... 440"C using graphite lubricant with a view to characterising the friction over
this ... graphite/oil suspension and fine graphite powder to create a
homogeneous ...
s o lvinglfor Frtc don Effec ts durtng Mechanical Compression Testing J. M'C. Buchanan, R.D. Knutsen and J.A. Basson
A visioplastic und finite element analysis of the hot plane struin compression test for commercial purity uluminium hus been investigated to determine the local strain and strain rate distribution within the plane strain compression test specimen. This paper reports the results of the veriftcation and validation of three-dimensional finite element simalations by compurison with visioplastic results. It is shown that the locul strain and strain rate distribation, within a certain ronge of nominal struin, cun be conftdently predicted on condition that the friction condition l's well characterised. The importunce of anulysing the friction condition as fanction of test temperature is highlighted und u set of ulgorithms is presented to predict the friction cofficient fo, graphite lubricution for temperatures ranging from room temperatare ap to 440oC, NOMENCLATURE Finite Element Analysis Plastic Equivalent Strain Plane Strain Compression Rolling, Transverse, and Normal Directions Temperature Zener -Ho I I omon p arameter Keywords
Plane strain compression, friction, strain distribution, visioplastic, aluminium.
lntroduction
transient specimen/tool configurationt'u. Much work has been performed in this area for deformation variables including nominal strain, strain rate, temperature, and friction condition, and the interpretation of flow stresses and strain patterns is well documentedT-n. As a consequence, it is quite feasible to map the
local strain and strain rate distribution within the tool gap using finite element analysis on condition that the deforming geometry and friction condition are well understood. By using this approach, the nominal strain and strain rate conditions can be translated to local conditions within apartrcular volume of the test specimen. Thus, more accurate interpretation of metallur-
gical transformations, including crystallographic texture,
is
possible when considering only a small portion ofthe deformed specimen volume, &s is usually the case when performing microstructural analysis. For example, a nominal strain value of 1.0 may yield a local strain of up to I .4 atthe centre of the PSC specimen, whereas the near surface will be much less deformed. Recognition of this deformation pattern is important when relating the microstructure evolution to the imposed nominal strain. Of the input variables required for numerical analysis ofthe strain and strain rate distribution, friction is possibly the most difficult to capture. Nevertheless, reasonable assumptions can be made on the basis of tool/specimen surface roughness and choice of lubricant. Conducting PSC tests over abroadtemperature range, however, requires much more attention than fixed temperature tests since the lubricant properties can change with temperature. Thus it may be improper to assume a single
friction value when attempting to predict the deformation pattern for tests conducted at different temperatures. The present study considers the influence of friction on the deformation pattern in comm€rcial purity aluminium deformed at temperatures ranging from ambient up to 440"C using graphite lubricant with a view to characterising the friction over this temperature range. Several combinations of strain, strain rate
Laboratory plane strain compression (PSC) testing is widely employed to replicate the deformation conditions that prevail in the roll gap during hot and cold rolling of metal flat product. The PSC test may be used to measure flow stress up to strains much greater than that possible from uniaxial tensile tests', but perhaps the most useful feature of this test method is to study microstructures that evolve during the therrnomechanical processing ofplate and sheet metal product2. Not only is it possible to study the metallurgical restoration processes that occur in response to the imposed deformation conditions, but the
strain and strain rate distribution can be calculated by finite element analysis for any combination of deformation conditions for the valid temperature range.
