CONTENT
VOLUME FIRST 
Mathematical modelling and information
technologies, model of a weld
pool 

INTRODUCTION  9  
Chapter 1  MATHEMATICAL
MODELING
and COMPUTER TECHNOLOGIES in
WELDING 

1.1.  Modern information technologies — major component in an advanced industry  20 
1.2  The mathematical models of welding processes and its application  30 
1.2.1.  The models for a mathematical support of CAD  32 
1.2.2.  The mathematical models for the expert systems  37 
1.2.3.  Models for the software of welding robots and robotic setups  42 
1.2.4.  Mathematical models in control systems of arc welding processes  49 
1.2.5.  A role of mathematical models in the automized systems of scientific researches  68 
Chapter 2  ANALYSIS of
INFORMATION STREAMS for TECHNOLOGICAL PROBLEM «WELD FORMATION DURING ARC WELDING» 
72 
2.1.  Main features and methods of investigation of information streams in scientific and technical literature on welding  73 
2.2.  The database for the publications in a scientific and technical area «Weld formation during arc welding»  81 
2.3.  Definition of an information core for the journals on welding  93 
Chapter 3  THE PHILOSOPHY and FEATURES of MATHEMATICAL MODELING the ARC WELDING PROCESSES  102 
3.1.  The system analysis — the main methodology of mathematical modelling the welding processes  102 
3.2.  The mathematical (computing) experiment  107 
3.3.  The characteristics and classification of the factors included in the mathematical models of objects and processes  118 
3.4.  Classification of the mathematical models of welding processes  128 
3.4.1.  Main types of models used for mathematical modelling of welding processes  131 
3.4.2. 
The regression mathematical models of welding processes and features of their usage 
138 
3.4.2.  The
neuronet mathematical models and their application for
simulation of welding processes 
153 
3.5.  Main types of
theoretical mathematical models of weld pool (weld formation) for fusion welding 
163 
3.5.1.  Capillaryhydrostatic models and their usage for modeling of a weld shape  169 
3.5.2. 
Volumetric thermal capillaryhydrostatic models (VTCHM)
and their application for simulation of weld formation 
184 
3.5.3. 
Thermal magnetohydrodynamic models (TMHDM)
and their usage for modelling of weld pool 
198 
3.5.4.  Mathematical models for oscillations of molten metal in a weld pool  205 
3.5.5.  Main features of models TMHDM and its application for simulation of a weld pool  211 
3.6.  Adequacy of the mathematical models  222 
3.7.  Optimization
of technological processes of welding by means of application the mathematical models 
225 
3.7.1.  Optimizations with the regression models and multifactor experiment design  231 
3.7.2.  Optimization by a design of experiments on the Taguchi method  232 
3.7.3.  The specialized problems of optimising  237 
3.7.4.  Peculiarities of optimization the technological processes of welding  239 
3.7.5.  The sinergetic approach to simulation of welding processes  244 
Chapter 4  SURFACE
TENSION PHENOMENA and ITS ROLE and SIGNIFICANCE for THE WELDING 
252 
4.1. 
EFFECT of SURFACE TENSION FORCES on WELD FORMATION DURING ARC WELDING 
259 
4.2. 
EFFECT of SURFACE TENSION PHENOMENA on WELDED METAL
PENETRATION 
263 
4.3.  THE THEORY of CAPILLARITY, CAPILLAR EFFECTS and SURFACE TENSION PROPERTIES of LIQUID METALS in the TECHNOLOGICAL PROCESSES  278 
4.3.1.  Fundamentals of the capillarity theory  278 
4.3.2.  Main mathematical models of the capillarity theory  291 
4.3.2.1.  The mathematical model of a sessile drop  292 
4.3.2.2.  The mathematical model of a pendant drop  302 
4.4.  THE VARIATION METHODS and THEIR USAGE for PROBLEM SOLVING in THE THEORY of capillarity  
4.4.1.  Advantages and features of the variation methods application  304 
4.4.2.  Application of variation methods for definition the equilibrium shape of capillary liquids surfaces  307 
4.5.  SURFACE (INTERPHASE) TENSION and CAPILLARY CONSTANT VALUE of LIQUID METALS in WELDING SYSTEMS  311 
4.6.  THE STATIC and DYNAMIC CAPILLARY EFFECTS 
316 
4.6.1.  Static capillary effects  316 
4.6.2.  Static capillary effects  318 
4.7. 
