Volume 20 - Materials Selection and Design Part 12 pdf

•

Steel Castings Handbook, 6th ed., ASM International, 1995


Deformation Processes
•

T. Altan, S I. Oh, and H. Gegel, Metal Forming: Fundamentals and Applications,
American Society
for Metals, 1983
•

T.Z. Blazynski, Ed., Plasticity and Modern Metal Forming Technology, Elsevier, 1989
•

T.G. Byrer, Ed., Forging Handbook, Forging Industry Association, Cleveland, 1985
•

Forming, Vol 2, Tool and Manufacturing Engineers Handbook,
4th ed., Society of Manufacturing
Engineers, 1984
•

Forming and Forging, Vol 14, ASM Handbook, ASM International, 1988
•

S.K. Ghosh and M. Predeleanu, Ed., Materials Processing Defects, Elsevier, 1995
•

W.F. Hosford and R.M. Caddell, Metal Forming: Mechanics and Metallurgy,

2nd ed., Prentice Hall,
1993
•

K. Lange, Ed., Handbook of Metal Forming, McGraw-Hill, 1985 (now SME)
•

O.D. Lascoe, Handbook of Fabrication Processes, ASM International, 1988
•

Z. Marciniak and J.L. Duncan, The Mechanics of Sheet Metal Forming, Edward Arnold, 1992
•

E. Mielnik, Metalworking Science and Engineering, McGraw-Hill, 1991
•

R. Pearce, Sheet Metal Forming, Adam Hilger, 1991
•

J.A. Schey, Tribology in Metalworking: Friction, Lubrication and Wear,
American Society for
Metals, 1983
•

D.A. Smith, Ed., Die Design Handbook, 3rd ed., Society of Manufacturing Engineers, 1990
•

R.H. Wagoner, K.S. Chan, and S.P. Keeler, Ed., Forming Limit Diagrams,
TMS, Warrendale, PA,
1989

•

R.A. Walsh, Machining and Metalworking Handbook, McGraw-Hill, 1994


Powder Processing
•

H.V. Atkinson and B.A. Rickinson, Hot Isostatic Pressing, Adam Hilger, 1991
•

G. Dowson, Powder Metallurgy: The Process and its Products, Adam Hilger, 1990
•

R.M. German, Powder Metallurgy Science, Metal Powder Industries Federation, 1985
•

C. Iliescu, Cold-Pressing Technology, Elsevier, 1990
•

H.A. Kuhn and B.L. Ferguson, Powder Forging, Metal Powder Industries Federation, 1990
•

M.H. Liebermann, Rapidly Solidified Alloys, Dekker, 1993
•

Powder Metallurgy, Vol 7, ASM Handbook, American Society for Metals, 1984
•

Powder Metallurgy Design Manual, 2nd ed., Metal Powder Industries Federation, 1995

Machining Processes
•

G. Boothroyd and W.W. Knight, Fundamentals of Machining and Machine Tools, 2nd ed., Dekker, 1989
•

Machining, Vol 1, Tool and Manufacturing Engineers Handbook,
4th ed., Society of Manufacturing
Engineers, 1983
•

Machining, Vol 16, ASM Handbook, ASM International, 1989
•

S. Malkin, Grinding Technology: Theory and Applications, Ellis Horwood, 1989
•

P.L.B. Oxley, The Mechanics of Machining, Ellis Horwood, 1989
•

M.C. Shaw, Metal Cutting Principles, 4th ed., Oxford University Press, 1984
•

D.A. Stephenson and J.S. Agapiov, Metal Cutting Theory and Practice, Dekker, 1996
•

R.A. Walsh, Machining and Metalworking Handbook, McGraw-Hill, 1994

Joining Processes
•

Adhesives and Sealants, Vol 3, Engineered Materials Handbook, ASM International, 1990
•

Brazing Handbook, 4th ed., American Welding Society, 1991
•

G. Humpston and D.M. Jacobson, Principles of Soldering and Brazing, ASM International, 1993
•

D.L. Olson, R. Dixon, and A.L. Liby, Ed., Welding Theory and Practice, North Holland, 1990
•

R.O. Parmley, Ed., Standard Handbook of Fastening and Joining, 3rd ed., McGraw-Hill, 1997
•

A. Rahn, The Basics of Soldering, Wiley, 1993
•

M. Schwartz, Brazing, ASM International, 1987
•

Welding, Brazing, and Soldering, Vol 6, ASM Handbook, ASM International, 1993
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Welding Handbook, 8th ed., American Welding Society, 1996

Ceramics Processing
•

Ceramics and Glasses, Vol 4, Engineered Materials Handbook, ASM International, 1991
•

Engineered Materials Handbook Desk Edition, ASM International, 1995
•

S. Musikant, What Every Engineer Should Know about Ceramics, Dekker, 1991
•

G.C. Phillips, A Concise Introduction to Ceramics, Van Nostrand-Rheinhold, 1991
•

J.S. Reed, Principles of Ceramics Processing, 2nd ed., Wiley, 1995
•

M.M. Schwartz, Ceramic Joining, ASM International, 1993
•

M.M. Schwartz, Handbook of Structural Ceramics, McGraw-Hill, 1992
•

R.A. Terpstra, P.P.A.C. Pex, and A.H. DeVries, Ed., Ceramic Processing, Chapman & Hall, 1995


Polymer Processing
•

R.J. Crawford, Ed., Rotational Moulding of Plastics, Wiley, 1992
•

Engineering Plastics, Vol 2, Engineered Materials Handbook, ASM International, 1988
•

R.G. Griskey, Polymer Process Engineering, Chapman & Hall, 1995
•

Handbook of Plastics Joining: A Practical Guide, Plastics Design Library, 1996
•

N.C. Lee, Ed., Plastic Blow Molding Handbook, Van Nostrand-Rheinhold, 1990
•

N.G. McCrum, C.P. Buckley, and C.B. Bucknall, Principles of Polymer Enginering,
Oxford University
Press, 1989
•

E.A. Muccio, Plastics Processing Technology, ASM International, 1994
•

Plastic Parts Manufacturing, Vol 8, Tool and Manufacturing Engineers Handbook,
Society of
Manufacturing Engineers, 1995
•

R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Tests for Design,

Hanser,
Munich, 1993
•

G.W. Pye, Injection Mold Design, Longman/Wiley, 1989
•

D.V. Rosato, D.P. DiMattia, and D.V. Rosato, Designing with Plastics and Composites: A Handbook,
Van
Nostrand-Rheinhold, 1991


Manufacture of Composites
•

Composites, Vol 1, Engineered Materials Handbook, ASM International, 1987
•

L. Hollaway, Handbook of Polymer Composites for Engineers, Woodhead, Cambridge, 1994
•

B.Z. Jang, Advanced Polymer Composites: Principles and Applications, ASM International, 1994
•

M.M. Schwartz, Composite Materials Handbook, McGraw-Hill, 1992
•

M.M. Schwartz, Handbook of Composite Ceramics, McGraw-Hill, 1992
•

M.M. Schwartz, Joining of Composite Matrix Materials, ASM International, 1994
•

W.A. Woishnis, Ed., Engineering Plastics and Composites, 2nd ed., ASM International, 1993


Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Introduction
MANUFACTURING PROCESSES typically involve the reshaping of materials from one form to another under a set of
processing conditions. To minimize the production cost and shorten the time to market for the product, all iterations in
terms of an appropriate set of operating conditions should not be done on the shop floor. Predictive models need to be
used generously to perform numerical experiments to give an insight into the effect of the operating conditions on the
properties of the final product.
Process models must be able to build the geometry of the product/process that is being modeled, accurately describe the
physics of the process, and be able to analyze the results in a way that is comprehensible by manufacturing engineers.
There are several types of models that are used by the industry. These include models used as a tool in the course of
scientific research, models that are very generalized in nature and can be applied to a wide variety of processes, but may
not be able to address the nuances of any one process, models that are very specific in nature and can address a narrow
range of operating conditions, models that rely on gross phenomena and are about 90% accurate but take only 10% of the
execution time or more accurate models. Irrespective of the complexity, most models are used to gain one or more of the
following advantages:
• Reduce iterations on the shop floor
• Optimize an existing process
• Understand an existing process better
• Develop a new process
• Improve quality by reducing the variability in the product and process
Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Classification of Models
The following points need to be considered in order to classify modeling problems:
• The physical phenomena affecting the process under consideration
• Mathematical equations describing the physical process
• Data needed to solve the equations
• Numerical algorithm to solve the equations given the boundary conditions and the constitutive behavior

• Availability of the software to provide answers
One of the most common classification methods is by type of process or physics. This means that one must identify the
major phenomena occurring in the process, for example, convection, radiation, chemical reaction, diffusion, deformation,
and so forth. Once the phenomena has been identified, the process needs to be defined in terms of mathematical
equations, typically partial differential equations. These equations are dependent on time, space, field variables, and
internal states. Ordinary differential equations can be used if the problem can be simplified so that the shape is not
important and a lumped-parameter model can be used. Several people have used lumped-parameter models for various
materials processes (Ref 1, 2).
The requirements for particular data and the way in which it is gathered is an important step in the construction of a
model. Researchers typically play down this step as an "industrial implementation detail." This means that the rest of the
model needs to be robust and accurate before data are needed. Once accurate data are available, the result of the modeling
effort will be good too. On the other hand, industrial practitioners place a greater emphasis on data gathering because they
know the difficulties and time involved in gathering data on production-scale equipment.
Numerical algorithms to solve the differential equations consist of meshed-solution methods and lumped-parameter
models. The major meshed-solution models consist of finite differences, finite elements, and boundary methods. Each of
them is more appropriate for different types of equations and boundary conditions. Within these methods, one can use a
structured or an unstructured mesh. Structured meshes are created by using rectilinear, bricklike elements. It is easy to use
this type of mesh; however, fine geometry details may be missed. Unstructured meshes can be of any shape tetrahedra,
bricks, hexahedral, prisms, and so forth. Many of the disadvantages of using a structured mesh are eliminated through this
type of a mesh.
Lumped-parameter models may help in understanding the effect of certain parameters on the process as along as the
problem formulation does not change. These models do not model spatial variation directly, and the parameters may or

may not be physically meaningful in themselves.
Choice of the appropriate software is an important aspect of the usefulness of the model. Almost without exception,
research process models and all commercial software can be written directly in a third-generation language (Fortran, Lisp,
Pascal, C, C++). User interfaces can be derived from various libraries. Because of the large number of calculations
necessary to get the desired degree of detail, models may require parallel computing hardware for cost-effective solutions.
Software developed for this has to be able to run and make use of the parallel-processing capabilities of the hardware.
Models for manufacturing processes can be classified in two primary ways, as shown in Fig. 1. One classification scheme
considers whether the model is on-line or off-line; the other considers whether the model is empirical, mechanistic, or
deterministic.

Fig. 1 Classification of models for manufacturing processes

Fully on-line models are part of the bigger process control system in a plant. Sensors and feedback loops are
characteristics of these models. They get their input directly from the system. These models implement changes in the
plant on a continuous basis (Ref 3). Fully on-line models are extremely fast and reliable; therefore, these models need to
be rather simple without the need to do any significant numerical calculations. For these models to be reliable, the physics
of the process that they are addressing must be understood thoroughly. A good example of this type of model is the spray
water system on a slab-casting machine, which is designed to deliver the same total amount of water to each portion of
the strand surface. The flow rate changes to account for variations in the casting-speed history experienced by each
portion as it passes through the spray zones.
Semi-on-line models are similar to fully on-line models. The distinguishing factor is that rather than the model taking
appropriate action, the process engineer analyzes the situation and performs any corrective action, if needed. These
models are typically slightly more complex than the fully on-line models because they are essentially run by an operator.
These models need to have an excellent user interface and should require minimum user intervention.
Off-line models are typically used in the premanufacturing stage, that is, during research, design, or process parameter
determination. These models help to gain an insight into the process itself and thereby help optimize it. There are many
general-purpose models as well as models designed to be used for very specific applications. These models are typically
very complex and therefore need to be validated thoroughly before being used for any real predictions.
Literature models are those that exist primarily in the literature and are seldom used in conjunction with experiments.
Typically these models are developed and run by the same individual. The advantage of literature models is that other

developers can benefit from them instead of starting from scratch.
Empirical models are developed through statistical data gathering on a number of similar events. The model does not
help one understand the process itself and may not be valid beyond the range of the available data.
Mechanistic models are based on the solution of the mathematical equations that represent the physics of the process
that is being addressed. These models are very extensible and study the effect of a variety of external factors on the
process.
Neural network models are based on artificial neural networks and provide a range of powerful techniques for
solving problems in pattern recognition, data analysis, and control. Neural networks represent a complex, trainable,
nonlinear transfer function between inputs and outputs. This allows an effective solution to be found to complex,
nonlinear problems such as heat distribution.

References cited in this section
1.

M.F. Ashby, Physical Modeling of Materials Problems, Mater. Sci. Technol., Vol 8 (No. 2), 1992, p 102-111

2.

H.R. Shercliff and M.F. Ashby, Modeling Thermal Processing of Al Alloys, Mater. Sci. Technol.,
Vol 7 (No.
1), 1991, p 85-88
3.