crystallographic textures that develop during plane strain deformation akin to rolling reduction, and during consequent
Experimental Approach
recrystallisation, can also be monitored3'4. Despite the simple specimen geometry, however, the specimen volume does not deform homogeneously and the aforementioned attributes need to be interpreted cautiously. The deformation pattern between the compression tools is controlled by the prevailing slip-line fields, which are dependent inter alia on friction and
Centre for Materials Engineering, Mechanical Engineering Department, University of Cape Town
and temperature have been employed and comparisons are made between visioplastic and finite element analysis (FEA) of the deformation patterns. This test approach allows a validated algorithm to be obtained that can reliably predict the coefficient of friction over a certain temperature range for a particular PSC apparatus. With the correct friction condition, the true local
Plane strain compression tests were performed at 25,250,325, 350, 37 5,400 and 440"C,and at strain rates of0. I and 10 s-r . The PSC apparatus employed in this study was purpose built to fit a250 kN load frame on an electro-servo hydraulic (ESH) universal testing machine'0. Computer control of the system ensures a constant nominal strain rate during the test. Specimens were machined from fully recrystallised commercial purity AA 1200 to produce a blank geometry measuring 10 mm (specimen height) x 3 3 rnm x 52mm. The compre ssion platens provide a cont act ar ea measuring 1 3 mm (platen width) x52ffiffi, which is sufficient to
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3
Solving for Friction Effects during Mechanical Compression Testing minimise lateral spread in order to generate a reasonable plane strain condition. Since the intention is to simulate deformation in the roll gap,the specimen thickness and platen width directions can be described as the norrnal direction (ND) and rolling direction (RD) respectively. The third orthogonal direction is naturally referred to as the transverse direction (TD). Direct resistance heating is employed such that the tool surfaces and test specimen attain the same temperature. Lubrication was effected by coating the contact surfaces with a mixture of graphite/oil suspension and fine graphite powder to create a homogeneous graphite film.
Visioplastic Analysis Localised strain distribution was measured with the aid of reference grids that were scribed on the specimen before testing. The PSC blank was split mid-way along the transverse direction and fine lines measuring 0.75 mm apartwere scribed on one of the surfaces of the RDAID plane, as shown in Figure 1. After scribing the grid, the split-specimen was clamped to retain the original contactarea( I 3 mm x52mm). Comparing the grid intersections before and after testing enabled calculation of the equivalent plastic strain distribution according to the method described by Beynon and Sellars' ' . Strain contour maps and plots of peak strain versus distance along specimen sym-
metry lines illustrate the visioplastic results obtained in this way. The in-plane (RD) and through-thickness (ND) symmetry lines are superimposed on Figure 1.
density is expected to be 0.5. In order to derive relationships for equivalent flow stress o as a function of equivalent strain e for a range in deformation conditions, the parameterS oo, or. and t, must be expressed in terms of strain rate, e , and temperature, T. For this purpose the Zener-Hollomon parameter Z combines these effects in equation 2, where R is the gas constant and Q is activation energy for deformatron.
z-;
'tnr/ -,4[sin "*p(gl
h(ao.)]*'
(2)
The parameter o* is the stress at any constant strain, and the parameters A, d and m' areconstants derived forthe appropriate o* value of interest, which in our case is o0 and o,,. The hyperbolic sine relationship is employed to cover the whole range of stresses't. In this way the determination of oo and o,, from equation 2 enables the stress-strain behaviour under specific strain rate and temperature conditions to be calculated from equation 1. The values for A, a, and m'were adapted from Shi et al't to match the measured flow behaviour for our alloy. These values for commercial purity aluminium are presented in Table l. Finally, the transient strain constant, t., is determined from the steady state flow stress't. The Young's Modulus used in the FEA is 70 GPa and temperature dependence is not employed since the elastic strain is negligible in comparison to the plastic strain. o*
CX,
mt
A
q
01
11 .0
4.00 x
1011
Q.
0.027
5.0
1
.96 x
1011
Table 1. Constants for constitutive flow stress relationship for aluminium alloy AA1 200. Figure 1: Deformed visioplastic PSC specimen indicating grid scribed
Results and Discussion Visioplastic Strain Analysis Physical analysis of the strain distribution has demonstrated the influence of test temperature on strain distribution in the tool gap, even when the nominal strain is relatively small. Figures 2 and 3 illustrate the measured strain distribution for tests conducted at 250"C and 44}"Crespectively, and atastrain rate of 10 s-r . Contours calculated from the visioplastic strain analysis indicate a greater dead-zone near the specimen surface
and concomitant increase in strain in the centre for the test
Where oo is the initial flow stress, o,, is the ste ady state flow Stress, e,. is the transient strain constant, and n rs a constant, which from the relationship between stress and dislocation
4
Figure 2: Visioplastic strain contours for a specimen deformed at 250"C to a nominal strain of 0.42 (RD/ND plane).
