COUPLED TRANSIENT THERMAL AND PULSED EC MODELING
FOR NOT OF MATERIALS SUBJECTED TO LASER BASED HEAT
TREATMENT
S. Veeraraghavan and Krishnan Balasubramaniam*
Center for Non Destructive Evaluation,
^Department of Mechanical Engineering,
Indian Institute of Technology-Madras, Chennai -600036, INDIA
ABSTRACT. The pulsed eddy current (PEC) technique is widely used to detect and quantify flaws in
conducting plates .This method has been used to test materials subjected to heat treatment. Material
properties such as conductivity show a marked change with temperature. As the heat diffuses into the
material, the probe output changes owing to the changing material properties. Hence, it becomes
difficult to obtain consistent results when searching for defects. Such problems can be effectively
tackled by combining thermal and pulsed eddy current analysis for getting a reference data for the
signals acquired from the probe. In this paper, a model is presented which combines transient thermal
analysis and PEC Finite Element Model to track the probe output changes occurring during Laser based
heat treatment of thin sheet conducting materials. The first step encompasses modeling of the laser
treatment process using a three dimensional Finite Difference method .Properties of the material such as
conductivity and specific heat capacity changes with temperature. Such parameters can be averaged over
the temperature range of the laser treatment process and used in the model. The thermal model can also
be extended to heterogeneous materials like Carbon reinforced composites. The temperature distribution
in the material which is obtained from the thermal model is used as the input data for the axisymmetric
PEC Finite element analysis. Hence the probe output can be tracked over the entire time interval. This
forms an effective basis for combining the physics of two diffusive phenomena, i.e. thermal and pulsed
eddy current in the NDE and control of such processes.
INTRODUCTION
This paper studies the use of pulsed eddy current (PEC) method for the monitoring
of laser surface heat treatment process. Surface transformation hardening is based on rapid
localized heating and cooling induced by a scanning laser beam on hardenable alloys. This
method creates a wear resistant zone with good fatigue properties on load-bearing
surfaces. The method is more flexible than induction hardening and more rapid than gas
carburising with less distortion. Surface melting is another process which allows wear,
corrosion and oxidation properties to be improved by refining, homogenizing or
transforming the microstructure of a wide range of engineering alloys. The composition
and properties of the surface can also be modified by the addition of small amounts of
alloy elements. The wear resistance of aluminum is increased by alloying with silicon,
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
351
and titanium is hardened by melting in a nitrogen atmosphere. Composite surfaces can be
produced by injecting hard ceramic particles into a laser melted surface[7].
The mechanics of laser material removal has been studied in detail in the past. The
defects that occur during laser treatment are primarily due to high energy laser material
interaction which leads to ductile fracture. High pressure shock waves are generated under
hydrodynamic conditions by the ablation of a skin layer. High strain rate (over 100 s"1)
shock compression leads to extensive crystallographic slip and twinning. A tensile state of
stress is generated on release and voids form due to strain localization and incompatibility
conditions. Material failure occurs due to void coalescence[7].
In the pulsed eddy current method, an air-core coil is placed over the conducting
material and a pulsed excitation of sufficient amplitude and frequency is given. The
signal is sampled and acquired through an A/D converter. In the case of surface
treatments where low power CW (continuous wave) lasers are normally used, the coil can
be placed above the region that is under treatment. However, when using high power
pulsed lasers, the place where the coil is placed has to be carefully selected in order to
avoid the high conducting plasma which may shield the coil.
PULSED EDDY CURRENT METHOD : MODELING
An axisymmetric pulsed eddy current finite element method is used for the
modeling work. The governing equation for the pulsed eddy current method is derived
from the Maxwell's law and is shown in equation (1)
V x l/ji(V x A) - Js - a(8A/8t)
(1)
where a is the conductivity of the material; A is the magnetic vector potential; Js is the
source current density; and ji is the permeability of the material. The objective of the
numerical analysis procedure is to solve the governing equation. Without a(8A/8t) term,
the equation is an elliptic partial differential equation which can be solved using spatial
discretisation (FEM method). The transient nature of the solution is due to the pulsed
excitation source. Due to the presence of this term, the solution evolves as a function of
time. Hence, the time is also discretized and the finite difference method is employed for
marching in time. For the finite element model, quadrilateral elements are used, therefore,
the trial function Ae within the element can be approximated by the nodal values AI, Aj,
Ak, and AI and the shape function N. In matrix form this can be written as,
A(r,z,t) = [N(r,z)][A(t)]e
(2)
The element matrix is derived using the weak formulation of the governing equation (1)
e
where,
e
[S]e =
352
(3)
FIGURE 1. Axisymmetric mesh for pulsed eddy current analysis (dimensions in meters).