B.G. Thomas, Comments on the Industrial Application of Process Models,
Materials Processing in the
Computer Age II, V.R. Voller, S.P. Marsh, and N. El-
Kaddah, Ed., The Minerals, Metals & Materials
Society, 1995, p 3-19
Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Important Aspects of Modeling
There are several important issues that need to be addressed to understand what is important in modeling in general and
what is important to the current problem in particular. Some of these are briefly discussed below.
Analytical versus Meshed Models. Developing an analytical or a closed-form solution model may be advantageous
in many instances. However, it may be necessary to construct a discrete meshed model for finite element or finite
difference calculations if the modeled volume:
• Has a complex shape (commonly found in many engineering applications)
• Contains different phases and grains, which are typically modeled by research groups (Ref 4)
•
Contains discontinuous behavior such as a phase change, which can be handled easily with meshes
using a volume-of-fluid (VOF) technique (Ref 5)
•
Has nonlinear process physics such as when the heat transfer coefficient is a nonlinear function of the
temperature
In many instances, meshed models are supplemented by some nonmeshed symbolic or analytical modeling. This is done
in order to decide on appropriate boundary conditions for the meshed part of the problem because it is the boundary
conditions that effectively model the physical problem and control the form of the final solution (Ref 6).
Analytic models are always useful for distinguishing between mechanisms that have to be modeled as a coupled set and
mechanisms that can be modeled separately. These models no longer need closed-form solutions. Even simple computers
can track evolving solutions and iterate to find solutions to implicit formulations (Ref 1).
Boundary Conditions and Multiphysics Models. Application of appropriate boundary conditions is a major part of
the activity of process modeling. Boundary conditions are statements of symmetry, continuity and constancy of
temperature, heat flux, strain stress, and so forth. Boundary conditions need to be set at a very early stage in analytical
models. In meshed models, these are typically represented separate from the main equations and are decoupled to some
extent from the model itself. Therefore, sensitivity analysis can be done much easier using meshed methods.
The type of boundary conditions used also determines what solving algorithm should be used for the partial differential
equations. This determines the speed, accuracy, and robustness of the solution.
Material Properties. All process models require material properties to be simulated. Acquiring these properties can be
difficult and expensive (Ref 7). A sensitivity analysis of the model with respect to these data provides information as to

the importance of minor changes in them. In many instances, it may be possible to use models with doubtful material
property information in order to predict trends, as opposed to determining actual values. A problem arises if the material
properties are extrapolated beyond the range of their applicability where one does not know the behavior of the material at
all. Related information is provided in the articles "Computer-Aided Materials Selection" and "Sources of Materials
Properties Data and Information" in this Volume.
Modeling Process Cycles. The process of modeling is done in different cycles. Figure 2 attempts to distinguish those
cycles (Ref 8). This figure shows three loops suggesting three levels of activities in any modeling effort. The outer loop is
managed by someone close to the process who understands the business context of the problem and can concentrate on
specifying the objective and providing the raw data. The innermost loop (shaded dark) requires mostly computational
skills while the middle loop (shaded light) consists of activities balancing the other two. It may very well happen that all
three of some combination of the activities can be done by the same person. However, very seldom is that the case. This
highlights the need for forming modeling teams where all aspects of the problem can be addressed rigorously. It also
emphasizes the importance of training and appropriate software tool development so that the input and output of the tools
can be easily understood by all involved in the process.

Fig. 2 Modeling cycles. Source: Ref 8

References cited in this section
1.

M.F. Ashby, Physical Modeling of Materials Problems, Mater. Sci. Technol., Vol 8 (No. 2), 1992, p 102-111

4.

M. Rappaz and Ch
A. Gandin, Probabilistic Modeling of the Microstructure Formation in Solidification
Processes, Acta. Metall. Mater., Vol 41 (No. 2), 1993, p 345-360
5.

J. Wang, S. Xun, R.W. Smith, and P.N. Hansen, Using SOLVA-

VOF and Heat Convection and Conduction
Technique to Improve the Casting Design of Cast Iron,
Modeling of Casting, Welding and Advanced
Solidification Processes VI, T. Piwonka, Ed., TMS, 1993, p 397-412
6.

L. Edwards and M. Endean, Ed., Manufacturing with Materials,
Materials in Action, Butterworth Scientific,
1990
7.

S.C. Jain, Recognizing the Need for Materials Data: The Missing Link in Process Modeling, J. Met.,
Oct
1991, p 6-7
8.

K J. Bathe, Some Issues for Reliable Finite Element Analysis,
Reliability Methods for Engineering Analysis:
Proc. First International Conference, K
J. Bathe and D.R.J. Owen, Ed., Pineridge Press, Swansea, U.K.,
1986, p 139-157
Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Modeling of Deformation Processes
Finite element analysis (FEA) of deformation processes can provide an insight into the behavior of the product under
various processing conditions and can help optimize the conditions to get the desired properties. It can also help
understand the performance of the product before the part is put in actual use. Common problems solved by FEA include
insufficient die filling, poor shape control, poor flow of material, cracks and voids that lead to fracture and poor final part
properties.

The occurrence of typical processes in a forging operation are shown in Fig. 3. Figure 4 (Ref 9) shows a schematic
representation of the interactions between the major process variables in metal forming. From Fig. 4 it can be seen that for
a metal-forming analysis, one needs to satisfy the equilibrium conditions, compatibility equations/strain-displacement
relations, constitutive equations, and, in some instances, the heat balance equation. In addition, one needs to apply
appropriate boundary conditions. These may comprise displacement/velocity imposed on a part of the surface while stress
is imposed on the remainder of the surface, heat transfer, or any other interface boundary condition.

Fig. 3 Typical physical phenomena occurring during a forging operation

Fig. 4 Interaction among major process variables during forming. Source: Ref 9
Relevant Equations. The equilibrium equations describing the various forces acting on the body are given as:


(Eq 1)
where is the normal stress component, is the shear stress component, and F is the body force/unit volume component.
Similarly, the strain-displacement relationships are given as:


(Eq 2)
where is the normal strain, is the shear strain, and u, v, and w are the displacements in the x, y, and z directions,
respectively.
Constitutive Theory. A constitutive equation relates the stress and strain behavior of a material. Schematic
stress/strain curves for idealized materials are shown in Fig. 5. In addition to this information, a yield criterion and flow
rules are also needed to adequately describe the material behavior. The stress/strain curve for a typical metal along with
its major features is shown in Fig. 6.

Fig. 5 Schematic stress/strain curves for various materials

Fig. 6 Stress/strain curve for a typical metal
Finite Element Analyses. One of the most commonly used techniques to solve the various equations (as shown

earlier) is finite element analysis (FEA). It is a numerical technique that approximates the mathematical equations that in
turn approximate reality. It is a discrete representation of a continuous system. It breaks the bigger problem into a number
of smaller ones and, therefore, is a piecewise representation of the problem.
Some of the common metal-forming problems that are solved by FEA are insufficient die fill, poor shape control, poor
flow of materials, prediction of cracks and voids that lead to fracture, and poor final part properties. Finite element
analysis has many advantages over the typical closed-form solutions. It provides greater insights into the behavior of the
product and the process, gives a good understanding of the performance of the product before the actual usage, is useful
in product and process optimization, is a powerful and mature design and analysis tool, and most importantly, gives
solutions to irregular shapes, variable material properties, and irregular boundary conditions. However, FEA is not to be
viewed as a solution to all problems or as a substitute for common sense and experience.
Typical costs of FEA can run into thousands of dollars. A rough estimate is that if brainstorming costs $1, then refined
hand calculations would cost $2, "quick and dirty" process models would cost $5, and detailed process models would cost
$30. However, the probability of success of detailed process models is much higher compared to that of brainstorming,
and great savings in costs of experimentation and rework may be achieved.
There are several steps involved in conducting these analyses. A brief description of each is provided below. More
detailed information is provided in the article "Finite Element Analysis" in this Volume.
Defining the Problem. Before a full-fledged FEA is undertaken, several questions need to be answered:
• Is FEA appropriate?
• What is desired from the FEA?
• When are results needed?
• What are the product and process limitations?
If it is determined that FEA would provide adequate answers, information is gathered to start the modeling process. The
required information includes the geometry of interest, initial and boundary conditions, material properties and material
behavior models, and an approximate solution to ensure that the finite element results are not physically absurd.
The preprocessing stage defines the physical problem and converts it into a form that the computer can solve. The
definition of the physical problem involves fully defining the geometry as well as the material of the part to be simulated.
Next comes defining the physics of the process. This involves using the appropriate set of mathematical equations and the
corresponding initial conditions and boundary conditions. Subsequently, the solution method needs to be determined.
This involves choosing the appropriate algorithms to solve the numerical approximations of the mathematical equations.
Finally, because FEA is a numerical approximation of reality, the problem needs to be discretized, (i.e., the continuum

