Analysis of Weld Puddle Distortion and Its Effect on Penetration

Transcription

1 Analysis of Weld Puddle Distortion and Its Effect on Penetration Weld puddle distortion contributes significantly to weld penetration characteristics, and the control of puddle surface tension is essential to the control of weld puddle shape change and the enhancement of penetration By E. FRIEDMAN ABSTRACT. An analytical finite element heat transfer model for the gas tungsten-arc (GTA) welding process is developed to account for distortion of the weld puddle due to pressure from the welding arc and gravitational forces in the puddle. The model is employed to assess the combined effects of heat flow in the weldment and changes in the puddle shape on weld penetration. The relationship between puddle configuration and penetration has not previously been studied from an analytical point of view. Introduction Extensive experimental data exist on the effects of such welding parameters as arc current, arc gap, weld speed and shielding gas on weld bead depth and width for the CTA welding process. What has been lacking, however, is a clear understanding of how these and other variables interact to produce the observed weld bead shapes. To help develop this understanding, a systematic effort has been undertaken to establish an analytical model of heat transfer in the molten weld puddle and the surrounding solid material, ll is the flow of heat in the weldment that ultimately determines the extent of the puddle and, therefore, the configuration of the solidified weld bead. Hence, characterization and study of the heat transfer phenomenon provide a vital link in gaining an improved comprehension of how weld penetration and shape can be controlled. Heat flow in a weldment for the GTA process is governed by the mechanisms of heat conduction in solid and liquid material and fluid motion in the puddle. These in turn are dictated by conditions at the weldment boundaries, which include heat input from the arc, surface energy losses to the environment and distortion of the weld puddle due to the force of the welding arc impinging on its surface, as well as by convective and magnetohydrodynamic interactions. The welding thermal cycle has been simulated numerically using finite difference approximations' as well as finite element methods of analysis for transient heat conduction.- ' ' The heat conduction mechanism for stationary GTA welds was studied extensively in an analysis and test program designed to assess the effects of both the magnitude and the distribution of heat input from the arc and of surface energy losses on penetration, weld bead width and temperatures for stationary GTA welds. ' The finite element procedures previously discussed were employed for the analytical part of this investigation. Implicit in all these efforts is the assumption that the molten weld puddle maintains a fixed shape as heat from the welding arc is being applied. Distortion of the weld puddle or weldment has not previously been considered in analyses of temperature transients and weld bead penetration and shape. The weld puddle is, however, Paper presented at the AWS 59th Annual Meeting held in New Orleans, Louisiana, during April 2-7, 978. E. FRIEDMAN is a Fellow Engineer in Reactor Technology, Bettis Atomic Power Laboratory, Westinghouse Electric Corporation, West Mifflin, Pennsylvania. locally distorted by gravitational forces and arc pressure" as well as by fluid flow induced by magnetic and convective forces in the puddle. The magnitude of this local distortion is most pronounced during the root pass of a full penetration weld and results, upon solidification, in the weld underbead. In the present work, the molten puddle is considered to be acted upon simultaneously by distributions of heat input from the arc, pressure from the arc and gravitational forces in the puddle. The effects of magnetic forces and convective motion are not addressed in the current treatment. The arc pressure and gravitational forces result in the top surface of the puddle being depressed below its original level prior to melting. If full penetration is achieved, the same forces cause the bottom surface of the puddle to be depressed. Surface tension at interfaces between the liquid puddle and the gaseous atmosphere serves to resist puddle distortion. For full penetration welds, the surface tension supports the molten material and prevents burn-through. The analytical model of weld puddle distortion considers a stationary arc heating the surface of a flat plate. The existence of a stationary arc implies that the distributions of both heat and pressure acting on the weldment surface are axisymmetric, thus simplifying the analysis. Extension of the model to moving welds would result in a non-axisymmetric puddle, which complicates the analysis procedure. This extension is unwarranted at this time since much information may be gathered on the effects of various welding and material variables on WELDING RESEARCH SUPPLEMENT I 6-s

