ISSN 1062-7391, Journal of Mining Science, 2012, Vol. 48, No. 3, pp. 429-435. © Pleiades Publishing, Ltd., 2012. Original Russian Text © V.N. Aptukov, S.A. Konstantinova, V.Yu. Mitin, A.P. Skachkov, 2012, published in Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, 2012, No. 3, pp. 35-43.

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Nano- and Micro-Range Mechanical Characteristics of Sylvite Grain V. N. Aptukova, S. A. Konstantinovab†, V. Yu. Mitina, and A. P. Skachkova a

Perm State National Research University, ul. Bukireva 15, Perm, 614000 Russia e-mail: [email protected] b GALURGIA JSC., ul. Sibirskaya 94, Perm, 614000 Russia Received March 12, 2012

Abstract—The paper discusses the indentation testing of sylvite grain using scanning probing microscope Dimension ICON and facility NanotTest-600. The authors present the approximated processing procedure for experimental curves and compare the obtained values of hardness and elastic modulus at different scale levels. Keywords: sylvite, elastic modulus, hardness, micro- and nano-range

INTRODUCTION

The interest in the determination of micro- and nano-range mechanical properties of salt rock fractions aims at finding the influence of mineral structure on its strength [1] and in the analysis of role of mineral nanoparticles in the natural and mining-induced processes in rock masses [2]. The authors have previously determined elastic modulus and microhardness of grains and grain-to-grain interfaces in sylvite, halite and carnallite on Nano-Test-600 facility [3]. The present study focuses on estimating mechanical characteristics of sylvite grains in nanorange, comparing the obtained estimates with the micro-range data and on the experimental data processing procedures. 1. NANOINDENTATION DATA PROCESSING TECHNIQUES

The tests were conducted at the Perm State National Research University on the scanning probing microscope Dimension ICON (Bruker AXS, Germany), a dedicated unique facility for surface analyses of semiconductor plates, magnetic media, bio-materials and other materials [4]. Indentation is one of the main tools of nano- and micro-range testing. A specimen is loaded via indentation a special micromechanical probe-indenter. The measurement parameters are penetration force and the current indentation depth. Reduction in size of a specimen, or it structural elements up to R * ≤ 1 µm greatly changes mechanical properties of the specimen [5-7]; high changes are observed when R * ≤ 100 − 1000 nm, and at R * ≤ 10 nm the changes become sweeping. The Meyer hardness is determined by the formula: P , (1) H= Ac where P is indenter’s force; Ac = f (u ) is projection of indentation surface area onto the specimen face; u is the current indentation depth. Difficulty arises in finding Ac = f (u ) as there is a diversity of the approaches and methods to this purpose [8, 9]. The most wide spread is the standard method by Oliver-Pharr [10]: elastic modulus of a specimen, E, is determined in terms of a reduced modulus E * and Poisson’s ratio ν : _____________________________ †Deceased.

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E * Ei , (2) Ei − E * (1 −ν i2 ) where Ei = 1141 GPa and ν i = 0.07 are elastic modulus and Poisson’s ratio of the Berkovich pyramid. E = (1 −ν 2 )

2. INDENTATION HARDNESS ESTIMATION ON THE DIMENSION ICON MICROSCOPE

The experimental data base includes the function d(z), where z is the fixed tip displacement of the indenter, nm; d is the indenter offsetting after interaction with the specimen, nm (refer to Fig. 1). Plotting the force and displacement curves P(u) requires drawing the function d(z) as follows. 1. Shift the plot vertically so that the offsetting of the push stroke goes out of zero point (Fig. 1). 2. The point of sharp growth in d is the probe indentation start. Shift horizontally presented data at this point. 3. Calculate deformation: at each downward step Δz the probe bends by Δd and at the same time dents into specimen by Δu. The push stroke shortening is calculated as u = z − d, аnd P = kd, where k = 440 N/m is the indenter stiffness. The backward stroke calculations are made in the same manner, the penetration depth P(u) is obtained as a result. The processing of the curves P(u) needs the geometrical sizes of the indenter (the probe tip specifications [3]). The tip of a diamond indenter is a trihedral angle with corner radius; to make symmetrical dints the diamond is mounted with the indenter axis to be normal to the specimen surface. On indentation the indenter is initially set off the specimen surface approx by 12° horizontally. In the present study testing, the indenter had the following sizes: the tip height—50 µm; front angle— α F = 55 ± 2.0° , back angle— α B = 35 ± 2.0° , side angle— α S = 51 ± 2.04° , tip radius—40 nm. The hardness determination needs the data on the dint side area A and its projection on the specimen face, Ac , versus current indentation depth u. The calculations assumed that the indenter is a triangular pyramid with side surface represented by three isoscale triangles. One triangle tip angle is divided by apothem into two equal parts (51°), the rest two triangles have their tip angles divided by apothems into unequal parts (35 and 55°). The indenter penetration depth was 120-150 nm and its tip corner radius was 40 nm in the experiment. The latter was disregarded in the calculation of the required areas, so was the indenter incline angle to the specimen surface.

