12

T H E R M O - V I S C O P L A S T I C A N A L Y S I S

12.1 Introduction The main concern here is the analysis of plastic deformation processes in the warm and hot forming regimes. When deformation takes place at high temperatures, material properties can vary considerably with temperature. Heat is generated during a metal-forming process, and if dies are at a considerably lower temperature than the workpiece, the heat loss by conduction to the dies and by radiation and convection to the environment can result in severe temperature gradients within the workpiece. Thus, the consideration of temperature effects in the analysis of metal-forming problems is very important. Furthermore, at elevated temperatures, plastic deformation can induce phase transformations and alterations in grain structures that, in turn, can modify the flow stress of the workpiece material as well as other mechanical properties.

Since materials at elevated temperatures are usually rate-sensitive, a complete analysis of hot forming requires two considerations---the effect of the rate-sensitivity of materials and the coupling of the metal flow and heat transfer analyses.

A material behavior that exhibits rate sensitivity is called viscoplastic. A theory that deals with viscoplasticity was described in Chap. 4. It was shown that the governing equations for deformation of viscoplastic materials are formally identical to those of plastic materials, except that the effective stress is a function of strain, strain-rate, and temperature. The application of the finite-element method to the analysis of metal-forming processes using rigid-plastic materials leads to a simple extension of the method to rigid-viscoplastic materials [1].

The importance of temperature calculations during a metal-forming process has been recognized for a long time. Until recently, the majority of the work had been based on procedures that uncouple the problem of heat transfer from the metal deformation problem. Several researchers have used the following approach. They determined the flow velocity fields in the problem either experimentally or by calculations, and they then used these fields to calculate heat generation. Examples of this approach are the works of Johnson and Kudo [2] on extrusion, and of Tay et al. [3] on machining. Another approach [4] uses Bishop's numerical method in which

222

Thermo-Viscoplastic Analysis 223

heat generation and transportation are considered to occur instantaneously for each time-step with conduction taking place during the time-step. This technique was used by Altan and Kobayashi [5] on extrusion; by Lahoti and Altan [6] on compression and torsion; and by Nagpal, et al. [7] on forging. Usually the temperature calculations are done by using finite differences, or finite elements, and the upper-bound technique is the most common method for determining flow patterns theoretically.

In order to handle a coupled thermo-viscoplastic deformation problem, it is necessary to solve simultaneously the material-flow problem for a given temperature distribution and the heat transfer equations. Numerical solutions of such forming problems were discussed by Zienkiewicz et al. [8] with examples of steady flow in extrusion, drawing, rolling, and sheet- metal forming. Zienkiewicz et al. [9] made a coupled thermal analysis of steady-state extrusion. Rebelo and Kobayashi [10, 11] developed the method for a coupled analysis of transient viscoplastic deformation and heat transfer. They applied the method to solid cylinder compression and ring compression.

12.2 Viscoplastic Analysis of Compression of a Solid Cylinder There are a number of materials that exhibit viscoplastic behavior. They include most metals at high temperature, superplastic materials, heated glass, and polymers. When the deformation is large, most of them can be considered to be rigid-viscoplastic.

Because of the importance of the application of viscoplastic behavior to the metal-forming processes, the treatment of time-dependent material behavior within the framework of the theory of viscoplasticity is the subject of several recent studies. Cristescu [12, 13] applied the theory to the upper-bound approach in drawing. Zienkiewicz and Godbole [14, 15] have shown the feasibility of the finite-element approach in the deforma- tion analysis of rigid-viscoplastic materials by treating them as non- Newtonian viscous fluids. Price and Alexander [16, 17] have applied this formulation to creep forming.

There are only a few references in the literature in which strain-rate effects in non-uniform deformation are explicitly in evidence. Among them, the work by Klemz and Hashmi [18] on simple upsetting of lead cylinders at room temperatures provides good experimental results for comparison with the finite-element solution, since lead is strain-rate sensitive at room temperatures. A cylinder having a diameter of 25.4 mm (1.0in.) and a height of 24.13mm (0.95 in.) was considered. The stress, strain, and strain-rate data for pure lead at room temperature were taken from Reference [18] and from experimental results by Loizou and Sims [19]. The static stress-strain curve was approximated by fitting experimen- tal data to curves of the form 0 = Y[1 + (~/y)m], and the obtained values of strain-rate exponent m and ym were interpolated in the program for intermediate strain values. The friction factor m was taken as 0.06 (m = a

224 Metal Forming and the Finite-Element Method

fraction of the static yield shear strength), in agreement with the assumed value in Reference [18].

For the finite-element formulation of viscoplastic deformation, a par- ticular attention should be paid to the strain-rate sensitivity of the flow stress of the material. The linearized stiffness equations (6.4) or (6.48) in Chap. 6 contains the second derivative of the functional given by eq. (6.44), in which the effective stress 0 is now a function of the effective strain-rate ~ as well as the effective strain g.