2 weld bead penetration and distortion from calculations based on the simpler stationary arc model. The analytical model for weld puddle distortion is presented first, followed by the results of calculations made to determine the effects of arc pressure and puddle surface tension on weld bead shape and penetration, and on puddle distortion. Analytical Model i i i r -i i r k i r WELD PUDDLE r-axis The analytical model is based on the weld puddle deforming under the action of pressure from the welding arc and gravitational forces in the puddle only at temperatures exceeding the melting point. For purposes of calculating these local deformations (as opposed to distortion resulting, for example, from shrinkage strains), material at temperature less than the liquidus is taken to be rigid. At any given time during which heat is being applied by the stationary heat source, a given volume of the weldment material may be in the molten state. This volume may or may not fully penetrate the thickness of the weldment. Nevertheless, its shape is axisymmetric with respect to the center of the axisymmetric heat source. The molten puddle is acted upon by axisymmetric distributions of thermal energy, arc pressure and gravitational forces. The surface tension at the interface of the weld puddle with the surrounding atmosphere supports the molten material and prevents burn-through. Surface tension at the liquid-solid interface (see Fig. ) is not considered in this model. The new features added to an earlier heat transfer model'' are:. Simulation of pressure from the welding arc and gravitational forces in the puddle. 2. Distortion of the puddle due to these forces. 3. Resistance to puddle shape change by surface tension at puddle interfaces with atmosphere. 4. Heat transfer in the weldment as the puddle continually changes shape. Calculational Procedure The computational process, which is incremental in time, is an outgrowth of the nonlinear transient finite element heat conduction analysis previously carried out for stationary arc welds.'' Although heat from the arc is applied at a constant rate, diffusion of heat through the weldment is a timedependent phenomenon requiring calculations at a number of time steps. The present analysis procedure deviates from that used previously"' when weldment material melts. LIQUID-SOLID INTERFACE 0 r-axis (mm) Fig. Weld bead distortion as depicted by deformed finite element mesh At any time step during which the temperature at one or more nodes of the finite element mesh exceeds the specified liquidus temperature, the motion of these nodes due to arc pressure and gravity forces is determined. The nodes at temperatures less than the melting point are completely restrained from any motion. As illustrated in Fig., the calculated nodal displacements are employed to update the undistorted mesh geometry. The resulting distorted mesh is used for temperature calculations in the next time step. An outline of the required calculations is as follows:. Calculate nodal temperatures in time increment. 2. If all nodal temperatures are less than the melting point (liquidus), return to Step for the next time increment. 3. If any nodal temperature exceeds the melting point, calculate displacements of nodes in the puddle. 4. Add displacements to undeformed grid coordinates to obtain distorted mesh geometry. 5. Return to Step for the next time increment using the distorted geometry. 6. Continue until solution becomes unstable. (This occurs subsequent to full penetration when the undersurface sag becomes very large, indicating the onset of burn-through.) In general, calculations during an increment of time involve:. Determinations of temperatures throughout the weldment, and 2. Distortions in that part of the weldment in which temperatures exceed the melting point. The BESTRAN finite element system 7 is used to make these computations. The output of the temperature calculations is used to determine what region of the weldment is in the molten state and therefore subject to distortion. The results of the distortion calculations are utilized to calculate the distorted finite element mesh required for the next set of temperature calculations. Temperature Calculations The application of heat from the arc to the workpiece and the distorted geometry used for temperature calculations is discussed briefly. Detailed procedures for calculating welding temperatures have been described in the literature.' The heat input from the welding arc is applied as a radially symmetric normal distribution function: q(r) = [3Q/(OT 8 )] exp [-3(r/r H () where q is the heat flux acting on the weldment surface, Q is the strength of the heat source (that is, the product of the arc current, voltage drop and arc efficiency), r is the distance from the weld centerline (see Fig. ), and r h is a dimensional parameter defining the region in which 95% of the heat from the arc is deposited. Thus, Q and r h completely define the steady application of heat from the stationary arc. 62-s I IUNE 978