needs to be broken into many smaller pieces, the sum of which will represent the whole problem). In this stage, the shape
of these smaller pieces, or elements, their size, and their complexity needs to be determined.
Discretization is the process of subdividing the whole geometry into discrete parts. The discrete parts are called
elements. Many different types of elements exist depending on their shape, linearity, and order. The factors that impact
the selection of the elements are the type of the problem, geometry, accuracy desired, availability within the algorithm,
nature of the physical problem, and user familiarity. The size and number of elements are primarily determined by the
various gradients (temperature, stress, etc.) in the system. For example, if the gradients are steep, a larger number of
smaller-sized elements should be used. An example of discretization is shown in Fig. 7. For clarity, the node and element
numbers are not shown in the mesh.

Fig. 7 The finite element analysis discretization process
The mesh can be made coarser (fewer elements) or finer (more elements) depending on the needs of the problem. Coarser
meshes use minimal computer resources in terms of storage and run time. However, because the representation is
approximate, the results can be crude. Finer meshes provide a more accurate representation with improved results.
Typical elements are either linear or quadratic and can be one-, two- or three-dimensional. Figure 8 schematically shows
some typically used elements.

Fig. 8 Linear and quadratic elements used in typical finite element analyses
Post Processing. Once the equations have been solved by the computer program, huge amounts of data are generated
that are typically not in a user-friendly format. These data need to be organized and processed so as to make sense and
give only the information that the user desires. The most frequently analyzed data are metal flow, time-history plots,
ductile fracture tendency, stress/strain plots, and load-stroke curves. A good visualization scheme is crucial to
understanding the outcome of the analysis.
Requirements of a Good FEA Code. For an FEA code to be useful, it must be easy to use and provide results that
are easy to understand. A well-written user's manual or other documentation and good software support are also
important. From a technical standpoint, the code must also be accurate, fast, and numerically stable under a wide range of
conditions. It must be sufficient to capture the physics of the problem and should be capable of being interfaced with
other codes (as appropriate).
Example 1: Use of FEA to Study Ductile Fracture during Forging.
Brittle fracture is sometimes observed after forging and heat treating large nickel-copper alloy K-500 (K-Monel) shafts

(Ref 10). Inspection of failed fracture surfaces revealed that the cracks were intergranular and occurred along a carbon
film that had apparently developed during a slow cooling cycle (such as during casting of the ingot stock). Cracks were
observed in both longitudinal and transverse directions and arrested just short of the outer bar diameter. Finite element
analysis was used to determine the cause and remedy for this cracking. In addition, new processing parameters were
determined to eliminate cracking.
A two-dimensional model of the bar was developed and discretized into a finite element mesh. NIKE2D, a public domain,
nonlinear FEA code from Lawrence Livermore National Laboratory (Ref 11), was used to calculate the stress plots under
the forging conditions. The stress/strain histories resulting from forging were required for ductile fracture analyses. In this
particular case, the Oyane et al. ductile fracture criteria (Ref 12) were used. These criteria attempt to predict ductile
fracture based on a porous plasticity model.
Figure 9 (Ref 10) shows in-process hydrostatic stress history at the center of a bar during forging with two rigid, parallel
flat dies. (Hydrostatic stress is the mean of any three mutually perpendicular normal stresses at a given point. It is
invariant with direction.) Various strain rate and temperature conditions are shown having similar trends. At very low
reductions (1.25%), the center of the bar is still within the elastic range and, therefore, does not experience large stresses.
At 2.5% reduction, the outward movement of the free sides of the bar produces a high tensile hydrostatic stress at the
center of the bar. As reduction continues, the compressive stresses acting in the vertical direction begin to grow. This acts
to reduce the hydrostatic stress at the center. When the bar has been reduced by 11%, the hydrostatic stress is compressive
in all cases. Beyond this reduction, the hydrostatic stress at the center of the bar continues to grow in a compressive
manner.

Fig. 9 Hydrostatic stress at the bar center during forging. Source: Ref 10

Figure 10(b) (Ref 10) shows a map of ductile fracture accumulation after one forging pass for air-melted material at 930
°C and a strain rate of 10 s
-1
. Notice that the maximum damage is located at the center of the bar. After this pass, the bar
would be rotated and further reductions would be taken. Damage would accumulate during these subsequent passes.
Ductile fracture was predicted to occur at the bar center from the FEA results after six reducing passes under the specified
conditions. One possible way of eliminating this ductile fracture is to use a V-shaped die assembly as shown in Fig. 10(a).
In this case, the fracture does not occur at the center, but at the edges and gets an opportunity to heal as the bar is rotated

during subsequent passes.

Fig. 10 Ductile fracture map of a nickel-copper alloy K-
500 bar at 10% reduction, 930 °C, and a strain rate of
10.0 s
-1
using the Oyane et al. ductile fracture criteria. (a) Three forging dies. (b) Two forging dies. Source:
Ref
10
Example 2: Use of FEA for Modeling Superplastic Forming of Aluminum
Assemblies.
Superplastic forming (SPF) technology is being increasingly used for aerospace applications. The major benefits of using
SPF include the ability to produce complex near-net shapes better than other forming operations and its good postformed
mechanical properties. Even though titanium components have been formed by SPF for some time, SPF of aluminum
components pose some challenges. These include the ability of the material to be superplastic only at elevated
temperatures and within a small window of strain rates, cavitation, and amount and rate of application of pressure.
Superplastic forming of aluminum is usually performed in a closed-die press capable of applying inert gas pressure to
both sides of a superplastic sheet. Initially, an equilibrium pressure is applied to both sides of the sheet. Slowly, according
to a predesigned forming schedule, pressure is released from one side of the sheet, with the pressure differential pushing
the sheet into the die cavity. As the sheet freely forms into the die cavity, thinning occurs relatively uniformly. Once the
sheet makes contact with the die, frictional effects begin to make the thinning less uniform.
The constitutive behavior of the superplastic material (aluminum alloy 7475) is expressed in the form of a simple power-
law equation:
= A
n


(Eq 3)
where is the strain rate, is the stress, and A and n (= 1/m) are material constants. The above equation assumes that
grain size remains constant during forging. Critical regions, where grain coarsening and an unacceptable amount of

cavitation take place, can also be modeling by expressing the material behavior in a more rigorous form:


(Eq 4)
where A' is a material constant, D is the diffusion coefficient, G is the shear modulus, b is the Burgers vector, k is the
Boltzmann's constant, T is the absolute temperature, d is the grain size, p is the grain size exponent, is the flow stress, n
is the strain-hardening exponent, Q is the activation energy, and R is the universal gas constant. Cavitation may be
expressed as a function of accumulated plastic strain as:
C = C
o
exp (K
3
)


(Eq 5)
where C
o
is the initial void volume, K
3
is a constant, and is the accumulated plastic strain.
Several subroutines were developed and implemented in a commercially available FEA code (Ref 13). Several trial
forming problems were solved to show the benefits of various enhancements to the FEA code. Simple geometries were
first taken to allow comparison of simple analytical solutions with the compared results. Subsequently, complex
geometries were discussed to show where the simple analytical solutions broke down and where the FEA results give a
significant insight into the forming process.