The calculations simulate the deformation of the cylinder by a drop- hammer, with a tup mass of 35.5 ib, hitting the specimen at a speed of 30 ft/s. The assumption of a quasistatic process is justified for these velocities [20]. Accordingly, the calculated work done by the contact forces between tup and workpiece in each step of deformation was subtracted from the kinetic energy of the tup, and the process ended when the tup came to a stop. For comparison, the calculations were repeated without introducing strain-rate effects (making y = o0 in the formulation).

This nonsteady-state deformation problem was analyzed in a step-by- step manner by treating it quasilinearly during each incremental deforma- tion. The reduction in height at each step was 1%. The solution of the velocity field for uniform compression was used as an initial guess. The solution obtained from the previous step was then used as an initial guess for the subsequent step. Normally, seven iterations were required for the first step to reach an accuracy of IIAvll/llvll _< 0.000 05. For subsequent steps, only two to four iterations were necessary to reach the same accuracy.

The calculated overall quantities are compared with experimental data and with the numerical dynamic analysis of Klemz and Hashmi [18] in Figs. 12.1 and 12.2. Some observations can be made from the figures. The results for nonstrain-rate-sensitive analysis (Fig. 12.1) tend to follow those obtained by the analysis of Klemz and Hashmi, which did not take into account strain-rate effects but was based on a dynamic analysis. The agreement of the two analyses indicates that the dynamic effect is

22,400

~ ' 17,920

I 3 , 4 0 0

8,960

I.i.I 4 , 4 8 0 _ , , , 9¢ , , ~ ' ~

0.2 0 .4 0.6

DEFORMATION ( in )

1.0

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I~ 0.4

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o

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0.8 1.6 2.4

TIME (msec) FIG. 12.1 Comparison of the finite-element rigid-plastic analysis (0) with dynamic analysis ( - - - ) [18] and experiment (--) [18].

Thermo-Viscoplastic Analysis 225

22,400

I 7,920

13,400

8,960

I.U 4)480

0.2 0.4 0.6 0 DEFORMATION ( in )

1.0

"~ 0.8

0.6

0.4

bOO.2 ' j

0.8 1.6 2.4 TIME (msec)

FIG. 12.2 Comparison of the finite element rigid-viscoplastic analysis (O) with dynamic analysis ( - - - ) [18] and experiment (--) [18].

negligible. However, both solutions show poor agreement with experi- ments. The analysis that includes strain-rate sensitivity shows good agreement with experiments (Fig. 12.2). No "rigid" zones were formed at such a low value of friction in either rate-sensitive or rate-insensitive cases.

12.3 Heat Transfer Analysis Basic Equations The basic equations and corresponding finite-element formulations for the deformation analysis are described in previous chapters. For the heat transfer analysis, we begin with the energy balance equation, expressed by

kl T, ii + ~ - pcT" = 0 (12.1)

where kl T . is the heat transfer rate, kl denotes thermal conductivity, ? is the heat generation rate, and pcT is the internal energy-rate. In eq. (12.1), the notation T . is used for T,/,,-, with the comma denoting differentiation and repeated subscript meaning summation (Laplace differential operator applied to temperature T). We consider that the heat generation in the deforming body is due only to plastic deformation

i- = roiykiy (12.2)

where the heat generation efficiency r , represents the fraction of mechani- cal energy transformed into heat and is usually assumed to be 0.9. The fraction of the remainder of the plastic deformation energy ( 1 - r ) is expended to cause changes in dislocation density, grain boundaries, and phases. This energy is usually recoverable by annealing.

Along the boundaries of the deforming material, either the temperature T is prescribed or a heat flux is given. The energy balance, eq. (12.1), can be written in the form

fvk , T i i6TdV- fvPCJ' tSTdV+~vrOOk~y6TdV=O (12.3)

226 Metal Forming and the Finite-Element Method

for arbitrary variation in temperature 6T. By using the divergence theorem, eq. (12.3) becomes

f k , T i6T, dV + fvOCT~STdV- lvrO,j~o6TdV- fs q. 6TdS=O (12.4)

where q. is the heat flux across the boundary surface Sq, n denotes the unit normal to the boundary surface, and

q. = k, T. (12.5)

Solutions to problems of this nature require the temperature field to satisfy the prescribed boundary temperatures and eq. (12.4) for arbitrary pertur- bation 6T.

Finite-Element Formulation The temperature field in eq. (12.4) is approximated by

T = ~, q,~T~, = N T T (12.6a)

where q,, is the shape function and T~ is the temperature at ath node. With the quadrilateral element shown in Fig. 8.1,

NT= {ql, q2, q3, q4} and

Tr={T~, T2, T3, T4} (12.6b)

where ql, .. •, q4 are given by eq. (8.2b) in Chap. 8. Putting

[" aq,/ax aq,/ay "]

I aq2/ax aqz/ay I M,j=N,,j or M= laq31a x aq31ay| (12.7)

L Oq J Ox Oq 4/ Oy J and substituting eq. (12.6) into eq. (12.4),

j= l

- fv, X(O,) 6TTN dV - fsq q. 6TTN dS] (12.8) I

Because of the arbitrariness of 6T, the following system of equations is obtained,

i~=~ [fvjk,MMT dV T + fvjpcNNT dVT- fv r( O~)N dV - fs,jq,N dS] = 0

(12.9)