Fig. 1. The function dz: curve 1 illustrates motion of the probe toward the specimen (push stroke); curve 2 shows the probe movement off the specimen (backward stroke).

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With cumbersome transformations omitted, the main formulas of the surface area calculation are: A = Ku 2 , Ac = K cu 2 , (3) where K =

sinψ sin γ + 2 sin β sin ϕ 4C 3 cos α * ; Kc = 1 , 2 D1D2 D3 D4 sin γ D1D2 D3 D4 D1 = C1 + C2 − C3 , D2 = C1 + C2 + C3 , D3 = C1 − C 2 + C3 , D4 = −C1 + C2 + C3 ,

C1 =

sin 2 ϕ sin 2 ψ sin β − , C2 = sin α * , C3 = , 2 2 * sin γ sin γ 4 sin α

α * = 90° − α S , β = 90° − α F , γ = 90° − α B , ϕ = α B + α F , ψ = 2α S . For the tip with the pre-set sizes ( α F , α B , α S ), the coefficients K = 5.43 and K c = 3.34 , wherefrom we have expressions to find the hardnesses by Meyer, Н, and Brinell, H * : H = 0.30

P P , H * = 0.18 2 . 2 u u

(4)

3. HARDNESS AND ELASTIC MODULUS UNDER THE BERKOVICH PYRAMID INDENTATION

An approximated approach to hardness and elastic modulus estimation is based on the dimension analysis. Let the elastic modulus of the pyramid greatly exceed the elastic modulus of the test material; so the pyramid is assumed absolutely stiff. The main variables of the penetration process are: the indentation force P and depth u, elastic modulus E, Poisson’s ratio ν , yield strength σ s , friction factor k f , tip angle of the four-sided pyramid (the angle between the pyramid and its side heights), α . According to the π-theorem [11], seven variables can form five independent dimensionless complexes P / Eu 2 , σ s / E , ν , α , k f , so: P = f (σ s / E,ν ,α , k f ) . Eu 2

(5)

Elastic deformation is neglectable at this stage of loading by high forces, since this is a plastic process governed mainly by the yield strength of the material. Poisson’s ratio and elastic modulus are eliminated from the analysis, and relationship (5) is transformed to: (6) P = σ s u 2 f L (α , k f ) . Consider the pyramid indented into a half-space by the depth u. The side surface is exposed to the normal stress σ n , that is the specimen material resistance to the pyramid penetration, and to the shear stress τ n determined by friction. The integral of the force distribution over the indented pyramid surface, projected to the vertical axis, is the total indentation resistance force P. It follows from (6) that P in the first approximation is proportional to the indentation depth square and σ n .The normal stress, or the indentation resistance of the specimen material, is a value physically close to the material hardness. Material hardness (e.g., Brinell’s hardness), in its turn, is proportional to the material yield strength σ s [12] and depends on the strain hardening of the material and on contact friction. The friction is possible to present as a separate term. The pyramid indentation into a half-space uses the work W spent to form a dint: W = σ dV , where σ d is energy density per dint unit volume formation; V is dint volume. The base of the Berkovich 1 pyramid is a regular triangle, therefore the following is valid: V = u S base = 3 u 3 tan 2α . The change 3 JOURNAL OF MINING SCIENCE Vol. 48 No. 3 2012

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in the work, dW, equals the product of the force P and the indentation depth increment du: dW = Pdu , which allows deriving the close to (6) expression: P = 3 3σ d u 2 tan 2α . (7) Thus, the expression to find the indentation resistance force P in terms of the assigned normal stress σ n and shear stress τ n on the pyramid side is:

P = Pn + Pτ = 3 3u 2σ n tan 2α + 3 3u 2τ n tanα . (8) Relationship between the friction resistance force and penetration resistance force is: Pτ τ n = C tanα = k f C tanα . With the Berkovich pyramid’s α ≈ 65°7′ and friction factor k f 0.1-0.3, Pn σ n we have Pτ / Pn ≈ 0.03 − 0.09 . Thus, friction can be eliminated from estimating the penetration resistance force in indentation of the Berkovich pyramid in the half-space. Comparing the force expressions (7) and (8) yields σ n ≈ σ d . Put it otherwise, average penetration resistance (energy density required to form a dint unit volume) σ d equals average normal indentation response of the medium and indenter interface, σ n . Having an experimental curve Pe (u ) allows finding the dint unit volume energy density that is proportional to the yield limit of the material: Pe (u ) P (u ) ≈ 0.087 e 2 . σd ≈ (9) 2 2 u 3 3 u tan α The relieving stage of the material is free from the influence of the yield limit σ s , which is thus withdrawn from the analysis. We have a new variable—u ′ —elastic displacement of the pyramid from the depth u at the relieving stage. Then, (5) transforms into: P = Eu 2 fU (u′ / u,ν , α , k f ) . (10) Let the force and displacement relation be linear in the first approximation at the relieving stage and the friction be disregarded. The, relation (1) changes to: P = Eu 2 (u′ / u ) fU* (ν , α ) = Euu′C *(ν ) . (11) Formula (11) is derived for the fixed α of the pyramid. The unknown member C * only depends on Poisson’s ratio ν . Using experimental data, the elastic modulus of relieving, E, is from (11): 2 P (u ) . (12) E= C * (ν ) u u ′ The unknown member C *(ν ) is found from solving elastic relieving problem in 2D axially symmetric formulation with replacing the pyramid by a cone with its apex angle equal to arithmetic mean value of angles of the inscribed and circumscribed cones of the pyramid. Below is the approximated relation obtained by solving a series of elastic problems with varied ν (the expression is valid for 0.15 < ν < 0.45 ): C *(ν ) = − 0.538 − 0.78ν + 2.20ν 2 . (13) 4. EXPERIMENT

The experimental analysis used a fragment of milk-white sylvite grain obtained from a fine- and mid-grain structure sylvite specimen (grain size 1-4 mm). Sampling site was the Solikamsk Mine-2 (seam VS, layers nos. 2 and 4). The grain fragment was obtained by cleaving the grain along the cleavage plane.

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The primary analysis was the optical study of the sylvite surface area 2×2 mm. Two areas 40×40 µm were distinguished: the area A—free from visible inclusions (crystal grains) and the area B—with crystal grains. The surface relief of the areas A and B is illustrated in Fig. 2: circles point at the further nanoindentation sites, and in Fig. 3.

Fig. 2. Surface relief of the area 40×40 µm: (a) area A; (b) area B.

Fig. 3. 3D image of the sylvite fragment surface relief: (a) area A; (b) area B.

Fig. 4. Curve of the force to indentation depth on the loading-relieving exposure of the area A, point 2 in Fig. 2.

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In any curve P(u) plotted based on the experimental data, distinguishable are three basic points: point 0 ( u0 , P0 )—the force is maximal; point 1 ( u1, P1 )—the displacement under relieving is maximal (visco-plastic relieving transforms to purely elastic relief); point 2 ( u2 , P2 )—the force is minimal (almost vanishes). Table1 gives the calculated values of hardness for all indentation points in the areas A and B. The hardnesses H 1 and H1* by Meyer and Brinell, respectively, correspond to point 1 in Fig. 4—the point of transfer to the elastic relieving domain. The hardnesses allow evaluating the tested material microyield limit [12]. Based on the analysis of the results, the average Meyer’s hardness Н at the test points in the area B (1.62 GPa) is higher than in the area A (1.40 GPa). This is probably due to the statistic properties of micro-relief, which is more disordered (chaotic) in the area B, and owing to the presence of crystal grain centers (in the area B). The maximal hardness is obtained for point 4 in the area B, in the crystal grain growth zone. Comparison of the presented data obtained on the scanning probing microscope Dimension ICON and the earlier results obtained using the cell NanoTest-600 (0.23 ± 0.03 GPа [3]) shows that the present values are 5-8 times higher, which is explained by the scale effect [5-7]. The elastic modulus Е was evaluated using (12) re-written based on the analysis of dimensions: P2 − P1 1 , (14) E= * u C (ν ) 2 (u2 − u1 ) where u1 and u2 —the indenter penetration depths for points 1 and 2, respectively (refer to Fig. 4). As we said earlier, the coefficient С * in (14) depends on Poisson’s ratio only and ranges from 0.47 (ν = 0.2 ) to 0.58 (ν = 0.4 ), the, we find to elastic modulus values for two values of С * (see Table 2). Table 1