CAPILLARYHYDRODYNAMIC
INSTABILITIES of LIQUID BOUNDARY SURFACES 
332 
4.7.1.  RayleighTaylor INSTABILITY  334 
4.7.2.  KelvinHelmholtz INSTABILITY  340 
4.7.3.  RichtmyerMeshkov INSTABILITY  344 
4.8. 
MARANGONI EFFECT
and the MATHEMATICAL MODELS of
THERMAL CAPILLARY FLOWS in 
348 
4.8.1.  Mathematical models of a Marangoni convection for simple regions  348 
4.8.2. 
Mathematical models of a Marangoni convection for
regions with complex forms with interphase boundaries moving 
359 
Chapter 5  MATHEMATICAL MODELLING of WELD FORMATION in a FLAT POSITION  374 
5.1.  General formulations of the task, main allowances and simplifications  376 
5.1.1.  Design of a conceptual model for a weld pool on the analysis of acting forces basis  378 
5.1.2.  Physical and mathematical models of a weld pool crystallization zone  382 
5.1.3.  Derivations of an equilibrium equation for an interphase surfaces of a crystallization zone of welding pools by a variationenergy method  386 
5.1.4.  Features of processes of wetting and spreading of molten metal at arc welding  393 
5.2. 
A mathematical model of WELD REINFORCEMENT FORMATION (BEADONPLATE)
DURING welding or surfacing 
403 
5.2.1.  Formulations of the boundary task  403 
5.2.2.  Integration of the differential equilibrium equation for a surface of weld pool tail zone  407 
5.3.  LinkS of physical and geometrical parameters of mathematical models with technological parameters of WELDING CONDITIONS  419 
5.3.1.  Technological methods of control of weld metal deposits and weld width and methods them calculation on parameters of welding conditions  419 
5.3.2.  Application of a mathematical model for different types of welds  435 
Chapter 6  EFFECT of WELDING POSITIONS on WELD FORMATION  438 
6.1.  Mathematical modelling and WELD FORMATION optimiSing for weld reinforcement in a ceiling position  440 
6.1.1.  Technological features of application for arc welding in a ceiling position  440 
6.1.2.  Physical and mathematical models of the welding process in ceiling position  445 
6.1.3.  Definition of stability range for ceiling weld formation  453 
6.1.4.  Experimental check of mathematical model for a ceiling weld formation  462 
6.2.  Mathematical modelling and optimization for weld formation during arc welding in DIFFERENT positions  467 
6.2.1.  Technological features of weld formation during arc welding in different positions  467 
6.2.2.  Features of simulation of creation of a seam at welding in different positions  468 
6.3.  Mathematical modelling of horizontal welds formation on a inclined plane  473 
6.4.  A mathematical model of horizontal welds formation on a vertical plane  487 
6.4.1.  Technological features of welding execution on a vertical plane  487 
6.4.2.  Mathematical models of horizontal weld formation on a vertical plane  492 
6.4.3.  Experimental checking of mathematical models  496 
ENCLOSURES  508  
REFERENCES for VOLUME 1  521  

Mathematical modeling and optimization of different types welds formation 

Chapter 7 
MATHEMATICAL MODELING and STRUCTURAL TECHNOLOGICAL OPTIMIZATION of FILLET and GALTEL WELDS FORMATION  8 
7.1.  THE FEATURES of APPLICATION of FILLET WELDS in WELDED STRUCTURES  8 
7.1.1.  Varieties and configurations of fillet welds  9 
7.1.2.  Principal dimensions of fillet welds and their choice  18 
7.1.3.  Effect of geometrical parameters of welded joints with fillet welds on their strength  28 
7.1.4.  Optimization of geometry of fillet welds by a criterion of decrease of stress concentration factor in welded joints  36 