3 The heat supplied by the arc is conducted through the weldment in its distorted configuration. Under the action of arc pressure and gravitational forces, the r«ated surface of the weld puddle in the vicinity of the weld centerline is depressed below its original elevation, as shown in Fig.. It is this depression that may significantly affect the penetration characteristics of the weld. Since all calculations are carried out in increments of time, the weld puddle distortion progresses in discrete steps, rather than continuous- ly- Distortion Calculations Puddle distortions are computed by specifying mechanical properties of the weldment to simulate the behavior of a nearly incompressible liquid with negligible shear resistance. The compressibility of the puddle is taken to be that of water at 0 C or 50 F (50 X 0"/atm). This is equivalent to a bulk modulus K, of 2 GPa (290 ksi). Since a perfectly incompressible material, corresponding to a Poisson's ratio v of 0.5, would generate a singular stiffness matrix in the finite element program (BESTRAN), a small parameter a, is introduced such that v = 0.5 (-a). Letting the elastic modulus E = 3Ka, the bulk modulus K = E/[3(-2v)J is independent of a. Results of a test problem yielded reasonable solutions for a = 2 x 0" (see Appendix). Although the behavior of the entire weldment is characterized in this manner, those finite element nodes at temperatures less than the melting point are constrained to have zero displacement. Surface Tension Distortion of molten material under the action of arc pressure and gravitational forces is inhibited by the surface tension existing at the interface between puddle and atmosphere. At a given time, there may exist one or two of these interfaces, depending upon whether partial or full penetration has been attained. Surface tension is simulated in the finite element model by a membrane with uniform isotropic tension equal to the surface tension N. Consider a circular membrane of radius b, supported around its edge and subject to a normal pressure q(r), which may vary radially. The normal deflection w, of the membrane satisfies the equilibrium equation:' (/r) (d/dr) (r dw/dr) = -q/n, (2) provided d'-w/dr'- <<. Integrating this equation with the conditions (dw/dr),, = = 0 and w(b) = 0, the following is obtained for the deflection: w(r) = In (b/r) / ' (q/n) p dp -/," (q/n) In (p/r) pdp. (3) This equation is used to establish a relationship between the restraining forces of the membrane and the membrane deflections at a discrete number n, of location's (corresponding to finite element nodes) on the puddle surface. The deflection w i = w(ri) at a location defined by r = t tl due to a ring load P,, applied at a distance r,, from the center of the membrane, is found by letting the pressure distribution q(r) = [Pj/(27r Tj)] S(r tj), where S is the Dirac-delta function, and i,j =,2 n (r,,i > r,). Substituting this expression into equation (3), the following linear flexibility equation is obtained: where w, = Qj P (4) C = ln(b/r.) j<i (5) y = ln(b/r,) j > i. Equation (4) is generalized in matrix notation by: {w} - [C] {P}, where the elements of the symmetric flexibility matrix [C] are given by equation (5). A numerical problem arises when the ring load is applied at the center of the membrane. In the limit, this is equivalent to an infinite pressure applied at the center, which results in an infinite deflection at the center. This problem is circumvented by calculating the flexibility coefficient at node j = (r, =0) by linear extrapolation of the corresponding terms evaluated at r, and r,. The inverse of the flexibility matrix is the membrane stiffness matrix [Kj. The constraint effect of puddle surface tension is incorporated into the finite element model as an equivalent elastic foundation with stiffness [K. The surface tension simulation procedure was verified by formulating stiffness matrices for a circular membrane of unit radius. The membrane is acted upon by a uniform pressure q, which is discretized by a series of ring loads applied at a given number of equally spaced nodes. The numerical solution for deflection at each node was compared with the exact solution, given by w(r) = q (I r 2 )/(4N). Errors of less than 0.4% were obtained with 0 nodes used in the discretization. The use of more nodes reduces the error. Problems run to assess the accuracy of the puddle distortion computational procedure, which couples the effects of forces transmitted through molten metal and of surface tension, are described in the Appendix. Arc Pressure The pressure applied by the arc to the surface of the weldment is taken to