References cited in this section
9. S. Kobayashi, S. Oh, and T. Altan, Metal Forming and the Finite Element Method,
Oxford University Press,

1989, p 27
10.

M.L. Tims, J.D. Ryan, W.L. Otto, and M.E. Natishan, Crack Susceptibility of Nickel-Copper Alloy K-
500
Bars During Forging and Quenching,
Proc. First International Conference on Quenching and Control of
Distortion, ASM International, 1992, p 243-250
11.

J.O. Hallquist, "NIKE2D
A Vectorized Implicit, Finite Deformation Finite Element Code for Analyzing
the Static and Dynamic Response of 2-
D Solids with Interactive Rezoning and Graphics," User's Manual,
UCID-19677, Rev. 1, Lawrence Livermore National Laboratory, 1986
12.

S.E. Clift, P. Hartley, C.E.N. Sturgess, and G.W. Rowe, Fracture Prediction in Plastic Deformation
Processes, Int. J. Mech. Sci., Vol 32 (No. 1), p 1-17
13.

D.K. Ebersole and M.G. Zelin, "Superplastic Forming of Aluminum Aircraft Assemblies
Simulation
Software Enhancements," NCEMT Report No. TR 97-
05, National Center for Excellence in Manufacturing
Technologies, Johnstown, PA, 1997

Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Modeling of Casting Operations
Of late, considerable developments have taken place in the field of solidification modeling of casting processes. In the
current state-of-the-art solidification simulation, several software packages are available to analyze the solidification
behavior in complex-shaped castings. These packages make use of several different approaches for solving the various
problems associated with casting processes.
An overall architecture of a comprehensive solidification modeling system is shown in Fig. 11. This figure (Ref 14)
depicts the various modules available in the current state-of-the-art solidification simulation of casting processes, the
information available from each module, and the interconnection between the modules. It is evident from the figure that
the initial casting design is linked to a module called the quick-analysis module. Here, one can make use of approximate
analysis schemes, such as the modulus approach (Ref 15), which uses geometry-based considerations to provide valuable
insights into the solidification times and, therefore, the propensity for defect formation during solidification.

Fig. 11 Overall architecture of the modeling of casting processes. Source: Ref 14
The next stage is to design the rigging system for the casting, which includes the design of the gate, risers, downsprue,
and so forth. This is currently based on the "rules of thumb" of foundry experts and empirical charts. Once the rigging
design is established, the stage is set for solidification simulation. Here, the continuum mechanics problem of heat, mass,
and momentum transfer are solved for the casting process simulation. Thus, one obtains the cooling history of the casting.
Subsequently, one can obtain information about the microstructure in the casting by coupling with the module for
microstructure evolution. Further, the simulation data can be postprocessed using special-purpose models for defect
prediction that enable one to visualize the defects under a given set of processing conditions. Apart from porosity-type
defects, prediction of other defects such as macrosegregation is possible. Because macrosegregation primarily occurs due
to the movement of solid phase by convection during solidification, solving the fluid-flow equations in the mushy zone
provides a solution.
Modeling the development of stresses in the casting has been another area of great challenge. Of late, several researchers
have addressed the issue of development of stresses during and after solidification, which is often the cause of distortion
in castings. This is especially the case for highly nonequilibrium processes, such as die-casting. Special numerical
algorithms and techniques are being developed for handling more complex casting processes. For example, in large
structural thin-walled castings, the normal solution methods would require an extremely large number of elements or
nodes in the mesh, which significantly increase the computation time.
Quick-Analysis Schemes. Traditionally, for sand-casting analysis, the use of geometric methods has been known as

the section modulus approach. The fundamental basis of geometric modeling is the relationship between the solidification
time (t
f
) and a geometric parameter, called the section modulus (given by volume-to-surface area ratio, V/A), as given by
Chvorinov's Rule (Ref 16):


(Eq 6)
where C is a constant for a given metal-mold material and mold temperature.
For simple shapes, the modulus in Eq 6 can be calculated from the ratio of volume and surface area involved in cooling.
However, for complex shapes discretized in a three-dimensional grid, the continuous distribution of modulus can be
determined using the concept of distance from the mold, as discussed in Ref 15. The modulus at each point in the casting
is determined by the relation:


(Eq 7)
Recently, this technique has been extended to model the investment-casting process, taking radiation loss into account
through the use of a novel approach for view-factor calculations (Ref 17).
Knowledge-Based Systems for Rigging Design. The starting step after the initial design of the casting is the
design of gates and risers. This consists of proper orientation of the part and the determination of the parting plane and the
size, number, and location of sprues, runners, gates, and risers. To this end, the use of knowledge-based design systems is
growing for foundry applications. A feature-based design system has been developed for casting applications (Ref 18).
Also, a system for design of gating and risering for light alloy castings has been developed (Ref 19). This system was
extended to investment castings (Ref 20). In a similar vein, strategies have been developed for shape-feature abstraction
in knowledge-based analysis of manufactured components (Ref 21).
However, the drawback of most of the currently available design tools is the lack of a fully integrated system for the
design of gates and risers, followed by a comprehensive process simulation. More recently, an attempt has been made to
close this gap by developing a system for automatic rigging design (Ref 22). The drawback of this system is that the final
design (which includes the part and rigging) is in the form of a finite difference mesh model and not a solid model, which
inhibits automatic pattern generation. Currently, efforts are underway to overcome this problem.

In these methods for rigging design, the first step is the generation of the solid model of the part. This solid model is used
to generate a discretized description of the part, in the form of a finite difference or finite element mesh. The gating
design is achieved by applying some empirical heuristics to the discretized solid model, as well as performing a geometric
analysis to determine the natural flow paths for liquid metal. These empirical heuristics include rules for design of
runners, sprues, and gates.
As a general rule, the casting should be gated and fed in a manner to ensure progressive solidification of the casting.
There should be an adequate supply of molten metal to feed every section as it solidifies. The solidification should start at
the location furthest from the ingate/casting junction and proceed toward the risers, which should solidify last. The design
of risers involves a geometric analysis of the casting using empirical relations such as Chvorinov's Rule to obtain the
solidification time profile, followed by determination of the size and location of the riser. The size of the riser is decided,
based on the feed metal requirements as well as the solidification time.
The Comprehensive Problem: Fluid Flow and Heat Transfer. Knowledge about fluid flow during the filling of
casting is important, for it affects heat transfer both during and after filling. The information obtained could help avoid
problems of cold shuts, where the melt solidifies before filling a void, or where a molten front of liquid comes in contact
with a solidified metal. A mold-filling simulation is, therefore, indispensable if a high-quality casting-solidification
analysis is desired. There are several computational techniques to simulate fluid flow during mold filling, some of which
follow.
Momentum Balance Technique: The Solution Algorithm-Volume of Fluid Approach. To obtain an accurate
profile of the velocity distribution in the mold during the filling of castings, one has to solve the governing equations. The
main governing equations that need to be solved to track the free surface and obtain the velocity distribution in the melt
are the continuity equation, the Navier-Stokes equation, and the equation for free-surface tracking. The continuity
equation is given by:
· v = 0


