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OVERVIEW OF THERMAL MODELING OF PULSED LASER

Pulsed lasers have also been used for welding. Thermal analysis of the pulse laser welding process can assist in prediction of welding parameters such as depth of fusion and residual stresses. The use of modeling to minimize defects such as spatter, porosity, and cracking will not be discussed here, but in a different article. This article is mainly concerned with modeling for the prediction of depth of fusion (i.e. where the base metal reached the melting temperature) given process variables such as laser beam power, pulse frequency, pulse duration, travel speed, etc. Due to the complexity of the pulsed laser process, it is necessary to employ a procedure that involves constructing a mathematical model, calculating a thermal cycle using numerical modeling techniques like either Finite Elemental Modeling (FEM) or Finite Difference Method (FDM) or analytical models with simplifying assumptions, and validating the model by experimental measurements. Often, temperature-dependent materials properties are necessary for reasonable accuracy. The physics of pulsed laser can be very complex and therefore, some simplifying assumptions need to be made to either speed up calculation or compensate for a lack of materials properties. Each of the simplifying assumptions have their consequences as will be discussed. The constitutive equations for laser physics are described mathematically, and the reasons for their use will be explained. This article also provides methods for validating the model.

Physics overview

Some laser physics need to be understood before exploring the number-crunching techniques. An overview of the physical phenomenon that must be accounted for will be given at first with more mathematical detail to follow. First, lasers can operate in one of two modes: conduction and keyhole. Which mode is in operation depends on whether the power density is sufficiently high enough to cause boiling and evaporation. Conduction mode occurs below boiling while keyhole mode occurs above the boiling point. The keyhole is analogous to an air pocket. The air pocket is in a state of flux. Forces such as the recoil pressure of the evaporated metal open the keyhole while gravity (aka hydrostatic forces) and metal surface tension tend to collapse it. At even higher power densities, the vapor can be ionized to form a plasma.

Second, unlike CW which involves one moving thermal cycle, pulsed laser involves repetitively impinging on the same spot, so it is like having overlapping thermal cycles. In one model, the thermal conduction model involves a step function to account for the pulsing nature by creating a relation where the heat flux goes on and off. This treatment assumes that when the

Third, not all radiant energy is absorbed and turned into heat for welding. Some of the radiant energy is absorbed in the plasma created by vaporizing and then subsequently ionizing the gas. In addition, the absorptivity is affected by the wavelength of the beam, the surface composition of the material being welded, the angle of incidence, and the temperature of the material. Radiant energy is also not uniformly distributed within the beam. Some devices produce gaussian energy distributions, whereas others can be bimodal. Understanding the absorption efficiency is key to calculating thermal effects. Often this is one of the calibration points of most models.

Details of Laser Physics For Modeling

The temperature-dependence of absorptivity of a metal can be determined by in the solid and liquid states can be described by

Consequences of Simplifying Assumptions on Calculation Results

The temperature-dependence of material properties such as specific heat are ignored to minimize computing time.

The liquid temperature can be overestimated if the heat loss due to mass loss from vapor leaving the liquid-metal interface is not accounted for.

Modeling Methodologies or Procedures

One method involves the following steps : ' Step 1. '
 * 1) Determining power absorption efficiency.
 * 2) Calculating the temperature distribution
 * 3) Calculating the recoil pressure based on temperatures and a Clausius-Clapeyron equation.
 * 4) Calculate the fluid flow velocities using the VOF (Volume of method).
 * 5) Increment time and repeat steps 1-4.

Rosenthal point source assumption leaves a infinitely high temperature discontinuity which is addressed by assuming a Gaussian distribution instead. This then allows for easier calculation of temperature-dependent material properties where the beam impinges. Absorptivity is one of those properties.

On the irradiated surface, when a keyhole is formed, Frensel reflection (the almost complete absorption of the beam energy due to multiple reflection within the keyhole cavity) occurs and can be modeled by $$\alpha_{\theta}=1-R_{\theta}=1-0.5{{1+(1-\epsilon \cos \theta)^2 \over {1+{1+\epsilon \cos \theta)^2}}}+ {{{\epsilon^2}-2\epsilon \cos \theta+2 \cos^2 \theta} \over {\epsilon^2}+2\epsilon \cos \theta+2 \cos^2 \theta}}$$, where ε is a function of dielectric constant, electric conductivity, and laser frequency.θ is the angle of incidence.

' Step 2. '

In order to determine the boundary temperature at the laser impingement surface, you'd apply an equation like this. $$k_n{\partial T\over \partial n}-q+h(T-T_o)+\sigma \epsilon (T^4-T^2_o)=0$$, where kn=the thermal conductivity normal to the surface impinged on by the laser, h=convective heat transfer coefficient for air, σ is the Stefan-Boltzman constant for radiation, and ε is the emissivity of the material being welded on, q is laser beam heat flux. This can be modified for pulse by including a step function that only makes it applicable when the

Next you would apply this boundary condition and solve for Fourier's 2nd Law to obtain the internal temperature distribution. Assuming no internal heat generation, the solution is $$\rho C_p ({\partial T \over \partial t}+v \bigtriangledown T)=k \bigtriangledown T$$, where k=thermal conductivity, ρ=density, Cp=specific heat capacity, v=travel velocity.

' Step 3. '

The recoil pressure is determined by using the Clausius-Clapeyron equation. $${dP \over dT}={d\Delta H_{LV} \over dT\Delta V_{LV}}\thickapprox {d\Delta H_{LV}\over T_{LV} V_{LV}}$$, where P is the equilibrium vapor pressure, T is the liquid surface temperature, HLV is the latent heat of vaporization, TLV is the equilibrium temperature at the liquid-vapor interface. Using the assumption that the vapor flow is limited to sonic velocities, one gets that $$P_r\approxeq0.54P_oexp(\Delta H_{LV})$$, where Po is atmospheric pressure and Pr is recoil pressure.

Validation of results

Results can be validated by specific experimental observations or trends from generic experiments.