Method for determining a heat source model for a weld

[0002] This application claims the benefit of prior provisional patent application Serial No. 60/157,244 filed Oct. 1, 1999.

[0003] This invention relates generally to a method for determining a heat source model for a weld and, more particularly, to a method for determining a heat source model for an arbitrarily shaped weld profile.

[0004] It is imperative when welding a joint between two metals, for example, in gas metal arc welding, to minimize distortions and residual stresses, and to maximize the strength of the welded joint and the surrounding structure.

[0005] One method for determining the residual stress of a weld joint is to model the stress based on a modeled weld heat source. For example, a common and very popular method for modeling a weld heat source is known as a double elliptical heat source model, disclosed by Goldak et al. (Goldak) in a paper entitled, A New Finite Element Model for Welding Heat Sources(Metallurgical Transactions, Volume 15B, June, 1984, pages 299-305).

[0006] The model disclosed by Goldak, however, may not be directly applied to an arbitrarily shaped weld profile. For example, welding procedures, parameters of the materials, and joint types, e.g., T-fillet joints, butt joints, lap joints, and the like, result in weld heat sources that are no longer elliptical. Therefore, it would be desired to incorporate the weld profile geometry of the joint into the heat source model to improve the accuracy of the model.

[0007] The present invention is directed to overcoming one or more of the problems as set forth above.

[0008] In one aspect of the present invention a method for determining a heat source model for a weld is determined. The method includes the steps of determining a double elliptical distribution of the heat density of the weld, modifying the double elliptical distribution as a function of a profile geometry of the weld, and determining the heat source model as a function of the modified double elliptical distribution.

[0009] FIG. 1 is a diagrammatic illustration of a butt joint;

[0010] FIG. 2 is a diagrammatic illustration of a T-fillet joint;

[0011] FIG. 3 is a diagrammatic illustration of a lap joint;

[0012] FIG. 4 is a diagrammatic illustration of a modified double ellipsoid heat source model of the T-fillet joint of FIG. 2; and

[0013] FIG. 5 is a diagrammatic illustration of a cross section of the weld portion of the T-fillet joint of FIG. 4.

[0014] The present invention provides a method for determining an enhanced heat source model for a weld by incorporating the weld profile geometry in the mathematical formulation for a double elliptical heat source model. The invention is described with reference to the accompanying drawings and illustrations.

[0015] The double elliptical heat source model is an advanced weld heat source model. The double elliptical heat source model assumes that the heat input from the weld arc is distributed in a volume with double ellipsoidal geometry. The major advantage of the double elliptical heat source model over other surface or spherical models is that it offers more flexibility in modeling weld penetration due to arc digging and stirring.

[0016] However, typical weld cross section profiles are not elliptical in shape. For example, with reference to FIGS. 1-3, exemplary weld profiles are shown for some commonly used welds. FIG. 1 illustrates a half-filled butt joint 100, in which a first piece 102 is attached to a second piece 104 by means of a butt joint weld 106. FIG. 2 displays a T-fillet joint 200, in which a first piece 202 is attached to a second piece 204 by means of a T-fillet weld 206. In like manner, FIG. 3 illustrates a lap joint 300, in which a first piece 302 is attached to a second piece 304 by means of a lap joint weld 306. It is to be understood that the weld joints of FIGS. 1-3 are merely examples of weld joints that may be used with the present invention. Virtually any type of weld joint used may benefit from use of the present invention.

[0017] Referring now to FIGS. 4 and 5, the T-fillet joint 200 of FIG. 2 is shown in greater detail.

[0018] The heat source model assumes the following Gauss distribution for the heat density distribution in the weld pool: $\begin{array}{cc}q\left(\xi ,r,\theta \right)=q_{max}{e}^{-{k\left(\frac{\xi }{c}\right)}^2}{e}^{-{k\left(\frac{r}{r_b\left(\theta \right)}\right)}^2}& \left(\mathrm{Equation}1\right)\end{array}$

[0019] where q_maxis the maximum heat density at an arc center 404 of the T-fillet joint weld; ξ is a moving coordinate in the welding direction, i.e., ξ=z−νt+z₀; v is an arc traveling speed; z₀is the distance from the arc starting position to the cross sectional plane of interest for a two dimensional cross sectional model, or to the plane z=0 for a three dimensional model; r_b(θ) is the distance from the heat center to points 1, 2, . . . k, k+1, . . . on the boundary of the weld profile; r and θ are local polar coordinates at the weld cross section; c is the semi-axis of the weld pool in the welding direction; and k is a parameter known as a heat concentration coefficient used to determine heat density at the boundary of weld cross section, i.e., ${k=\ln \left(\frac{q_{max}}{q_b}\right)|}_{\xi =0}.$