7.1.5.  Mathematical modelling and optimization of fillet welds formation during welding by a way «in a corner»  51 
7.1.6.  Technological methods of control for the fillet welds shape  67 
7.2.  MATHEMATICAL MODELLING and OPTIMIZATION of FLUTE WELDS FORMATION  75 
7.2.1.  Effect of geometrical parameters of welded joints on their strength characteristics  75 
7.2.2.  The purposes and features of application the flute welds for rise of serviceability welded joints and structures  80 
7.2.3.  The technological features of flute welds execution  83 
7.2.4.  A mathematical model of flute welds formation for optimization their geometrical parameters  97 
7.2.5.  Optimization of flute welds formation by a criterion of the stress concentration factor lowering  103 
Chapter 8 
MATHEMATICAL MODELLING and OPTIMIZATION of WELD FORMATION DURING ARC WELDING of a THIN METAL WITH BURNING THROUGH and ROOT WELDS WITH FREE FORMATION  109 
8.1.  PROBLEMS of OPTIMIZATION for THIN METAL WELDING PROCEDURES  109 
8.1.1.  The technological features of through welds execution  116 
8.1.2.  The technological features of root welds execution  125 
8.2.  MATHEMATICAL MODELS of THROUGH WELDS AND ROOT ONES WITH FREE FORMATION  139 
8.2.1.  The mathematical models on the basis of pressure balance  142 
8.2.2.  Twodimensional capillaryhydrostatic models of through welds formation  147 
8.2.3.  The mathematical models on the basis of concentrated forces balance  153 
8.3.  TWODIMENSIONAL CAPILLARYHYDROSTATIC MODEL of THROUGH WELD FORMATION  157 
8.3.1.  Optimization of weld formation during welding with burning through on sizes of backing weld  168 
8.3.2.  Application of the mathematical model for the analysis of weld formation
during arc welding 
172 
8.3.3.  Usage of volumetric thermal capillarhydrostatic models of weld pools 
177 
8.4.  TECHNOLOGICAL WAYS of IMPROVEMENT for WELD FORMATION DURING ARC WELDING WITH BURNING THROUGH  179 
8.5.  STABILITY of a WELD POOL SURFACES AT WELDING WITH BURNING THROUGH and FREE FORMATION  204 
Chapter 9 
OPTIMIZATION of WELD FORMATION for a BUTT WELD DURING ARC ORBITAL WELDING of TUBES  210 
9.1.  TECHNOLOGICAL FEATURES of ORBITAL WELDING of TUBES  214 
9.1.1.  Mathematical models of weld formation during arc orbital welding  236 
9.1.2.  An analytical solution of the volumetric task for definition of the
surface form a fluid phase at fusion welding 
250 
9.2.  TECHNOLOGICAL
METHODS of OPTIMIZATION of WELD FORMATION DURING ARC WELDING 
262 
Chapter 10 
MATHEMATICAL MODELLING and OPTIMIZATION of WELD FORMATION DURING MULTIPASS ARC WELDING of a THICK METAL  273 
10.1.  MAIN
TECHNOLOGICAL FEATURES and PROBLEMS of MULTIPASS WELDING 
274 
10.1.1.  Process control of automatic multipass arc welding  285 
10.1.2.  Choice of welding conditions and optimization the process of multipass arc welding of a thick metal  295 
10.2.  MATHEMATICAL MODELS of METAL LAYER FORMATION DURING ARC SURFACING  306 
10.2.1.  The empirical and regression mathematical models for optimization of
layer formation 
308 
10.2.2.  An analytical mathematical model of layer metal formation  310 
10.3.  MATHEMATICAL MODELLING and OPTIMIZATION of WELD FORMATION for NARROW GAP WELDING  320 
10.3.1.  Basic versions of the technology of multipass narrow gap welding  320 
10.3.2.  Features of weld formation during narrow gap welding  333 
10.3.3.  Mathematical modelling and optimization of weld formation narrow gap welds  337 
10.3.4.  Mathematical model of the process of weld formation on scheme «one bead in 
339 
