Final Report

Steven R. Schmid William R.D. Wilson Jiying Liu

November 12, 2002

1.

Introduction

Simulation of metal forming processes has become a standard step in the design and procurement of tooling. A designer will evaluate a design and ensure that the material will flow as intended in blocking and forging stages to the desired shape. Simulation allows the designer to make sure that the metal flows as intended, does not develop flaws, such as folds or underfills the die. In addition, quite advanced models are available to predict evolution of the material microstructure as well as fracture of the workpiece. Very advanced material models can be used, with any number of iteration points taken from a stress-strain curve, and strain rate and thermal effects can be incorporated. It is very surprising that, with all of the effort invested into the development of computational tools, very little attention has been directed towards improving the approaches used to model friction and heat transfer in forming. This is especially surprising since it is well known that friction strongly affects the flow of metals, and the temperature history affects the microstructure of the metal. Designers overcome these difficulties by running a large number of simulations under different friction conditions and heat transfer coefficients until a design is achieved that appears to work satisfactorily under all conditions. This process is time consuming, computationally inefficient, and under no circumstances does it ensure proper design. Friction and heat conduction are evolutionary phenomena; they change value at different locations and times of a forging operation. Simulations that limit friction or heat transfer coefficients to a constant value are inherently flawed. Certainly a designer recognizes that a somewhat flawed analysis still can give qualitative feedback and some assurance that a design is reasonable, but the value of the simulations conducted is limited. The tribology and heat transfer module developed under this research and described herein is the first attempt at a more complete, evolutionary model for friction and heat conduction.

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2.

Report Organization

The main deliverable for this research was a user routine for DEFORM 2D that would allow evolution of friction based on the physics of forging. The theoretical framework for the program is contained in this report. This user subroutine was delivered electronically to Concurrent Technologies Corporation in October, 2002. The theoretical background for the user routine is contained in Appendix A. Execution of the user routine requires specification of a series of variables contained in the file “para.ini”, which is summarized in Appendix B. Taken together, this report is intended to be a guide to users of the subroutine to assist them in performing finite element analysis of metal forming using adaptive friction and heat transfer approaches. 3.

Summary of Progress

The routine developed in 2002 built upon the basic friction model developed in the previous year. The major accomplishments were: •

Incorporation of a heat transfer module and demonstrated integration with DEFORM. It was determined that the DEFORM code was not properly accessing the user routines, so evaluation of this model needed to wait until Scientific Forming released version 7.2 of DEFORM.

•

The user routine currently allows more than two contact patches. The initial trials were restricted to ring compression tests, which allowed consideration of two contact patches (workpiece-upper die and workpiece-lower die). With more complicated geometries, a large number of contact patches may be present, presenting additional bookkeeping challenges.

•

Die surfaces may be of any shape and orientation. In 2002, the die surface was assumed to be linear and horizontal, consistent with the ring compression test geometry. The current subroutine allows for curved, inclined and vertical surfaces, as demonstrated by the trials in Appendix A using spike test and emboss test configurations.

•

Difficulties in the DEFORM interface with user routines were identified and software patches were prepared. For example, there is a tendency for DEFORM to query the user routine for friction and heat transfer coefficients

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for elements that do not exist. That is, the element defined by the nodes passed by DEFORM to the user routine is not an element in the DEFORM discretization. The code as it exists is stable and reasonably fast. It provides process modelers with a new tool for increased efficiency and more accurate simulations than previously possible, 4.

Future Work

The routine still has some areas that could be improved. Some of these include: •

The tribology routine is written for DEFORM2D, thus it is restricted to plane strain or axisymmetric problems. Ideally, a routine would eventually be written for three-dimensional cases, but this also requires that DEFORM3D integrate user routines.

•

The heat transfer module works well for fractional contact areas below 0.5 or so, but gives high predictions for heat transfer coefficients for larger fractional contact areas. This is not surprising, since a basic assumption of the heat transfer theory was that the contact patches are far enough apart that they do not interfere with each other. When the fractional contact area becomes large, this is clearly not the case. A thermal interface model for large fractional contact areas is needed.

•

The thermal interface is currently restricted to circular contact patches, or isotropic surfaces. It would be beneficial to develop the theory for other surface lays.

•

No oxide layer has been considered in heat transfer or lubrication.

•

It would be beneficial for a database to be constructed of all wellcharacterized metalworking lubricants as a guide to process modelers.

5.

Conclusions

This project has progressed very quickly from concept to demonstration to working code. A large number of process conditions can be accommodated by this code, making it versatile enough for most process simulators. The use of this module provides, for the first time, a modern, evolutionary treatment of friction and heat transfer in metal forming, and is presented as a user

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routine for a popular commercial code. This is a large step in the development of accurate finite element simulation ability.

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Appendix A: Theoretical Background for the Tribology Module 1.

Introduction The working of metals is at least as old as recorded history, although some of the

processes for doing it are not. Coining, forging and hammering, drifting or making holes, cutting or parting or indenting are clearly among the oldest processes. Rolling, swaging and drawing were known and well developed by A.D. 1500. A brief summary of the history of metalworking is given in Kalpakjian and Schmid [1999]. Metal forming is a multi-disciplinary topic, involving solid mechanics, fluid mechanics, heat transfer, mathematics, tribology, chemistry, and materials science. Metal forming is based on large strain plasticity, which is a unique sub-discipline in itself. Manufacturing operations present extremely demanding applications of tribology. Elevated temperatures, high processing speeds, demanding surface finish and reliability requirements, and environmental considerations all play a role in determining tooling life and product quality. Tribological phenomena are important to the viability and optimization of most manufacturing processes, but relatively little attention has been paid to this subject compared to the efforts directed towards tribology in machine elements, for example. Lubricants are expensive and difficult to apply, remove and dispose of. They also create problems in subsequent operations such as welding, adhesive bonding and painting. However, in many operations, lubricants are critical to control surface quality, tool wear and power consumption, and must be tolerated as a necessary evil. Metal forming is usually conducted at elevated temperatures in order to increase the ductility of the workpiece and reduce the forces and energies required. Workpieces are usually preheated to within 50-90% of their absolute melting temperature before hot forging, and placed into tooling that is not as highly preheated. Since the material properties of the workpiece are strongly affected by local temperature, heat transfer

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between the workpiece and tooling is essential; thus, metal working simulation is a coupled plasticity and heat transfer problem. Simulation of metal forming is essential for a number of reasons, including elimination of workpiece defects, selection of proper equipment, and proper design of tooling. Tooling design is an extremely important issue, since the dies for large aerospace hot forgings can cost more than $1.5 million and require nine months to produce. Tooling that leads to defects such as underfills, folds, internal cavities or improper surface finish is extremely difficult to replace. This proposal presents an improved approach to the simulation of forging operations, incorporating a tribology model that more realistically simulates metal forming, and incorporates this model into modern finite element software. 1.1

Simulation Approaches One concern regarding forging operations is a prediction of the externally applied

loads needed to cause the metal to flow and deform to the desired shape. Because of uncertainties introduced by frictional effects, non-homogeneous deformation, and the true manner by which strain hardening or softening occurs during complex deformations, exact values of force requirements are difficult to predict. Several techniques have been developed that enable an approximation of load requirements. To varying degrees, each of the uncertainties mentioned above is introduced into these analytical methods, thereby permitting an estimation of the deformation forces, the constraining forces, and the manner in which metal flow occurs. Each technique has its own advantages and limitations, but all have been proven valuable in the simulation of metal forming operations. 1.1.1 Ideal Work of Deformation The simplest approach for metal forming simulations is referred to as the method of ideal work of deformation. The method is derived from the simple idea that the external work applied by machinery is equal to the energy consumed in deforming the workpiece. The

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process is assumed to be ideal in the sense that external work is completely utilized to cause deformation only. Effects of friction and non-homogeneous deformation are ignored. To employ this technique, it is necessary to envision an ideal process, whether practical or not, that produces the desired shape change by homogenous deformation. This approach is useful for low friction applications, or situations where there is no appreciable redundant work. When either situation is violated, however, the ideal work of deformation approach gives erroneous predictions. This can be best seen by considering the workpiece in Figure 1.1, which depicts a workpiece undergoing idealized deformation to achieve a desired reduction in thickness. This can be achieved in practice through direct tension, extrusion, drawing, or rolling. For an ideal situation, the deformation

Figure 1.1: Deformation of grid patterns in a workpiece: (a) original pattern; (b) after ideal deformation; (c) after inhomogeneous deformation, requiring redundant work of deformation. Note that (c) is basically (b) with additional shearing, especially at the outer layers. Thus (c) requires greater work of deformation than (b). Source: From Kalpakjian and Schmid [2003].

pattern as seen in Figure 1.1b is achieved. In reality, the pattern of Figure 1.1c results, which clearly has additional deformations required to achieve the desired reduction in thickness. The additional work of deformation is referred to as redundant work. In addition, there is some energy lost to friction, so that the total energy required in a metal deformation process is given by

utotal = u ideal + u friction + u redundant

where utotal is the total energy required, uideal † is the ideal energy, ufriction is energy needed to overcome friction, and ureduntant is the redundant work done in a process. Clearly, neglecting all but the ideal work can lead to erroneous results in simulations.

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(1.1)

1.1.2 Slab Method The slab method is a statics-based approach that entails a force balance on a slice or slab of metal. This produces a differential equation for die pressure as a function of tooling geometry, friction, material properties, etc. Using pertinent boundary conditions, integration of this equation then provides a solution. For example, the plane strain compression situation depicted in Figure 1.2a can be analyzed by considering slabs as shown in Fig. 1.2b. This results in the equilibrium equation

Figure 1.2: (a) Definition of a slap in plane strain compression; (b) definition of stresses acting on the slab; (c) resulting die pressure profile. Source: Kalpakjian and Schmid [2003].

(s x + ds x )h + 2ms y dx - s x h = 0

(1.2)

This can be solved to obtain the die pressure † (see Kalpakjian and Schmid [2003]) Ê 2m (a - x ) ˆ p = Y ¢ expÁ ˜ h Ë ¯

where Y’ is the flow stress, µ is the friction coefficient, a is the width of the workpiece, h † is the workpiece thickness and x is the location. This stress is plotted in Figure 1.2c. Some key assumptions in the slab method are:

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(1.3

•

The direction of the applied load and planes perpendicular to this direction define principal directions;

•

The principal stresses do not vary on these planes.

•

Although effects of surface friction are included in the force balance, these do not influence the internal distortion of the metal or the orientation of principal directions.

•

Plane sections remain plane, thus the deformation is homogeneous with respect to the determination of induced strain.