Point

Н *, GPa

Н , GPa

Н1*, GPa

Н1 , GPa

A1 A2 A3 B1 B2 B3 B4

1.07 0.82 0.70 1.01 0.93 0.79 1.24

1.74 1.33 1.14 1.65 1.51 1.29 2.01

0.30 0.25 0.21 0.29 0.22 0.25 0.29

0.48 0.41 0.35 0.47 0.35 0.40 0.48

Table 2

Point A1 A2 A3 B1 B2 B3 B4

Elastic modulus, GPa С * = 0.47 ( ν = 0.2 )

С * = 0.58 (ν = 0.4 )

21.5 21.9 19.3 30.2 23.4 19.4 21.7

17.4 17.7 15.6 24.5 19.0 15.7 17.6

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The average elastic moduli (for instance, at С * = 0.47 ) are 20.9 GPa in the area A and 23.7 GPa in the area B, i.e., they are close. Comparison with the previously obtained data [3] shows sufficiently satisfactory correspondence: Е = 27.4 ± 3.7 GPa. In one of the areas on the tested specimen surface, analogous to the area A, indentation was conducted at varied indenter penetration rates of 50, 500 and 5000 nm/s. The tests showed that the two orders change in the penetration rate did not affect the force-displacement curves. We assume the average penetration depth of 120-150 nm. This means that the relieving periods in the material at the set scale are out of the range of 0.02-3 s. CONCLUSION

The authors have compared the new and previous indentation test results on a milk-while sylvite fragment using scanning probing microscope Dimension ICON and the facility NanoTest-600. Based on the comparison, the hardness and elastic modulus are approximately estimated using the experimental curves of force-penetration depth for different areas of sylvite grain with different nanostructures. It has been shown that the indenter penetration rate change in the range of 50-5000 nm/s weakly influences the force-penetration depth curve. ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 09-08-99075 r-ofi, and by the Analytical Departmental Target Program “Development of Sceintific Potential of the Higher School,” project no. 2.1.2/5135. REFERENCES 1. Proskuraykov, N.M., Permyakov, R.S., and Chernikov, A.K., Fiziko-mekhanicheskie svoistva solyanykh porod (Physico-Mechanical Properties of Salt Rocks), Leningrad: Nedra, 1973. 2. Trubetskoy, K.N., Viktorov, S.D., Galchenko, Yu.P., and Odintsev, V.N., “Technogeneous Mineral Nanoparticles as the Problem of the Subsoil Development,” Vestn. RAN, 2006, vol. 76, no. 4. 3. Aptukov, V.N., Konstantinova, S.A., and Skachkov, A.P., “Micromechanical Characteristics of Carnallite, Sylvinite and Salt Rocks at Upper Kama Deposit,” Journal of Mining Science, 2010, vol. 46, no. 4. 4. http://www.bruker-axs.com/dimension-icon_atomic_force_microscope. 5. Golovin, Yu.I., “Nanoindentation as a Means for Complex Estimating Physico-Mechanical Properties of Submicro-Volume Materials (Review),” Zavod. Lab. Diagnost. Mater., 2009, vol. 75, no. 1. 6. Andrievsky, R.A. and Ragulya, A.V., Nanostrukturnye materialy (Nanostructure Materials), Moscow: Akademiya, 2005. 7. Gusev, A.I., Nanomaterialy, nanostruktury, nanotekhnologii (Nanomaterials, Nanostructures, Nantechnologies), Moscow: Fizmatlit, 2005. 8. Panich, N. and Yong, S., “Improved Method to Determine Hardness and Elastic Module Using NanoIndentation,” KMITL Sci. J., 2005, vol. 5, nо. 2. 9. Sun, Y., Zheng, S., Bell, T., and Smith, J., “Indenter Tip Radius and Load Frame Compliance Calibration Using Nanoindentation Load Curves,” Philosophical Magazine Letter, 1999, nо. 79. 10. Oliver, W.C. and Pharr, G.M., “An Improved Technique for Determining Hardness and Elastic Module Using Load and Displacement Sensing Indentation Experiments,” J. Materials Research, 1992, nо. 7. 11. Sedov, L.I., Mekhanika sploshnoi sredy (Continuum Mechanics), Moscow: Nauka, 1973. 12. Aptukov, V.N., “Expansion of a Spherical Hole in an Elastic-Plastic Medium under Finite Deformations. Report I: Effect of Mechanical Characteristics, Free Surface, Cleavage,” Porbl. Prochn., 1991, no. 12.

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