenon is thus, an electro - thermal problem
which is also influenced by plastic flow in the sheet. Coupled with this
are various types of nonlinearities present in the system. For example,
thermal conductivity and bulk electrical resistivity vary with the
temperature. Besides, the interface resistance along sheet -- sheet
interface and sheet -- electrode interface varies with various
parameters in a very uncertain manner. Hence, a finite element code for
simulating the spot welding process which includes all those features
mentioned above is developed in the present work for modeling resistance
spot welding process of aluminium alloys.

######### PREVIOUS INVISTIGATIONS

Since the physics of the process is so complicated, it is quite
understandable that very little was published in the open literature on
the finite element modeling which covers these many aspects. In 1984,
Nied \[3\] had reported a two-dimensional simulation model for analysing
resistance spot welding process of uncoated steel using commercial FEM
package ANSYS. A coupled thermo-electrical and thermo-mechanical
analysis has been tried. However, the contact resistance along the
sheet-to-sheet and sheet-to-electrode interfaces was neglected. Gould
\[4\] reported a one-dimensional numerical model to calculate weld
nugget development during spot welding of uncoated steel. However, the
model being one-dimensional, failed to account for the radial heat loss
into the surrounding sheet. Cho \[5\] had reported a two-dimensional,
finite difference method based heat transfer model for resistance spot
welding process. It has been concluded from the publications cited above
that the resistance spot weldability of aluminium alloys is not yet
fully explored although there is now tremendous demand of the these
materials to be used in the automobile industry. The purpose of this
present work is to develop finite element based numerical model which
also consists a nonlinear thermo -- mechanical coupling to provide a
more realistic simulation of the resistance spot welding process of
aluminium alloys.

##### FINITE ELEMENT MODEL  {#finite-element-model .unnumbered}

**[Geometric Modeling]{.underline}**

Considering a typical arrangement for resistance spot welding of two
pieces of aluminium sheets, the geometric representation of two
identical electrodes simplifies the geometry of two -- dimensional
axisymmetric model. Fig. 1 shows the finite element mesh structure used
for the modeling purpose. The mesh structure consists of 358 nodes and
297 elements. The element mesh size at the end of the electrode and for
the work piece is sufficiently refined to account for thermal gradients
in that region. A coarser mesh is considered in the upper region of the
electrode where the gradients are shallower because of heat conduction
to the water -- cooling channel. Only one quadrant of the complete
geometry has been analyzed considering the axial symmetry of the
sheet-electrode system in spot welding process.

**[Heat Transfer Analysis]{.underline}**

Heat transfer in resistance spot welding process involves convective
heat transfer as well as heat conduction in bulk of the sheet-electrode
system. The transient heat flow in resistance spot welding process has
been modeled as a case of axisymmetric heat conduction problem.

$\frac{1}{r}\frac{¶}{¶r}\text{  }\left( rK\frac{¶T}{¶r} \right)\text{  +  }\frac{1}{r}\frac{¶}{¶\text{ z}}\text{  }\left( rK\frac{¶T}{¶z} \right)\text{   +   }\overset{\text{.}}{Q}\text{   =   }sc\text{  }\frac{¶\text{ T}}{¶\text{ t}}$
(1)

where *s*, *c* and *K* are density, specific heat and thermal
conductivity respectively. All the material properties are considered to
be temperature dependent. The term $\overset{\text{.}}{Q}$ refers to the
rate of internal heat generation per unit volume.

**[Electrical Field Analysis]{.underline}**

The current density distribution in the sheet-electrode geometry (in
two-dimensional cylindrical coordinate system) can be represented by the
following relationship,

> $\frac{1}{r}\frac{\partial}{\partial r}\text{  }\left( r\frac{1}{\rho}\frac{\partial J}{\partial r} \right)\text{  +  }\frac{1}{r}\frac{\partial}{\partial\text{ z}}\text{  }\left( r\frac{1}{\rho}\frac{\partial J}{\partial z} \right)\text{   } = \text{  0}$
> (2)

where, *ρ* is the electrical resistivity and **J** is the current
density vector. The electrical resistivity is considered temperature
dependent in the present work.

**[Internal Heat Generation]{.underline}**

The internal heat generation at every point in the sheet-electrode
geometry is calculated by the following relationship,

$\overset{\text{.}}{Q}\text{  =  }\rho\text{  }J^{2}$ (3)

**[Internal Heat Generation]{.underline}**

The finite element discretisation of the complete sheet-electrode
geometry has been carried out using four-noded ring type isoparamtric,
solid element with rectangular cross-section. Within an element,
temperature (T) can be expressed as,

$\left\{ T \right\}\text{  =  }\left\lbrack N_{i}\text{  }\text{.}\text{.}\text{.}\text{.}\text{.}\text{  }N_{l} \right\rbrack\text{  }\left\{ T^{e} \right\}$
(4)

where, *N~i~*, .... *N~l~* are the shape functions (based on nodal
co-ordinates of the element) of the element. The transient heat
conduction equation (eq.1) is first discretised and the discretised
equation can be stated as,

$\lbrack H\rbrack\left\{ T \right\}\text{  +  }\lbrack S\rbrack\frac{¶}{¶t}\text{  } + \text{  }\left\{ F \right\}\text{  =  0}$
(5)

where \[*H*\] is the thermal conductivity matrix, \[*S*\] is the thermal
capacity matrix and {*F*} is the load vector due to internal heat
generation. Equation (6) is further discretised in time domain following
Galerkin's Principle. The solution of electrical analysis represents the
elemental current density distribution throughout the sheet-electrode
geometry. These results are then used to calculate the internal heat
generation in each element and subsequently the heat transfer equation
is solved to obtain the nodal temperature distribution in the complete
geometry. The total weld time has been divided into a number of small
time steps. Within each time step, the electrical field analysis is
carried out first to obtain the elemental current density and the heat
transfer analysis is done next considering the internal heat generation.