10.3.5.  Experimental researches of weld formation during narrow gap welding  358 
10.3.6.  A mathematical model of the process of bead formation in narrow gap
welding under the technological schemes 
362 
Chapter 11  MAIN PECULIARITIES and MATHEMATICAL MODELLING of METAL PENETRATION at ARC WELDING  370 
11.1.  Main features of welded metal penetration during arc welding  371 
11.1.1.  Effect of welding parameters and conditions on metal penetration  372 
11.1.2.  Dependence of metal penetration with geometrical parameters of edges preparation  389 
11.1.3.  Effect of chemical composition of welded
metal, welding materials (fluxes,
shielding gases 
400 
11.2.  WAYS of CONTROL for the FORM and SIZES of a PENETRATION ZONE at ARC WELDING  408 
11.2.1.  Using the activating fluxes  408 
11.2.2.  Using of gas mixtures with activating additions  427 
11.2.3.  The analysis of reasons of rise of penetration depth with activating welding materials usage  430 
11.2.4.  Technological ways of rise penetrated ability of welding arc and control of the penetration form at arc welding  433 
11.3.  THE METHODS of SIMULATION and CALCULATION of the PENETRATION FORM and SIZES AT ARC WELDING  447 
11.3.1.  Empirical models of metal penetration  448 
11.3.2.  Empirical computational methods of parameters of penetration zone on the welding conditions  453 
11.3.3.  Models of metal penetration on the basis of the theory of a thermal conduction in solids  457 
11.3.4.  Mathematical models of weld pool crater surface  465 
11.3.4.1.  Models of derivation of crater form on the basis of acting forces balance and pressures  466 
11.3.4.2.  Models of derivation of crater form on the basis of solution of a
differential equilibrium equation 
468 
11.4.  CONCEPTUAL, PHYSICAL
and MATHEMATICAL MODELS of CRATER PART 
480 
11.4.1.  A mathematical model crater zone
of weld pool 
483 
11.4.1.1.  An equilibrium equation of crater surface  483 
11.4.1.2.  Solution of a differential equilibrium equation for crater surface of weld pool  489 
11.4.2.  A numerical solution of a nonlinear differential equilibrium equation for a crater surface  493 
11.4.3.  Effect of filler material and edge preparation on
the form and sizes of a crater of weld pool 
504 
11.5.  AN EXPERIMENTAL RESEARCH of EFFECT of WELDING PARAMETERS of the FORM and SIZES of CRATER and THICKNESS of a LIQUID INTERLAYER UNDER THE ARC  512 
11.5.1.  Present methods of experimental definition of depth and form of a weld 
512 
11.5.2.  An experimental research of effect of parameters of the mode of welding
on depth of a crater and width 
521 
ENCLOSURES  533  
REFERENCES for VOLUME 2  547  
Arc pressure, the defects of the welds, electrode metal transfer 

Chapter 12  FORCE ACTION of an ELECTRIC ARC ON WELDED METAL  7 
12.1.  The mechanism of arc presuure on welded metal origin  8 
12.2.  Methods of experimental definition for the arc pressure characteristics  13 
12.1.1.  Application of a weight method for measurement of arc pressure integral force of  17 
12.1.2.  Effect type of an arc and main welding conditions on integral force of arc pressure  54 
12.1.3.  Application of a manometric method for measurement of arc pressure distribution  67 
12.1.4.  Effect of external magnetic fields on arc pressure distribution  106 
12.1.5.  The characteristics of pressure for electric arcs used in special electrometallurgy  13 
12.2.  Methods of arc pressure characteristics calculations for welding arc on the basis of its mathematical models  116 
