LIST OF TABLES
CHAPTER 4: 3-D THERMO-MECHANICAL NUMERICAL SIMULATION OF LASER BENDING PROCESS USING FINITE ELEMENT METHOD
4.2 Thermo-Mechanical Modeling of Laser Bending Process
4.2.2 Governing equations and boundary conditions
Primary mechanism of laser bending is the deformation occurred due to the stresses induced into the worksheet by the laser heating. Therefore, both thermal and mechanical analyses are need to be carried out. The governing equations and boundary conditions for laser bending process are presented in details as follows,
A. Thermal analysis: governing equations and boundary conditions Governing equations
Laser bending is a result of non-uniform temperature distribution into the worksheet material.
The laser heat transfers into the worksheet material as a result of thermal conduction and the heat flows into the surrounding by convection and radiation heat flow. This heat transfer problem is solved by the law of conservation of thermal energy. The transient temperature field generated in isotropic material due to the laser beam irradiation is determined by using,
. 3 2 th ii
c T k T T
t
, (4.1)
where T is the temperature (°C), which is a function x, y, z and time t(s), ρ is the density of material (kg/m3), c is the specific heat (J/kg-°C), k is the thermal conductivity vector of the material (W/m-°C), th is the coefficient of thermal expansion,
iiis the volumetric strain rate and λ and ν are the Lame’s parameters. The last term of Equation (4.1) is the heat generated due to volumetric change of the worksheet and it is neglected due to its very less value incomparison with the laser heat input into the worksheet surface. is the gradient operator which is given as
ˆ ˆ ˆ
i j k
x y z
. (4.2)
Boundary conditions
The initial thermal condition of the worksheet was set to room temperature,
0 0
Tt T . (4.3)
The thermal boundary conditions are modeled by using natural convection and radiation heat losses. The convection heat loss (
q
c) is given as(
0)
c s
q h T T
, (4.4) where h( 25 W m - C ) 2 ° is the convective heat transfer coefficient, Ts is the worksheet temperature andT
0( 20 C)
is the environmental temperature. The heat loss due to radiation( ) q
r is calculated as4 4
(
0)
r s s s
q T T
, (4.5) where s( 5 .6 7 1 0 8 W m - C )2 ° 4 is Stefan-Boltzmann Constant, and
s( 0.2)
is the surface emissivity of the worksheet.The effect of melting was incorporated by considering the latent heat near to the melting temperature. The solidus and liquidus temperature of the material is 648 °C and 649 °C, respectively (Avedesian and Baker 1999). Incorporation of the latent heat in such a small range of melting temperature destabilizes the process of solving the numerical problem. Therefore, the latent heat was considered to be distributed over the temperature range of 645 °C to 655
°C. The latent heat of the magnesium alloy M1A is 370 kJ/kg-°C, and the distributed latent heat for the range 645 °C to 655 °C was taken as 37 kJ/kg-°C.
Heat flux model
The laser beam was assumed to be circular with Gaussian distribution of heat flux. The beam diameter is controlled by changing stand-off distance (H) between laser head and worksheet
surface. A mathematical model based on standard beam propagation equations was used to calculate the beam diameter.
Stand-off Distance (H) Laser Head
Focal Lens
Focal Point
Focal Length (f) Beam Diameter
before Lens,
Worksheet Surface R Gaussian Distribution of
Laser Heat Flux
θ
DL
D RL
Beam waist, w0
Figure 4.1. Schematic of laser beam profile.
Figure 4.1 shows schematic of the laser beam passing through the focal lens. The beam radius (R) is calculated from the stand-off distance (H) as (Sun 1998)
2 12 2
0 2
0
1 M H
R w w
, (4.6)
where w0(=0.05 mm) is the laser beam waist which is the minimum beam radius at focal point, λ(=10.6 µm) is the wavelength of CO2 laser beam, H is the distance between focal point to the worksheet surface and adjusted as equal to the stand-off distance, M2 is the beam quality factor. It is equal to one for a perfect Gaussian beam but in actual it always has a greater value.
