LIST OF TABLES
CHAPTER 2: LITERATURE REVIEW ON LASER BENDING PROCESS
2.9 Process Modeling of Laser Bending Process
2.9.1 Analytical models
Observations
The laser bending process involves localized high temperature and plastic deformation which significantly affect the mechanical and microstructural properties. Variation in these properties depends on process conditions and the type of workpiece material. Investigation on these properties is important to assess the feasibility of application of laser bending for various materials.
Vollertsen (1994) derived an expression for the bend angle in TGM. The analytical expression is given by,
2
3 th
b
p s
P c Vt
, (2.8) where αth is the coefficient of thermal expansion of the work piece, P is the laser power, is the absorptivity, ρ is the density, cp is the specific heat capacity, V is the scan speed and ts is the sheet thickness. Magee et al. (1997) modified the Equation (2.8) for metal matrix composites by multiplying with a constant on the right hand side. The value of constant was 0.5 at slow scan speed and it was decreased linearly with the increase in scan speed.
Vollertsen’s models did not include the effect of yield stress and Young’s modulus of elasticity of worksheet material. Yau et al. (1998) included this effect in their model. The bend angle is given by,
2
21 36 2 p s
th y
b c Vt h
P l
E
, (2.9)
where E is the Young’s modulus and y is the yield stress.
Kyrsanidi et al. (2000) considered non-uniform temperature distribution throughout the thickness of the plate and applied the concept of basic mechanics of materials. Their model, although computationally efficient, requires programming and includes iterative steps. Cheng et al. (2006a) proposed analytical model for plate with varying thickness. The bend angle of the plate at the location with thickness, ts(x), is,
max
( ) ( )
( )
( ) ( )
b 2 3
s s
f x f x
b 1 t x t x
, (2.10)
where
1
/
b c P V
, (2.11) and/ ( )
2 s
f c P Vt
, (2.12) where c1 and c2 are constants dependent on materials properties, ts(x) is the thickness of the sheet and μ is the Poisson’s ratio. In Equation (2.10), max is the maximum plastic strain at the heated surface. It is given by,m a x th m ax y /E
, (2.13) where Tmax is the maximum temperature attained.
The model of Shen et al. (2006c) is based on the assumption that the plastic deformation is generated only during heating, while during cooling the plate undergoes only elastic deformation. According to this model, the bend angle is given by,
2
12 2
4 ( ) ( )
y p s p
b v
s s p s p
s a R t a
E t t a t a
, (2.14)
where v is the bend angle provided by Equation (2.8), R is the laser beam radius, s is the reduction coefficient to account for the variation of yield strength and Young’s modulus of elasticity with temperature and ap is the characteristic length of the plastic zone. The constants s and ap need to be evaluated empirically, which is a limitation of this model. The model is valid for TGM as well as BM.
Lambiase (2012) proposed an expression for the bend angle based on assumption of elastic-bending theory without considering the plastic deformation during heating and cooling phases. The bend angle is given by,
1 2
1 1
3 ( )
( 3 3 )
s th
s s s
p
b vc
P t t t t t t t
, (2.15) where ts is the sheet thickness and t1 is the thickness of heated volume, which is invariably estimated empirically. Lambias and Ilio (2013) developed a more rigorous analytical model to predict the deformation of thin sheets.
Kraus (1997) provided a closedform expression for estimating the bend angle during upsetting mechanism. The bend angle is expressed as
2
4
(2 )
b th
p s s
lP D
c V Dt t E
, (2.16)
where D is the beam diameter.
Shi et al. (2007a) provided a model for estimating the bend angle in an in-plane axis perpendicular to the scan direction. The bend angle is given by,
1/ 2 3/ 2 2 1/ 2
6.92 th
e
s
P WR L t ckV
, (2.17)
where W and L are the width and length of worksheet respectively, k is the thermal conductivity.
Hu et al. (2013) expressed a model as a function of laser parameters, worksheet material parameters and worksheet geometry parameters to calculate the bend angle between various points on the scan line (edge effect).
Vollertsen et al. (1995) derived an expression for estimating the bend angle in buckling mechanism dominated process conditions, i.e. for bending of thin worksheets with high ratio of thermal conductivity to worksheet thickness. The proposed analytical model was dependent on important influencing parameters such as worksheet thickness (ts), absorptivity (η), laser power (P), scan speed (V), coefficient of thermal expansion (
th), specific heat (c), density (ρ), elastic modulus (E) and temperature dependent flow stress (
y). The expression is given as1/3 2
36 th y 1
b
s
P c E V t
. (2.18)
Cheng et al. (2005b) introduced an another analytic model to capture the effects of change in width and length dimensions of the sheet,
3 2
b h th s
l T
t
, (2.19)where lh is the width of heated zone, ΔT is the temperature rise in the heated region.
Liu et al. (2005) introduced analytical model for laser bending of metal matrix composites. The model is given as
2
( )
9(1 2 )
[ ( ) ][1 ( 1)
m th p
b
m m s p p
b c b V AP
Vt b e b V d V
c
, (2.20)
where 1 2
, , ,
1 2
p thp p p p p
m thm m m m m
c
v K E
b c d c d
v K E
are the dimensionless quantities, A is a
correction factor for energy dissipation and counter-bending effect, K is the bulk modulus. The indices m and p mean the corresponding quantity belongs to the matrix and particle, respectively.
Several other analytical models have also been proposed. Cheng and Lin (2001) considered that the final bend angle is a sum of angle induced during heating and cooling
cycles. The bend angle during heating process was assumed to be a function of the temperature distribution at the moment when the highest temperature was achieved, while the bend angle during cooling process was assumed to be a function of the surrounding temperature. McBride et al. (2004, 2005) presented an analytical model to describe the local curvature in terms of the interaction time and area energy induced. This model was used for the iterative laser forming of non-developable surfaces. Ueda et al. (2005) measured temperatures at both top and bottom surfaces using two color pyrometers with an optical fiber to determine the relationships between bend angle and beam diameter, surface temperature and sheet thickness. Shen et al.
(2008) developed an analytical model based on force and moment balancing to predict the bend angle in metal/ceramic bi-layer material system of laser forming. Gollo et al. (2008) derived formulae based on laser process parameters using regression analysis to predict the bend angle.
Gollo et al. (2011) developed a relationship based on the sheet thickness, material properties and laser parameters including the number of scans to predict the bend angle. Recently, Eideh et al. (2015) developed an analytical model based on the elastic-plastic bending of sheet to evaluate the bend angle in laser bending of metal sheets.
Several models are developed to find out the temperature distribution during laser bending process. Cheng and Lin (2000a) proposed a model to calculate the temperature field induced by laser scanning. Shi et al. (2007b) derived a simple analytical model to calculate the temperature distribution in the workpiece. They explored similarity of temperature distributions about the scan line axis during laser forming of sheets. Chen et al. (2010) proposed an analytical model to calculate the temperature distribution based on the similarity of temperatures at different thicknesses. Based upon the proposed temperature model, they developed an analytical model for the estimation of bend angle as a function of laser process parameters and dimensions of the plate.
The accuracy of the closed form expressions to predict the laser bending is not high. It typically ranges from 10 to 50%. Moreover, the closed form expressions provide limited information. Several FEM methods have been proposed to get better accuracy and deeper insight into the process. The important studies on numerical modeling of the laser bending process are presented in the following subsection.