The main difference between the slab method and the ideal work of deformation approach involves friction. The slab method simply extends the information provided by the ideal work method to include frictional effects. 1.1.3 Upper bound methods The mathematical approaches to modeling large strain plasticity problems discussed to this point have not incorporated the effects of redundant work. Inclusion of these effects are actually quite difficult to accomplish using stress-based methods. A valuable approach is to use the upper bound method based on strains of deformation. As discussed by Drucker et al [1952], there are two basic classifications of metal forming analysis, lower bound methods, usually stress-based approaches, guarantee that the structures under consideration will support, at a minimum, the energies calculated by this approach. The alternative method, an upper bound approach, is based on strain analysis. Drucker et al. showed that the rate of work done by actual surface tractions with prescribed velocities is less than or eequal to the rate of work done by the surface tractions corresponding to any other kinematically admissible velocity field. Thus, any kinematically admissible velocity field, i.e., a strain field without overlaps, die penetration, voids, etc., will over predict the rate of work done by a metal forming operation, and the resulting predictions of energy can be seen as upper bounds to the energy required in the actual operation.

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Upper bound approaches have been extremely valuable in metal forming. A simple form of upper bound approaches, referred to as the velocity discontinuity fields approach, restricts deformations to shear planes. Integral to the approach is the construction of a velocity plot, or hodograph, of the kinematically admissible displacement field. Outstanding references to this approach are the texts by Lange [1985] and Hosford and Cadell [1983]. The key elements to upper bound approaches are: •

An internal flow field is assumed and must account for the required shape change. As such, the field must be geometrically self-consistent.

•

The energy consumed internally in this deformation field is calculated using the appropriate strength properties of the work material.

•

The external forces (or stresses) are calculated by equating the external work with the internal energy consumption.

•

Friction is incorporated along workpiece-tooling interfaces.

Upper bound approaches can therefore inherently accommodate large amounts of frictional and redundant works. 1.1.4 Slip Line Field Theory The slip line field theory applied the method of characteristics to solve large strain metal plasticity problems. While beyond the scope of this proposal, slip line field approaches are worthy of mention because they represent an exact solution, and can incorporate advanced friction models and redundant work effects. Detailed information on this approach can be found in Johnson et al [1982]. It should be noted that the slip line field approach, while extremely powerful and suitable for a wide range of metal working operations, has some basic limitations, namely: •

It is limited to plane strain problems.

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•

The material’s constitutive law is restricted to rigid-perfectly plastic behavior, although strain hardening can be incorporated at the price of additional mathematical complexity and is usually not done.

•

The deformation field must be known a priori.

•

It is difficult to incorporate numerically for general purpose applications.

For these reasons, the slip line field approach has become less popular, and this has been exacerbated by the proliferation of inexpensive and user-friendly finite element software. 1.1.5 The Finite Element Method The finite element method (FEM) is an extremely powerful numerical technique for the solution of complex problems in solid mechanics, including metal forming [Kobayashi et al 1989]. The essence of FEM is to break large, complex structures into smaller interconnected components, called elements. Since the differential equations of metal flow are only solved over the volume of the element, this leads to a piecewise representation of the actual response of the overall structure. In the terminology of Drucker et al. [1952], the finite element method is an upper bound approach, and exact solutions can never be achieved. However, with a large number of elements, FEM gives results that are very close to the actual solution, and for this reason the approach has become widespread in engineering practice. A number of commercial codes are available for finite element simulation of metal working operations. The commercial software DEFORM2D, produced by Scientific Forming, Inc., is the finite element code used in this research. 1.2

Friction Models As discussed above, friction can play a very large role in metal forming

operations, and can seriously affect workpiece deformation and machinery force and power requirements. Friction is sometimes essential or beneficial, such as in metal rolling or in the flash sections of impression forgings. For example, the ring compression

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experiments shown in Figure 1.3 give very different deformations, with the only process variable that has been modified being the friction. An understanding of the mechanics of friction is therefore clearly essential to understanding metal deformation simulation.

Figure 1.3 (a) Effect of lubrication on barreling in ring compression test: (a) with good lubrication, both the inner and outer diameters increase as the specimen is compressed; and with poor or no lubrication, friction is high and the inner diameter decreases. The direction of barreling depends on the relative motion of the cylindrical surfaces with respect to the flat dies. (b) Test results (1) original specimen and (2-4) increasing friction. Source: Kalpakjian and Schmid [2003].

Physically, there are a number of friction sources, namely adhesion, abrasion, lubricant shear, hysteresis, etc. Adhesive friction arises due to the shearing of microwelds at contacting asperities between tooling and workpiece. It can be best understood by noting that even in elastically loaded machinery elements, the contact pressures at contacting asperities are very large and are sufficient to cause cold welding. With metal forming, the pressures are sufficient to cause plastic deformation and are often at elevated temperatures, so adhesion between workpiece and tooling asperities is unavoidable. Abrasive friction arises when one material of two contacting objects is harder than the other, so that asperities on the hard surface penetrate and plow through the softer material. This mode of friction is usually important only in rare cases of metal working, Page 13

because the surface finish of tooling is small enough to limit the penetration of asperities, and adhesive friction will dominate. It is therefore useful to obtain models for the friction force that can be achieved in metal working. The most popular and simple friction model is the Coulomb or Amontons Law, given by: F = mN

where F is the frictional force, N is the normal force between contacting surfaces, and µ is the coefficient of friction. The complete statement of † the Coulomb friction law will also include admonitions that friction is independent of the contact area and sliding speed. The Coulomb law works well for low pressure contacts, such as bearings, gears, brake pads, shoes on hard flooring, etc. However, as can be seen in Figure 1.4, this approach is not valid for metal forming operations, and this is attributable to the very high real areas of contact encountered in metal forming operations. In engineering design applications, including brake pads, bearings, gears, etc., the real area of contact between the surfaces is a small fraction of the apparent area of contact. For elastic or plastic surfaces, the contact area can then be shown to increase linearly with normal force, and therefore the friction force needed to shear the microwelds increases linearly with normal force [Johnson]. In metal forming, the contact pressures are high enough to force the real area of contact to approach the apparent area of contact, and the friction force cannot increase above a saturation level, as shown in Figure 1.4.

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Figure 1.4 Schematic illustration of the relation between friction force F and normal force N. Note that as the real area of contact approaches the apparent area, the friction force reaches a maximum and stabilizes. Source: Schmid and Wilson [2001]. Another common friction model, and one that works quite satisfactorily for many situations, is the Tresca model: F = mtA a

where F is the friction force, m is the friction factor, t is the workpiece shear strength, and Aa is the apparent area of contact between the surfaces. † As will be discussed later, the Tresca model does not allow for the effect of a lubricant on the real area of contact, and therefore can be improved upon further. 1.3

Heat Transfer In metal forming, there is usually a temperature difference between the workpiece

and tooling. Much research has been directed towards prediction of the heat transfer across surfaces in contact because it is recognized that a chilled layer, or skin, can have as large an effect as the friction in affecting material flow. Most previous researchers have emphasized elastic surface contacts, with the realization that two surfaces in contact are not perfectly flat, so most of the heat or energy passes through a limited number of actual contact spots, as shown in Figure 1.5. This allows researchers to define a heat transfer partition between asperity surfaces and across non-contacting valleys, but this is only defined after experiments. Also, previous researchers have emphasized elastic contacts for machine element applications. Whereas the real area of contact in static or slowly moving contacts is made up of a large number of widely distributed extremely small contact patches, at higher pressures and especially with the substrate strain rate the situation can be quite different [Korzekwa, et al 1992]. The usual technique for estimating heat transfer coefficients in metal forming is to place thermocouples in the workpiece and/or die and measure the temperature evolution during a process. Correlation with a mathematical model, often finite element-based, allows extraction of an average heat transfer coefficent. Examples of previous research

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(1.4)

using these techniques are Kellow et al [1969], Dadras and Wells [1984], Boer, et al [1985], Malinowski et al [1994], and Nshama and Jeswiet [1995]. This proposal presents a new heat transfer routine for metal working operations, and is discussed in detail in Chapter 3 of this Appendix. 1.4

Advanced Tribology Models As can be understood from the previous discussion, friction plays an essential role

in metal forming. However, the incorporation of advanced tribology models into plasticity approaches is fairly difficult, but a few noteworthy examples are worth discussion. Wilson et al [1995] presented a friction model for sheet metal forming and incorporated it into a finite element solver to simulate operations such as deep drawing and stretch forming. The model was based on the different lubrication regimes which may occur at the sheet/tooling interface. An outline of their approach is given in Figure 1.6, and examples of their results in Figure 1.7.

Figure 1.7 Measured and prediction strain distributions for stretch forming of a brass sheet over a spherical punch using the tribology model of Figure 1.6 Sheet metal forming always takes place at or near room temperature, so heat transfer to the tooling has a less critical effect on workpiece material properties. In addition, the real area of contact is much lower in sheet metal forming than in bulk

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forming operations like forging, as can be seen in Figure 1.4. However, Wilson, et al [1995] were the first to demonstrate the importance of advanced friction models in metal forming simulation.

2.

Isothermal Friction Model

2.1

Fundamentals of Lubrication

From the viewpoint of the nature of the contact, there are four regimes of lubrication, which will determine the role of a lubricant film between contacting surfaces [Wilson 1979]. Commonly the regimes are defined by a film parameter or dimensionless film thickness defined by:

Ht =

h 2 2 Rqa + Rqb

=

h Sq

where h is the average film thickness, and R†qa2 and Rqb2 are the roughnesses of the workpiece and tooling, respectively. Sq is commonly referred to as the composite surface roughness. The dimensionless film thickness given by Eq. (2.1) is a direct indicator of the lubrication mechanisms that will take place in any lubricated contact. For large values of Ht, usually greater than ten or so, the contact is said to be in a state of thick film lubrication, as shown in Figure 2.1a. In thick film lubrication, surfaces are completely separated, can be taken as perfectly smooth, and all friction arises from the shear resistance of the lubricant. Friction coefficients are very low, typically near 0 to 0.001. Direct contact between asperities is almost nonexistent, and all load is supported by the pressurized lubricant film. Thick film lubrication is sometimes encountered, but rarely intended in manufacturing, because of the detrimental effects of orange peel, as will be discussed later.

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For lower values of Ht, typically between 3 and 10, the contact is said to be in a state of thin film lubrication. In the thin film regime, as illustrated in Fig. 2.1(b), only subtle differences exist with the thick film regime. Mainly these are associated with the asperity contact; asperities do occasionally contact but carry negligible load, whereas in thick film lubrication asperities almost never come in contact. Often, the thick and thin film regimes are classified together as full film lubrication. For full film lubrication, it is

Figure 2.1 Regimes of lubrication. (a) Thick film lubrication; (b) thin film; (c) mixed or partial lubrication; (d) boundary lubrication. Source: From Kalpakjian and Schmid [2003].

relatively easy to model the friction, as a direct expression for interfacial shear stress arises from the definition of viscosity In the mixed lubrication regime shown in Fig. 2.1(c), Ht is taken to be between one and three. Both the pressurized lubricant and asperity contact will support the interface load. It should be noted that asperity contact rarely involves metal-to-metal contact, because lubricants are formulated with boundary and extreme pressure additives that produce a protective film on asperities.