12.3.  Calculation of integral values of volumetric electrodynamic forces in a welded workpiece  132 
Chapter 13  DEFECTS of WELDS and MODELS of THEIR ORIGIN  145 
13.1.  Undercuts of welds and character of their origin  149 
13.1.1.  Definition of undercuts as defects of weld formation and their classification  152 
13.1.2.  Effect of undercuts on strength of welded joints and constructions  154 
13.2.  Main technological factors influential in origin of undercuts  156 
13.3.  Present models of undercuts origin in the welds  171 
13.3.1.  Models of undercuts origin during arc welding  172 
13.4.  The analysis of reasons of undercuts appearance  201 
13.5.  The technological methods of undercuts preventing in the welds  205 
13.6.  Unmeltings, reasons of their appearance and methods of preventing  222 
13.7.  Gas and slag inclusiongs in the welds  226 
13.7.1.  Technological features and forms of appearance of gas concavities in welds  227 
13.7.2.  Models and mechanisms of gas concavities origin in welds  232 
13.7.3.  Methods of identification and elimination of gas concavities in welds at welding in shielding gases  242 
13.8.  Origin of slag concavities at a submergedarc welding  247 
13.8.1.  Effect of technology factors on derivation of slag concavities and inclusions in the welds  248 
13.8.2.  Models of derivation of slag concavities  251 
13.8.3.  Ways of slag concavities preventing in the welds  257 
13.9.  Nonuniformity of weld formation, reasons of appearance and methods of elimination  258 
13.9.1.  Nonuniformity of edge weld formation at welding of thin metal  265 
13.9.2.  Models of edge weld formation during welding thin metal  265 
13.9.2.  Models of edge weld formation edge at welding thin metal  270 
13.9.3.  Optimization of parameters for edge welded joints and welding conditions  275 
Chapter 14  ELECTRODE METAL TRANSFER in ARC WELDING: MAIN FEATURES, the MATHEMATICAL MODELS and CONTROL METHODS  279 
14.1.  ROLE and SIGNIFICANCE of METAL TRANSFER TYPES for ARC WELDING PROCESSES  279 
14.2.  CLASSIFICATION of METAL TRANSFER TYPES  284 
14.3.  DROP TRANSFER of ELECTRODE METAL and EFFECT of WELDING PARAMETERS upon the TYPES of METAL TRANSFER  289 
14.3.1.  Stream metal transfer and its features  297 
14.3.1.1.  Effect of the different factors on the critical current of transition to stream transfer  298 
14.3.1.2.  Calculating of the critical current  306 
14.3.2.  Peculiarities of drop transfer during of consumable electrode welding  307 
14.3.3.  Features of metal transfer during pulsearc welding  315 
14.3.4.  Effect of external magnetic fields upon the metal transfer and spatter at arc welding  318 
14.4.  SPATTER and LOSSES of ELECTRODE METAL and FACTORS INFLUENTIAL on it  326 
14.5.  MATHEMATICAL MODELING of the DROP METAL TRANSFER  337 
14.5.1.  Static Force Balance Method (SFBM)  339 
14.5.1.1.  Forces acting on a drop of electrode metal  339 
14.5.2.  Method of surface pressures balance on a drop  355 
14.5.3.  Models of Pinch Instability Theory (PIT)  371 
14.5.4.  Variationalenergetical methods of drop shape modeling  379 
14.5.5.  Thermal hydrodynamic models of drop growth on the edge of an electrode  384 
14.6.  MATHEMATICAL MODELS of METAL TRANSFER with SYSTEMATIC SHORTCIRCUITS of an ARC GAP  392 
14.6.1.  Technological features of welding with the shortcircuits of an arc gap  394 
14.6.2.  Model of surface tensional forces acting during drop bridge breakup  399 
14.7.  METHODS of SPATTER LOWERING and METAL TRANSFER CONTROL  418 
ADDENDUM  430  
REFERENCES  434 
This monograph is written in Russian language