Thermo-Viscoplastic Analysis 227

Equations (12.9) can be expressed in the form

CT + ~ T = Q (12.10)

where C is the heat capacity matrix, I~ is the heat conduction matrix, Q is the heat flux vector, T is the vector of nodal point temperatures, and t the vector of nodal point temperature-rates. The heat flux vector Q in eq. (12.10) has several components and is expressed with the interpolation function N by

Q= fvx(?r~:)NdV + fs Oe(T4- T4)NdS + fs h(T~- T~)NdS

+ fSc hlub(Td-- Tw)NdS + fsc qrNdS (12.11)

The first term on the right is the heat, generated by plastic deformation inside the deforming body. The second term defines the contribution of the heat radiated from the environment to the element, where o is the Stefan-Boltzman constant, e is the emissivity, and Te and T~ are environ- ment and surface temperatures, respectively. The third term describes the heat convected from the body surface to the environment with heat convection coefficient h. The fourth term represents the contribution of the heat transferred from the workpiece to the die through their interface. Ta and Tw are die and workpiece temperatures, respectively, and htub is the heat transfer coetticient for the lubricant. The last term is the contribution of the heat generated by friction along the die-workpiece interface, qr being the surface heat generation rate due to friction.

The theory necessary to integrate (12.10) can be found in numerical analysis books; see for instance Dahlquist and Bjorck, [21]. The conver- gence of a scheme requires consistency and stability. Consistency is satisfied by an approximation of the type

Tt+A, = Tt + A t [ O - f l ) ~ + [JT,+A,] (12.12)

where fl is a parameter varying between 0 and 1, and t denotes time. For unconditional stability, fl should be greater than 0.5, and a value of

0.75 was chosen. Selection of a proper value of fl is an important factor in situations where it is desirable for the time step to be as large as possible, provided that the increments in strain are compatible with an infinitesimal analysis.

12.4 Computational Procedures for Thermo-Viscoplastic Analysis We treat the workpiece and the die separately, assuming that the die properties do not change. Thus we greatly reduce the number of equations to be solved simultaneously, which, in turn, reduces the cost of the solution. There is no internal heat generation in the die and therefore deformation calculations are not necessary.

228 Metal Forming and the Finite-Element Method

The heat generated through friction, qf, is calculated as

qf =f~ lusl (12.13)

where f~ is the friction stress, and lusl the relative velocity between die and workpiece. The heat qf is evenly distributed between the die and deforming material.

It should be noted that the nodes on the die do not generally coincide with those on the deforming material along the interface, and that the calculation of nodal point temperatues requires interpolation. When boundary conditions such as the convection term in eq. (12.11) apply, they are split into two parts, wherein one containing the unknown temperatures is added to the matrix g~. Boundary conditions such as the radiation term in eq. (12.11) are applied using previous iteration values for body temperatures.

The equations for the flow analysis and the temperature calculation are strongly coupled, making a simultaneous solution of their finite-element counterparts necessary.

Considering Tt+a, as a primary dependent variable, we have, from eq. (12.12), with t = 0 initially,

Ta, = Tat To /~At /~At

The circumflex over "I" denotes

(L- )to Ta, (12.14)

c . _ _ , . +

"'~ " f l A t / " " ' - " ~ " " " for

/~ At

Substituting eq. (12.14) into eq. (12.10), gives

(go + ~AAt)TA, = Q,,, - b'~i? (12.15)

The coupling procedure makes use of eq. (12.15) through the following sequence.

1. Assume the initial temperature field To. 2. Calculate the initial velocity field u corresponding to the tempera-

ture field To. 3. Calculate the initial temperature-rate field To from eq. (12.10) using

values from (1) and (2); 4. Calculate the quantity T. 5. Update the nodal point positions and the effective strain of elements

for the next step. 6. Use the velocity field at the previous step to calculate the first

approximate temperature T such as

Thermo-Viscoplastic Analysis 229

7. Calculate a new velocity field with the solution of (6). 8. Use the new velocity field to calculate the second temperature field

such as

( +--~-C ITS) = Q~t ~ - ~ for T(2) Kc fl At/ ' iA ,

9. Repeat steps for (7) and (8) until both have converged. 10. Calculate the new temperature rate field Ta,. 11. Repeat steps from (5) to (10) until the desired deformation state is

reached.

The iteration process for temperature calculations is not likely to require much computing time, because only the heat input vector Q is changed during iterations and, as a result triangularization of the matrix is necessary only once. Moreover, additional iterations necessary to obtain a velocity field after a new temperature field is obtained should be relatively few, because the velocity field does not show much sensitivity to small variations of the temperature field.

12.5 A p p l i c a t i o n s

Applications of the thermo-viscoplastic analysis include compression of a solid steel cylinder [11, 22], hot nosing of a steel shells [23], and forging of titanium alloys [24, 25].