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A lot of research has been focused on the mixed lubrication regime because it is the most common lubrication condition in metal forming. From a microscopic standpoint, friction arises from two mechanisms: the asperity valleys containing a pressurized fluid and the contacting asperities. The friction coefficient in the mixed regime can have values around 0.05 to 0.25 or more. In the boundary regime shown in Fig. 2.1(d), the asperity peaks in contact carry the entire interface load. A lubricant can still exist in the valleys of the contact, but this lubricant is under low pressure and supports negligible load. The friction behavior of this system is dominated by the phenomenon occurring at the asperity peaks. Nevertheless, direct metal-to-metal contact is rare in metalworking. A lubricant is usually formulated with boundary and extreme pressure additives that form a strong surface adhering layer. An example is stearic acid added to lubricating oils to form iron stearate on steel surfaces, as shown in Figure 2.2. The chemisorbed iron stearate film can be as thin as a few hundred nanometer, and this film prevents adhesion between the tooling and workpiece.

Figure 2.2: Formation of iron stearate on a steel surface from stearic acid additive in a lubricant [Schey 1984].

Forging operations are quite complicated from a lubrication standpoint. Clearly, applying friction rules requires an understanding of lubrication regime that is present in

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an operation. However, in forging, all four lubrication regimes can exist simultaneously in the same die. 2.2 Evolving film thickness 2.2.1Reynolds Equation As discussed above, the lubrication regime is an important concept in modeling the friction in metal forming. To determine the lubrication regime from Ht, one needs to calculate or otherwise obtain both the lubricant film thickness and surface roughness. The lubricant film thickness is calculated from the Reynolds equation, given by [Hamrock, 1994]: d Ê rh 3 dp ˆ (u1 + u 2 ) dh u1 + u 2 df h d (u1 + u 2 ) dh ÁÁf x ˜˜ = + Sq s + + dx Ë 12h dx ¯ 2 dx 2 dx 2 dx dt

(2.2)

where f x and f s are flow correction factors due to surface roughness effects on lubricant † entrainment, as first developed by Patir and Cheng 1978. Wilson and Marsault (1998) expressed the flow correction factors suitable for metal forming as:

Ê 1 fx = Á Ë Ht

where

ˆ3 È 2 3˘ ˜ ÍÎa 2 (H t - H tc ) + a 3 (H t - H tc ) ˙˚ ¯

(2.3)

† -0.25007 ˆ Ê Ê 0.47476 ˆ ˜ Htc = 3ÁÁ1 - Á + 1˜ ˜ g Ë ¯ Ë ¯

(2.4)

a2=0.051375 ln3 (9g) – 0.0071901 ln4 (9g)

(2.5)

a3 = 1.0019 – 0.17927 ln g + 0.047583 ln2 g – 0.016417 ln3 g

(2.6)

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f s is calculated from fs = b0 + b1H t + b2 Ht2 + b3 Ht3 + b4 Ht4 + b5 Ht5

(2.7)

b0 = 0.12667 g-0.6508

(2.8)

b1 = exp(-0.38768 – 0.44160 ln g – 0.12679 ln2 g + 0.042414 ln3 g)

(2.9)

b2 = -exp(-1.1748 – 0.39916 ln g – 0.11041 ln2 g + 0.031775 ln3 g)

(2.10)

b3 = exp(-2.8843 – 0.36712 ln g – 0.10676 ln2 g + 0.028039 ln3 g)

(2.11)

b4 = -0.004706 + 0.0014493 ln g + 0.00033124 ln2 g – 0.00017147 ln3 g

(2.12)

b5 = 0.00014734 – 4.255 x 10-5 lng – 1.057 x 10-5 ln2 g + 5.0292 x 10-6 ln3 g

(2.13)

where

In forging, u 2 = 0 because the punch moves vertically, and, simplifying the stretch term, Eq. (2.2) becomes:

d dx

Ê r h3 dp ˆ u1 dh u1 dfs h ˙ dh + Sq + ex + Áf x ˜= 12 h dx 2 dx 2 dx 2 dt Ë ¯

(2.14)

It has been shown by Wilson [1999] that the Poiseuille term (the term on the left side of Eq. (7)) is negligible in the contact region of metal forming. Therefore, Eq. (2.14) becomes:

0=

u1 dh u1 df h dh + Sq s + e˙ x + 2 dx 2 dx 2 dt

(2.15)

Eq. (2.15) is the governing equation of film thickness evolution during forging simulation for plane strain. The equation for lubricant flow in polar coordinates is the same as for Cartesian coordinates, and yields:

0=

u1r dh u1 df hr dh + Sq s + e˙r + 2 dr 2 dr 2 dt

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(2.16)

It should also be noted that the surface asperity lays are defined as circumferential, not linear for axisymmetric cases (see Figure 2.3).

Figure 2.3: (a) Typical contact areas for linear-layed surfaces. From Patir and Cheng [1978]. (b) Typical contact areas for radial or circumferentially-layed surfaces.

2.2.2 Initial Film Thickness Mathematically, the Navier-Stokes or Reynolds equation could be used to obtain the film thickness generated at the initial contact of the workpiece with a die. However, this approach is extremely burdensome from a computational standpoint. Fortunately, the Reynolds equation has been solved for special cases that are useful in determining film thickness. A complete discussion of the film thickness equations given in this section is contained in Wilson [1986], and the equations are merely summarized here. For an isoviscous lubricant, the central film thickness for a flat patch of width x first coming into contact is:

Ê 3h Vx 2 ˆ h0 = Á 0 i ˜ Ë s ¯

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1/ 3

(2.17)

where h0 is the lubricant viscosity, V is the approach velocity and s is the material flow strength. For the isothermal case with a piezoviscous (pressure-dependant viscosity) lubricant,

Ê 3h0gVx 2i ˆ h0 = Á -gs ˜ Ë1 - e ¯

1/ 3

(2.18)

where g is the pressure exponent of viscosity. As the workpiece flattens, the macroscopic or apparent contact area between the workpiece and tooling grows, as shown in Figure 2.4. The film thickness that is entrained at the edge of contact is given for an isoviscous case by

h0 =

3h0 (u1 + u2 )

(2.19)

s tan q

where u1 is the rate of expansion of the contact patch, u2 is the horizontal velocity of the die surface, and q is the contact angle between the workpiece and tooling surface. For the isothermal piezoviscous case, the film thickness is

h0 =

3h0g (u1 + u2 )

(

tan q 1 - e -gs

)

For forging with flat dies, where the stroke is perpendicular to the die surface (as in ring compression or upsetting), u2 can be taken as 0.

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Figure 2.4: Schematic illustration of evolving surface contact in upsetting. (a) Initial contact with face of cylindrical slug. (b) 75% reduction in height, showing that the initial contact expands and new surface 'rolls' into contact with the punch and die. 2.3 Surface Roughness Evolution It is well-known that plastic deformation causes roughening of surfaces, a phenomenon known as orange peel because the grainy appearance of sheet metals is close to that of an orange. An excellent discussion of surface roughness evolution is given by Tong, et al. (1997). Osakada and Oyane (1971) found that the roughening increases with strain and is greater for coarser grained materials and for metals with a small number of slip systems. Thus, HCP metals roughen most, FCC materials less, and BCC materials the least. Tokizawa and Yosikawa (1973) supports these findings but includes the influence of two material phases on the roughening process. Chen et al. (1990) found that grain rotation is the most important source of surface roughening. For small strains, the workpiece surface roughness is linearly proportional to applied strain. As strains become larger, the rate of roughening decreases, and as reported by Lee (1996), surface roughness can actually decrease with strain when the total strain is very large (greater than 300% in his experiments). Lee suggests an expression for roughness of Ra=(-0.276+0.352Hg-0.059Hg2)le

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(2.21)

where Hg is the hardness ratio between adjacent grains, l is the spacing between grains, and e is the total strain. For the purposes of this research, this has been simplified to Ra=fe where f is a constant of proportionality. This can be done in this case since a single material with a single grain size is considered in the testing, so that Hg and l in Eq. (2.21) are constant. The surface roughness that arises in the presence of tooling is very different from unconstrained roughening. Wilson and Sheu [1983], Sutcliffe [1988] and Korzekwa, Dawson and Wilson [1992] have all shown that workpiece surface asperities flatten readily against smooth tooling when the bulk material is straining. The surface finish of the workpiece, the lubrication regime, and the real area of contact between the workpiece and the tooling must therefore be seen as evolutionary and not constant during a manufacturing process. The surface finish of the workpiece depends greatly on the degree to which asperity flattening has progressed and the separation of the surfaces. If the discussion is limited to bulk deformation processes, then strains and strain rates are sufficient to suggest that asperities will be flattened readily, as proven by Wilson and Sheu [1983]. In this case, the workpiece surface finish depends on the surface strain and the lubricant film thickness. Figure 2.5 shows the results of Wilson and Schmid [1992], who investigated the evolution of roughness in metal rolling. For very thin lubricant films, the tooling surface finish is impressed onto the workpiece. At very large film thicknesses, the tooling leaves the workpiece unaffected, while at intermediate films, the workpiece surface roughness is strongly dependent on the film thickness. Wilson and Schmid gave an approximate relationship between surface roughness and film thickness in rolling 5052-O aluminum strip as

Page 25

(2.22)

Figure 2.5 Surface roughness versus film thickness in rolling of 5052-O aluminum. From Wilson and Schmid (1992). Ra=Rat+Ch where Ra is the surface roughness of the workpiece, R at is the tooling roughness, C is an experimental constant for a given material, and h is the film thickness. Wilson and Schmid reported a value of C=0.154 for their experiments, while Wilson and Stilleto [1980] reported a value of C=0.25 in forging of 6061 aluminum alloy. Surface roughness is therefore calculated based on the lubricant film. If the lubricant film is very thin, then the tooling surface finish is impressed onto the workpiece. A thicker film gives a roughness according to Eq. (2.23), unless this prediction would exceed the natural roughness of the material, in which case Eq. (2.22) is used. 2.4 Real Area of Contact

Page 26

(2.23)

The real area of contact is needed in order to obtain estimates of friction in the interface, as discussed below. If the film thickness is greater than three times the composite surface roughness, then fluid film lubrication occurs and the contact area is essentially zero. In this case, the friction will arise from the lubricant alone. If the film thickness is less than three times the surface roughness, Wilson and Marsault [1998] show that the real area of contact can be calculated for a sinusoidal surface from A = 0.5 - 0.3183sin -1( 0.7071H )

(2.24)

and for a Gaussian surface from the model of Christensen [1970]:

A=

35 Ê 16 3 1 ˆ Á - z + z3 - z 7 + z7 ˜ 32 Ë 32 5 7 ¯

(2.25)

where A is the contact ratio and z is a surface parameter given by

z=

h

h 3Sq

(2.26)

is the nominal surface separation, that is, the distance between mean plane of

undeformed surfaces. However, the Reynolds equation calculates h, that is, the mean film thickness or the volume of lubricant divided by the area. h and h are related for a sinusoidal surface by

Ht = 0.5H + 0.4502 1 - 0.5H 2 + 0.3183H sin -1( 0.7071H )

(2.27)

where Ht=h/Sq and H= h /Sq. For a Gaussian surface h and h are related by Ê ˆ 3Á35 + zÊÁ128 + zÊË140 + z 2 -70 + z 2 28 - 5z2 ˆ¯ ˆ˜˜ Ë ¯¯ Ë Ht = 256

(

Page 27

(

))

(2.28)

Figure 2.6 compares the contact area ratio predicted as a function of dimensionless film thickness for the two cases. A sinusoidal surface is slightly easier to implement numerically because a realtionship expressing H in terms of Hc can be obtained, whereas for a Gaussian surface a numerical solution scheme is necessary.