R & D Journal, 2004, 20 (2) incorporsted into The SA Mechunical Engineer
Solving for Friction Effects during Mechanical Compression Testing 0.8
r 0.05
o tu
. 0.1 a 0.15
H o6 F
1il.i.i1$r-
ztu
04 E J UJ
tf i: OA
IU
L z
.0.2
o
0.2
conducted
0.6
z tF
a
c-) 0.4
tr
r 250
oo
o 325 a 350
I io * t
tr 375 A 400
.440
@
I(L 0.3 I9. O.2
A
o
0
0.0
1.0
2.0
3.0
4.0
DISTANCE ALONG ND PATH (mm)
at 440"C. The strain pattern in Figure 3 is not
consistent with predicted slip-line field theory for frictionless conditions and highlights the breakdown in lubrication at the higher temperature. The peak strain measured at the centre (intersection of symmetry lines) of the 440"C test specimen is 0.58 as opposed to 0.46 for the lower temperature. In order to further illustrate the influence of test temperature on strain distribution, a comparison is made of equivalent plastic strain distribution for several test temperatures between 250oC and 440"C. These results are presented in Figure 4 by plotting the measured visioplastic strain along the symmetry line in the specimen norrnal direction (ND) for each test temperature. In the same way that the contour maps show differences in strain distribution, the gradients for the curves in Figure 4 vary as a function of test temperature. As expected, the difference in strain between centre and surface is greatest for the highest test temperature, with a gradual change in slope from the highest to lowest test temperature.
A
o
tr Figure 3: Visioplastic strain contours for a specimen deformed at 440"C to a nominal strain oI 0.42 (RD/ND plane).
Il
ao
Figure 5: Simulated equivalent strain along node path in different friction coefficients.
N
D for
modelled for arangein assumed friction coefficients. Since the visioplastic analysis and FEA are best compared by reference to deformation along straight-line trajectories, the finite element plastic equivalent strain (PEEQ) was also plotted as function of distance along ND path. Examples for simulated friction coefficients between 0.05 and 0.2 areillustrated in Figure 5. The trend towards steeper gradient with increasing friction is consistent with the suggestion that friction increases with temperature for the visioplastic analyses (Figure 4). At this point it should be emphasised that the visioplastic strain values cannot be matched
directly with the finite element strain analysis. In the case of FEA, the equivalent strain value includes the small component of transverse strain that arises from a slight deviation from the plane strain condition. For the measured visioplastic strains, only the ND-RD strain is calculated since the grid itself is only two-dimensional. Nevertheless, the strain gradients can be directly compared, and in this way matching strain gradients allow the friction value corresponding to the FEA to be assignedto the actual measuredtest condition. Several visioplastic tests were performed between 25"C and 440"C for strain values of 0.22, 0.42 and 0.60. The strain gradients along ND were determined and compared to the gradients that were predicted by FEA using certain assumed friction coefficients. This approach is demonstrated in Figure 6 where the gradients for visioplastic analyses (nominal strain :0.42) are plotted as bars, 0.12
0.1
0.0
1.0
2.0
3.0
4.0
DISTANCE ALONG ND PATH (mm) Figure
4: Visioplastic strain along node path in ND for nominal strain
of 0.42 at different test temperatures (strain rate =
10s-1).
tU
o-
o J a uJ o
tuJ
Friction Measurement Calculation of the strain gradient in Figure 4 can give a sense of the variation in friction condition as a function of test temperafure. However, appropriate friction coefficient values are required if FEA is to be employed routinely to predict local
strain distribution for a range of nominal strains and test temperatures. One such way of achieving this is to match the strain distribution predicted by FEA with the measured visioplastic analysis. Accordingly, the strain distribution was
0.10 0.08 0.06 0.04
- - -I-
- - J0.05
- - l-
- -f-
-
0.02 0.00
250 325 350 375 400
440
TEMPERATURE (degrees Celcius) Figure 6: Comparison of visioplastic (vertical bars) and FEA (horizon-
tal lines) average strain gradient along ND symmetry line. All values were measured for nominal strain = 0.42.