We note that the conductivity is contained in the Ce term. The resistivity for
Aluminum is 2.65e-6 ohm-cm. The temperature coefficient for aluminum is 0.004308
ohm/deg Celsius. The conductivity decreases with an increase in temperature. Modeling
of the Laser heating process by finite difference method can give accurate estimates of the
temperature profiles [2,8]. These temperature values are converted to the conductivity
matrix using the temperature coefficient and are given to the axisymmetric model.
Equation (3) contains a time dependent A' term. Crank-Nicholson scheme is used
for the evaluation of this term. This is a time-stepping procedure in which the derivative is
obtained using a recurrence relation between the time steps.
A(n)}/At
+ A(n)}/2
(4)
(5)
After substituting the above equations in (3),the element matrix equation is
obtained. The contributions from each of the elements are summed to form the global
matrix for the entire solution domain. The magnetic flux potential can then be computed.
After solving for the potential, the signal from the coil is extracted and analyzed for
detecting and characterizing defects.
Voltage induced in the coil is given by,
V = - 27i n (8(pA)/8t)
(6)
Where n is the number of turns in the coil, and p is the radius at the considered
node.
The mesh employed for modeling the geometry is shown in Fig. 1. A single turn
coil whose outer radius is 3.03 cm and inner radius is 0.61 cm is used for modeling
purpose. Thickness of the coil and the material are taken as 3.84 cm and 3.1 cm
respectively.
1.2
£,
0.8
1
°4 i
0.2
O
0.5
-0.2
Time (ms)
Time (ms)
FIGURE 2. Excitation current density (A/m2)
versus time (milliseconds).
FIGURES. Output voltage of the coil: coil
voltage(V) versus time (milliseconds)
Dirichlet boundary conditions are used in the analysis with the magnetic vector
potential zero along all four boundaries of the solution space. A gaussian excitation pulse
with a duration of 0.5 ms and peak current density of 107 A/m2 has been used Fig. 2. The
output voltage of the coil is shown in Fig. 3. During the initial period when the current
applied to the coil increases, the flux increases and the induced EMF is negative. Later the
polarity of the induced EMF becomes positive due to the decrease in flux. The peak
amplitude and the zero crossing time contain a significant amount of information relating
to the specimen [4].
The diffusion process has also been studied by way of animation of the flux plots.
The applied magnetic field does not penetrate into the material during the initial period;
therefore the induced eddy current is weak. Subsequently, when the applied magnetic field
penetrates deep into the metal the eddy current magnitude reaches a maximum. Finally the
eddy current decays to zero due to the losses in the specimen.
The pulse contains a broad range of frequency components. As in the case of
conventional EC methods, the PEC method is also limited by the skin depth phenomenon
(Equation 7).
Skin depth = (1
vl/2
(7)
The decrease in conductivity due to laser heating gives an increased skin depth
which aids in identification of defects. Also the above equation implies that for far side
defects, the low frequencies are affected and for surface defects both high and the low
frequencies are affected by the presence of the defect [1].
THERMAL MODEL
Figure 4 shows the schematic representation of a laser based material processing
method. The laser head contains a rotating mirror which changes the path of the laser. A
focusing lens is used to concentrate the laser power to the desired region in the material.
The parameters that can be varied in the process are the laser power, the focal spot radius
and the laser scanning velocity. The process can be controlled by changing these
parameters.
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Laser Head
Gaussian
Laser Beam
FocussingLens
Test Piece
loving Carriage
Grooved
region
FIGURE 4. Schematic representation of a laser
based material processing method.
FIGURES. Heat balance within a
differential element of size dx*dy*dz at
)* node.
The objective of the thermal model is to obtain the distribution of temperature
within the material which is subjected to CW laser. For obtaining this, a three dimensional
heat transfer model is developed with the following assumptions,
1. Distribution of power within the focused spot is assumed to be Gaussian, operating
in a TEMoo mode.
2. Plasma effects are neglected and the ejected/vaporized material will not affect the
incoming laser beam.
3. Latent heat of melting and solidification are neglected since they are much smaller
than the latent heat of vaporization.
4. Convection and radiation losses to the surroundings are neglected [2].
For heat transfer analysis the considered portion of the geometry is divided into a
number of elements. Fig. 5 shows the heat balance within a small differential element of
size dx*dy*dz i.e. say at (m^pf* node where m,n,p are chosen along the length, width
and thickness direction of the considered portion. The general heat transfer by conduction
is formulated using finite difference technique. The rate equation for energy balance in a
typical differential control volume can be represented by the Equation 8.,
Q +Q =Q
ZZ-
in
2Z-
g
+
2^ out
(8)
E
^s
where Qin is the rate of heat input, Eg is the energy generation rate, Qout is the rate of
heat output, Es is the energy storage rate at (m^pf* node respectively.