Figure 2.6: Contact area ratio as a function of film thickness for sinusoidal and Gaussian surfaces. 2.5 Friction Calculation In full film lubrication, asperities never contact each other, and it is reasonable to use a constant friction factor that corresponds to the shear strength of the lubricant. Therefore, for full film lubrication, the friction stress is given by

tl=mlk where tl is the friction stress associated with the lubricant and k is the material shear strengh. m is the friction factor, and the l subscript means that the friction factor is associated with the lubricant only. In mixed lubrication, the friction stress depends on both the lubricant and the asperities. Wilson, Huang and Tsu [1995] suggest a friction stress expression of

Page 28

(2.29)

tf=ckA+qtkHA+tl(1-A)=mmkA+mlk(1-A)

(2.30)

where c is an adhesion coefficient, A is the fractional contact area, k is the material shear strength, qt is a plowing coefficient (and is proportional to the surface slope and therefore related to the roughness), and H is the surface hardness. mm is the combined effect of adhesion and plowing, given by mm=c+qtH In the boundary regime, Wilson [1999] proposes that the lubricant’s effect is negligible, and the friction stress can be calculated from the asperity contact only.

3.

Thermal Interface Model

Part 1 presented a model for friction in isothermal forging operations. While useful for some limited applications, forging usually takes place with preheated workpieces and tooling, with the tooling significantly cooler than the workpiece. This has important implications for simulations, namely that heat transfer at the workpiece/tooling interface serves to chill layers of the workpiece near the tooling. Since strength and stiffness increase with temperature decreases, this temperature drop can restrict deformation just as much as friction does [Schey 1985]. This can also be clearly seen in Figure 3.1, which compares predicted billet shapes for upsetting operations at different heat transfer coefficients but a constant friction factor. Modern finite element codes have a built-in capability for calculating the heat generated in plastic deformation and frictional sliding at workpiece tooling interfaces. In addition, they have heat convection and conduction options readily available for workpiece surfaces. Along tooling/workpiece interfaces, a constant heat transfer coefficient can be defined to simulate heat transfer. Just as friction should be an evolutionary variable, it should be recognized that heat transfer coefficients are not constant in forging. Heat transfer across the tooling/workpiece interface is a complex combination of conduction across asperity contacts and heat transfer across

Page 29

(2.31)

a )

b )

c )

d ) Figure 3.1: Deform‰ FEM meshes at 50% reductions for various heat transfer coefficients using Rene-88 as the workpiece material; a) 1.0 kW/m2K, b) 5.0 kW/m2K, c) 10.0 kW/m2K, and d) 20.0 kW/m2K. The initial conditions were; Ring Temperature - 1900°F (1037°C), Die temperature 500°F (260°C), Strain Rate - 1.0s-1, and friction factor (m) - 0.10.

the lubricant film. Although the lubricant can be thermally conductive, more typically the lubricant conducts heat with far less efficiency than metal on metal contacts. Much research has been directed towards prediction of the heat transfer across surfaces in contact. Most previous researchers have emphasized elastic surface contact, with the realization that two surfaces in contact are not perfectly flat, so most of the heat or energy passes through a limited number of actual contact spots, as shown in Figure 3.2. However, heat does flow through the lubricant or gas present between the contact spots as well. For unlubricated elastic contacts, Mentes et al [1981] showed that the gasses filling the voids may significantly affect the overall heat transfer.

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Figure 3.2: Heat transfer partition between asperity and lubricant.

There are a number of techniques available for the prediction of thermal contact conductance for elastic contacts. Snaith et al [1983] recommended a number of special criteria for thermal contact conductance testing and analysis including a specimen form ratio, the reduction of transverse heat losses in the measurement system, and temperature measurement, among others. Antonetti and Eid [1987] proposed a procedure for measuring steady-state thermal contact conductance that relies on monitoring the contact resistance rather than specimen temperatures. Tsai and Crane [1992] developed an analytical solution which provides a theoretical basis for transient measurement of thermal contact conductance. Taylor [1998] used the laser flash method to measure the contact conductance of coatings and thin films. However, all of these approaches are suspect when considering plastic contacts with large fractional contact areas. The usual technique for estimating heat transfer coefficients in metal forming is to place thermocouples in the workpiece and/or die and measure the temperature evolution during a process. Correlation with a mathematical model, often finite element-based, allows extraction of an average heat transfer coefficient. Examples of previous research using these techniques are Kellow et al [1969], Dadras and Wells [1984], Boër, et al [1985], Malinowski at al [1994], and Nshama and Jeswiet [1995]. The paper by Nshama and Jeswiet [1995] is of special interest because it uses multiple thermocouples and a finite element based model based on Deform2D in upsetting experiments. The authors found that an average value of heat transfer coefficient, U, is close to 100 kW/m2C for cold forging of aluminum and 50kW/m2C for hot forging of aluminum. However, the authors clearly noted that the results from the finite element model did not match the temperature profiles

Page 31

well, and that this was clearly because the heat transfer coefficients and friction factors were not constant. Whereas the real area of contact in static or slowly moving contacts is made up of a large number of widely distributed extremely small contact patches, at higher pressures and especially with the substrate strain rate the situation can be quite different [Korzekwa, et al 1992]. This paper addresses the nature of surface contact in metal forming with respect to heat transfer at workpiece/tooling interfaces. The goal is to obtain a model of simultaneous heat transfer through asperity contacts and fluid films, and to use a formulation and implementation that is consistent with the friction module described in Chapter 2 of this appendix.

3.1

Thermal Interface Model Formulation

As discussed above, bulk deformation operations such as forging have unique characteristics that make constant heat transfer coefficient models suspect. However, forging usually takes place with hot tooling, albeit at a temperature a few hundred degrees cooler than the workpiece. Thus, the workpiece can cool significantly at the surface, leading to resistance to deformation that is not attributable to friction. Heat transfer can take place at an interface due to [Incropera and Dewitt 1996]: 1. Conduction across asperities in contact; 2. Conduction across a lubricant film; 3. Convection within the lubricant; 4. Radiation across the lubricant. In metal forming, the lubricant is very thin compared to characteristic lengths in directions normal to the thickness direction; this suggests that convection will be much lower than conduction across the interface. Boër et al [1985] showed that radiation effects are minor, so that heat transfer in metal forming is dominated by conduction effects. Thus, the first two forms of heat transfer will be considered here. 3.1

Heat flow across asperities

Consider heat flow through an asperity contact within the workpiece. Assume all the asperity contacts are circular with radius a spaced a distance 2L apart. Heat flow away from the

Page 32

asperity contact is roughly radial over a conical region as shown in Figure 3.2. At a depth d these regions merge and thereafter heat flow is uniform and normal to the surface. If the heat flow from the asperity contact is Qc , then the temperature at distance r from the apex of the conical region approximately obeys 2

dT Qc c =dr pkw a 2r 2

(3.1)

where k w is the conductivity of the workpiece material. It should be noted that thermal conductivity, density, specific heat, etc., can all be expressed as functions of temperature - see for example Boyer and Gall [1985] - but are taken as constants in this analysis. Equation (3.1) approximates the spherical area for heat flow by the projected plane. From geometry

c=

ad (L - a)

(3.2)

Substituting for c in equation (3.1) and integrating yields

T=

Qc d 2 1 +C 2 pkw ( L - a) r

(3.3)

where C is a constant of integration. We will arbitrarily assume the temperature is zero at the base of the conical region and Tc at the asperity contact: r = d+c, T=0

(3.4)

r= c, T=Tc

(3.5)

Substituting these conditions yields

Tc =

Qc d 2 1 ˆ Ê1 Á ˜ 2 pk w (L - a) Ë c c + d ¯

Page 33

(3.6)

The fractional contact area A for a conical contact is given by a2 L2

(3.7)

Qc d pk w L2 A

(3.8)

A=

Using Equations (3.2) and (3.7) yields

Tc =

If the heat flow were not constricted by the constriction at the asperity contact, Tc would be given by

Tc =

Qc d

pk w L2

(3.9)

Thus, the constriction elevates Tc by an amount DTc given by

DTc =

Qc d pkw L2

Ê1 - A ˆ Á ˜ A ¯ Ë

(3.10)

This quantity may be associated with the extra temperature difference due to the constriction resistance in the workpiece. Of course, the heat flow in the tooling is also constricted. Thus the total additional temperature drop due to the heat flow is

DTc =

Qc d pL2

Ê 1 - Aˆ Ê 1 1ˆ Á ˜Á k + k ˜ A ¯Ë w Ë t¯

(3.11)

It is convenient to continue by using the idea of a film heat transfer coefficient or film conductivity U defined by

U=

heat flow rate area ¥ temperature difference

(3.12)

Thus, from Eq. (3.11), the relevant film conductivity Uc for conduction through the asperity contacts is given by

Page 34

Uc =

1 A k t kw d (1 - A ) (kt + k w )

(3.13)

The distance d remains to be estimated. As a first approximation, it seems reasonable to write

d=L

(3.14)

but a better estimate could undoubtedly be derived by numerical simulation. 2.2

Heat flow across lubricant

Heat flow also occurs through the lubricant film. This is also subject to a constriction resistance. In practice, heat flow through the lubricant is probably only important when heat flow through the asperities is small. This implies very small fractional contact area A and hence a small constriction of the heat flow through the lubricant film. If Q f is the heat flow through the lubricant film in the area surrounding an asperity then we can write

DT f =

Qf hf

1 pL2 (1 - A) k f

(3.15)

where DTf is the average temperature difference across the film, hf is the average film thickness and kf is the lubricant conductivity. If the lubricant were replaced by equal thicknesses of tooling and workpiece material, then the heat flow would be given by

DT f =

Qf hf

pL2 (1 -

Ê 1 1 ˆ Á 2k + 2k ˜ A) Ë t w¯

(3.16)

Thus the extra temperature drop DTf is given by

DT f =

Ê 1 1 1 ˆ Á ˜ pL2 (1 - A) ÁË k f 2k t 2k w ˜¯ Qf hf

Page 35

(3.17)

As before this can be converted into a film conductivity

Uf =

2 k f kt k w (1 - A) h f (2k t kw - kw k f - k f kt )

(3.18)

The average film thickness h f remains to be estimated. This is obtained from the tribology module, and the approach is described in Chapter 2 of this appendix. The total interface film conductivity U1 is the sum of the film conductivities associated with heat flow through the asperity contacts and through the lubricant film

U1= UC + Uf

(3.19)

This can be used in the numerical simulation of the thermal problem with appropriate substitutions form Equations (3.13), (3.17) and (3.18). The heat transfer coefficients as a function of contact area is shown in Figure 3.3a for the circumstance of forging of aluminum with a typical surface produced by sand blasting and a film thickness of 1 µm. Figure 3.3b shows the effect of film thickness on total heat transfer coefficient for the interface. As can be seen from the plot, the film conductivity is dominated by the heat flow across the lubricant film at small contact areas, but is increasingly dominated by heat flow across asperities as the fractional contact area increases. It should be noted that area of contact and film thickness are not independent in a forging operation, so Figure 3.3 is intended merely to indicate typical values predicted and qualitative trends that occur. Many improvements could be made to this simple model. Numerical simulation of the contact problem could be used to tune the form of the functions which have been derived. Other features of the real system such as an oxide layer could also be incorporated.