Compression of Steel Cylinder [11] Pohl [26] conducted temperature measurements in order to test his uncoupled analysis, in which approximate stream functions were used for the deformation, and finite differences for the heat balance. A solid cylinder of a carbon steel AISI 1015 was compressed between flat dies at room temperature. Thermocouples were inserted in the cylinder at different locations. Upon deformation, their readings indicated the tem- perature increases due to the heat generation. Figure 12.3 shows the dimensions and locations of measuring points. The conditions used in computations with the finite-element method [11] were as follows. The deformation took place at room temperature, until a reduction of 33% in height was achieved. The finite-element grid was composed of 132 four-node quadrilateral elements in the workpiece, and 119 in the die. Because of symmetry, only one-quarter of the cylinder needed to be analyzed. The friction factor m was taken as 0.65.

The die velocity was changed at each time-step to simulate a mechanical press. Each step corresponded to 1% reduction in height, which was equivalent to time-steps of up to 0.03 s. The flow stress was considered to be independent of strain-rate and temperature and its values were given by Pohl. The heat transfer characteristics, other than the thermal conductivity and the heat capacity of the AISI 1015, which were given by Pohl, were taken from handbooks.

I-- 0

230

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ro

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0 I'-0

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=.

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0 B

d E

d 8

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0 u o 0

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0 :"

t~

o ~ 0

.

o "~ F

0-o

Thermo-Viscoplastic Analysis 231

The dimensions of the die were such that at the outside boundaries a constant temperature was imposed. The temperature values measured by Pohl are compared with the calculated values in Fig. 12.3. The agreements are excellent for the internal points. For the three points near the billet surface, however, the computed results indicate that the temperature difference for these points is small, while experiments show larger differences. This discrepancy may be attributed to inaccuracies in the material constants used in computations or to inaccuracies in experimental measurements.

Hot Compression of Steel Cylinder Wu and Oh [22] developed an FEM-based computer program (ALPIDT) that is capable of simulating nonisothermal forming operations with arbitrarily shaped workpieces and dies. This program consists of two independent FEM programs, ALPID for viscoplastic deformation analysis [27] and the program for the heat transfer analysis. They are coupled in an efficient manner in the program ALPIDT for simulation of thermo- viscoplastic deformation. To demonstrate the capability of ALPIDT, a hot compression process was simulated. The temperature changes during the initial resting and dwelling periods were included in the simulation. The simulation also accounted for the changes in the heat transfer coefficients between the workpiece and the die during the process.

A cylindrical billet of AISI 1020 steel was compressed between two flat dies. The initial billet temperature was 1232°C (2250°F) and the initial die temperature was 204°C (400°F). In order to estimate the workpiece temperature accurately, simulation was performed in the process consisting of three periods as follows:

1. The heated billet is placed on the lower die for 6 seconds without deformation (free resting period).

2. The workpiece is compressed to 67% in height (deformation period), 1.5 seconds).

3. The deformed workpiece stays on the lower die for 3 seconds after the upper die is retracted (dwelling period).

Detailed computational conditions are given in Reference [22]. In Fig. 12.4 photographs of grid-distortions and temperature distribu-

tions are shown at various stages of the process. The temperature scale (°F) is also shown in the figure.

The predicted temperatures at the end of the resting period are shown in Fig. 12.4a. Owing to relatively small loss of heat to the environment, the temperatures of the upper die and of the upper portion of the workpiece remained almost unchanged. However, heat loss from the bottom portion of the billet to the lower die was considerable. The temperature at the bottom of the workpiece dropped by 280°C (536°F) during the free resting period of 6 s. At the same time, the surface of the lower die was heated to nearly 600°C (1112°F). It is to be noted that the flow stress of the

232 Metal Forming and the Finite-Element Method

workpiece material was doubled when the temperature was reduced from 1230 ° to 950°C (2246 ° to 1742°F).

The predicted temperature distributions and grid distortions during the deformation period are shown in Figs. 12.4b to e. At the beginning of the deformation period, the temperature of the workpiece at the upper die interface was higher than that at the lower die interface. However, the temperature of the top surface dropped quickly after the upper die contactec" the workpiece. The bottom and top surface temperatures of the workpiece become almost the same, reaching 700-800°C (1292-1472°F) at 0.375 s after deformation started. However, it is noted that the tempera- ture gradient at the top was higher than at the bottom of the workpiece. These differences between the temperature distributions in the upper and lower regions of the workpiece became less as the deformation proceeded further. The temperature distributions shown in Figs 12.4b to e suggest that the flow stress of workpiece near the die workpiece interface could have been three times that in the mid-height region.

The predicted grid distortions reflect the effect of temperature on the flow stress of the workpiece, influencing metal flow. During the early stage of deformation (see Figs. 12.4b and c), the barreling near the top surface was more pronounced than near the bottom surface. It is also noticed that the upper surface moved radially more than the bottom surface, suggesting that the average flow stress was higher near the bottom surface. It can be also seen that, throughout the deformation period, the deformation occurred mainly in the mid-height region, while the chilled regions near the die workpiece interfaces remained almost rigid (Figs. 12.4d and e).