(Eq 8)
where v represents the velocity. The Navier-Stokes equation is:


(Eq 9)

where P is the fluid pressure, is the stress tensor, g is the acceleration due to gravity, and is the density. The equation
for free-surface tracking is:


(Eq 10)
where F is a function that defines the fractional volume of the control element occupied by the fluid, is the volume, S is
the surface area, and n is the unit normal vector on the surface.
Several boundary conditions are applied to solve the equations, such as no-slip condition at the solid surfaces and
atmospheric pressure at the free surface. The solutions to these equations, especially for three-dimensional cases, can be
computationally very demanding. Over the past decade, effective algorithms have been developed for simulation of mold
filling and solidification using both finite difference as well as finite element methods. Calculation of the location of the
liquid and the orientation of its free surface is an integral part of this computational technique.
The thermal history inside the casting and mold is obtained by solving the energy equation:


(Eq 11)
where C
P
is the specific heat, q is the heat flux, and is the rate of heat generation.
The boundary conditions can be of three types. In the first type, the temperature at the boundary is specified, and in the
second, the heat flux is specified. More popularly, a third type of boundary condition is used, expressing the heat loss at
the interface through a heat transfer coefficient:


(Eq 12)
where h is the effective heat transfer coefficient across the mold-metal interface, T is the casting surface temperature, and
T
o
is the mold surface temperature.
Recently, an alternative method of solving the volume-of-fluid equation through an analogy between the numerical

treatment of filling and solidification has been developed (Ref 23). In this method, an alternative volume-of-fluid
equation has been proposed, based on an enthalpy-type variable to determine the function F. Encouraging results have
been obtained with this approach.
Modeling Microstructural Evolution. The solution to Eq 11 requires knowledge of the term (the rate of latent
heat evolution during solidification). This term can be described in two ways: specific heat method or latent heat method.
Specific Heat Method. In this classical approach, the latent heat is released by assuming that the solid forms in a
specified temperature range. The specified heat is modified as:


(Eq 13)
where L is the latent heat and df
s
/dT is the rate of change of solid fraction with temperature, obtained from the phase
diagram. Alternatively, one could use the Scheil equation to express solid fraction evolution as:


(Eq 14)
where T
f
is the fusion temperature of the pure metal, T
l
is the liquidus temperature, and k is the partition coefficient of the
alloy. These parameters are obtained from the phase diagram for the alloy. Using the above equation, one can estimate the
latent heat released due to the evolution of the primary phase.
The limitation of the above approach is that one cannot obtain information regarding the microstructure, such as grain
size. New approaches were developed in the last decade to overcome this limitation (Ref 24, 25). These approaches
incorporate the metallurgy of solidification into the simulations. In one such approach, known as the latent heat method, it
has been shown that, in order to obtain microstructural information, the evolution of fraction of solid should take into
account not only temperature but also the kinetics of nucleation and growth.
Latent Heat Method. In the latent heat method, the term in Eq 11 is evaluated based on the solidification kinetics of

nucleation and growth as applicable to the transformations occurring in the system. The expression for is given by:


(Eq 15)
where L is the latent heat of the solidifying phase and is the rate of evolution of fraction of solid. Latent heat generation
can be determined through the use of mathematical expressions to describe the evolution of the solid phase. The
temperature history in the casting can then be obtained by solving Eq 11. Depending on the nature of the alloy, the
expressions can vary. The following section describes the solidification kinetics for equiaxed eutectic structures, which
are commonly found in cast iron systems and aluminum-silicon systems.
Probabilistic Models. Useful as they are, the deterministic models suffer from several shortcomings. They neglect any
aspect related to crystallographic effects. So they are unable to account for the grain selection near the mold surface,
which leads to the columnar region. Furthermore, they do not account for the random nature of equiaxed grains. One
cannot visualize the actual evolution of the grains; one can only get an idea of the size of the grains.
To overcome these limitations, several researchers have used an altogether different track to model microstructural
evolution (Ref 4, 26). They make use of probabilistic models to simulate the evolution of grain structure. These
simulations are more like numerical experiments. In one such work (Ref 26), a Monte Carlo procedure was used to
simulate evolution of grain structure. This type of method is based on the principle of minimization of energy, where the
energy of a given structural configuration is evaluated considering the present state of the various sites (whether solid or
liquid). Transitions are allowed to take place according to randomly generated numbers. Using this technique, the
researchers were able to compute two-dimensional microstructures that closely resembled those observed experimentally.
The main drawback of such an approach was the lack of a physical basis.
In a more recent investigation (Ref 4), a new approach for modeling grain structure formation during solidification was
proposed. Based on a two-dimensional cellular automata technique, the model includes the mechanisms of heterogeneous
nucleation and of grain growth. Nucleation occurring at the mold wall, as well as in the liquid metal, is treated by using
two distributions of nucleation sites. The location and the crystallographic orientation of the grains are chosen randomly
among a large number of cells and a certain number of orientation classes. The model has been applied to small
specimens of uniform temperature. The columnar-to-equiaxed transition, the selection and extension of columnar grains
that occur in the columnar zone, and the impingement of equiaxed grains are clearly shown by this technique.
Prediction of Defects: The Porosity Problem. Analysis of the conditions leading to the occurrence of casting
porosity has been the focus of a number of investigations in the past few decades. With the advances in computer

modeling of the casting process in recent years, there has been considerable interest in the usage of numerical heat
transfer and solidification models to predict casting porosity.
As far as the solidification parameters are concerned, the variables that control porosity may be narrowed down to the
thermal gradient, the rate of solidification, the cooling rate, and the solidification time. Based on these, various
approaches have been suggested to predict casting porosity, the oldest being the empirical criteria. Thermal parameters
have also been formulated recently, from Ref 27 and 28. Many of these criteria are based on d'Arcy's Law, approximating
the mushy region to a porous medium. The pressure drop in the mushy region is then expressed in terms of thermal
criteria functions to predict the onset of porosity.
The current modeling practice is to calculate these criteria functions using the solidification model to predict porosity.
Some of these functions are quite successful in predicting porosity in short-freezing-range alloys, though there are many
difficulties in applying them for long-freezing-range-alloys. Figure 12 (Ref 14) shows the Niyama distribution for a
ductile iron plate casting. The figure clearly shows the defects ending up in the riser, demonstrating adequacy of feeding.