R & D Journal, 2004, 20 (2) incorporated into The SA Mechanical Engineer
Solving for Friction Effects during Mechanical Compression Testing whilst horizontal lines indicate the FEA average gradient for assumed friction coefficients. The negative of the strain gradi-
z
ent is used for ease of graphical representation and the position-
E
ing of the horrzontal lines for each friction coefficient corresponds to the FEA average strain gradient plotted on the
@
vertical axis.
Analysis of the results presented in Figure 6, as well as similar analysis for visioplastic strains of 0.22 and 0.60, has enabled approximation of friction coefficients over the current temperature range. Regression analysis has given rise to the following two equations for approximating the friction coefficient for PSC tests conducted between 25"C and 440"C. For test temperatures from 25 to 250'C:
It
12
O
tr
a
J 0.6 IL F
ztU J f
o tu
0.4 0.2 0.0
5
For temperatures above 250"C:
lt -2.86 x 10-6 T2 - 1.33x10-3 T + 0.2137
= VISIOPLASTIC N=FEA
0.8
(4)
Where T is the temperature at which the PSC test is performed in degrees Celsius. The effor is approximately 15% above and below the value of the coefficient of friction. The Coulomb friction condition described by the two equations is illustrated graphically in Figure 7 for the range of test temp eratures investigated. Error bars indicate the expected variance in friction coefficient at each temperature.
z f
I - VISIOPLASTIc
1.0
X-FEA
U)
P U)
08
d 06 F
fr
o4
f
o uJ
00
1234 DISTANCE ALONG ND PATH (mm)
F
o
Figure 8: Visioplastic and FEA predictions of the equivalent plastic strain for PSC specimens deformed at 250'C at a strain rate of 0.1 sec-
0.20
lr
LL
[U
0.1 5
o O
z
o
strain along the node paths in the RD and ND. A best-fit polynomial is used to include the transverse strain at the visioplastic nodal points in the RD and ND by employing the
0.1 0
tr
O 0.05
E
LL
0.00
0
1
00
200 300 400
500
TEMPERATURE ("C) Figure 7: Variation in coefficient of friction for the range of temperatures investigated. Error bars indicate a 15% variation above and
Validation In order to validate the friction analysis, the FEA predictions are directly compared with modified visioplastic results for the range of test temperatures investigated. The visioplastic results are modified to include the relatively small transverse strain that occurs as a result of the less than perfect plane strain conditions for the present specimen geometry. As mentioned already, the FE model includes transverse spread, whereas the direct visioplastic analysis only includes ND/RD strain. Incorporating the transverse strain allows a direct comparison of the visioplastic results with FEA prediction assuming the coefficient of friction has been well characterised over the range of test temperatures investigated. The inclusion of the transverse
6
15
1-2
0.25
zLu
10
DISTANCE ALONG RD PATH (mm)
(3)
- 0.05
|
F 1.0
method described by Beynon and Sellars". Comparing strain along both symmetry lines, namely RD and ND, funher substantiates the validation process. Figures 8 and 9 demonstrate the good data fit for several strain increments at 250"C and 400'C. Good agreement is shown between visioplastic analysis and FEA at both temperatures. Unfortunately the friction analysis could not be validated over the strain range normally applied during PSC testing. The visioplastic grids scribed onto the PSC specimen were 0.75 mm apart, which in comparison to the FE nodal distance in the deformation zone is significantly larger. The FE mesh is refined to more accurately approximate the material behaviourwhereas the visioplastic grid allows poor calculation of the real state of
deformation. The visioplastic technique could not be used above nominal strains of 0.8 because of large mesh distortion making the intersection of the grid lines difficult to determine. However, it is reasonable to assume constant coefficient of friction at a particular temperature and thus the finite element model can be used to determine the strain and strain rate distribution within the PSC specimen for nominal strains greater than that investigated in the present study.
strain in the calculation of the equivalent plastic strain is
Gonclusions
achieved by incorporating the FEA prediction ofthe transverse
Detailed visioplastic and FE analysis has been carried out to
R & D Journal, 2004, 20 (2) incorporated into The SA Mechanical Engineer
Solving for Friction Effects during Mechanical Compression Testing
z E.
0.9
lU)
0.8
C)
0.7
tr U)
2.