Figure 12.4f shows the temperature distributions at 3 s after the upper die was retracted from the workpiece at the end of deformation. It is seen from the figure that the top surface of the workpiece has warmed up again, while the average temperature of the workpiece had dropped. This was due to the redistribution of heat within the workpiece, without die chilling at the top surface of the workpiece. The predictions shown in Fig. 12.4 agree with general observations made in nonisothermal forging of cylindri- cal billets.

Forging of Titanium Alloy Ti6242

The preform considered is a cylindrical composite material with a central core of (c~ + fl) phase and an outer ring of (fl)-transformed phase, the two diffusion-bonded together. Its dimensions are given in Fig. 12.5a. The preform was forged isothermally (dies and workpiece at the same initial temperature) at 1227 K (1750°F) at a constant ram speed of 5.08 mm/min. (0.2in./min). The total reduction in height was 60%. At these high temperatures, a glass-type lubricant is very effective; a friction factor of m = 0.2 was used in computer simulations.

In isothermal forging, very slow speeds are usually employed in order to (1) avoid increasing the flow stress of the material, which is strain-rate dependent, and (2) allow the heat generated during deformation to spread

i i l ; i i i i • i i i i • i i •

i ~ ¸

(a)

~ i ~ : ~ - i i ~ ......

(c)

l~,,~, ,̧ ,, ~~~i~! i-iii i ̧̧

(b)

(d)

m l ii~

(e) (f)

iiiii!iiii!i) . . . . . . .

FIG. 12.4 Predicted temperature dislribulio.s a,ld grid distortions at various stages of hot compression process [22]. (a) At the end of free resting (elapsed time t -- 6s); (b) 16.67% reduction in height (t ---- 6.375s); (c) 33.34% reduction in heigh! (t =~ 6.750s)1 (d) 50.00% reduction in height (! = 7.125s); (c) 66.67% rcduclion i . height (t = 7.500s); (l) al the end of 3s free resting after deformation (t : 10.Ss).

Thermo-Viscoplastic Analysis 235

5

4

3

2

I

(a)

(b)

E;I ! 5.625

1229"-------~122 oK

\

E ~ - ' ~ 2 . 5 "~' E , 0 7< I- ( c )

4

3

2

I

1228K ~1230

1232

I / 1235 ": I I I I 1239 I / I I /

0 2 4 6 8 I0 12 14 X I0 mm

FIG. 12.5 Die and workpiece temperatures in forging a t i tanium 6242 alloy composite billet at 60% reduction in height with different ram speeds: (a) preform dimensions; (b) ram speed = 0.2 in . /min. ; (c) ram speed = 2.0 in . /min.

uniformly throughout the workpiece, assuring uniformity in temperature and deformation. If higher speeds can be used without adverse effects, the increase in productivity is obvious. To investigate this aspect, the whole process was simulated at 5.08 mm/min. (0.2 in./min.) and at 50.8 mm/min. (2.0 in./min).

The difference in speed of deformation did not produce any noticeable differences in the overall deformation process, or even in the local deformation histories. The only changes found were in the temperature fields, also shown in Figs. 12.5b and c. An average calculated temperature increase of 6.5 K in the slow-speed deformation increased to 12.5 K for the faster deformation. However, the temperature gradients were such that the greatest difference between two points in the workpiece was roughly 5 K at the end of deformation. This indicates that the speeds of deformation can be increased (at least to 2.0in./min) without loss of uniformity of properties, provided the average temperatures reached are not critical.

236 Metal Forming and the Finite-Element Method

The resulting microstructures at various locations after forging are shown in Fig. 12.6. The strain, strain-rate, and temperature variations during forging were found to be almost the same at all locations, and typical variations for 5.08 mm (0.2 in./min.) ram speed are shown for the elements near the outer periphery and near the center in Fig. 12.6.

~ .~i,,

• , . . . . . . . . . ~ '" ...... ~ : ,~ii"" : '~ ~"~'~!~

I' ~ ~.~h I pancake macrostructure

) resultant microstructure

A B C (Interface) D E

1,2 I,,.~.~; '

1.0

0.8

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0 IO 20 50 40 50 60

f !

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A { E L E M E N T 5 { 5 )

I0 20 30 40 50 60

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L

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1200

I180

I160

0

% REB.L"rK~

FIG. 12.6 Local strain, strain-rate, and temperature variations during (or +//)//~ composite forging and corresponding microstructures after forging. (Microstructures courtesy of C. C. Chen, Chen Tech. Industries.)

Thermo-Viscoplastic Analysis 237

The temperature histories indicate that no appreciable changes took place. Therefore, any microstructural modifications must be due to maintaining the workpiece at an elevated temperature for some time under pressure and also to the amount of deformation imposed.

The information given in Fig. 12.6 allows further qualitative interpreta- tion of the relationships between local strain, strain-rate, and temperature histories and corresponding microstructures.

