For welding situations, where the effective depth of penetration is small, the surface heat source model of Pavelic, Friedman and Krutz has been quite successful. However, for high power density sources such as the laser or electron beam, it ignores the digging action of the arc that transports heat well below the surface of the weld pool. In such cases a hemispherical Gaussian distribution of power density (W/m^) would be a step toward a more realistic model. The power density distribution for a hemispherical volume source can be written as:

,(*,*£) = 4^e-^e-3,■/,■,-.№ (2_5)

С 71^71

where q(x, y,%) is the power density (W/m*). Eq. (2-5) is a special case of the more general ellipsoidal formulation developed in the next section.

Though the hemispherical heat source is expected to model an arc weld better than a disc source, it, too, has limitations. The molten pool in many welds is often far from spherical. Also, a hemispherical source is not appropriate for welds that are not spherically symmetric such as a strip electrode, deep penetration electron beam, or laser beam welds. In order to relax these constraints, and make the formulation more accurate, an ellipsoidal volume source has been proposed.

Ellipsoidal Power Density Distribution

The Gaussian distribution of the power density in an ellipsoid with center at (0, 0, 0) and semi-axis a, b, с parallel to coordinate axes x, у, с can be written as:

= q(0)e~Ax2 e~By2 e~c^2 (2-6)

where q (0) is the maximum value of the power density at the center of the ellipsoid.

Conservation of energy requires that:

£ у x

2Q = 2rjVI = sj J J q(0)e-Ax2 e~Bv* е~сЄ dxdyd% (2-7)

о о о

where;

rj = Heat source efficiency V= voltage I = current

Evaluation of Eq. (2-7) produces the following:

л/ ABC

яф)__204Ш (M)

7ГлІ 7Г

To evaluate the constants, А, В, C, the semi-axes of the ellipsoid a, b, с in the directions x, у, і are defined such that the power density

falls to 0.05 q(0) at the surface of the ellipsoid. In the x direction:

q(a, 0,0) = q( 0)e~A“2 = 0.05q(0) (2-10)

Hence

Similarly

в = jr (2-12)

С = Л" (2-13)

c

Substituting Л В, С from Eqs. (2-11) to (2-13) and q (0) from Eq. (2-9) into Eq. (2-6):

Я(Х, У,4) = - Me (2.,4)

abc n^Jn

The coordinate transformation, Eq. (2-3), Figure 2-8, can be

substituted into Eq. (2-14) to provide an expression for the ellipsoid

in the fixed coordinate system.

= <2-15)

abcKyjn

If heat flow in the z direction is neglected, an analysis can be performed on the z-y plane located at z = 0 which is similar to the ‘disc’ source. The power density is calculated for each time increment, where the ellipsoidal source intersects this plane.