[0020] It is assumed, with reference to FIG. 4, that the heat source model 402 forms double elliptical distributions for the heat density in the welding, i.e., longitudinal, direction with front semi-axis c₁and rear semi-axis c₂. Therefore, $\begin{array}{cc}{q}_1\left(\xi ,r,\theta \right)=q_{max}{e}^{-{k\left(\frac{\xi }{c_1}\right)}^2}{e}^{-{k\left(\frac{r}{r_b\left(\theta \right)}\right)}^2}\mathrm{for}\xi \ge 0& \left(\mathrm{Equation}2\right)\\ \mathrm{and}& \\ q_2\left(\xi ,r,\theta \right)=q_{max}{e}^{-{k\left(\frac{\xi }{c_2}\right)}^2}{e}^{-{k\left(\frac{r}{r_b\left(\theta \right)}\right)}^2}\mathrm{for}\xi <0.& \left(\mathrm{Equation}3\right)\end{array}$

[0021] The continuity condition for the heat density at ξ=0

[0022] q₁(0,rθ)=q₂(0, r,θ)  (Equation 4)

[0023] holds from Equations 2 and 3.

[0024] The total heat input in the model, preferably in Joules per second, can be related to the welding parameters as follows: $\begin{array}{cc}\begin{array}{c}Q=\eta \mathrm{VI}={\int }_0^{\infty }{\int }_0^{{\theta }_n}{\int }_0^{\infty }{q}_1\left(\xi ,r,\theta \right)rd\theta drd\xi +\\ {\int }_0^{\infty }{\int }_0^{{\theta }_n}{\int }_0^{\infty }{q}_2\left(\xi ,r,\theta \right)rd\theta drd\xi \\ =q_{max}\frac{\sqrt{\pi }\left(c_1+c_2\right)}{4k\sqrt{k}}{\int }_0^{{\theta }_n}{r}_b^2\left(\theta \right)d\theta \\ =q_{max}\frac{\sqrt{\pi }\left(c_1+c_2\right)}{4k\sqrt{k}}\Psi \end{array}& \left(\mathrm{Equation}5\right)\end{array}$

[0025] where Ψ is a function of weld profile geometry only. Preferably, the weld profile can be represented by a discrete number of points (1, 2, . . . k, k+1, . . . ) as shown in FIG. 5. The function Ψ, in the preferred embodiment, is expressed as $\begin{array}{cc}\begin{array}{c}\Psi =\sum _{k=2}^n{\int }_{{\theta }_{k-1}}^{{\theta }_k}{r}_b^2\left(\theta \right)d\theta \\ =\sum _{k=2}^n\left(2A_k\right)=2A_w\end{array}& \left(\mathrm{Equation}6\right)\end{array}$

[0026] where A_wis the area of the fusion region. From Equations 5 and 6, it may be determined that $\begin{array}{cc}{q}_{max}=\frac{2k\sqrt{k}}{\sqrt{\pi }{A}_w\left(c_1+c_2\right)}\eta \mathrm{VI}.& \left(\mathrm{Equation}7\right)\end{array}$

[0027] In the special case of an elliptical weld cross section, A_w=πab/2, and therefore $\begin{array}{cc}{q}_{max}=\frac{4k\sqrt{k}}{\pi \sqrt{\pi }\mathrm{ab}\left(c_1+c_2\right)}\eta \mathrm{VI}.& \left(\mathrm{Equation}8\right)\end{array}$

[0028] In the thermal analysis, the initial nodal temperature in the weld area is often assigned to be the melting temperature. Thus, the heat density distribution becomes $\begin{array}{cc}q\left(\xi ,r,\theta \right)=\frac{2k\sqrt{k}\left(\eta \mathrm{VI}-Q_w\right)}{\sqrt{\pi }{A}_w\left(c_1+c_2\right)}{e}^{-{k\left(\frac{\xi }{c}\right)}^2{e}^{-{k\left(\frac{r}{r_b\left(\theta \right)}\right)}^2}}& \left(\mathrm{Equation}9\right)\\ \mathrm{where}& \\ Q_w=\rho A_{w}v{\int }_{T_o}^{T_m}\stackrel{\_}{c}dT& \left(\mathrm{Equation}10\right)\end{array}$

[0029] and {overscore (c)} is the temperature dependent specific heat of the weld material.

[0030] The present invention is an enhanced welding heat source model that can be used for an arbitrarily-shaped fusion profile. The model is based on a modification of a double elliptical heat source model with an explicit representation of the weld profile geometry in the mathematical formulation. The shape of weld profile is discretized by a set of points that can be coincided with nodes in a finite element model. The Gaussian distribution function is modified in such a way that the heat density at a point inside the weld is proportional to the radial distance from the arc center to the weld profile point. Based on these assumptions, mathematical formulations are derived. The heat source model is determined from these mathematical formulations, and is used to determine distortions, residual stresses, and potential reductions in strength of the weld.

[0031] Other aspects, objects, and features of the present invention can be obtained from a study of the drawings, the disclosure, and the appended claims.