Isothermal and Hot-Die Disk Forging

Using the program developed by Oh [27], coupled with the heat transfer finite-element calculation, the compressor disk forging of a (fl)-phase Ti 6242 alloy was analyzed under two sets of conditions, isothermal forging and hot-die forging [25].

In isothermal forging, the initial temperatures of both die and preform were chosen to be 1171.9K (1650°F) and the die velocity was 5.08 mm/min. (0.2 in./min). Again, a glass-type lubricant was used at the die-preform interface.

In addition to isothermal forging, hot-die forging was simulated for the same die and preform geometries in order to determine the effects of temperature and strain-rate on the details of deformation behavior. The initial temperature of the preform was 1171.9 K (1650°F), but the die and environmental temperatures were assumed to be 644 K (700°F) and 293 K (68°F), respectively. The die speed was 76.2 mm/min. (3.0 in./min).

The grid distortions at 70% reduction under the two forging conditions are compared in Fig. 12.7a. The difference in metal flow, perhaps mainly due to temperature effects, can be seen qualitatively. Distortion is more severe in hot-die forging than in isothermal forging. Nonuniformity of deformation due to temperature gradients within the workpiece in hot-die forging can be seen from the shape of the grid distortions. The contact area at the rim region is different, being larger in isothermal forging. It is evident that the bulge at the outer surface is greater with hot-die forging. Although the observation is qualitative, it is clear that the effect of temperature, possibly combined with the strain-rate effect, causes the metal flow to differ under the two forging conditions.

Figure 12.7b shows a comparison of temperature distributions under the two forging conditions. The temperature variation within the workpiece for isothermal forging is only a few degrees, while a severe temperature gradient can be seen near the contact region between the die and the workpiece in hot-die forging.

The strain distributions, on the other hand, are almost the same for the two forging conditions, as shown in Fig. 12.7c. It appears that the strains are predominantly determined by the preform geometry and the die configuration.

Hot Nosing [23]

The nosing process was described in Chap. 2. To study the nature of the nosing process of shells, Carlson [28] conducted hot nosing on a small

(a)

I s o t h e r m a l

(mm)

Hot - Die

0 ' 2 0 ' 40 ' 6 0 8 0 I00 120 (ram)

(b)

I-.._ ,,~ ~ / / ~.~,,,o

o l - , .---~.~,~,,,,,, ,, . ~ 3 ''°' 0 I 2 3 4

Hot Die

2--

670 I.

I ..... 750 x,,,~ J,f'x------~ II17

o ~i-~~, i l l 'l'' ~ r"i /~ , ,I 0 I 2 3 4

FIG. 12.7 (a) Grid-distortions; (b) temperature distributions; and (c) strain distributions in isothermal and hot-die forging. Material: Ti 6242 alloy; /]-phase. Isothermal, die and workpiece temperature 1227K; speed 0.2in./min., hot-die, die temperature 624 K, speed 3.0 in./min.

238

I s o t h e r m a l

(c)

- .6 .5

.~__~-~- u "

o, , 4 3 I , ' I 2 3 4

Hot Die

Thermo-Viscoplastic Analysis 239

~.o ,.o ~ ~ 7 / ~ ' x ~

/I,I ., x,1 I 2 3 4

FIG. 12.7 (cont'd)

model of a 105-mm shell. Commercial open-hearth, cold-rolled steel was used in making the specimens. Geometric details of the specimens are given in Fig. 12.8.

The die that was employed throughout the experiments had a curvature radius of 300 mm (11.85 in.) for an ogive profile. To simulate the amount of nosing, 36.8 mm (1.45 in.) of the specimen was made to enter the die, giving a maximum reduction in the mean diameter of about 33%.

In preparation for the test, the die was heated for about 1 hour to bring it to a temperature of 811 K (1000°F). The shell temperature varied from about 1144K (1600°F) to 1311 K (1900°F) at the tip, depending upon

2.31

28.57 . . . . . . .

95.25

FIG. 12.8 Hot nosing specimen [28].

240 Metal Forming and the Finite-Element Method

o K

130(:

. i 120( OE

E I 1 0 0 i , i o .

i , i F- I 0 0 0

9 0 0

8 0 0

I

1600

1200

0

I 0

I

I / 4 I /2 3 /4 t l / 4

I I I I 0 2 0 30

DISTANCE FROM NOSE TIP

in.

mrn

FIG. 12.9 Temperature distributions obtained by induction heating for a shell [28].

heating conditions. To measure the temperature distribution, six chromel- alumel thermocouples were placed along the shell specimen. The measured temperature distributions are shown in Fig. 12.9.

The heating of the shell was done in about I min by an induction coil. The length of time for nosing was in the order of I s. For the finite-element simulations, the flow stress expressions were obtained from experimental data of Altan and Boulger [29], and the heat transfer characteristics were taken from standard handbooks.

