An alternative to the conventional triaxial compression test

Powder Technology 161 (2006) 220 – 226 www.elsevier.com/locate/powtec

An alternative to the conventional triaxial compression test M.S. Nielsen, N. Bay *, M. Eriksen, J.I. Bech, M.H. Hancock Department of Manufacturing Engineering and Management, Produktionstorvet - bygning 425, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Received 30 April 2004; received in revised form 9 November 2005; accepted 11 November 2005 Available online 4 January 2006

Abstract A new test for measurement of the mechanical properties of granular powders is proposed, consisting of upsetting the powder inside a metal tube. Varying the tube material as well as its wall thickness allows superimposing of a variable radial pressure. By pre-compacting the powder inside the tube in a closed die it is possible to test the powder at different densities. The radial pressure is found by correlating measurements of radial bulging of the tube with numerical analysis of tube bulging. Estimates of the error on the determination of the radial pressure are given and it is found that this error may be kept less than T 4%. The coefficient of friction between powder and tube for a specific case is evaluated and found to be between 0.05 and 0.1 at p c > 400 MPa. Data from the test with axial pressures up to 1100 MPa and radial pressures up to 500 MPa are presented. Data on the yield surface for BSCCO (ceramic powder (Bi,Pb)2Sr2Ca2Cu3Ox for manufacturing of superconductors) at 74% density are given and found to be in good agreement with previously published data determined by closed die compaction and fracture tests. The results obtained show that the new test may be a good alternative when high pressures are required or when pressurizing by fluid is impractical. D 2005 Elsevier B.V. All rights reserved. Keywords: BSCCO; Constitutive equations; Mechanical testing; Drucker – Prager; Powder metallurgy

1. Introduction This paper suggests a new way of making a cylindrical triaxial compression test for granular or geo-technical materials. Manufacturing superconductors, such as Fe/MgB2 or Ag/ (Bi,Pb)2Sr2Ca2Cu3Ox (Ag/BSCCO), by the OPIT or OxidePowder-In-Tube method, implies triaxial loading with variable hydrostatic pressures during mechanical processing. The maximum hydrostatic pressure may exceed 500 MPa. When modelling these processes by e.g. the finite element method, it is required to test the MgB2 or BSCCO precursor powder under similar conditions. A large amount of research on triaxial testing of powder has been reported in literature. In the following examples are given on this literature and general principles of triaxial testing. In a true triaxial test two of the three principal stresses should be independently adjustable, which makes the device very complex to build and use. These designs generally rely on multiaxial piston, flat-jack or fluid-bag loading of cubical

* Corresponding author. Tel.: +45 4525 4764; fax: +45 4593 0190. E-mail address: [email protected] (N. Bay). 0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2005.11.004

samples (Hojem and Cook [1], Mogi [2], Gau et al. [3], Amadei and Robinson [4], Handin et al. [5]). The equipment used for this kind of test is usually designed for pressures smaller than 100 MPa. However, in a true triaxial cell designed by Esaki and Kimura [6], cubic samples were subjected to pressures in the Giga – Pascal range. Smart et al. [7] succeeded in designing a true triaxial cell using a simpler test procedure than that used in ordinary square true triaxial cells. Radial stresses reached a magnitude of 55.2 MPa (8000 psi); the author’s ambition being to reach 68.9 MPa (10,000 psi) in the future. An alternative way of obtaining true triaxial data consists of subjecting a hollow cylinder to varying internal and external pressures along with torsion (Alsayed [8], Handin et al. [5]). As with the tests involving cubical samples, equipment used to test hollow samples is complex to build and use. The pressures applied are generally smaller than 100 MPa. Since the typical pressures reached in the true triaxial tests are too low for the testing purpose in the present work, one may look at simpler tests. An example of a simpler test is the typical rotational symmetric triaxial test (Hoek and Franklin [9]) which only controls radial stress r r and thus the hydrostatic pressure by a

M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226

D0 d0

D1 d1

221

F Punches

Stress Ring

∆H/2 ∆H/2

Powder F

Split die

Fig. 1. Outline of PFD test: tube with powder before and after loading.