Fig. 12 Distribution of Niyama values in a ductile iron plate casting, showing propensity of
defects only in the
riser. Source: Ref 14
A major limitation of the criteria functions (discussed previously) to predict porosity is that they ignore the effects of
casting macrostructure and grain size on porosity. The resistance to liquid feeding in the mushy region depends on the
available surface area of solid in the interdendritic region, which is dictated by the macrostructure and grain size.
Recently, Suri et. al. have proposed a new number called the feeding resistance number (Ref 31), which takes into
account the effect of casting macrostructure on final porosity. The validity of this proposed criterion to predict porosity is
still under investigation.
Modeling Special Casting Processes: The Investment-Casting Process. Many critical and value-added
components in automotive, aerospace, and other key industries are manufactured by special casting processes, such as the
investment-casting process, lost-foam process, tilt-pour (Cosworth) process, and so forth. Simulation of such processes
requires the application of suitable submodels to handle the phenomenological aspects specific to each process. In this
section, some of the research efforts in modeling investment casting are reviewed.
A comprehensive solidification simulation of the investment-casting process involves a number of computationally
intensive steps, particularly the calculation of view factors (to model the radiation loss), and the three-dimensional
analysis of mold-filling and solidification.

The external heat loss is either purely radiative, or radiative as well as convective. For the general case, the heat transfer
coefficient can be given by:
h = h
r
+ h
c


(Eq 16)
where h
r
and h
c
are the radiative and convective heat transfer coefficients, respectively. The radiative heat transfer
coefficient is given by (Ref 32):
h
r
= F
m - a
( + )(T
s
+ T
o
)


(Eq 17)
where , , and F
m - a
are the Stefan-Boltzmann constant, the mold emissivity, and the view factor of the mold with

respect to air, respectively.
The convective heat transfer coefficient valid for natural convection at high temperature is given by:
h
c
= c(T
s
- T
o
)
1/3


(Eq 18)
where c is a constant dependent on the surface geometry.
View factor is defined as the fraction of the radiation that leaves surface i in all directions and is intercepted by surface j.
When two surfaces, dA
1
and dA
2
undergo radiation exchange, the view factor can be mathematically expressed as (Ref
32):


(Eq 19)
were R
1-2
is the distance between the two surfaces, and and are the angles of the two surface normals with the line
joining the two surfaces.
The calculation of view factors could become very complicated when multiple surfaces are involved in the radiation
process. The presence of multiple surfaces can create a partial or full obstruction in the view path between any two

surfaces. Thus, the view factors will now depend not only on the two surfaces exchanging heat, but also on the shadows
cast by the other surfaces present in the model. Additional calculations are needed for determining the shadowing effects
for any realistic three-dimensional geometries.
More recently, another technique has been proposed that enables quick calculation of the view factor distribution at the
mold surface (Ref 17). This is a modified ray-racing technique, where a scheme is devised to send rays in various
directions, and the number going into air without mold interception is estimated. The view factor is then computed by
calculating the fraction of rays that go into air. This scheme has been successfully applied to three-dimensional finite
difference geometries.

References cited in this section
4. M. Rappaz and Ch
A. Gandin, Probabilistic Modeling of the Microstructure Formation in Solidification
Processes, Acta. Metall. Mater., Vol 41 (No. 2), 1993, p 345-360
14.

G. Upadhya and A.J. Paul, Solidification Modeling: A Phenomenological Review, AFS Trans.,
Vol 94,
1994, p 69-80
15.

G. Upadhya, C.M. Wang, and A.J. Paul, Solidification Modeling: A Geometry Based Approached for
Defect Prediction in Castings, Light Metals 1992,
Proceedings of Light Metals Div. At 121st TMS Annual
Meeting (San Diego), E.R. Cutshall, Ed., TMS, 1992, p 995-998
16.

N. Chvorinov, Theory of Solidification of Castings, Giesserei, Vol 27, 1940, p 17-224
17.

G. Upadhya, S. Das, U. Chandra, and A.J. Paul

, Modeling the Investment Casting Process: A Novel
Approach for View Factor Calculations and Defect Predictions, Appl. Math. Model., Vol 19, 1995, p 354-
362
18.

S.C. Luby, J.R. Dixon, and M.K. Simmons, Designing with Features: Creating and Using Features
Database
for Evaluation of Manufacturability of Castings, ASME Comput. Rev., 1988, p 285-292
19.

J.L. Hill and J.T. Berry, Geometric Feature Extraction for Knowledge-
Based Design of Rigging Systems for
Light Alloys, Modeling of Casting, Welding and Advanced Solidification Processes V, TMS, 1990, p 321-
328
20.

J.L. Hill, J.T. Berry, and S. Guleyupoglu, Knowledge-
Based Design of Rigging Systems for Light Alloy
Castings, AFS Trans., Vol 99, 1992, p 91-96
21.

R. Gadh and F.B. Prinz, Shape Feature Abstraction in Knowledge-
Based Analysis of Manufactured
Products, Proc. of the Seventh IEEE Conf. on AI Applications,
Institute of Electrical and Electronics
Engineers, 1991, p 198-204
22.

G. Upadhya, A.J. Paul, and J.L. Hill, Optimal Design of Gating and Risering i
n Castings: An Integrated

Approach Using Empirical Heuristics and Geometric Analysis,
Modeling of Casting, Welding and
Advanced Solidification Processes VI, T.S. Piwonka, Ed., TMS, 1993, p 135-142
23.

C.R. Swaminathan and V.R. Voller, An "Enthalpy Type" F
ormulation for the Numerical Modeling of Mold
Filling, Modeling of Casting, Welding and Advanced Solidification Processes VI,
T.S. Piwonka, Ed., TMS,
1993, p 365-372
24.

D.M. Stefanescu and C.S. Kanetkar, Computer Modeling of the Solidification of Eutecti
c Alloys: The Case
of Cast Iron, Computer Simulation of Microstructural Evolution, D.J. Srolovitz, Ed., TMS-
AIME, 1985, p
171-188
25.

D.M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Heat Transfer-
Solidification Kinetics Modeling of
Solidification of Castings, Metall. Trans. A, Vol 21A, 1990, p 997-1005
26.

J.A. Spittle and S.G.R. Brown, Acta Metall., Vol 37 (No. 7), 1989, p 1803-1810
27.

E. Niyama, T. Uchida, M. Morikawa, and S. Saito, A Method of Shrinkage Prediction and Its Application to
Steel Casting Practice, AFS Cast Metal Res. J., Vol 7 (No. 3), 1982, p 52-63
28.

Y.W. Lee, E. Chang, and C.F. Chieu, Metall. Trans., Vol 21B, 1990, p 715-722
31.

H. Huang, V.K. Suri, N. El-Kaddah, and J.T. Berry,
Modeling of Casting, Welding and Advanced
Solidification Processes VI, T.S. Piwonka, Ed., TMS Publications, 1993, p 219-226
32.

G.H. Geiger and D.R. Poirier, Transport Phenomena in Metallurgy, Addison Wesley, 1975, p 332
Modeling of Manufacturing Processes
Anand J. Paul, Concurrent Technologies Corporation

Modeling of Fusion Welding Processes
In fusion welding, parts are joined by melting and subsequent solidification of adjacent areas of two parts. Welding may
be performed with or without the addition of a filler metal.
Figure 13 is a schematic diagram of the fusion welding process. Three distinct regions in the weldment are observed: the
fusion zone, which undergoes melting and solidification; the heat-affected zone, which experiences significant thermal
exposure and may undergo solid-state transformation, but no melting; and the base-metal zone, which is unaffected by the
welding process.