Page 36

(a)

(b) Figure 3.3 Predicted heat transfer coefficients. (a) Asperity and lubricant partitions for aluminum forging. (b) Effect of film thickness on heat transfer coefficient.

Page 37

3.

Results and Discussion

3.1

Comparison to Previous Research Results

Correlation of the simple model derived above to experimental results is fairly difficult, because very few good experiments have been conducted to date. Nshama and Jeswiet [1995] reported that the average film heat transfer coefficients for unlubricated ring compression tests of aluminum was 100 kW/m2C and 50kW/m2C for cold and hot forging, respectively. However, the “hot forging” performed by Nshama and Jeswiet was not actually a forging operation, as the dies were allowed to touch the workpiece with sufficient pressure to provide good thermal contacts without undergoing plastic deformation. Using the thermal properties of air for the lubricant with a film thickness equal to the workpiece roughness yields the results in Figure 3.4. It is difficult to directly apply the experiments of Nshama and Jeswiet, because the fractional contact area in their experiments were not measured and insufficient information was given to obtain an estimate. However, the heat transfer coefficients suggested correspond to reasonable values of fractional contact area.

Figure 3.4: Heat transfer prediction for the situation of Nshama and Jeswiet [1995].

Page 38

Figure 3.5: Heat transfer coefficients predicted from temperature measurements and model of Malinowski et al [1994].

Malinowski, et al [1994] conducted contact experiments of 303 stainless steel tooling, with embedded thermocouples. Using an axisymetric heat transfer analysis, they were able to obtain the average heat transfer coefficient as a function of time. Examples of their results are shown in Figure 3.5. By performing non-linear regression analysis, they were able to express the heat transfer coefficient as

[

]

ÏÔ 1000 ( A - 2BQ )t + Bt 2 , for 0 < t £ Q U=Ì 2 ÔÓ1000 At - BQ , for Q < t < 40 seconds

[

]

(3.20)

The constants A, B, and Q are expressed as functions of temperature for different initial billet temperatures. For example, at an initial billet temperature of 700°C and 90 MPa, A, B, and Q are reported as: 2 2 3 2 [ ] 2 2 3 2 B = [-0.27807 + 6.04270 p + 0.16841T - 4.73161p + 4.00936T - 12.2651T ]

A = -3.70828 + 4.44037 p + 5.59705T - 3.75139p - 3.58267T - 1.62445T

Page 39

(3.21) (3.22)

[

Q = -2.02052 + 14.04297p - 0.00530T + 1.26445p2 - 5.54230T 2 + 2.02646T 3 + 0.59486 p3 - 0.04034T 4

2

]

(3.23) + 20

where p is the pressure in megapascals divided by 100 and T is the absolute temperature divided by 1000. Using stainless steel as the workpiece, with air as the separating fluid and a film thickness equal to the material surface roughness leads to a heat transfer coefficient of 4 kW/m2°C for a fractional contact area of zero, and which increases steadily to 20 kW/m2°C at around Ar=0.2. Since the experiments of Malinowski et al are for elastic contacts at progressively higher pressures, the results from our model give reasonable predictions for the plateau pressure in Figure 5. The initial increase in heat transfer coefficient as a function of time is difficult to correlate because there are no plastic deformations in the experiments of Malinowski, et a [1994], so the fractional contact area should be low and should not change appreciably. Figure 3.6a shows the results of ring compression tests conducted by Barker, et al [2000]. The figure depicts load/stroke calibration curves for different heat transfer coefficients in ring compression tests of Rene-88, a nickel-based superalloy of interest in aerospace forgings. The figure also shows an example of experimental results, indicating that the heat transfer coefficient is initially around 5.0 kW/m2°C, and eventually attains an average value of 20 kW/m2°C. Figure 3.6b shows the results for the heat transfer model developed in this paper using steel tooling (k=60.5 W/mK), Rene-88 workpiece (k=12 W/mK), and glass lubricant (k=0.05 W/mK). The conductivity values were obtained from Incropera and Dewitt [1996]. Figure 5b was generated using a glass film thickness of h=10µm. As can be seen, the predictions for low heat transfer coefficients during initial contact are supported, and as the film thins and regions of thin films are developed along the ring periphery, the increase in heat transfer coefficient is also predicted. The values predicted are also very close to the experimental values.

Page 40

(a)

(b) Figure 3.6: (a) Load-stroke results for ring compression tests of Rene-88, from Barker, et al [2000]. (b) Predicted heat transfer coefficients from Eq. (19).

Page 41

4.

Implementation

4.1

Tribology Module Variables

A theoretical framework has been presented for friction and heat transfer in forging. This framework was implemented on the commercial finite element software package DEFORM2D, using the user subroutine capability available in release 7.2. The implementation is not straightforward, and a number of issues needed to be overcome to develop a stable user routine. First of all, it is useful to define a new tribology module discretization for the tribology and heat transfer routines. The reason for this is that within the contact zone, tribological variables (film thickness, surface roughness, heat transfer coefficient, etc.) change very slowly, whereas these same variables change rapidly at the edge of contact. In general, a discretization performed by DEFORM using automeshing routines will concentrate too many nodes within a contact zone, and too few along the edges. Even if a user manually defines mesh density in the initial discretization, this is not preserved for remeshing. A typical forging simulation commonly involves three to four remeshings. There are two basic problems to consider. First, it is possible for workpiece material to come into contact with the tooling along a finite width, defining a new contact patch. Second, as patches expand, the book keeping associated with tribology nodes must be maintained. For initial contact, the approach followed here is to define tribology nodes in the initial patch based on the existing finite element discretization. The number of nodes defined is somewhat arbitrary, but the implementation described here defined one tribology node for every three finite element nodes in contact patches, but always defined at least two nodes. For evolving contacts, two new nodes are “launched” at each contact patch edge every timestep. This requires calculation of the contact patch edge, a routine for which was written and sample patch evolution is shown in Figure 4.1 for a ring compression test. For complicated forging simulations, or for situations where a very large number of timesteps is desired, this may lead to an overly burdensome number of tribology nodes,

Page 42

and the associated calculations may unnecessarily extend the simulation time. A solution is to run the tribology module every few timesteps, with the number of timesteps being a user defined variable. Both friction and heat transfer models depend on the computation of film thickness and real contact area. In calculating a film thickness for new tribology nodes, the main difficulty is in the value of entrainment velocity used. As can be seen from Figure 4.1, the derivative of the edge of contact is not continuous. This is due to nodes from the DEFORM discretization contacting the tooling, resulting in a local jump in the contact patch. As a result, the velocity of the contact patch needs to be averaged over a number of timesteps, and the new contact points are given the initial value of film thickness for the center of the patch for the first few timesteps.

Figure 4.1: Contact patch evolution for a ring compression test. (a) Outer and inner edges of the ring as a function of timestep. (b) Evolution of tribology nodes for the initial stages of a ring compression test simulation.

Lubricant rheology is also somewhat unknown, especially for hot forging lubricants at relevant temperatures. Chapter 2 provided a number of film thickness relationships for different lubricant rheology models, all of which have their shortcomings and difficulties in application. However, it is reasonable to use piezoviscous approaches for most cold forging liquid lubricants, isoviscous methods for Page 43

certain cold forging lubricants or hot forging lubricants such as glasses, and constant shear stress models for solid lubricants such as waxes, soft metals, polmers, or for hot forging, graphite or molybdenum disulfide. Unfortunately, neither the high-pressure nor high-temperature properties of metal forming lubricants is usually well defined. In this proposal, cold working simulations presented in this proposal use lubricants with the piezoviscous properties as defined by Schmid and Wilson [1995]. Hot forging lubricants are defined experimentally. Concurrent Technologies Corporation (CTC) measured the residual lubricant on a Rene-88 ring compression test specimen at (1850°F) and tooling and obtained a film thickness of 10µm (394µin.). For the isoviscous case, the film thickness developed is: Ê 3m Vx 2 ˆ h0 = Á 0 i ˜ Ë s ¯

1/ 3

An expression for viscosity can be obtained from this relationship as follows:

m0 =

h03s 3Vx 2i

At 1850°F, the strength of the Rene-88 was measured by CTC as 13.5 ksi. Using V=0.02 in./s, and x=0.375 in., the viscosity of the glass is determined to be

m0 =

( 394min )3 (13.5ksi) 2

3(0.02)( 0.375in)

= 9.786 ¥ 10 -5 lb - s / in 2

or m0=0.675 Pa-s. This can be compared to the viscosities of a variety of glasses shown in Figure 4.2, and is a reasonable value for glasses at the temperatures considered. In its implementation, a user can define lubricant properties and use the theoretical framework of Chapter 3 to determine the film thickness. An alternate approach is to specify a known film thickness from ring compression or upsetting tests and define the viscosity based on this thickness. The roughness and real contact area can then be computed, leading to estimates of friction and heat transfer as described in Chapters 2 and 3.

Page 44

Figure 4.2 Viscosity of glass lubricants as a function of temperature, as reported by Schey [1985].

It is assumed for all lubricants that a limiting shear stress can be modeled using a viscoplastic approach (see Hamrock 1994). Values of friction factors ml are determined from ring test experiments (ml=0.15) where a very small reduction in height has taken place. For these cases, the film thickness should be very large, and this value will be a reasonable approximation. For the asperity/tooling contact, a value of m a=1 has been assumed. It is recognized that at the elevated temperatures involved in hot forging, the workpiece material will strongly adhere to the tooling. This is especially clear when it is recognized that nascent material from the workpiece substrate is exposed during deformation, and this oxide- and contaminent-free surface is extremely chemically reactive, Further, CTC reported that a dry friction test results in the ring welding to the tooling, suggesting that very high friction is involved. 4.2

Tribology Module Organization The tribology module is called by DEFORM every time an element edge requires a

friction coefficient and/or heat transfer coefficient. A flowchart of the tribology module is given

Page 45

in Fig. 4.3. For a new patch, the routine undergoes a different set of calculations as for an evolving patch.