superimposed liquid pressure. Using this principle, special test equipment for characterization of powders for powder metallurgy may be designed for fluid pressures as high as 200 –700 MPa (Park and Kim [10], Doremus et al. [11], Sinka and Cooks [12]) and 1000 MPa (Massat et al. [13]). These devices are, however complex and expensive to build and they cause limitations due to sealing and friction problems. Even simpler tests may be used to evaluate the powder yield surface: Axial compression (see, e.g. Brown and Abou-Chedid [14]), diametral compression of pre-compacted samples (see, e.g. Fahad [15]) along with data from closed die compaction (see, e.g. Kim et al. [16]) may provide three points on the yield surface as described in Bech [17]. This strategy has also been used by Allais et al. [18] although with a different set of tests. In order to obtain a test method of moderate complexity the PFD (Powder in Flexible Die) test was proposed by Bech [17]. In this test the idea is to induce the radial pressure passively by a plastically deformable metal tube. This makes the experimental set-up quite simple and easy to operate (see Fig. 1). The axial and the radial pressure are determined by measuring corresponding values of the axial load, displacement and degree of bulging and calculating the relationship between bulging and radial pressure by FE analysis. 2. Powder in Flexible Die — basic testing method Before testing, the powder is pre-compacted according to the following procedure, see Fig. 2: 1) lubrication of the inner tube wall, 2) inserting of the tube in a split die clamped in a stress ring, 3) filling the tube with a weighted amount of powder, and 4) loading to a pre-determined density in a 600 kN hydraulic press. The die is mounted in such a way, that it is floating in the axial direction to minimize axial density variation caused by die friction. Lubrication is applied to minimize friction and decrease scattering compared to dry friction conditions. When testing BSCCO a liquid polymer is applied by stippling and when testing MgB2 a thin film of ZnS grease is used or no lubrication is applied at all. All tests are carried out with tubes having the bore diameter f10 mm. In order to vary the superimposed radial pressure the tube wall thickness and the tube material are varied. Nominal outer

Fig. 2. Set-up for pre-compaction of powder in the PFD test.

diameters of the tubes are chosen as f11, f12 or f13 mm, respectively, using three sets of split dies with corresponding bore diameters. The tube materials applied were steels of types: AISI M3:2, A2, P20 modified (denoted P20m), H13, see furthermore Table 1. The punches are designed to withstand a maximum pressure of 2000 MPa. After pre-compaction of the powder the split die is disassembled to take out the tube/powder test specimen. The test specimens are subsequently loaded stepwise with the same punches recording corresponding values of axial load and tube diameter with PC based data acquisition. The maximum diameter of the tube bulge is measured with two opposing electronic length transducers (Tesa gauges) with flat heads. The resolution of the gauges is 1 Am. A combined experimental and numerical analysis applies the measured tube bulge and axial punch displacement to calculate the radial stress o´r on the powder billet by FE analysis and the measured axial load to determine the axial pressure o´z on the powder billet. Assuming a uniform radial stress exerted by the powder sample on the tube wall during the PFD test, an FE model is set-up calculating the radial stress as a function of the tube bulging. This is explained in detail in Section 3. Fig. 3 illustrates the data handling. Following the arrows with one, two and three arrowheads indicate the sequence of the procedure. Curve I, showing the axial pressure p z as a function of the axial strain ( z = ln(1 + DH / H), is determined experimentally. The load is taken as the mean value of the signal from two force transducers on which the punches are mounted; the axial displacement DH is measured by a length transducer. Table 1 Tube material properties AISI code

DIN W. Nr.

E/[Gpa]

M3:2 A2 P20 mod. H13

1.3344 1.2363 1.2738 mod. 1.2344

230 190 205 210

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M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226

pz ; pr

II: pz(εθt)

compression, so instead of using the mean stress it is in the present work chosen to use pressures instead of stresses. Thus the mean pressure becomes p m =  r m, the axial pressure p z =  r z, etc. The decomposition then yields the stress deviator defined by

I: pz(-εz)

p z0 powder yield point

p r0

sij ¼ rij  dij rm

III: pr(εθt)

ð3Þ

where d ij is Kronecker’s delta. The second invariant of this stress deviator J 2 defined by

tube yield point

J2 ¼ sij sij =2 -εz ; εθt Fig. 3. Outline of PFD test data interaction.