**RESULTS AND DISCUSSIONS**

A Simulation model has been developed and extensive numerical
calculations were carried out to find out the nugget diameter,
penetration, etc. for resistance spot welding of aluminium alloy sheets
using the FEM Software ANSYS. Fig. 1 shows the geometry used for the
modeling purpose. Only one quadrant of the complete geometry has been
analyzed considering the axial symmetry of the sheet-electrode system in
spot welding process.

The sheet thickness used for the present analysis is 0.8mm and the
temperature dependent material properties for aluminium and copper
electrode are shown in Fig.2 -- 5. The sample temperature distributions
in the sheet-electrode system for welding current of 45 KA at different
instant of times are shown in Fig.6. It can be observed that the
temperature isotherms are more concentrated along the faying surface and
the maximum temperature occurs along the sheet-to-sheet contact zone
only. The time histories of the highest temperature experienced by the
sheet-electrode system at two different welding currents (35 KA and 45
KA) are plotted in Fig.7. The development of nugget diameter with time
for two different weld currents (40 KA & 45 KA) for the same material is
shown by the Fig.-8. It is observed that the nugget development process
is complete within 0.06 sec and with further increase in time no more
radial growth occurs. It has been observed in Fig.9 that within 0.02
sec., the maximum temperature generated is above the liquidus
temperature of the sheet and with further increase in time there is no
more rise in maximum temperature. This occurs presumably as the faying
surface resistance decreases and heat generation becomes more heavily
dependent on bulk resistivity and further due to the higher conductivity
of aluminium alloy heat dissipation becomes more with longer weld time.
It has been shown by the Fig.9 that the current density is not uniform
throughout the sheet-electrode geometry and hence hosts the importance
of the electrical field analysis in case of resistance spot welding
process.

### 

# CONCLUSION

A comprehensive simulation model using FEM for the analysis of
resistance spot welding process has been developed. It has been observed
that the finite element modeling of the resistance spot welding process
can provide good simulation, if the model includes the electro-thermal
mechanical interaction and good temperature -- dependent material
properties. The results presented herein indicate that there is another
alternative: the use of a realistic analytic model. This finite element
model so developed can calculate most of the resistance spot welding
responses in terms of nugget diameter, depth of penetration, the extent
of heat affected zone, rate of heating and cooling, electrode face
heating etc. Finally this FEM model will certainly help in optimizing
process parameters combinations in any industrial application of
resistance spot welding process. The authors are also intending to
perform real time experiments so as to validate the theoretical results
with in-house experimental data since such data are scarcely available
in the literature.

**REFERENCES**

1.  Johnson, K. I. 1976. Aluminium in vehicle bodies, [Metal
    Construction & British Welding Journal]{.underline} 8(9): 392 to
    395.

2.  Ostgard, E. 1980. Spot welding of aluminium as delivered, [Metal
    Construction]{.underline}12 (2): 78 to 86.

3.  Nied, H. A. 1984. The finite element modeling of the resistance spot
    welding process. [Welding Journal]{.underline} 63(4): 123 to 132.

4.  Gould, J. E. 1987. An examination of nugget development during spot
    welding using both experimental and analytical techniques. [Welding
    Journal]{.underline} 66(1): 1 to 10

5.  Cho, H. S. 1989. Study of the thermal behavior in resistance spot
    welding. [Welding Journal]{.underline} 68(6): 236 to 244.

6.  Brown, D. J. 1995. Computer simulation of resistance spot welding in
    aluminium. [Welding Journal]{.underline} 74(10): 339 to 344

7.  Brown, D. J. 1995. Computer simulation of resistance spot welding in
    aluminium. [Welding Journal]{.underline} 74(12): 417 to 422.

8.  Murakawa, H.; Kimura, H.; and Ueda, Y. 1995. Weldability analysis of
    spot welding on aluminium using FEM. [Transaction of
    JWRI.]{.underline} 24: 101 to 111.

9.  De. A.; Dorn. L.; and Gohil. A. V. 2001. Numerical Modeling of
    Resistance spot welding of aluminium alloys. [Proc. International
    Conf. on Advances in Welding and Cutting Technology.]{.underline}
    Eds. The Indian Institute of Welding.

###  Fig. 2 Temperature dependent resistivity

of copper

## 

## Fig. 1. Finite Element Model for equal

## sheet thickness 

##  Fig. 3 Temperature dependent resistivity

> heat of aluminium

## Fig. 4 Temperature dependent thermal Fig. 5 Temperature dependent thermal

## conductivity of copper conductivity of aluminium

## 

## 

## Fig. 6(i) Temperature Distribution as Resistive heating progress (after 0.2 sec)

![](media/image13.png){width="4.750694444444444in" height="3.5in"}

## Fig. 6(ii) Temperature Distribution as Resistive heating progress (after 0.8 sec)

##  Fig. 7 Maximum temperature generated Fig.8 Nugget development at different

##  at different instant of time (sec) instant of time (sec.)

## 

## Fig. 9 Current Density Distribution as Resistive heating progress

( For sheet thickness 0.8 mm and current I = 45 kA)

[^1]: ^^ Department of Production Engineering, Shantilal Shah
    Engineering College, Bhavnagar, India

    \# Department of Production Engineering, Shantilal Shah Engineering
    College, Bhavnagar, India