be distributed as a radially symmetric normal distribution function, such that p(r) = p exp [-3(r/r f )! ] (6) where p is the pressure acting on the surface, p 0 is the maximum pressure at the weld centerline, r is the distance from the weld centerline, and r, is a dimensionless parameter defining the region in which 95% of the electromagnetically-induced arc force is applied. p and r, completely define the pressure distribution from the stationary arc. This is analogous to the steady distribution of heat applied to the weldment see equation (). Results of Analysis Since the purpose of this work is to assess the effects of arc pressure, gravity forces, and surface tension on penetration, weld bead dimensions, and puddle distortion, all computations were made for a heat input of fixed magnitude and distribution applied to the center of a 50.8 mm (2 in.) diameter, 6.35 mm (0.25 in.) thick circular Alloy 600 plate. The thermal input parameters used are Q = 000 W and r = 3.8 mm (0.5 in.). The magnitude of heat input is equivalent to an arc current of 25 A, voltage drop of 0 V, and arc efficiency of 0.8, which are consistent with parameters used previously." The effects of arc pressure and gravity forces on penetration and weld bead dimensions are illustrated by determining the growth of the weld puddle with time neglecting puddle distortion (i.e., neglecting arc pressure and gravity forces or assuming very large surface tension), and then comparing the results with those obtained using representative puddle distortion parameters. For this purpose, a reasonable estimate of the maximum arc pressure is taken to be 00 mm H,0, 6-9 or 000 Pa (0.5 psi). The arc pressure distribution parameter r f, is taken to be 3.8 mm (0.5 in.) (this is consistent with the limited number of measurements reported in the literature), and a surface tension of 000 dynes/cm, or N/m ( lb/in.) is used. The results plotted in Fig. 2 show that the distortion of the puddle contributes significantly to the penetration characteristics of the weld especially when the depth of penetration approaches the weldment thickness. When including the effects of puddle distortion due to arc pressure and gravity forces, full penetration occurs after 32 s of heating; neglecting puddle distortion, full penetration is not yet attained at 40 s. WELDING RESEARCH SUPPLEMENT I 63-s

4 I P0 500 Pa P0 000 Pa P0 =500 PO ' ~-""~ WIDTH AT TOP SURFACE FULL PENETRATION / ^ ^ *AAT7~~~/ / WIDTH AT / ~T''~' / A BOTTOM S ~. SURFACE 7 / / *' / / ' '' Fig. 2 Effect of puddle distortion on puddle dimensions. Maximum pressure = 000 Pa (0.5 psi); surface tension = N/m ( lb/in.) JT DEPTH OF PENETRATION 7 / ( if " Fig. 3 Effect ot arc pressure on puddle dimensions. Maximum arc pressure - p ; surface tension = I N/m ( lb/in.) MAXIMUM ARC PRESSURE(Pd) Fig. 4 Effect of arc pressure on time to lull penetration. Surface tension = I N/m ( lb/in.) The maximum arc pressure resulting from electromagnetic forces is proportional to the square of the current." It is thus of interest to determine the effects of arc pressure on the growth and penetration of the weld puddle. (Although an increase in current produces increases in both heat input and arc pressure, only the latter is considered here. The effects of heat input on penetration and weld bead shape are found elsewhere.') The results are presented in Fig. 3, in which the growth of the puddle dimensions with time are plotted for a number of values of the maximum arc pressure. Additional information appears in Fig. 4, which shows how the time to full penetration is influenced by the magnitude of the arc pressure. Figure 3 also demonstrates that arc pressure significantly influences weld penetration when the puddle penetrates to approximately 60% of the weldment thickness. For these penetrations, the depression of the heated surface of the puddle is of sufficient magnitude that 0 IOOO 2000 MAXIMUM ARC PRESSURE (Pa) fig. 5 Weld puddle distortion as a lune tion of arc pressure for various durations of heating. Duration of heating = t*; surface tension = I N/m ( lb/in.) the rate of further penetration depends strongly on the degree to which the surface is depressed. The effect on bead width (at the heated (top) 64-s I JUNE 978