In order to compensate for elastic deflections of punches, press and sub-press, a calibration curve is determined measuring the load-displacement curve when loading the punches directly towards each other. Curve II shows the axial pressure as a function of the experimentally measured circumferential strain during testing, ( ht = ln(1 + DD / D). Curve III shows the relationship between radial pressure and circumferential strain, which is based on the above-mentioned FE model. It should be noted that the powder yielding is not necessarily initiated simultaneously with the tube yielding. This is not a problem if the tube yields simultaneously with or slightly before the powder; but if the powder yields before the tube, a large error is introduced because the evaluation of the radial stress then will be based on elastic deformation of the tube. At some point in the test both the powder and the tube will be fully plasticised and as such the data from this point on will be valid for determination of the yield surface. This requires, however, precise determination of the actual powder density at this point and later in the test to be of any use. Having the axial and radial pressure at yield p z0 and p r0 for a specific density, a point on the yield surface for the powder is known. In case the test could be extended to precisely monitor the actual density during the testing, this point could be expanded to an interval yielding information about hardening properties of the powder.

ð4Þ

expressing the magnitude of the stress deviator is then assumed to describe the yield sufficiently. The two invariants thus define the yield surface of the tested powder in the invariant plane, see Fig. 4. The definitions may for the PFD test be reduced to: 1 ðpz0 þ 2pr0 Þ 3 1 qJ2 ¼ pffiffiffi ðpz0  pr0 Þ 3 pm ¼

ð5Þ ð6Þ

Here p z0 is the axial pffiffiffiffiffipressure and p r0 is the radial pressure. The term q J 2 denotes J2 , not to be confused with the geo-technical q. q J 2 has the unit of stress and is proportional to the equivalent stress or v. Mises stress r eq implying: pffiffiffi ð7Þ req ¼ 3qJ2 : As mentioned in the introduction determination of the Drucker –Prager capped cone surface in Fig. 4 may be based on three tests determining the three points A, B and C. The two points A and B are determined by fracture tests. Point A is thus found by a diametrical compression test [12] loading a circular cylindrical disc diametrically between plane anvils and measuring the fracture load, whereas B is determined by uniaxial compression of a circular cylindrical specimen measuring the fracture load for this situation [11]. Point A and B together determine the failure line or the slope of the cone part of Drucker –Prager’s capped cone model. Point C is determined by powder compaction in a closed die [13] measuring corresponding values of load and density. Assuming associated flow potential the strain increments are perpendic-

3. Calculation of invariants The stress situation at initial yielding of the powder is given by the Cartesian stress tensor components r ij : 2 3 0 rr0 0 rij ¼ 4 0 rr0 0 5 ð1Þ 0 0 rz0 Stresses are here negative in compression. This stress tensor is decomposed into hydrostatic and deviatoric parts: the first part being the first invariant of the stress tensor, i.e. the mean stress r m given by the scaled trace of the stress tensor as rm ¼ rkk =3

ð2Þ

adopting Einstein’s summation convention. In geo-technical tradition it is more common to use positive values for

1 ( p + 2pr 0 ) 3 z0 1 (p − p ) qJ2 = 3 z0 r 0 pm =

qJ2

PFD test limits

C B A

D pm

Fig. 4. Outline of invariant plane for PFD test, with Drucker – Prager capped cone model.

M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226

ular to the yield surface implying that the cap part is determined by point C and the direction of the strain path in C both measured in the closed die powder compaction test. The Drucker – Prager model deviates somewhat from the actual shape of the yield surface of ceramic powder material close to the hydrostatic axis and in the region between point B and C. An improvement of the former part of the yield surface may be obtained by a fourth point D determined by isostatic compression in a liquid pressure chamber, see Fig. 4. The region between point B and C may be covered by the PFD test. Performing this test with a thin tube wall in a soft material yields points close to B on the yield surface, which corresponds to p r0 = 0, whereas PFD tests with a thick tube wall in a hard material yields points close point C, which corresponds to compaction in a closed die. Choosing tubes of intermediate thickness and yield stress will give intermediate points between B and C. 4. FEM model of tube deformation In order to determine the radial pressure as a function of the circumferential strain (curve III in Fig. 3) a parametric study of tube bulging was set-up in the commercial FEM program ELFEN [19]. Fig. 5 shows the basic set-up of the FEM model. The parameters used were:

223

σ [MPa] 1200 1000 800 M3:2 P20m fit fit H13 fit A2 fit M3:2 P20m data d t H13 data A2 data

600 400 200 0 0

0.05

0.1

0.15

ε 0.2

Fig. 6. Stress – strain curves for selected tube materials with interpolating fits.

the internal pressure and a shear stress due to Coulomb friction on the tube wall. The pressure and the corresponding shear stress loading were increased in 50– 100 increments. The results were post-processed via custom software inverting the data to find an interpolated function f for the radial pressure as a function of the radial displacement, the powder sample height, the friction coefficient and the wall thickness: pr ¼ f ð DD; H; l; t Þ

ð8Þ

This function was then used to determine the radial pressure from measured data in the PFD test. 5. Estimates of radial pressure error

& & & &

Tube material — see Fig. 6 and Table 1 Tube wall thickness: t Coulomb friction coefficient: l Powder sample height: H

Due to plastic flow under well lubricated conditions and the fact that the radial stress influencing friction increases gradually from small values, sliding between powder and tube wall is assumed to appear. Coulomb’s friction law is therefore adopted for the powder/tube interface. The output of the FEM model was the increase of outer tube diameter as a function of

t

r

∆D/2

τ =µpr pr

H/2

a) undeformed

b) deformed

Fig. 5. FEM model of PFD bulging.

The stress – strain curves of the tube materials were measured directly by upsetting tests, assuming the tube material to be isotropic. The FEM-analysis showed very small or no variations in p r with varying modulus of elasticity, Poisson’s ratio and yield stress (r 0.2). The powder height was directly monitored by length gauges implying an error corresponding to the resolution of measurement of the order 0.1%, i.e. negligible in comparison to other sources of errors such as those of bulging, friction and wall thickness variations. The influence on p r of errors in these free parameters, DD, l and t was analysed by FEM. The curves in Fig. 7 show the radial pressure as a function of the bulging of a tube made of AISI P20m steel (with t = 1 mm, l = 0 and H = 10 mm) at different friction coefficients. A close examination of this figure reveals that if the friction coefficient used is T0.05 off target, the error on the determined p r is of the order T2%. Further simulations show the error to decrease for smaller H. Other analyses show the error in radial pressure to be proportional to variations in the wall thickness. The test tube must be precisely manufactured with a quite tight fit between the punch and the tube (ISO 286-1 tolerance H7/g8) implying rather small variations in wall thickness. Errors due to these variations are estimated to be less than T 1%. The slopes of the curves in Fig. 7 give the error in p r due to measurement errors in DD. In the linear-elastic region of the curve, the error is 3.5 MPa/Am, whereas in the fully plastic region (above DD = 0.055 mm and 140  p r  190 MPa) the slope is less than 0.5 MPa/ Am. With a measurement accuracy of T 1 Am this leads to an

224

M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226

250

pr [Mpa]

200 µ = 0.00

150

µ = 0.05 µ = 0.10 µ = 0.15 µ = 0.20

100

50

0 0

0,1

0,2

die, (see Fig. 2), and compacting the loose powder inside the PFD tube in two different set-ups, i.e. with 1) floating die and 2) die fixed with respect to the lower punch allowing no sliding of the tube along the die. The difference in load-displacement curves in the two set-ups determines the friction load on the tube wall. Fig. 8 shows the estimated friction stress in the powder/tube interface as a function of the axial pressure in compaction of BSCCO. Assuming a radial pressure of 35– 45% of the axial pressure as discussed in Section 7, the friction coefficient lies between 0.05 and 0.10 for p c > 400 MPa which according to the previous analysis introduces an acceptably low error on the estimated radial pressure.

