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For the first time main equation of the theory of capillary attraction circumscribing the form of an equilibrium surface of a capillary fluid in a field of gravity, for the analysis of weld formation during arc welding is applied by Voloshkevich G.Z. in [Ref.377]. Having written it for a cylindrical surface of a welding pool as follows

, (1)

author [377] by a method of graphic integration had obtained the set of integral curves of this equation circumscribing the form of a surface of welds, forming at arc welding in different positions. Having imposed an appropriate curve on the profile of real fillet weld, Voloshkevich G.Z. has received good enough correspondence of the calculated and real forms of the weld ( Fig.1).

Fig.1
Matching of the real and theoretical shapes of a fillet weld
[377,Voloshkevich G.Z.,1951]

The solution of the equation (1) in quadratures for calculation of the bead shape in a flat position, had obtained Emelyanov I.L. in 1972 year [471]. In this solution the equation of a integral curve for the profile of a weld is written as follows

, (2)

where h - height of the weld; R_o - curvature radius in the top of a weld. All calculations were fulfilled in [471] for concrete value of a capillary stationary value а_к=23.2 мм².
In 1973 Emelyanov I.L. had spread the solution (2) to the fillet welds [472].
From this model is follows that such defects of fillet weld as undercut on a vertical wall or overlap on a horizontal one, are explained by unfavorable combination of the fillet weld leg length , square of a deposited metal and capillary stationary value. With the model the quantitative assessment of conditions of such defects origin is possible.

Fig. 2
The theoretical shapes of fillet welds with a different leg length K and deposited metal square
F_н and К=3 mm (a), К=5 mm (b), К=10 mm (c)
[472,Emelyanov I.L.,1973]

In 1975-1976 Nishiguchi K. [474,475] had obtained a solution of the equation (1) as follows

, (3)

where the function f (y) is determined by an expression

.
This solution is reduced in [474] to known functions - elliptic integrals of 1^st and 2^nd kind (F and E)

This solution utilised also for the description of the shapes of fillet welds at a position «in corner» and horizontal welds on a vertical plane [474, Nishiguchi K., 1976]. In fig.3 the comparison of calculated and experimentally obtained weld's shape is exhibited.

Fig.3
Comparison of the calculated and theoretical shape of a fillet weld (а) and a horizontal weld on a vertical plane (б)
[475, Nishiguchi K., 1976]

In 1977 Berezovsky B.M. [479] had obtained a solution of the equation (1) in more convenient parametric kind

That had allowed to find complete closed system of the equations linking main geometrical parameters of weld reinforcement (deposited bead)

where

, , , ,

   where B,C - width and height of weld reinforcement, mm; Z₀ - parameter of curvature, mm; F_н - square of a weld metal, мм².
       This system from three equations links main geometrical parameters of weld reinforcement and contains 3 unknowns parameters: Z₀, C and Fik; by a method of exception of unknowns the given system was reduced to one equation with one unknown parameter, to solve and to receive values of all parameters of weld reinforcement.
      In a Fig. 5.9 are presented cross sections of beads with the calculated curves, imposed on them.

Fig.4
Matching of the substantial and designed form of cylinders

Fig.5
Matching of the substantial form of a cylinder, deposited
By electrode fillet, with the designed form (file lenta1.bmp from 19.11.99, Size 2 273 760) z₀=0.07 mm

In 1979 Berezovsky B.M. [667] has applied the given approach to mathematical modelling of creation of a seam in a ceiling position and has received solution by the way

And the lower sign takes at build-up of an integral curve circumscribing the form of convexity of a seam in a lower position, and upper sign - in a ceiling position.
The form of integral curves being solution of the differential equation (6.38), is reduced in a Fig. 6.22.
The set of equations linking main specifications of convexity looks like

And the upper sign corresponds(meets) to a ceiling position, and lower sign - lower position.
As an example in a Fig. 6.9 the comparison of theoretical curves circumscribing the form of convexity of seams, formatived in lower and ceiling positions and dimensionless values, having identical value, of square deposited fо and width of a seam bо is reduced.

Experimental check of model of weld formation in overhead positions

For check of the obtained designed equations the experiments on a surfacing of cylinders in a ceiling position were conducted. The surfacing was produced by electrodes with different types of coatings of brands МР-3, АНО-4, ОЗС-4, УОНИ-13/55 a dia of 4 mms on slices from mild become a size 400х200х10 mm. As the power supply have utillized a welding rectifier VDU-504. With the purpose of exception of the factor of qualification of the welder (or with reference to conditions of automatic welding) a surfacing realized on an automaton ADS-1000-2, equipped by special accommodating for welding by the plated electrodes (Fig. 6.13).