5 i i 0 f 8 e z o S 6 a -i a H 4 2 * - '/ /, Y j /y N N - N/m = 0.5 N/ = 0. N/ ^ DEPTH ^ OF PENETRATION y * NUMERICAL SOLUTION BECOMES ILL- i BEHAVED I I I I r T i ^ - H - " " WIDTH AT TOP SURFACE FULL PENETRATION / / l^ ^ 7.'WIDTH AT BOTTOM / SURFACE y^ / / / i i I I i li I i S! _ Fig. 6 Effect of surface tension on puddle dimensions. Surface tension = N; maximum arc pressure = 7000 Pa (0.5 psi) Fig. 7 Top surface puddle depression as a function ol surface tension for various durations of heating. Duration of heating = t*; maximum arc pressure = 7000 Pa (0.5 psi) surface) is less pronounced, since the influence of puddle depression on the radial transfer of heat in the weldment is less than on the heat transfer through the thickness. Arc pressure enhances penetration because of its ability to depress the molten weld puddle, thus resulting in thermal energy from the arc applied at a level below that of the undistorted weldment surface. The shape of the distorted puddle is, however, of interest in its own right, since it contributes to the configuration of the solidified weld bead. This is an especially important effect for full penetration welds, as illustrated in Fig.. The variation of the magnitude of puddle depression at the weld centerline with arc pressure is shown in Fig. 5. Here the maximum depression of the puddle at both the top and, for full penetration welds, bottom surfaces are plotted against pressure at the weld centerline for a number of durations of heating. If the magnitude of weld depression at the top surface is less than 0% of the weldment thickness, depression varies linearly with arc pressure. For greater amounts of puddle distortion, energy from the arc is applied at a level sufficiently below that of the undistorted weldment such that the resulting depth of puddle penetration is significantly greater than the penetration that would have occurred had the puddle surface not been depressed. Since enhanced penetration is accompanied by increased weld puddle width (see Fig. 2) and, therefore, volume, and since it is only molten weld metal that is subject to distortion by arc pressure, the greater puddle volume that results from pressureinduced puddle distortion brings about a greater degree of distortion. An increase in arc pressure, therefore, produces both an increase in puddle distortion and an increased puddle volume. The latter, in turn, effects a further increase in puddle distortion, thus causing the deviation from linearity of the top surface puddle depression shown in Fig. 5. For sufficiently high magnitudes of puddle distortion (subsequent to full penetration), the rates of increase of puddle depression at both top and bottom surfaces are very sensitive to increases in pressure. The final parameter study performed was on the influence of surface tension on puddle dimensions and puddle distortion. Calculations were made for values of surface tension N =,0.5, and 0. N/m (0.0057,0.0029, and lb/in.) to illustrate the sensitivity of this physical property. Figure 6 shows the effect of surface tension on penetration when the puddle depth exceeds about 60% of the weldment thickness. Puddle width at the heated surface is relatively insensitive to changes in surface tension. For the case of N = 0. N/m ( lb/in.), the restraint forces introduced by the surface tension are so small that the puddle distorts so severely that an ill-behaved finite element solution results. Similar effects of low surface tension are shown in Fig. 7. For a very small exposure time of.6 s, the magnitude of puddle depression is extremely high when N = 0. N/m ( lb/in.). For the range 0.5 I.O SURFACE TENSION (N/m) 0.5<N<.0 and a rather shallow puddle (exposure times of.6 and 3 s), the depression of the puddle is inversely proportional to the surface tension. For higher exposure times (e.g., 26 s), enhanced puddle depression results in energy from the arc being applied at a level sufficiently below that of the undistorted weldment that the depth of penetration and volume of the puddle are significantly greater than they would be if the puddle were not depressed. Thus, a decrease in surface tension causes increases in both puddle depression and volume, with the latter effecting a further increase in depression, as explained previously. This phenomenon yields the more pronounced effect of surface tension on puddle depression that is observed in Fig. 7 for a heating time of 26 s. For values of surface tension much higher than N/m ( lb/in.), the restraint effect becomes very large and the limiting case of no distortion is approached. Conclusions The employment of an analytical model to simulate weld puddle distortion and sink due to electromagnetically induced arc pressure acting on the weld puddle and gravitational forces in the puddle has produced the following results and conclusions:. Distortion of the weld puddle contributes significantly to the penetration characteristics of the weld. For example, it is demonstrated that the propensity of a weld bead to fully penetrate the thickness of the weld- WELDING RESEARCH SUPPLEMENT I 65-s