The half divergence angle (θ) of the laser beam is given as (Sun 1998)
2
0
M w
. (4.7)
Multiplying both sides by focal length (f), Equation (4.7) becomes,
2
0
f M f w
(4.8)
where f(=127 mm) is the focal length of the lens used in this work. Now f RL, therefore, Equation (4.8) can be written as
2 0
L
M f
w R
, (4.9)
where
R
L is the laser beam radius before lens. The Equation (4.9) was used to calculate the M 2and it was found to be 1.4.Figure 4.2. Burn prints at various stand-off distances (in mm) on photographic paper.
Table 4.1. Comparison between experimental and numerical laser beam diameter.
S.
No.
H (mm) Experimental Beam Diameter, D(mm) Mathematical Value (mm)
Error Major Axis Minor Axis Average (%)
1 10 1.989 1.916 1.953 1.938 0.74
2 20 3.898 3.873 3.886 3.872 0.35
3 30 5.852 5.830 5.841 5.807 0.58
4 40 7.5 7.280 7.39 7.743 4.78
5 50 9.903 9.617 9.76 9.678 0.84
6 60 11.691 11.253 11.472 11.337 1.18
7 90 17.546 16.557 17.052 17.006 0.27
8 120 23.528 22.173 22.851 22.674 0.77
9 150 29.355 27.931 28.643 28.342 1.05
H = Stand-off Distance Average error = 1.17%
Beam diameter obtained from mathematical model was validated by taking burn prints on the photographic paper at various stand-off distances. During this process, the laser power was selected in such a way that a clear circular impression of the laser beam generates on the photographic paper, but the paper does not burn out completely. Images of burn prints on the
photographic paper are shown in Figure 4.2. It can be observed that the burn prints are slightly elliptical in shape. However, the difference between major and minor axes is small, and therefore, the beam can be considered as of circular shape. The measurement of major and minor axes of the laser beam was carried out on an optical surface profile projector. The comparison between experimental and calculated values of the beam diameter is shown in Table 4.1. The average error between experimental and calculated values of beam diameter is about 1.17%. Therefore, it can be concluded that the mathematical results are in good agreement with the experimental results, and Equation (4.6) can be used to compute the beam diameter at various stand-off distances.
Figure 4.3. Heat flux distribution and beam diameter terminologies.
The heat flux into the laser beam obeys a normal distribution as shown in Figure 4.3. It has peak value (
I
max) at the center of the beam. The heat flux distribution in Gaussian distributed circular shaped laser beam as a function of beam radius is defined as (Holzer et al.1994)
2
2 2
2 2
( ) Pexp r
I r R R
, (4.10)
where η is the absorptivity, P is the laser power, r is the distance from center of laser beam and R is the effective beam radius which is defined as the radius in which power density is reduced from the peak value (
I
max) by a factor of the square of natural exponent (1/ )e2 . The mean heat flux density within the area of laser beam can be calculated as (Hu et al. 2001)2
2 2 2 2 2
0 0
1 2 2 2 0.865
(2 )d exp d
R R
m
P r P
I I r r r r
R R R R R
. (4.11)B. Mechanical analysis: governing equations and boundary conditions Governing equations
Thermal expansion caused by the laser heating induces thermal stresses in the heated region.
These stresses lead to the plastic strains, and further, distortions in the worksheet. Total strain and strain rate can be decomposed into elastic, plastic, creep and thermal components of strain and strain rate. However, deformation occurs at relatively short time scale, and hence, the contribution of creep can be neglected. Total strain rate as a sum of elastic, plastic and thermal strain rate is given as
total elastic plastic thermal
. (4.12)Elastic strains are calculated through isotropic Hook’s, law and yielding is determined by using von-Mises criterion as
2 2 2 2
1 2 2 3 1 3
1 ( ) ( ) ( )
2
y, (4.13)where
y is the temperature and strain rate dependent flow stress and 1, 2 a n d 3 are x, y and z components of the stresses, respectively. The material properties are significantly affected by temperature, strain and strain rate. Therefore, temperature, strain and strain rate dependent yielding should be considered in the numerical modeling of laser bending process.The strain rate dependent flow stress is given as
m
y C
(4.14) where C is the strength coefficient and m is the strain rate sensitivity exponent. The C and m are the temperature dependent parameters.
Boundary conditions
To avoid the rigid body movement, one side (clamped side) of the worksheet was fully constrained in mechanical analysis. The boundary conditions of zero displacement and zero rotation were applied on the clamped side of the worksheet. All other sides were free of mechanical boundary conditions.