Timothy,,S.P. , Yiu, H.L., Fine, J.M., Ricks, R.A., Simulation of single pass of rolling deformation of aluminium alloys by plane strain compression, Mater. Sci. Technol., 1991, 7, 225-
1.0
O
-
VISIoPLASTIC
X-FEA
26r.
3. Duckheffi, A., Knutsen, R.D., Asymmetricflow duringplane strain compression of aluminium alloys, Mat. Sci. Eng., /,998,
0.6 (L 0.5 F 0.4 zul 0.3
I
A256,220-226.
4. Duckhoffi, A., Knutsen, R.D., Engler, O., Influence ofdeformation variables on the formation of copper-type shear bands in Al-IMg, Acta Mater., 2001, 49, 2739-2749. 5. Hill, R., Lee, E.H., Tupper, 5.J., A method of numerical analysis of plastic flo. in plane strain and its application to the compression of a ductile material between rough plates, J. Appl. Mech., 1951, 18, 46-52. 6. Green, A.P., A theoretical investigation of the compression of a ductile material between smooth flat dies, Phil. Mog., 1951, 42, 900-918. 7. Gelin, J.C., Ghouati, O., Shahani, R., Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys, Int. J. Mech. Sci., 1994, 36, 773-796. 8. Mirza, M.5., Sellars, C.M., Modelling the hot plane strain
J
0.2
5 g
0.1
UJ
0.0 10
15
DISTANCE ALONG RD PATH (mm)
z 1'o *oe 6
O
-
VISIOPLASTIC
X-FEA
o.B
o 0.7 'a 0.6
d 05
i
compression test: Part I - Effrct of specimen geometry, strain rate, andfriction on deformation, Mater. Sci. Technol., 2001,
0-4
5 0.3
17, 1133-II4l.
1 o-2
9.
ao1 tlJ
0.0
1234 DISTANCE ALONG ND PATH (mm) Figure 9: Visioplastic and FEA predictions of the equivalent plastic strain for PSC specimens deformed at 400'C at a strain rate of 10 sec1. Nominal strains include 0.10,0.24, 0.40 and 0.65.
investi gate lubrication breakdown during PSC testing of AA1200. The friction condition for PSC testing using graphite lubrication has been successfully derived by direct comparison of visioplastic and FE results. The characterisation of the friction condition over the range of temperatures investigated allows a more accurate determination of the real state of deformation within the PSC specimen. The FE prediction of equiva-
Mirza, M.5., Sellars, C.M., Modelling the hot plane strain compression test: Part 2 - Effect of friction and specimen geometry on spread, Mater. Sci. Technol., 2001, 17, I142I 148. I0. Duckham, A., Theformation of copper-type shear bands in Al-1Mg and their influence on recrystallisation behaviour, PhD Thesis, University of Cape Town, 1998. I I. Beynon, J.H., Sellars, C.M., Strain distribution patterns during plane strain compression, J. Test. Eval., 1985, 13, 2838.
I2. Shi,
H., Mclaren, A.J., Sellars, C.M., Shahoni, R.,
Bolingbroke, R., Constitative equations for high temperature flow stress of aluminium alloys, Mater. Sci. Technol., 1997, I3, 2
10-2 16.
lent strain along symmetry lines incorporating the friction characterisation shows good agreement between experimental and theoretical results. The finite element model incorporating the friction characterisation can therefore provide more accurate approximations of strain and strain rate distribution within the PSC specimen, andwill consequently leadto better interpre-
tation of microstructure evolution. Acknowledgements The authors wish to acknowledge the financial support provided by Hulett Aluminium (Pietermantzburg, RSA), the National Research Foundation (Pretoria, RSA), and the University of Cape Town Research Committee. Gratitude is also extended to the UCT Centre forResearch in Computational Mechanics for providing technical support and access to computational software.
References
I.
Shi, H., Mclaren, A.J., Sellars, C.M., Shahani,
R.,
Bolingbroke, R., Hot plane strain compression testing of aluminium alloys, J. Test. Eval., 1997, 25, 61-73. R & D Journal, 2004, 20 (2) incorporsted into The SA Mechanical Engineer
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