Laser Metal Processing

Abstract

Laser material processing includes the application of lasers for achieving different manufacturing goals. The advantages include an accessible spacing between the sample surface and the laser source, enabling real-time inspection of the process, simplifying the quality control process, and relatively lower costs. In particular, selective laser sintering is a manufacturing process that uses laser and powder to generate a three-dimensional model based on the concept of rapid prototyping. This chapter covers basic working principles and instrumentation settings of these laser processes.

Problems

1.1 Problem 8.1

You are drilling titanium, 1.5 mm thick, hole diameter d = 0.5 mm, absorbance 50%, pulse length 2 × 10⁻⁵ s. Determine the instantaneous power P (W).

1.2 Problem 8.2

Considering line welding of stainless steel using a laser source with a power (P_a) of 1 kW and moving velocity (v) of 2 mm/s, (i) please determine the width of welded steel w over the steel surface, given that the melting temperature of steel (T_M) is 1425 °C and the environmental temperature (T_o) is 20 °C. The thermal properties of steel include thermal diffusivity α = 4.4 mm²/s, thermal conductivity k = 15 °CN/s, and density × specific heat of capacity ρc = 3.6 N/(mm² C). Please be noted that the cross-hatched area enclosed by the 1425 °C isotherm over the steel material indicates the cross section of the weld pool. (ii) Further, please express w with a symbolic expression, as a function of P_a, v, T_M, T_o, and material properties of the welded material.

1.3 Problem 8.3

Recalling Eq. 8.19 with the origin (x = 0 and r = 0) of coordinates moving with the laser source, i.e.,

$$ T\left(x,r\right)=\frac{q}{2\pi \rho {C}_{\mathrm{p}}\alpha r}\exp \left(-\frac{v\left(r+x\right)}{2\alpha}\right) $$

Please find the steepest cooling gradient and the corresponding location behind the moving laser source. The cooling gradient can be considered as –dT/dτ = v × dT/dx_S.

1.4 Problem 8.4

Considering that laser welding of alloy steel with thermal diffusivity α = 10.2 mm²/s and thermal conductivity k = 38 °CN/s, melting temperature T_M = 1510 °C, and environmental temperature T_o = 20 °C, the laser source is moving with a traversing velocity v = 5 mm/s for generating a weld width (w) on the material surface of 10 mm.

• (a) Determine the power input q.

• (b) Determine the distance behind the arc (x_c), at which the temperature cools down to 550 °C.

• (c) Determine the cooling rate, i.e., −dT/dτ, at this (moving) point.

1.5 Problem 8.5

Consider the laser welding along a line on carbon steel, stainless steel, and aluminum, as three individual processes. The traversing velocity for carbon steel is v = 3 mm/s, for stainless steel is 2 mm/s, and for aluminum is 5 mm/s. The weld width on material surfaces is 6 mm in all these three cases. The thermal properties are given in Table 8.P1.

• (a) Determine the power input q (mW) for welding.

• (b) Calculate the heat input H (kJ/mm) for three cases as well as the thermal efficiency η, defined as the ratio between the effective energy inducing material melting and the heat input.

• (c) Plot the heat input H (kJ/mm) versus speed v (mm/s) in the range 0.1–10 mm/s for a weld width of w = 6 mm, for the three materials. Plot all three cases in one plot.

Table 8.P1 Thermal properties of carbon steel, stainless steel and aluminum

1.6 Problem 8.6

Let’s consider that case of the “moving-point” laser welding of two very thin stainless steel plates. Accordingly, if we consider a large enough time scale, Eq. 8.7 would be simplified as a “planar” relation:

$$ \frac{1}{\alpha}\frac{\partial {T}_i}{\partial t}=\frac{\partial^2{T}_i}{\partial {x}^2}+\frac{\partial^2{T}_i}{\partial {y}^2}. $$

This is because the temperature variation along the z-direction can settle much faster than along x- and y-directions. We may neglect the derivative of temperature with respect to z as a reasonably good approximation.

• (a) Please derive the temperature profile as a function of x and r for the case of a continuous moving point source on a plane, similar to Eq. 8.19. (You answer might be expressed with the Bessel function.)

In the following parts, please use any missing parameters as defined in Problem 8.4.

• (b) Repeat Problem 8.4 as the planar laser welding of two alloy plate, by assuming each the plate has a thickness of 100 μm, to obtain the power input q, the distance behind the arc (x_c), and the cooling rate at x_c.

• (c) Please adopt the power input q you obtain from (b). Then, find out and plot the corresponding weld width w against different plate thicknesses from 50 to 200 μm.

• (d) Please fix the weld width w = 10 mm, and calculate the power output q for different plate thicknesses. If we further consider a plate to be “thin” when the welding process for the same weld width w requiring only ≤10% of the power output q for the thick materials as what you calculated in Problem 8.4(a), please suggest the condition for the “thin plate” approximation (and also the planar welding simplification).

1.7 Problem 8.7

The governing equation for conduction heat flow in a solid with a coordinate system fixed at a stationary origin relative to the solid may be expressed as

$$ \frac{\partial T}{\partial t}=\alpha \left(\frac{\partial^2T}{\partial {x}^2}+\frac{\partial^2T}{\partial {y}^2}+\frac{\partial^2T}{\partial {z}^2}\right) $$

Show that for the moving heat source case with a coordinate system moving with the heat source along the x-axis at a velocity u_x, as shown in Fig. 8.P1, the corresponding governing equation obtained by a coordinate transformation from the plate to the heat source, with x replaced by ξ, that is, ξ = x − u_xt, is given by

$$ \frac{\partial T}{\partial t}=\alpha \left(\frac{\partial^2T}{\partial {\xi}^2}+\frac{\partial^2T}{\partial {y}^2}+\frac{\partial^2T}{\partial {z}^2}\right)+{u}_x\frac{\partial T}{\partial \xi } $$

Fig. 8.P1

The coordinate system moving together with a translating heat source long the x-axis

1.8 Problem 8.8

The temperature distribution in a semi-infinite plate for a moving point heat source is given as in Eq. 8.19. Derive an expression for the cooling rate at points of the plate where the cooling rates are highest. Where do these occur?

1.9 Problem 8.9

Considering the laser processing on thin materials such as welding of two thin plates, we may assume that ∂T/∂z ≈ 0 for all material positions. Derive the temperature profile of a continuous moving laser source on a plate with a speed v along x. If we let ξ = x − vt, the temperature profile should be in the form

$$ T\left(\xi, r\right)={\Upsilon}_1\exp \left({\Upsilon}_2v\right){K}_0\left({\Upsilon}_3r\right) $$

where Υ₁, Υ₂, and Υ₃ can be expressed by other variables, r is the radial distance from the current position with laser exposure, and K₀(·) is the modified Bessel function of the second kind of order zero.