4.2.1 New Contact Patches A “new” patch refers to a contact in the first few timesteps of contact, and is not restricted to the first timestep of contact. A patch is considered “new” until it has been in contact sufficiently long to allow estimation of a patch spread velocity. As described above, this is problematic unless the spread velocity is averaged over a number of timesteps. When a patch is new, the complicated Reynolds equation calculations for film thickness are greatly simplified. Since the spread velocity is unknown, the tribology module assigns the initial film thickness at the patch center to the edge tribology nodes for the first few timesteps. 4.2.2 Old Contact Patches After several timesteps, the steady state routine can be applied since an entraining velocity can be accurately estimated. A flowchart of the calculations in the steady state routine for existing contact patches is given in Figure 4.3. It should be noted that the mapping steps referred to in Fig. 4.3 are required to transfer values between the DEFORM discretization and that of the tribology module. This is accomplished using simple first order LaGrange interpolation in the current user routine, although second order interpolation can be easily implemented in future versions if this is deemed necessary. 4.3

Heat Transfer Routine

The heat transfer routine is straightforward once the tribology module has been developed. That is, the heat transfer routine has the same concerns of patch evolution and film thickness calculation, since these are necessary inputs to the simulation.

Page 46

Figure 4.3: A flowchart for the tribology module.

However, it should be noted that the heat transfer routine presented in Chapter 3 used the fundamental assumption that contact patches are far enough apart that they do not interfere with each other. During evaluation of the routine, it was found that the heat transfer coefficients are very reasonable for low fractional contact areas, and this was the basis for the results presented in Chapter 3. However, as the fractional contact area becomes large, the heat transfer coefficient is clearly excessive. The heat transfer coefficient approaches infinity as the fractional contact area approaches unity. While it can be argued that asperities are persistent and the fractional contact area never reaches this problematic value, it was also seen in simulations that the heat transfer coefficient predictions are unreasonable for fractional contact areas above 0.5 or so. Since the heat transfer from adjacent contact patches are no longer independent at such large contact area ratios, this result is not too surprising. The long-term solution is to develop a contact model for large fractional contact areas, and this is discussed below. In the current implementation, an upper bound on the heat transfer coefficient is defined in the heat transfer routine to avoid errors associated with large contact areas.

Page 47

5.

Examples Forging is the controlled deformation of metal using compressive stresses. There is

usually a distinction between open die forging (where flat or other simple tooling shapes are used) and closed or impression die forging (where quite intricate dies can be used). Open die forging is always performed hot, while closed die forging may occasionally be done on cold workpieces.

It should be noted that “hot” and “cold” are relative terms, defined by the

homologous temperature ratio [Kalpakjian and Schmid, 2003]. Hot working refers to situations where the workpiece temperature is greater than one-half the melting temperature, cold working to situations where the workpiece temperature is less than one-third the melting temperature. Cold working will commonly involve significant preheat of the workpiece to take advantage of softening and ductility changes in the material, and heat transfer remains an important effect. However, for the purposes of this proposal, “hot forging” refers to situations where a temperature difference exists between workpiece and tooling. Isothermal or cold forging refers to situations where negligible heat transfer occurs between workpiece and tooling, a situation that reflects either a very high preheat or room temperature forging.

5.1

Cold/Isothermal Forging

Ring compression tests (RCT) are very popular for examining friction in forging operations. It has long been recognized that under low friction conditions, a ring will expand when reduced in height, while under high friction conditions, the inner diameter will decrease, as shown in Fig. 5.1. A number of analytical models have developed predictions of friction coefficient or friction factor in RCTs as a function of internal diameter, as shown in Fig. 5.2. It is long recognized that a constant value of friction is an inherently flawed approach for most forging operations, but because of its popularity, RCT simulations provide a valuable benchmark for simulations.

Page 48

(a)

(b)

Fig. 5.1 Deformation of a ring in upsetting with (a) low

Fig. 5.2

friction and (b) high friction.

coefficient friction from DEFORM

5.1.1

Calibration curve for extracting

Cold Ring Compression Test (RCT) Forging with Constant Friction

Factor

A popular geometry is a ring of 6 (outer diameter): 3 (inner diameter): 2 (height) proportion. The common practice of outer diameter = 25 mm (1 in.), inner diameter = 12.5 mm (0.5 in.) and height of 8.33 mm (0.33 in.) was assumed. Aluminum alloy 6061 was first analyzed at room temperature conditions. A number of simulation case with constant friction factor m were performed, Figure 5.3 shows the strain evolution of the case with m = 0.2. The inner diameter increases by a little first for small height reduction (less than 50%), and begin to decrease when height reduction is bigger than 50% and both internal and external surfaces barrel (Fig. 5.1b). Table 5.1 shows a comparison between simulations with different friction factors but all other process parameters are held constant. As can be seen, the maximum strain and load usually increase with increasing friction factor, while strain rate is maximum at an intermediate value of friction factor.

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(a)

(b)

(c)

(d)

Fig. 5.3 Strain evolution for cold forging of aluminum alloy 6061 in a ring compression test geometry, using constant friction factor m = 0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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Table 5.1

Max strain

Max strain rate

Max load

Friction factor

(m/m)

(m/m)/sec

(klbf)

m = zero

1.5433

3.5063

85.754

m = 0.05

1.9230

4.9738

107.192

m = 0.10

2.1984

5.9473

123.908

m = 0.15

2.2924

8.4605

146.220

m = 0.20

2.3432

9.7003

163.448

m = 0.25

2.5009

11.783

191.461

m = 0.30

2.5747

6.6792

213.596

m = 0.40

2.6518

7.9271

260.099

m = 0.50

2.6667

8.6152

298.522

Result for cold forging of aluminum alloy 6061 in a ring compression test with

constant friction factors.

5.1.2

Cold Ring Compression Test (RCT) Forging with Tribology Model

For cases with the tribology model of Chapter 2, a lubricant shear factor of ml=0.2 and an asperity friction factor of ma=1.0 were assumed. Figure 5.4 shows the resulting strain evolution. The strain is fairly uniform until a height reduction of around fifty percent is attained. The inner diameter increases until around a sixty percent reduction. Figure 5.5 shows the strain rate distribution for the same simulation. Figure 5.4 is nearly identical with Figure 5.3, especially when reduction is less than 50%, as seen in Figure 5.6. Figure 5.6 depicts a contour plot of friction values at the surface. This type of plot is the best available method of examining user-defined variables in the DEFORM post processor, and it is at times awkward and counterintuitive. It is important to examine surface values and not put much emphasis on the shapes of contours, as these can be affected by a non-uniform mesh below the surface. Figure 5.7 was prepared from Figure 5.6 and the results for film thickness to show the evolution of each variable. At first, the lubricant used generates a very large film thickness, resulting in a uniform low value of friction across the ring compression test geometry. As discussed in part 2.4, friction is a function of nondimensional film thickness. When film thickness decreases on the outer surface in Figure 5.7a and b, the roughness decreases as well, so the friction remains constant. As the specimen is reduced in thickness, some new FEM nodes will

Page 51

(a)

(b)

(c)

(d) Fig. 5.4 Strain evolution for cold forging of aluminum alloy 6061 in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(b)

(c)

(d)

Fig. 5.5 Strain rate evolution for cold forging of aluminum alloy 6061 in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(b)

(c)

(d)

Fig. 5.6 Friction coefficient evolution for cold forging of aluminum alloy 6061 in a ring compression test geometry, using ma=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.7 Film thickness & Friction coefficient evolution for cold forging of aluminum alloy 6061 in a ring compression test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60 % reduction in thickness; (d) 80% reduction in thickness

Page 55

“roll” onto tooling, and the lubricant film for these nodes are much smaller (as seen from Eq. 2.20), and friction increases markedly. Figure 5.8 shows the load-stroke results for the ring compression test using the tribology module. Load-stroke curves are also common benchmarks, since these are fairly easy to obtain in experiments. The value of m l=0.2 in Fig. 5.8 is somewhat arbitrary. Ideally, the value should be estimated from lubricant rheology measurements, or at the very least, from ring compression tests performed at very low reductions in thickness in order to minimize the effects of asperity contact. However, the results are somewhat sensitive to the lubricant shear stress, as shown in Table 5.2. The effects of lubricant shear stress can also be seen from Fig. 5.9, where the predictions for internal diameter in a ring compression test are shown as a function of reduction in thickness. For comparison purposes, results from constant friction factor simulations are also depicted. Using a high film thickness (as described in Chapter 4), the departure from constant friction factor simulations does not occur until large reductions in height have occurred. It should be noted that this departure can occur at smaller reductions in height for different lubricants.

Fig. 5.8 Load-Stroke for cold forging of aluminum alloy 6061 in a ring compression test geometry, using ma=1.0 and ml=0.2.

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Table 5.2 Comparison when ml = 0.15 or 0.2 and ma = 1.0. Max

Max strain rate

Max load

strain

(m/m)/sec

(klbf)

ml = 0.15

2.5823

14.853

162.6

ml = 0.2

2.6668

12.854

186.4

Figure 5.9 Results from cases using constant friction factor and tribology model and experiment by CTC.

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The results shown in Figures 5.4 through 5.9 are intuitive; it is expected that the friction should increase as height reduction increases. This can be attributed to the sides of the ring compression test ‘rolling’ into contact with the tooling, and is in agreement with the mechanisms described by Wilson [1986].

5.1.3 Isothermal Ring Compression Test (RCT) Forging with Tribology Model

Nickel-based superalloys are commonly used in the aerospace industry. However, these materials are difficult to form, and forging is commonly performed at very high preheat and with hot tooling. Lubricants are a serious concern, and usually solid lubricants and glasses are the only options available. At elevated temperatures, it is reasonable to consider glass as an isoviscous liquid. Figures 5.10 through 5.15 depict the results of an isothermal ring compression test simulation for Rene 88 (a nickel-based superalloy commonly used for turbine disks and other aerospace parts) using a glass lubricant. The stress-strain curve for Rene 88 is given in Fig. 5.10. The geometry for this simulation is: outer diameter = 19 mm (0.75 in.), inner diameter = 9.5 mm (0.375 in.) and height of 6.35 mm (0.25 in.), still in 6:3:2 proportion.

Figure 5.10

Stress-strain curve from compression testing for Rene 88. Source: Concurrent Technologies

Corporation.

