LIST OF TABLES
CHAPTER 2: LITERATURE REVIEW ON LASER BENDING PROCESS
2.9 Process Modeling of Laser Bending Process
2.9.2 Numerical models
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.
improve the fundamental knowledge of the process. Vollertsen et al. (1993) obtained the temperature distribution and bend angle using finite difference method (FDM) and finite element method (FEM). However, the models have limited applicability due to non-availability of the temperature dependent material properties. Holzer et al. (1994) conducted the FEM simulation of buckling mechanism using ABAQUS. Ji and Wu (1998) carried out FEM analysis of transient temperature field produced in the laser bending process. Kyrsanidi et al.
(1999) numerically simulated the laser bending process to generate a sine shape from a flat workpiece. Hu et al. (2001, 2002) carried out computer simulation and on-line experimental investigation for laser bending of the workpiece. Hsieh (2004a) investigated the vibration phenomenon during pulsed laser bending of thin metal plates. Shen et al. (2006a, b) developed an FEM model for two parallel laser beams scanning over the workpiece. Shi et al. (2007a) predicted the temperature field by using a three-dimensional FEM model. They considered temperature dependent thermal properties of the workpiece. Liu et al. (2007) developed a numerical model for laser forming of aluminum matrix composites with different volume fractions of reinforcement. Shen et al. (2009) used FEM to investigate the laser bending for metal/ceramic bi-layer materials.
Many researchers worked on the improvement of the numerical models. Shichun and Jinsong (2001) introduced weight coefficients to handle the transition from elastic zone to plastic zone in an element of FEM. Zhang and Michaleris (2004) compared 3-D Eulerian and Lagrangian approaches of FEM formulation. It was found that the Eulerian approach takes lesser time than the Lagrangian approach in the prediction of bend angle. However, the results of Lagrangian approach were in better agreement with the experimental results. Cheng and Yao (2005) incorporated the material anisotropy in the FEM model. Fan et al. (2005) developed thermal–microstructural–mechanical numerical model for the laser bending of Ti–6Al–4V alloy. They incorporated the effect of phase transformations by considering phase transformation kinetics. The flow stress was predicted by the rule of mixtures. Liu et al. (2008) developed an FEM model integrated with a multi-particle cell model to examine the effect of particle spatial distributions on deformation behavior of a composite in laser bending. Sowdari and Majumdar (2010) developed an enthalpy based computational model to analyze the temperature distribution, solid-liquid interface location, and shape and size of the molten pool.
Numerical models considering the strain rate and temperature effects usually give unsatisfactory results, when applied to the multi-scan laser bending operations. This is mainly due to the inadequate constitutive models employed to describe the hot deformation behavior.
Cheng and Yao (2002) developed a numerical model by considering the effects of microstructural change on the flow stress in multi-scan laser forming of low carbon steel.
Incorporation of the effect of microstructural change on the flow stress increased the accuracy of the developed numerical model.
Various efforts have been put to reduce the computational time for simulations of the large workpiece. Yu et al. (2001) adopted a rezoning technique to reduce the simulation time.
They studied the effects of mesh refinement on the temperature distribution and final distortion.
Zhang et al. (2004) studied the effects of temporal and spatial discretization and mesh density on angular distortion to reduce the computational time. It was concluded that to obtain an accurate solution, the temporal discretization requires at least four time increments to pass through the beam radius and the spatial discretization requires at least two elements per beam radius and three elements along the workpiece thickness. Reutzel et al. (2006) developed a computationally efficient method based on the concept of differential geometry to analyze the thermal forming process. The developed numerical model was reasonably accurate and provided the errors less than 12%. The computational efficiency was found to be improved and the computational time was reduced by about 99.9%. Shi et al. (2006b) derived similarity theory to predict the temperature field and deformation behavior of much larger plates through the analysis of much smaller ones. Pitz et al. (2010) introduced the moving mesh approach to save the computational time. Hu et al. (2012) developed a simple, robust and accurate FEM model using multi-layered shell elements. This intelligent mesh reduced the number of elements which increased the simulation efficiency significantly. The model could predict the laser bending process with good accuracy and was suitable for the simulation of the large workpiece. Eideh (2014) showed by FEM results that the cooling rate after completion of the laser scan has insignificant effect on the final bend angle. Therefore, it is a good idea to apply forced cooling on the workpiece after laser beam irradiation for the numerical analysis of the process. It reduces the simulation time.
The numerical simulations offered satisfactory results in terms of temperature, stress, strain and displacement distributions. However, advanced analysis is required for laser bending to be a viable process for rapid prototyping, shape correction and micro-adjustment as FEM simulation takes several hours. The numerical simulations are not suitable for the online optimization and control (Kyrsanidi et al. 2000). Researchers developed some soft-computing models to overcome long simulation time of numerical models and poor predictability of the mathematical models.