Fig. 13 Schematic representations of the fusion welding process

The interaction of the material and the heat source leads to rapid heating, melting, and vigorous circulation of the molten
metal driven by buoyancy, surface tension, impingement or friction, and when electric current is used, electromagnetic
forces. The resulting heat transfer and fluid flow affect the size and shape of the weld pool, the cooling rate, and the
kinetics and extent of various solid-state transformation reactions in the fusion zone and heat-affected zone. The weld
geometry influences dendrite and grain-growth selection processes. Both the partitioning of nitrogen, oxygen, and
hydrogen between the weld pool and its surroundings, and the vaporization of alloying elements from the weld-pool
surface greatly influence the composition and the resulting microstructure and properties of the weld metal. In many

processes, such as the arc welding and laser-beam welding, an electrically conducting, luminous gas plasma forms near
the weld pool.
Energy Absorption. During welding, the workpiece absorbs only a portion of the total energy supplied by the heat
source. The absorbed energy is responsible for the outcome of the welding process. The consequences of the absorbed
energy include formation of the liquid pool, establishment of the time-dependent temperature field in the entire weldment,
and the structure and properties of the weldment. Therefore, it is very important to understand the physical processes in
the absorption of energy during the welding process. The physical phenomena that influence the energy absorption by the
workpiece depends on the nature of the material, the type of heat source, and the parameters of the welding process.
For arc welding, the fraction of the arc energy transferred to the workpiece, , commonly known as the arc efficiency, is
given by (Ref 33):


(Eq 20)
where q is the heat absorbed by the workpiece, I and V are the welding current and voltage, respectively, q
e
is the heat
transferred to the electrode from the heat source, q
p
is the energy radiated and convected to the arc column per unit time
(of which a proportion n is transferred to the workpiece), and q
w
is the heat absorbed by the workpiece (of which a
proportion m is radiated away). For a consumable electrode, the amount of energy transferred to the electrode is
eventually absorbed by the workpiece. Thus, the above equation is simplified to:


(Eq 21)
Fluid Flow in the Weld Pool. The properties of the weld metal are strongly affected by the fluid flow and heat
transfer in the weld pool. The flow is driven by surface tension, buoyancy, and, when electric current is used,
electromagnetic forces (Ref 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45). In some instances, aerodynamic drag forces of

the plasma jet may also contribute to the convection in the weld pool (Ref 46). Buoyancy effects originate from the spatial
variation of the liquid-metal density, mainly because of temperature variations and, to a lesser extent, from local
composition variations. Electromagnetic effects are a consequence of the interaction between the divergent current path in
the weld pool and the magnetic field that it generates. This effect is important in arc and electron-beam welding,
especially when a large electric current passes through the weld pool. In arc welding, a high-velocity plasma stream
impinges on the weld pool. The friction of the impinging jet on the weld-pool surface can cause significant fluid motion at
high currents. Fluid flow and convective heat transfer are often very important in determining the size and shape of the
weld pool, the weld macrostructures and microstructures, and the weldability of the material.
Marangoni Force. The spatial gradient of surface tension is a stress, known as the Marangoni stress. The spatial
variation of the surface tension at the weld-pool surface can arise owing to variations of both temperature and
composition. Frequently, the main driving force for convection is the spatial gradient of surface tension at the weld-pool
surface. In most cases, the difference in surface tension is due to the temperature variation at the weld-pool surface. For
such a situation, the Marangoni stress can be expressed as:


(Eq 22)
where is the shear stress due to temperature gradient, is the interfacial tension, T is the temperature, and y is the
distance along the surface from the axis of the heat source. If a boundary layer develops, the shear stress can also be
expressed as (Ref 47):


(Eq 23)
where is the density, is the viscosity, and u is the local velocity.
Buoyancy and Electromagnetic Forces. When the surface-tension gradient is not the main driving force, the
maximum velocities can be much smaller. For example, when the flow is buoyancy driven, the maximum velocity, u
m
,
can be approximated by the following relation (Ref 48):
u
m

(Eq 24)
where g is the acceleration due to gravity, is the coefficient of volume expansion, T is the temperature difference, and
d is the depth. For the values of T = 600 °C, g = 981 cm/s
2
, = 3.5 × 10
-5
/°C, and d = 0.5 cm, the value of u
m
is 3.2
cm/s. The existence of electromagnetically driven flow was demonstrated by Woods and Milner (Ref 49), who observed
flow of liquid metal when current was passed in the metal bath by means of a graphite electrode. In the case of
electromagnetically driven flow in the weld pool, the velocity values reported in the literature are typically in the range of
2 to 20 cm/s (Ref 50). The magnitude of the velocities of both buoyancy and electromagnetically driven flows in the weld
pool are commonly much smaller than those obtained for surface-tension-driven flows.
Convection Effects on Weld-Pool Shape and Size. Variable depth of penetration during the welding of different
batches of a commercial material with composition within a prescribed range has received considerable attention. Often,
the penetration depth is strongly influenced by the concentration of surface-active elements such as oxygen or sulfur in
steels. These surface-active impurity elements can affect the temperature coefficient of surface tension, d /dT, the
resulting direction of convective flow in the weld pool (Ref 51), and the shape of the weld pool. The interfacial tension in
these systems could be described by a formalism based on the combination of Gibbs and Langmuir absorption isotherms
(Ref 52):
= - A(T - T
m
) - RT
s
ln (1 + k
1

a
i
e )


(Eq 25)
where is the interfacial tension as a function of composition and temperature, is the interfacial tension of the pure
metal at the melting point T
m
, A is the temperature coefficient of surface tension for the pure metal, R and R' are the gas
constants in appropriate units, T is the absolute temperature,
s
is the surface excess of the solute at saturation solubility,
k
1
is the entropy factor, a
i
is the activity of the solute, and H
o
is the enthalpy of segregation. The calculated values (Ref
53) of surface tension for Fe-O alloys are shown in Fig. 14. It is seen that for certain concentrations of oxygen, d /dT can
change from a positive value at "low" temperature to a negative value at "high" temperature. This implies that in a weld
pool containing fairly high oxygen contents, d /dT can go through an inflection point on the surface of the pool. Under
these conditions, the fluid flow in the weld pool is more complicated than a simple recirculation.

Fig. 14 Calculated values of surface tension for Fe-O alloys. Source: Ref 53

The calculated gas tungsten arc weld (GTAW) fusion-zone profiles (Ref 53) for pure iron, and an Fe-0.03O alloy are
shown in Fig. 15. The results clearly show the significant effect of oxygen concentration on the weld-pool shape and the
aspect ratio. Near the heat source, where the temperature is very high, the flow is radially outward. However, a short

distance away from the heat source, where the temperature drops below the threshold value for the change in the sign of
d /dT, the flow reverses in direction. The flow field is not a simple recirculation. Although the qualitative effects of the
role of surface-active elements are known, the numerical calculations provide a basis for quantitative assessment of their
role in the development of weld-pool geometry.