Page 58

Figure 5.11 shows that internal surface stays nearly unchanged when reduction is less than 20%, then begins to decrease. Figure 5.12 shows the strain rate distribution for this same simulation. Figure 5.12c shows the possible damage area in the ring. Figure 5.13 shows a contour plot of friction values on the surface. A noticeable increase in friction occurs at height reductions of around 60%. At lower reductions, friction remains the initial value. Figure 5.14 shows the film thickness and friction on the surface.

5.2

Hot Forging

Many materials are difficult to form at room temperature, but have acceptable forgeability at elevated temperatures, and do not need the extremes of isothermal forging. Tooling is rarely heated to the same temperature as the workpiece, so cooling effects can be important. This section examines a number of hot forging circumstances, and incorporates both the friction model from Chapter 2 and the heat transfer model from Chapter 3. 5.2.1

Ring Compression Test with Constant Friction and Heat Transfer

Coefficient The geometry for the following RCTs is the same as described in Section 5.1.3. Figure 5.16 depicts the strain evolution for ring compression test simulations using titanium alloy Ti-4V-6Al at a workpiece temperature of 1800°F and tooling temperature of 400°F, with a constant friction factor of m = 0.2 and constant heat transfer coefficient of U = 5 kW/m2K. Figure 5.17 shows the strain evolution for a simulation where the heat transfer coefficient U has been changed to 20 kW/m2K

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(a)

(b)

(c)

(d)

Fig. 5.11 Strain evolution for isothermal forging of Rene 88 in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.12 Strain rate evolution for isothermal forging of Rene 88 in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.13 Friction coefficient evolution for isothermal forging of Rene 88 in a ring compression test geometry, using ma=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.14 Film thickness & Friction coefficient for isothermal forging of Rene 88 in a ring compression test geometry, using ma=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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Figure 5.15 Load-stroke for isothermal forging of Rene 88 in a ring compression test geometry, using ma=1.0 and ml=0.2.

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(a)

(b)

(c)

(d)

Fig. 5.16 Strain evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using constant m = 0.2 and U = 5 kW/m2K. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.17 Strain evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using m = 0.2 and U = 20 kW/m2K. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

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Because of the temperature difference between specimen and tooling (1400 oF in this case), the heat transfer is clearly important for hot forging and can significantly affect deformation. Schey [1985] claims that the cooling and workpiece hardening associated with heat transfer between the workpiece and the tooling can be as important as friction. The results shown in Figs 5.16 and 5.17 suggest that heat transfer is more important than friction.

5.2.2

Ring Compression Test with Tribology and Heat Transfer Models

Figures 5.18 through 5.25 depict the results for ring compression test simulations using titanium alloy Ti-4V-6Al hot at a workpiece temperature of 1800°F and tooling temperature of 400°F. Figure 5.18 shows the classic barreled profile where a surface layer has been cooled due to heat transfer to the tooling. Figure 5.20 shows high levels of heat transfer at the corners because of the thin film thickness, so that the temperature of the workpiece drops much more rapidly than other areas. Figure 5.22 shows the heat transfer coefficient evolution for this simulation. Because the heat transfer coefficient changed very much, from an initial value of 5 to a peak of 100kW/m2°C we cannot see the contour line in the initial contact area in Figs 5.22c and d. The heat transfer coefficient is a function of real contact area and dimensional film thickness, and is more sensitive to film thickness than friction coefficient, which can be seen in Figs. 5.23 and 5.24. Clearly, heat transfer is a very important factor for hot forging, possibly even more important than friction. When tribology and heat transfer models were coupled in the simulations, they work together very well.

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(a)

(b)

(c)

(d)

Fig. 5.18 Strain evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using ma=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

Page 68

(a)

(b)

(c)

(d)

Fig. 5.19 Strain rate evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

Page 69

(a)

(b)

(c)

(d)

Fig. 5.20 Temperature evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.21 Friction coefficient evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.22 Heat transfer coefficient evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using ma=1.0 and m l =0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.23 Film thickness and friction coefficient evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.24 Film thickness and heat transfer coefficient evolution for hot forging of Ti-4V-6Al in a ring compression test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 76% reduction in thickness.

Page 74

Fig. 5.25 Load-stroke curve for hot forging of Ti-4V-6Al in a ring compression test geometry, using ma=1.0 and ml=0.2.

Fig 5.26 Correlation between height reduction and decrease in internal diameter.

Page 75

5.2.3

Spike Test

Spike tests are valuable because they are more demanding for the lubricant; significantly more shear and less opportunity for thick film formation exists than with ring compression tests. A number of spike test simulations have been performed. Figures 5.27 through 5.32 depict results for Ti-6Al-4V at 1800°F and with tooling temperature of 500°F. Figures 5.27 through 5.32 were valuable for a number of reasons. First, they demonstrate the utility of the tribology module for situations where a thick lubricant film is not dominant, as in the ring compression test. This is obviously of greater industrial relevance than a ring compression test. Also, this demonstrated the stability of the tribology module for more complicated geometries, and demonstrated that the tribology patches do not require horizontal tooling.

Page 76

(a)

(b)

(c)

(d)

Fig. 5.27 Strain evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 90% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.28 Strain rate evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 90% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.29 Temperature evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 90% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.30 Friction coefficient evolution for hot forging of Ti-4V-6Al in a spike test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 90% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.31 Heat transfer coefficient evolution for hot forging of Ti-4V-6Al in a spike test geometry, using m a=1.0 and m l=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 90% reduction in thickness.

Page 81

Fig. 5.32 Load-Stroke for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=0.2.

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5.2.4

Emboss Test

A more demanding geometry is the emboss test geometry. This geometry also is valuable because of the sharp die radius and associated strain concentration. The material used for the simulations in Fig. 5.33 through 5.39 was Ti-4V-6Al under the same conditions as in Section 5.2.2 above. Because of the small die radius, the finite element discretization of the workpiece in the vicinity of the die has to be very fine to obtain good results. As the workpiece strains, this forces DEFORM to perform a number of remeshings. The results shown in this section were for simulations that encountered twelve remeshings. An embossing test’s main value is that the height of the emboss is thought to correlate with lubricant effectiveness. Qualitative comparisons between experiments are possible, but with the tribology module, more elaborate quantitative evaluations of lubricants can be performed.

Page 83

(a)

(b)

(c)

(d)

(e) (f) Fig. 5.33 Strain evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=0.2. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 70% reduction in thickness; (e) 75% reduction in thickness; (f) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.34 Strain rate evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=015. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.35 Temperature evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=015. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.36 Friction coefficient evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and m l=015. (a) 20 % reduction in thickness; (b) 40% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

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(a)

(b)

(c)

(d)

Fig. 5.37 Heat transfer coefficient evolution for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and m l=015. (a) 20 % reduction in thickness; (b) 30% reduction in thickness; (c) 60% reduction in thickness; (d) 80% reduction in thickness.

Page 88

Fig. 5.38 Load-Stroke for hot forging of Ti-4V-6Al in a spike test geometry, using ma=1.0 and ml=015.

Fig. 5.28 Friction coefficient evolution for Ti_4V-6Al Emboss thermal forging.

Fig. 5.39 Correlation between total reduction and boss height, simulation and experiment by CTC.

Page 89

References

Antonetti, V. W., and Eid, J. C., 1987, “A Technique for Making Rapid Thermal Contact Resistance Measurements”, Proc. Int. Symp. Cooling Tech. Elect. Equip., W. Aung and P. Cheng, eds., Honolulu,, Mar. 17-21 1987, pp. 449-460. Barker, D., Goetz, R.L., and Fagin, P., Characterization of the Forging Die-Workpiece Interface Friction and Heat Transfer Between Alloy Steel and Alloy Ti-6Al-4V”, Report to Concurrent Technologies Corp., August 31, 2000. Boër, C.R., Rydstad, H., and Schröder, G., 1985, “Choosing Optimal Forging Conditions in Isothermal and Hot-Die Forging,” J. Applied Metalworking, v. 3, pp. 421-431. Boyer, H.E., and Gall, T.L., eds., Metals Handbook, 1985. Metals Park (OH), American Society for Metals. Chen, G., Shen, H, Hu, S., and Baudelet, B., “Roughening of the Free Surfaces of Metallic Sheets during Stretch Forming,” Mat. Sci. Eng., 1990, pp. 33-38. Christensen, H., “Stochastic Models for Hydrodynamic Lubrication of Rough Surfaces,” Proc. IMechE, 1970, pp. 1013-1022. Dadras, P. and Wells, W.R., 1984, “Heat Transfer Aspects of Nonisothermal Axisymmetric Upset Forging,” J. Engineering for Industry, v. 106, pp. 187-195. Drucker, D.C., and Prager, W, and Greenberg, H.J., “Extended limit design theorems for continuous media”, Q.J. Appl. Math., v. 9, 1952, pp. 381-389. Hamrock, B.J., Fundamentals of Fluid Film Lubrication. New York, McGraw-Hill, 1994. Hosford, W., and Caddell, R.M., Metal Forming. Prentice-Hall, 1983. Incropera, F.,P., and DeWitt, D.P., 1996, Fundamentals of Heat Transfer. New York, Wiley & Sons. Johnson, W., Plane Strain Slip Line Fields for Metal Deformation Processes. Oxford, Pergamon Press, 1982. Kalpakjian, S., and Schmid, S.R., Manufacturing Processes for Engineering Materials, 4th ed. Prentice-Hall, 2003. Kobayashi, S., .and Altan, T., Metal Forming and the Finite Element Method. Oxford University Press, 1989. Kellow, M.A., Bramley, A.N., and Bannister, F.K., 1969, “The Measurement of Temperatures in Forging Dies,” International Journal of Machine Design and Research, v. 9, pp. 239-260. Korzekwa, D.A., Dawson, P.R., and Wilson, W.R.D., “Surface Asperity Deformation during Sheet Forming,” Int. J. Mech. Sci., 1992, pp. 521-540.