The predicted load-displacement curves for hot nosing at various friction conditions, with initial tip temperature at 1269 K (1825°F) and die speed of 41 mm/s (1.62 in./s), are given in Fig. 12.10a. It can be seen that the simulation results, obtained with a friction coefficient of # = 0.1, are very close to the experimental results. In Fig. 12.10b, the load- displacement curves are computed for initial tip temperatures of 1158K (1625°F), 1227 K (1750°F), and 1269 K (1825°F), respectively, at a nosing speed of 41 mm/s and for /~ = 0.1. All of the finite-element simulation results showed good agreement with the experimental data.

12.6 Concluding Remarks Using the capabilities of a coupled thermo-viscoplastic analysis, attempts were made to predict metal flow and forming loads, and to correlate the metallurgical changes with the information obtained through simulation. Gegel et al. [30] provided a detailed interpretation of the microstructure

KN

80

( :3

o .J

4 0 z Or) o Z

Thermo-Viscoplastic Analysis

) TIP T E M P . : 1 2 6 9 ° X

NOSING SPEED =41 ram/see

s I /%. E X P E R I M E N T

c

5

0 0.5 1.0 I I I I

o DIE PENETRATION

A FEM /~.p. =0.1

. - 0 . 0 8 - ~ 0 . 0 5

1.5 in. I

40 rnm

(a)

241

KN i

8 0

t ~

o _J c~ 40 z

Z

xl,

I I I

NOSING SPEED =41 ram/see

EXPERIMENT 0 1 1 4 4 ° K

• 1200 °K

C

I FEM,/.L =0.1

//0111S8°K

r2z?°K 1269°1(

, / /

0 0 .5 1.0 1.5 in. I I I I I

O 20 4.0 mm

DIE PENETRATION

(b) FIG. 12.10 Load-displacement curves for (a) various friction conditions and (b) different initial tip temperatures, and comparison with experiment [28].

that develops during hot forging. Furthermore, Gegel et al. [31] proposed a new method of modeling the dynamic material behavior that explicitly described the dynamic metallurgical processes occurring during hot deformation.

Recently, Dawson [32] developed the numerical solution formulation for

242 Metal Forming and the Finite-Element Method

simulation of hot or warm metal forming under steady-state conditions. Of particular importance was the incorporation of a methodology for using particle stress-temperature trajectories in conjunction with deformation mechanism maps. Thus, the assumptions made regarding the assumed constitutive equations could be evaluated. The analysis of slab rolling of aluminum was given as an example of possible applications. A significant advance, made by Dewhurst and Dawson [33], is the development of a finite-element program that models steady-state viscoplastic flow and heat transfer in three dimensions.

References 1. Oh, S. I., Rebelo, N. M., and Kobayashi, S., (1978), "Finite-Element

Formulation for the Analysis of Plastic Deformation of Rate-Sensitive Mate- rials in Metal Forming," IUTAM Symposium, Tutzing/Germany, p. 273.

2. Johnson, W., and Kudo, H., (1960), "The Use of Upper-Bound Solutions for the Determination of Temperature Distributions in Fast Hot Rolling and Axisymmetric Extrusion Processes," Int. J. Mech. Sci., Vol. 1, p. 175.

3. Tay, A. O., Stevenson, M. G., and Davis, G. V., (1974), "Using the Finite-Element Method to Determine Temperature Distributions in Orthogo- hal Machining," Proc. Inst. Mech. Engr. Vol. 188, p. 627.

4. Bishop, J. F. W., (1956), "An Approximate Method for Determining the Temperature Reached in Steady-State Motion Problems of Plane Plastic Strain," Q. J. Mech. Appl. Math., Vol. 9, p. 236.

5. Altan, T., and Kobayashi, S., (1968), "A Numerical Method for Estimating the Temperature Distributions in Extrusion Through Conical Dies," Trans. ASME, J. Engr. Ind., Vol. 90, p. 107.

6. Lahoti, G., and Altan, T., (1975), "Prediction of Temperature Distributions in Axisymmetric Compression and Torsion," Trans. ASME, J. Engr. Materials Technology, Vol. 97, p. 113.

7. Nagpal, V., Lahoti, G. D., and Altan, T., (1978), "A Numerical Method for Simultaneous Prediction of Metal Flow and Temperatures in Upset Forging of Rings," Trans. ASME, J. Engr. Ind. Vol. 109, p. 413.

8. Zienkiewicz, O. C., Jain, P. C., and Onate, E., (1978), "Flow of Solids During Forming and Extrusion: Some aspects of numerical solutions," Int. J. Solids Structures Vol. 14, p. 15.

9. Zienkiewicz, O. C., Onate, E., and Heinrich, J. C., (1978), "Plastic Flow in Metal Forming--I. Coupled Thermal Behavior in Extrusion--II. Thin Sheet Forming." Applications of Numerical Methods to Forming Processes, ASME, AMD, Vol. 28, p. 107.

10. Rebelo, N., and Kohayashi, S., (1980), "A Coupled Analysis of Viscoplastic Deformation and Heat Transfer--I. Theoretical Considerations," Int. J. Mech. Sci., Vol. 22, p. 699.