0,3

7. PFD results

∆D[mm] Fig. 7. Radial pressure in bulging of a P20m tube, H = 10 mm, t = 1 mm for l = 0.00 – 0.20 as a function of diameter change due to bulging.

error in p r of less than 0.4%. The total error of the radial pressure may then be determined within T 4% (2% + 1% + 0.4% < 4%) even though this depends on precisely measured parameters. Pre-compaction of the powder in the tube in the split die leads to a slight increase of the tube diameter, after disassembly of the die, due to elastic recovery. The amount of deformation is, however very small and has not been taken into account in the present investigation. Eventually a precise evaluation of the density of the deformed powder has to be included in the analysis. The geometry of the pre-compacted sample and the precise weight of the powder are sufficient for an evaluation of the test density, but if data from stress states after the initial yielding of the powder are being used, the evaluation of current density is more difficult. This remains however for future work on development of the PFD test. 6. Measurement of friction An estimate of the friction between the powder and the PFD tube is done by mounting the tube inside the pre-stressed, split

Fig. 9 shows a PFD test of BSCCO precursor powder with an AISI H13 steel tube, t = 1.5 mm, initial density q rel = 73.7%. The bend on the p z(( z) curve indicates the onset on plastic yielding of the powder material. The fact that the p z(( h ) curve bends at the same axial pressure indicates that plastic deformation of the tube is initiated simultaneously with the powder yielding, which is required for accurate estimation of the radial pressure p r. In the figure is also shown an approximate relative density for the powder during testing. This approximation is done by calculating the volume in the PFD tube under the assumption of a geometrical similar bulgeform only depending on DD and H. Inhomogeneous distribution of the density is not taken into account, and the test is only valid for small DD because the powder will escape between punch and tube wall at a larger degree of bulging. Larger DD will also cause problems as regards the assumption of constant radial stress over the total sample height used in the FEM modelling of the tube deformation. Fig. 10 shows the results from Fig. 9 in the invariant plane along with a Drucker –Prager capped cone model from Bech [17], which is based on the three experimental test points determined by fracture tests, A and B and by compaction in a closed die, C. It is seen that the data from the PFD test, which are determined at a density 73.5% < q rel < 74% similar to the

70

700

50

600

40

500

30 20

2000

Fig. 8. Shear stresses as function of axial pressure for BSCCO in three closed die tests with PFD inner tube. Final height H = 10 mm, sample diameter d = f10 mm.

78%

74%

100 1600

80%

300

0 800 1200 pc [MPa]

ρrel=ρrel(εθ)

76%

10

400

pr=p r(εθ)

400

200

0

pz =pz(εθ)

ρ rel

Shear stress

pz,pr [MPa]

τ [MPa]

60

pz=pz(εz)

72%

0

70% 0

0,02

0,04 0,06 εθ,−εz/2

0,08

0,1

Fig. 9. PFD test of BSCCO with an AISI H13 steel tube — t=1.5 mm, initial relative density 73.7%.

M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226 Fracture and compaction tests Drucker Prager 74% Drucker Prager 65% PFD test relative density

600

73.5% < ρrel < 74.0%

78%

100

76%

C

ρ rel

qJ2 [MPa]

150

85%

450

250 200

B

75%

B 150

70%

0 -50

72%

65% 450 pm [MPa]

A

950

70% 150

250 350 pm [MPa]

450

Fig. 12. Results from test shown in Fig. 11 in invariant plane with Drucker – Prager capped cone models based on fracture and compression tests.

550

density of the Drucker –Prager test points A, B and C fits very well with these. In the figure is marked the interval where the density data may be considered sufficiently close to that of the Drucker – Prager model and the assumptions for the FEM model hold true. The results indicate that the PFD test results are in good agreement with the results from Bech. Another example of results from the PFD test is shown in Fig. 11 depicting a PFD test of BSCCO precursor powder with a P20m tube, t = 1.5 mm, initial density q rel = 62.7%. Bending of the p z(( z) curve occurs at p z = 200 MPa, i.e. much earlier than bending of the p z(( h ) curve occurring at p z = 420 MPa implying that the powder yields before the tube. This means that the test initially will perform like a closed die upsetting test. When the tube eventually yields, the powder is fully plasticised and as such the data from this point forward may be used as yield data for the powder. Fig. 11 furthermore verifies the earlier mentioned assumption in Section 6 on the ratio between radial and axial pressure equals to 35 –45%. This assumption is based on literature and own experiments with the PFD test of insufficiently precompacted precursor powder like the one presented in Fig. 11, pz=pz(εz)

pz=pz(εθ)

pr=pr(εθ)

ρ rel= ρ rel(εθ)