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Lange, K., ed., Handbook of Metal Forming. Society of Manufacturing Engineers, 1985. Lee, W., Surface Roughening in Metal Forming Processes. Ph.D. Thesis, Northwestern University, 1996. Malinowski, Z., Lenard, J.G., and Davies, M.E., 1994, “A Study of the Heat-transfer Coefficient as a Function of Temperature and Pressure,” J. Materials Processing Technology, v. 41, pp. 125-142. Mentes, A., Veziroglu, T. N., Samudrala, R., Sheffield, J. W., and Williams, A., “Effects of Interface Gases on Contact Conductance”, AIAA Paper No. 81-0214, 1981. Nshama, W., and Jeswiet, J., 1995, “Evaluation of Temperature and Heat Transfer Conditions at the Metal-Forming Interface,” Annals of the CIRP, v. 44, pp. 201-203. Osakada, K., and Oyane, M., “On the Roughening of Free Surface in Deformation Processes,” Bulletin of the JSME, v. 14, 1971, pp. 171-177. Patir, N., and Cheng, H.S., “An Average Flow Model for Determining Effects of ThreeDimensional Roughness on Partial Hydrodynamic Lubrication,” J. Lubrication Technology, v. 100, 1978, pp. 12-17. Schmid, S.R., and Wilson, W.R.D., “Lubrication of Aluminum Rolling by Emulsions,” Tribology Transactions 38 (1995): 452-458. Schmid, S.R., and Wilson, W.R.D., “Tribology in Manufacturing” in Handbook of Modern Tribology, CRC Press, 2001, pp. 1385-1411 Snaith, B., O'callaghan, P. W., and Probert, S.D.,

“Can Standards Be Set for Reliable

Measurements of Thermal Contact Conductance”, AIAA Paper No. 83-0533, 1983. Sutcliffe, M.P.F., “Surface Asperity Deformation in Metal Forming Processes,” Int. J. Mech. Sci., v. 30, 1988, pp. 847-868. Taylor, R.E., “Thermal Transport-Property and Contact-Conductance Measurements of Coatings and Thin Films”, Int. J. Thermophysics, v. 19, 1998, pp.931-939. Tokizawa, M., and Yosikawa, Y., “The Mechanism of Lubricant Trapping Under the Cold Compression of Metals,” J. Jpn. Inst. Met., v. 37, 1973, pp. 19-25. Tong, W., Hector, L.G., Weiland, H., and Wieserman, L.F., “In-Situ Surface Characterization of a Bnary Aluminum Alloy During Tensile Deformation,” Scripta Materialia, 1997, pp. 1339-1344. Tsai, Y. M. and Cran, R.A., “An Analytical Solution of a One Dimensional Thermal Contact Conductance Problem With One Heat Flux and One Insulated Boundary Condition”, J. Heat Trans., 1992, pp. 503-505.

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Wilson, W.R.D., “Friction and Lubrication in Bulk Metal Forming Processes,” J. Appl. Metalworking, 1979, pp. 1-19. Wilson, W.R.D., Modeling Friction in Sheet-Metal Forming Simulation,” in Zabaras, N., et al., eds., “The Integration of Material, Process, and Product Design,” A.A. Balkema, 1999, pp. 139-147. Wilson, W.R.D., 1986, “Friction Modeling in Forging,” in Arsenault, R.J., Beeler, J.R., and Esterling, D.M., eds., Computer Simulation in Materials Science, American Society for Metals, pp. 237-267. Wilson, W.R.D., Huang, X.B., and Hsu, T.C., 1995, “A Realistic Friction Model for Computer Simulation of Sheet Metal Forming Processes,” J. Eng. Ind., pp. 202-209. Wilson, W.R.D., and Marsault, N., “Partial Hydrodynamic Lubrication with Large Fractional Contact Areas,” J. Tribology, v. 120, 1998, pp. 16-20. Wilson, W.R.D., and Sheu, S., “Flattening of Workpiece Asperities in Metal Forming,” Proc. NAMRC XI, Society of Manufacturing Engineers, 1983, p. 172-178. Wilson, W.R.D., and Schmid, S.R., “Surfaces in Metal Rolling,” in PED-v. 62, Engineered Surfaces. Ehmann, K.F., and Wilson, W.R.D., eds, New York, ASME, 1992, pp. 91-100. Wilson, W.R.D., and Stilleto, J.G., “Surface Roughening in Liquid Lubricated Upsetting,” in Metalworking Lubrication. New York, American Society of Mechanical Engineers, 1980, pp. 87-90.

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Appendix B: Structure of File PARA.INI PARA.INI is the data file that contains the initial values of tribology module variables. The default version is listed in Figure A.1. It is a compact file, but contains enough information to direct the tribology module through a wide variety of cases. In practice, some of the variables in PARA.INI will be modified to fit the particular needs of the user. This Appendix contains notes regarding each of the variables in PARA.INI. The main variables in PARA.INI are the following:

•

Line 1, Strain gain. This parameter reflects the fact that materials roughen under the effects of strain, and that this roughening is a complicated function of grain size and morphology, crystal structure, oxide presence, etc. This is explained in Section 2.3 of Appendix A, and defined in Eq. (2.22). This parameter must be obtained experimentally. However, as reference values, it can be noted that the values of 0.54 µm was obtained in cold rolling of Aluminum 5052-O and a value of 1.1 µm was obtained in ring compression testing of Rene-88.

Line No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Contents 0.15 The strain gain 0.2 Friction factor for lubricant 1.0 Friction factor for asperities 0.00003937 Initial tooling roughness in inches 0.00009843 Initial workpiece roughness in inches 0.0003937 Initial film thickness in inches 0.0003937 Spacing between asperities 0.0003937 Characteristic length for heat flow into workpiece 0.0000006692 Thermal conductivity for lubricant 0.0008545 Thermal conductivity for tooling 0.0006686 Thermal conductivity for workpiece 0.000048 Viscosity of lubricant 0.0033 Pressure coefficient of viscosity 3.0 Peklinik surface pattern parameter 13.5 Material flow stress for workpiece in Ksi 100 Runge-kutta parameter 2 The plastic object number, 1 2 or 3 2 1: isoviscous ; 2: isothermal 1 1 : for Sinusoidal ; 2 : for Christensen's surface 2 1 : without output data ; 2 : output data in case remesh

Figure A.1 Contents of the file PARA.INI.

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It is usually the case that this variable is not too important; the surfaces roughen during deformation, but very quickly contact the tooling. Thus, workpiece roughness is usually controlled by tooling surface roughness. This behavior is incorporated into the tribology model.

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Line 2, Friction factor for lubricant. This can be thought of as the friction factor that would result when very large lubricant films are generated. The rational for this approach varies with the type of lubricant used. For liquid lubricants, it is well-known that there is a departure from Newtonian behavior at high shear strain rates, and that the shear strain rates in metal forming are certainly high enough to cause this departure. A number of models have been proposed for this non-Newtonian behavior, as discussed in Hamrock [1994], the simplest of which is to use a pseudo-plastic model that has a limit in the shear stress that can develop. The value of the limiting shear stress can be obtained through high-pressure and shear strain rate experiments, but there are no such experiments known at typical hot forging temperatures. Another approach can be used to experimentally determine this value, namely the performance of ring compression tests at high speeds and low reductions in thickness. As can be seen from Section 5 of Appendix A, the friction for this case is almost fully developed in the fluid; the resulting friction factor is a reasonable estimate for the limiting shear stress in the liquid. For solid lubricants, the lubricant friction factor can be thought of as the ratio of the shear stress that develops in the solid and the shear strength of the workpiece material. These also can be measured on special test equipment, or ring compression tests can be conducted at low reductions in height.

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Line 3, friction factor for asperities. This represents the friction stress that develops in direct workpiece-tooling contact in the absence of lubricant. In theory, this can be directly evaluated through dry ring compression test experiements, but in practice, this will lead to welding of the workpiece to the tooling, especially for hot forging. The value of unity represents full development of the workpiece shear strength, which is reasonable for complete welding.

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Line 4 and Line 5, initial roughnesses. These can be directly measured on the workpiece and tooling using a surface profilometer. They can also be estimated from the following chart, obtained from Kalpakjian and Schmid [2003]:

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Line 6, Initial film thickness. A lubricant is applied to the workpiece before forging; the thickness of the lubricant can be very large, and in practice, this lubricant layer will be difficult to remove from a flat section coming in contact with the tooling. This can be measured in a number of ways, and the default value of 10 µm was obtained from a measurement by Concurrent Technologies Corporation.

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Line 7, The spacing between the asperities is used by the heat transfer module. This value should be set between 10 and 50 times the composite workpiece and tooling roughness.

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Line 8, Characteristic length for heat flow into the workpiece. As defined in Eq. (3.14) of Appendix A, this is the distance into the workpiece at which heat flow is uniform and

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normal to the surface. It is defined as equal to the asperity spacing. Further research needs to be conducted to obtain better values, but this setting is reasonable.

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Line 9-11, Thermal conductivities. These are physical constants of the materials. Some examples are:

METAL Aluminum Aluminum alloys Berylium Columbium (niobium) Copper Copper alloys Gold Iron Steels Lead Lead alloys Magnesium Magnesium alloys Molybdenum alloys Nickel Nickel alloys Silicon Silver Tantalum alloys Titanium Titanium alloys Tungsten LUBRICANT Graphite Fused silica or glass Teflon Paraffin (wax form)

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THERMAL CONDUCTIVITY (W/m K) 222 121-239 146 52 393 29-234 317 74 15-52 35 24-46 154 75-138 142 92 12-63 148 429 54 17 8-12 166 5.70 1.38 0.35 0.240

Line 12-13, Viscosity of lubricant and pressure exponent of viscosity. These are the required variables for use in a Barus law, where the viscosity is given by

h=h0exp(gp)

(B.1)

The values in lines 12-13 are ignored for isoviscous lubricants. Some values for cold metal working fluids are:

Designation

h0 (Ns/m2)

g (m2/N)

Normal paraffin (approx 15

0.331

4.74 x 10-7

carbons per molecule) 3.4067 x 10-8

Thick petroleum (Thuban 250, 7.08 Texaco) Vacuum-distilled petroleum

6.3073 x 10-7

0.687

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Line 14, Peklinik surface pattern parameter. This is described in Figure 2.3 of Appendix A.

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Line 15, Material flow stress for workpiece in ksi. DEFORM does not make this material property available to the user routines. However a rough estimate of material flow stress is needed for film thickness calculations, so this value needs to be entered here. This should represent the flow stress at the forging temperature.

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Line 16, Runge-Kutta parameter. This is the maximum number of steps that will be taken for each internal tribology node to update the film thickness. The selection of the proper value is the classic one for numerical modeling; very large values will lead to more accurate simulations, but also will drastically increase the computation time. Very small numbers will give poor simulations.

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Line 17, Plastic object number. The user routine is not informed by DEFORM which of the objects defined in the preprocessor is the workpiece. This must be indicated here.

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Line 18, Lubricant behavior. The film thickness that develops at the edge of contact can be isoviscous or piezo-viscous. An isoviscous model is better suited for small or shortchained molecules where there is negligible increase in viscosity with pressure. A piezoviscous model is better suited for long chain molecules such as oils or polymers. Glasses are though to be far more dependant on temperatures than pressures, so these are approximated as isoviscous in this report.

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Line 19, Surface Type. A number of surface types have been described in Section 2.4 of Appendix A. The Christensen model was derived for Gaussian surfaces, and is probably acceptable for most simulations.

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Line 20, Remeshing Expectations. For complicated die geometries, DEFORM will remesh often, leading to difficulties in maintaining the data for the tribology module. The solution is to store the tribological data in a file in case of remeshing. However, for simple dies, this precaution can unnecessarily increase run times, especially if batch jobs have been defined. It is probably wise to use remeshing precautions unless absolutely certain it will not occur.

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