6 ment is enhanced by weld puddle depression. 2. Arc current influences weld penetration from two standpoints: (a) increased current results in more heat input from the arc to the workpiece, causing more material to be melted; and (b) arc pressure increases with the square of the arc current, producing a greater degree of puddle depression. Application of heat at surfaces beneath the undistorted surface of the weld further enhances penetration. Penetration becomes sensitive to small changes in current when the depth of the puddle exceeds 60% of the weldment thickness. 3. For partial penetration welds, depression of the top surface of the weld puddle varies linearly with arc pressure. When full penetration is achieved, weld bead shape changes are significantly more sensitive to changes in pressure. 4. Both puddle depression and weld penetration are strongly dependent on surface tension at the puddle surfaces. The understanding and controlling of surface tension is essential to controlling weld puddle shape change, as well as enhancing penetration. 5. Efforts are required to measure and control pressure in the welding arc and to investigate surface tension in molten metals. For example, a determination of how minor elements affect surface tension'" would be a major step in controlling both puddle distortion and weld penetration. In addition, analytical and experimental work on fluid motion in the puddle need to be pursued. Acfcnow/edgmenl The numerous consultations with Dr. S. S. Glickstein are gratefully acknowledged by the author. References. Paley, Z and Hibbert, P. D., "Computation of Temperatures in Actual Weld Designs," We/ding journal, 54 (), Nov. 975, Research Suppl., pp. 385-s to 392-s. 2. Clickstein, S. S., Friedman, E., and Yeniscavich, W., "Investigation of Alloy 600 Welding Parameters," Welding lournal, 54 (4), April 975, Research Suppl., pp. 3-s to 22-s. 3. Hibbitt, H. D., and Marcal, P. V., "A Numerical Thermo-Mechanical Model for the Welding and Subsequent Cooling of a Fabricated Structure," Computers and Structures, Vol. 3, No. 5, September 973, pp Friedman, E., "Thermo-Mechanical Analysis of the Welding Process Using the Finite Element Method," Trans. ASME, I. Pressure Vessel Techn.. Vol. 97, August 975, pp Friedman, E., and Glickstein, S. S., "An Investigation of the Thermal Response of Stationary Gas Tungsten-Arc Welds," Welding lournal 55 (2), Dec. 976, Research Suppl., pp. 408-s to 420-s. 6. Schoeck, P. A., "An Investigation of the Anode Energy Balance of High Intensity Arcs in Argon," Modern Developments in Heat Transfer, Academic Press, New York, 963, pp o -io - -. I I KNOWN SOLUTION K = 2 GPa \ ( 7. Friedrich, C M., "BESTRAN-A Technique for Performing Structural Analyses," WAPD-TM-40, Bettis Atomic Power faboratory, February Timoshenko, S., and Woinowsky- Krieger, S., Theory ol Plates and Shells, 2nd Edition, McGraw Hill, New York, 959, Ch Stepanov, V. V., and Nechaev, V. I., "On the Pressure of the Plasma Arc," We/ding Production, 974, No., pp Glickstein, S. S., and Yeniscavich, W., "A Review of Minor Element Effects on the Welding Arc and Weld Penetration," Welding Research Council Bulletin No. 226, May 977. a = 2 X IO" 9 LOG K (GPa) - KNOWN SOLUTION Fig. 8 Effect of puddle parameters a and K on maximum deflection. Surface tension = N/m ( lb/in.) Appendix: Test Problem to Confirm Distortion Model A test problem that includes the effects of both surface tension and forces transmitted through molten material was run and compared with its known solution to ascertain the reliability of the analytical model. Consider the weldment configuration of Fig.. Let the molten region be subject only to gravitational forces. For a weldment 6.35 mm (0.25 in.) thick, this is equivalent to a uniform pressure q, acting on the bottom surface of the puddle. q is given by: q = P g h where p = density = 845 kg/m' (0.304 lb/in.'); g = unit gravitational force = N/kg ( Ibf/lbm); h = height of molten region = 6.35 mm (0.25 in.). Therefore, q = 524 Pa (0.076 psi) is the pressure applied to the circular membrane, which simulates the constraint effects of the surface tension, at the bottom surface of the weld puddle. The maximum deflection w(0), at the weld centerline is given by: w(0) = q b/(4n) where q = pressure = 524 Pa (0.076 psi); b = bottom surface puddle, or membrane, radius = 3.75 mm (0.25 in.); N = surface tension = I N/m ( lb/in.). This expression is derived by letting q = q = constant in equation (3), evaluating the integrals, and letting r = 0. The resulting deflection w(0) =.32 mm (0.052 in.), is compared with finite element solutions for various values of the parameter a, and the bulk modulus K. The finite element results are given in Fig. 8, which shows maximum deflection plotted against a for fixed K, and against K for fixed a. The values finally selected -a = 2 x 0 ", K = 2 GPa (290 ksi) yield a maximum underbead deflection of.02 mm (0.043 in.) which is 6% less than the known deflection. The bulk of this error is attributed to the coarse finite element mesh employed within the puddle region (see Fig. ). Consistent with the objectives of this investigation and the uncertainties of the various input parameters (arc pressure, surface tension, compressibility), a 6% discrepancy for the puddle distortions does not alter any of the results and conclusions. 66-s IUNE 978