The rates of heat input Qin and heat output Qout can be obtained by summing the
rates of heat inputs and outputs along all the directions.
(9a)
fi f c = f i / f i , + f i .
(9b)
Here, x, y and z axes are chosen along the length, thickness and the width of the
considered portion of the volume respectively.
355
The rates of heat input and heat output along the x direction in a differential
volume can be obtained from the Fourier's law of heat conduction and by using the
element formulation. The rates of heat input (Qx) and output (Qx+dx) along y and z
directions can also be obtained in a similar fashion. The rate of energy generation ( Eg )
depends on the distribution of power within the focused spot size. Since this distribution
is Gaussian in nature, the power varies from point to point and the rate of heat generation
is given by:
•
=
p
p
at x = 0,y = 0 and z = w/2
in the plane y = 0, where r = d/2
= 0
at all other locations
(10)
Where P is the incident power in watts and r is the focused spot size in mm
The rate of energy storage can be expressed as,
where V is the scanning velocity.
The resulting equation is obtained as,
^P *m*P +K AxAy
Ax;
"**
'P~^
+
^^
Az
MA/>
m/tf
The boundary conditions employed are:
1) Kx(dT/dx) = 0 along x = 0 and x = 1
2) Ky(dT/dy) = 0 along y - 1
3) Kz(dT/dz) = 0 along z = 0 and z = w
£m,n,P, Km5n,p, pm,n,P, Cp m5n,P and Lm5n,p denote the emissivity, thermal conductivity, density,
specific heat and latent heat of vaporization of the differential element at the (m^pf*
node.
356
r700
0
De
(m m )
'0
6
Width(mm)
FIGURE 6. Results of the coupled model: The Steady state temperature distribution as a deformed
mesh around the region of laser impact on the material.
The equation (12) results in N number of algebraic equations when the considered
portion of the material is divided into N number of elements. The equations are solved
using the LU decomposition method. The solution gives the temperature distribution in the
laser affected region. From this temperature distribution the conductivity variation is
found. The temperature distribution is shown in Fig. 6 as a deformed mesh. The power of
the laser chosen is 100 W and the scanning velocity of 400 m d s is used.
RESULTS
The results of the coupled model are shown in the Fig. 7 as the variation of the coil
voltages obtained over time. A variation in the voltage response of the coil is seen when
the thermal effect is taken into account. This effect mainly comprises of a decrease in the
overall magnitudes of the voltages obtained and also a negative shift in the zero crossing
time of the signal. Further information about the thermal effect is obtained by looking at
the frequency domain. The Fourier transform of the coil voltage obtained by the coupled
model shows a shift in the low frequency components as shown in Fig 8. This is because
the laser heating primarily affects the surface, hence, the low frequency components
change appreciably. Apart from this frequency shift, we can also see that the amplitudes
variations are smaller when compared to the model which does not take the thermal effect
into account.
CONCLUSIONS
This paper is directed towards the modeling of a novel method concerning the use
of pulsed eddy current in the monitoring of the laser surface heat treatment process. It has
been shown in principle that it will be possible to interpret the PEC data obtained during
the process accurately if the thermal effect due to laser heating is also taken into account.
Future work involves development of inverse models which can be used to automate the
monitoring of the laser treatment method. Such models should effectively reconstruct the
357
0.000008 -i
0.000007
E 0.000006
0.000005
^ 0.000004
:> o.ooooos
~ 0.000002
0.000001
0
120
°
Time (ms)
170
220
Frequency(Hz)
FIGURE 8. Comparison at lower
frequencies of the coil voltage obtained by
the coupled model and the model which does
not take the heating into account (shown as
the dotted line).
FIGURE 7. Results of the coupled model:
variation of the coil voltages obtained over
time.
conductivity profiles ,thus allowing one to interpret the phenomenon of void formation and
spalling during laser treatment. This coupled model serves to provide reference data for
such inverse models. In addition to carrying out such studies on metals, studies may be
carried out with anisotropic materials. In such cases, modeling of the material structure
also becomes critical since the material properties like thermal conductivity ,density etc.
vary in each differential element. The equations have to be changed to accommodate this
variation of material properties in space.
ACKNOWLEDGEMENTS
The authors like to thank Dr Satish Udpa, Michigan State University for his
guidance during the initial stages of this work. The authors acknowledge the members of
Center for NDE, IIT-Madras for their support during this work.
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