1200

PFD tests Fracture and compaction tests Drucker Prager Soft rock model

84% 81%

800 600

78%

300

75%

250

72%

ρrel

1000

69%

400

66% 200 0 0,00

where test conditions are comparable to those in closed die upsetting. In Fig. 12, representing the invariant plane, it is noticed that the yield data of the test with insufficient pre-compaction corresponding to Fig. 11 are lying on a straight line passing through origin and the closed die upsetting point, C, confirming that the initial part of the test performs like the closed die upsetting test, although it is also noticed that the densities obtained with the PFD test at the intersection points with the cap, q rel, PFD = 77% and 84% are somewhat higher than the corresponding values obtained with the closed die upsetting test q rel, PFD = 75% and 80%. This may be due to elastic deformation of the tube, which is not accounted for. Again the data for large DD, i.e. large ( h, and high densities must be taken as approximate. Additionally it must be mentioned that there is a higher risk of powder escaping between the punch and the tube wall because of the longer punch travel for this particular test. Another extreme of the PFD tests may occur, when the radial pressure is too small to impede fracture of the powder. This leads to points on the failure line, which are qualitatively similar to those obtained by an ordinary upsetting of precompacted powder. A mapping of the yield surface of BSCCO powder determined by the PFD test at a relative density of 75% is

0,05

0,10 εθ,−εz/2

qJ2 [Mpa]

50

Fig. 10. Results from test shown in Fig. 9 in invariant plane with Drucker – Prager capped cone models based on fracture and compression tests.

pz,pr [MPa]

80%

C

300

A

74%

50 0 -50

90%

ρrel

300

Fracture and compaction tests Drucker Prager 80% Drucker Prager 75% PFD test relative density

80% qJ2 [MPa]

350

225

150 50

60%

0

Fig. 11. PFD test of BSCCO with a P20m tube, t = 1.5 mm, initial relative density 62.7%.

B

100

63% 0,15

C

200

A 0

D 100

200 300 pm [MPa]

400

500

600

Fig. 13. Results from PFD test of BSCCO for a relative density of 75% fitted to Drucker – Prager capped cone and Soft rock geo-technical yield surfaces.

226

M.S. Nielsen et al. / Powder Technology 161 (2006) 220 – 226

shown in Fig. 13 along with fitted Drucker – Prager capped cone and Soft rock models [15]. As seen from the figure, the points show the correct order of magnitude but are scattered. This scatter is probably due to poor control of the friction between the powder and the tube wall. The inner surfaces of the tubes were finished by grinding, which led to rather large variation in roughness. Another factor is the plastic coating, which was seen to penetrate the BSCCO powder sample to some degree, thus changing the material response. Generally is must be concluded that the radial pressures in the experimental points depicted in Fig. 13 are not determined within the optimum T4%, but also that the method shows promising results if the friction conditions may be better controlled. Note that the points from the fracture and compaction tests are earlier data from Bech [17] showing good agreement with the PFD data obtained in the present study. The data achieved is fitted to the Soft rock constitutive model given by:   ðpSRt þ pm Þsin/ pm  pSRc 1=n qJ 2 ¼ ð9Þ 1 p  pSRc cosh  pffiffiffi sin/sinh SRt 3 where n, p SRc, p SRt and / are material parameters, and h is the Lode angle given by ! pffiffiffi 3 3 sij sjk ski =3 cosð3hÞ ¼ ð10Þ 2 q3J2 using negative stresses in compression. In the PFD test the Lode angle is 60- corresponding to a stress state on the meridian of uniaxial compression. The model thus extrapolates the yield surface outside these meridians. 8. Conclusions Analysis of the suggested PFD test shows that the test may be a good alternative to an ordinary triaxial powder test with rotational symmetric stress state, especially when testing at high pressures or when pressurizing by fluid is impractical. Data from the test with axial pressures up to 1100 MPa and radial pressure of up to 500 MPa is shown. The test combines a rather simple experimental technique with numerical modelling using elastic – plastic FE analysis. Systematic FE analysis has enabled evaluation of the radial pressure as a function of the main test parameters showing that the error in determination of the radial pressure may be kept less than T 4%. The presented data is in agreement with data previously published by Bech [17] but provides additional data enabling more complete modelling of the yield surface. Acknowledgments The authors would like to express their thanks to Mr. J. R. Lete for carrying out experimental tests on BSCCO powder.

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