Fig.8 [667, Berezovsky B.M. et al, 1979].
Experimental installation on the basis of automaton ADS-1000-2 for welding with coated electrodes in a overhead position

Fig. 6 [#667,Berezovsky B.M., 1979]
The set of integral curves defining equilibrium
The form of a surface of convexity of a ceiling seam

Fig. 6.9 [#667,Berezovsky B.M. et al, 1979]
Matching of theoretical curves circumscribing the form bead reinforcement. Welds at welding in a flat (2,4) and overhead (1,3) positions: 1,2 - fo=1.5; bo=2.5; 3,4 - fo=3.5; bo=3.5

Fig.7
Cross sections of beads, deposited in a flat (а) and
an overhead (b) positions

In 1983 Berezovskii B.M. [482] had applied the given approach to mathematical modelling of horizontal welds on a vertical plane, and in 1988 - on an inclined plane [480].

Fig.9 [687,Berezovskii B.M., 1988].
The designed scheme of definition of the weld shape for horizontal weld on a inclined plane (a) and physical
model (b)

For this scheme obtained the follows equations:

	,
	,
	,

Fig.11


Fig.10 Cross sections of welds fulfilled in a vertical plane without a filler metal for stainless steel Х18Н9 [693, Berezovskii B.M., 1980] (a) and titanium alloy (b)


Fig.12 Matching of the real and calculated shape of weld reinforcement

The author of [484] experimentally and theoretically had studied weld formation in a flat and overhead positions. The physical analog of this model had represented by the two slices, between which backs the fluid phase is posed. For the analysis of the process of root weld formation the capillary-hydrostatic model for a special case was built, when width of a root seam В₁ and width it В₂ are identical.

In the paper [485] the model explaining a reconfigurating of a welding pool at welding of thin metal with burning through on weight in different space positions is offered. For obtaining equilibrium equations of weld pool surface the variational energetical method was utilised which has allowed not only to obtain the equilibrium form of a through seam at its shape in different positions, but also to estimate stability of a surface of a fluid phase between hard ridges.

Fig.13 [485, Andrews J.G. et al, 1980].
Effect of ambient pressure on the weld pool shape surfaces

References
377. Voloshkevich G.Z. Welding of vertical seams by a method of forced formation // The Anniversary issue dedicated 80-years of E.O.Paton. - Kiev: Ukraine Sciences Academy Publishing, 1951. p.p. 371-395.
471. Emelyanov I.L. Effect of surface tension forces and external pressure on the shape of a deposited bead // Trans. of Leningrad Institute of Water Transport. - 1972, Vol.135. - p.p.135-145.
472. Emelyanov I.L. Effect of surface tension forces on the shape of fillet welds // Trans. of Leningrad Institute of Water Transport. - 1973, Vol.142. - p.p.120-126.
474. Nishiguchi K., Ohji T., Matsui H. Fundamental research on bead formation in overlaying and fillet welding processes ( Report 1). Surface tensional analysis of bead surface profile // J. of the Jap. Welding Soc. - 1976, Vol.45. - No.1. - P.82-87 ( jap.)
475. Nishiguchi K. Fundamental researches on bead formation in overlaying and fillet welding process ( 2^nd Report ). Surface tensional analysis of surface profile of horizontal fillet weld // J. Jap. Welding Soc. - 1976, Vol.45. - No.2. - P.143-149 ( jap.)
479. Berezovskii B.M., Stikhin V.A. Influence of forces of surface tension on formation of reinforcement of weld.// Welding Production — 1977. — No. 1. — p.p. 51—53.
Berezovskii B.M., Stikhin V.A. Calculated definition of weld reinforcement shape and critical dimensions of weld pool at overhead welding // Proceedings of Chelyabinsk Polytechnical Institute. — Vol.207. — 1979.— p.p.103-111.
480. Berezovskii B.M. Mathematical modelling of weld formation on a inclined plane // Automatic Welding. — 1988. — №1. — p.p.26—31.
482. Berezovskii B.M., Suzdalev I.V., Stikhin V.A. et al Mathematical modeling and optimising of weld formation on vertical plane // Automatic Welding. — 1983. — No.3. — p.p.21—24.
484. Kureishi M. Correlation among parameters affecting on the formation of penetration beeds // J. of the Jap. Weld. Soc. - 1980, Vol.49. - №5. - P.297-304. (jap.)
485. Andrews J.G., Atthey D.R., Byatt-Smith J.G. Weld pool sag // J. of Fluid Mech. - 1980, Vol.100. - No.4. - P.785-800.