11. Rebelo, N., and Kobayashi, S., (1980), "A Coupled Analysis of Viscoplastic Deformation and Heat Transfer--II. Applications." Int. J. Mech. Sci., Vol. 22, p. 707.

12. Cristescu, N., (1975), "Plastic Flow Through Conical Converging Dies, Using a Viscoplastic Constitutive Equation," Int. J. Mech. Sci., Vol. 17, p. 425.

13. Cristescu, N., (1976), "Drawing Through Conical Dies--An Analysis Com- pared with Experiments," Int. J. Mech. Sci., Vol. 18, p. 45.

14. Zienkiewicz, O. C., and Godbole, P. N., (1975), Viscous, Incompressible Flow with Special Reference to Non-Newtonian (plastic) Fluids, Chap. 2 in "Finite Elements in Fluids" (Edited by R. H. Gallagher et al), Wiley, New York.

Thermo-Viscoplastic Analysis 243

15. Zienkiewicz, O. C., and Godbole, P. N., (1974), "Flow of Plastic and Viscoplastic Solids with Special Reference to Extrusion and Forming Proc- esses," Int. J. Num. Meth. Eng., Vol. 8, p. 3.

16. Price, J. W. H., and Alexander, J. M., (1976), "A Study of Isothermal Forming or Creep Forming of a Titanium Alloy," Proc. of the 4th NAMRC, Columbus, Ohio, p. 46.

17. Price, J. W. H., and Alexander, J. M., (1976), "The Finite-Element Analysis of Two High-Temperature Metal Deformation Processes," 2d Int. Symposium on FEM in Flow Problems, p. 717.

18. Klemz, F. B., and Hashmi, S. J., (1977), "Simple Upsetting of Cylindrical Billets: Experimental Investigation and Theoretical Prediction," 18th MTDR Conference, London, p. 323.

19. Loizou, N., and Sims, R. B., (1953), "The Yield Stress of Pure Lead in Compression," J. Mech. Phys. Solids, Vol. 1, p. 234.

20. Lippman, H., (1966), "On the Dynamics of Forging," Proc. 7th Int. MTDR Conference, Birmingham, England, p. 53.

21. Dahlquist, G., and Bjorck, A., (1974), "Numerical Methods," Prentice-Hall, Engiewood Cliffs, NJ.

22. Wu, W. T., and Oh, S. I., (1985), "ALPIDT: A General Purpose FEM Code for Simulation of Non-Isothermal Forming Processes," Proc. NAMRI--XIII, Berkeley, California, p. 449.

23. Tang, M.-C., and Kobayashi, S., (1982), "An Investigation of the Shell Nosing Process by the Finite Element Method. Part 2: Nosing at Elevated Tempera- tures," Trans. ASME, J. Engr. Ind., Vol. 104, p. 312.

24. Rebelo, N., and Kobayashi, S., (1981), "Thermo-Viscoplastic Analysis of Titanium Alloy Forging," ASME Publications PED--Vol. 3, Manufacturing Solutions Based on Engineering Sciences, p. 151.

25. Oh, S. I., Park, J. J., Kobayashi, S., and Altan, T., (1983), "Application of FEM Modeling to Simulate Metal Flow in Forging a Titanium Alloy Engine Disk," Trans. ASME, J. Engr. Ind., Vol. 105, p. 251.

26. Pohi, W., (1972), " A Method for Approximate Calculation of Heat Genera- tion and Transfer in Cold Upsetting of Metals," Doctoral Dissertation, University of Stuttgart.

27. Oh, S. I., (1982), "Finite Element Analysis of Metal Forming Process with Arbitrarily Shaped Dies," Int. J. Mech. Sci., Vol. 24, p. 479.

28. Carlson, R. K., (1943), "An Experimental Investigation of the Nosing of Shells," Forging of Steel Shells, presented at the Winter Annual Meeting of ASME, New York.

29. AItan, T., and Boulger, F. W., (1973), "Flow Stress of Metals and its Applications in Metal Forming Analysis," Trans. ASME J. Engr. Ind., Vol. 95, No. 4, p. 1009.

30. Gegel, H. L., Nadiv, S., Malas, J. C., and Morgan, J. T., (1980), "Application of Process Modeling to Analysis of Microstructural Changes During the Hot Working of a Two-phase Titanium Alloy," Appendix K, AFWAL-TR-80- 4162, p. 403.

31. Gegel, H. L., Prasad, Y. V. R. K., Malas, J. C., Morgan, J. T., and Lark, K. A., (1984), "Computer Simulations for Controlling Microstructure During Hot Working of Ti-6242," ASME, PVP, vol. 87, p. 101.

32. Dawson, P. R., (1984), "A Model for the Hot or Warm Forming of Metals with Special Use of Deformation Mechanism Maps," Int. J. Mech. Sci., Vol. 26, p. 227.

33. Dewhurst, T. B., and Dawson, P. R., (1984), "Analysis of Large Plastic Deformation at Elevated Temperatures Using State Variable Models for Viscoplastic Flow," Proc. Symp. Constitutive Equations: Micro, Macro, and Computational Aspects, ASME., p. 149.