Elsevier

Acta Materialia

Volume 56, Issue 11, June 2008, Pages 2476-2487
Acta Materialia

Elastic and hardness anisotropy and the indentation size effect of pyrite (FeS2) single crystal

https://doi.org/10.1016/j.actamat.2008.01.022Get rights and content

Abstract

The Knoop hardness of single crystal pyrite, FeS2, was investigated on the (1 1 1) and the (1 0 0) planes by indentation testing from 10 to 500 gf (0.10–4.90 N). An energy balance was applied to analyse the hardness and the indentation size effect (ISE). The hardness anisotropy is directly attributed to the anisotropy of the Young’s modulus (E). The isotropic force-independent hardness for these two planes are 962 kgf mm−2 (94.34 N mm−2) for the (1 1 1) plane and 900 kgf mm−2 (88.26 N mm−2) for the (1 0 0) plane. When the ISE is prevalent at low indentation test forces, the surface area to volume ratio of the indentation is high because a part of these phenomena is attributable to the friction between the indenter and the specimen.

Introduction

The stress–strain relationships of linear elasticity are described by the generalized Hooke’s law:σij=Cijlkmεkmεij=Sijkmσkmwhere Cijlkm and Sijkm are the stiffness and compliance tensors, respectively. Their components of anisotropic media depend on the orientation of the reference axes. The general transformation equations for the compliance and stiffness coefficients areSijkl=cosimcosjncoskpcoslqSmnpqCijkl=cosimcosjncoskpcoslqCmnpqwhere the indices of the direction cosines have values 1, 2 and 3. For cubic symmetry, Johnson [1] has demonstrated that the compliance transformation takes the form:Sijkl=S1122δijδkl+S1212(δikδjl+δilδjk)+(S1111-S1112-2S1212)cosincosjncoskncoslnwhere δij is the Kronecker delta, with summation on U = 1, 2, 3. The angular dependence is contained here in the sum of only three terms instead of the 81 terms of Eq. (3). The commonly used elastic coefficients are related to the components as follows:Young’s modulus,Ei=1SiiiiShear modulus,Gij=14Siijj(ij)Poisson’s ratio,Vij=-SiijjSiiii(ij)

The elastic anisotropy of a crystal is the orientation-dependence of the elastic moduli or sound velocities. Essentially, all known crystals are elastically anisotropic. A proper description of such an anisotropic effect has, therefore, important implications in engineering science as well as in crystal physics. Chung and Buessen [2] have provided a convenient method of describing the degree of elastic anisotropy in a given cubic, hexagonal, trigonal or tetragonal crystal by showing its actual values. It is well known that the elastic anisotropy of crystals controls their mechanical properties subject to stiffness or compliance concentration. A convenient method of describing the degree of elastic anisotropy for a cubic crystal has been defined asAcubic=3(A-1)2[3(A-1)2-25A]whereA=2C44(C11-C12)=S11-S122S44where A is the usual anisotropy factor. A is zero for elastically isotropic crystals (i.e. A = 1), and for an anisotropic crystal A is a single valued term of the elastic anisotropy regardless of whether A > 1 or A < 1. Further, it was shown that A gives the relative magnitude of the actual elastic anisotropy possessed by a crystal [2]. An example of this is tungsten at about 200 K where A is exactly equal to one. This material has elastic moduli independent of crystallographic direction. For anisotropic crystals, such as pyrite, the anisotropic factor A as calculated from the single crystal elastic constants is 0.13. The actual magnitude of the elastic anisotropy for pyrite is difficult to determine on the basis of A values if one compares values for other cubic crystals, e.g. 0.58 of CaF2 and 1.03 of BaF2. However, one can find the percentage elastic anisotropy specified by Eq. (7), which is 39% for pyrite, 3.53% for CaF2 and 0.01% for BaF2. Thus, one must conclude that pyrite is much more anisotropic than CaF2 and BaF2 in accordance with the actual data on the orientation-dependent shear modulus in these crystals.

In previous investigations [3], [4], [5], [6], [7], [8] researchers have obtained the compliances of pyrite at different temperatures. This allows the Young’s modulus and the shear modulus to be easily calculated as a function of the crystalline orientation of a co-zonal plane of pyrite single crystal. Fig. 1 shows the results for the (1 1 1) plane and Fig. 2 for the (1 0 0) plane. If the term S = (S1111  S1112  2S1212) has a negative or positive value, it may seen from Eq. (6) that both the Young’s modulus and the shear modulus depend on the crystalline directions 〈h k l〉.

Fig. 1 illustrates the Young’s modulus of pyrite (1 1 1), showing a minimum on [12¯1] [2¯11] and a maximum on [1¯1¯2]. The shear modulus has a maximum on [12¯1] [2¯11] and a minimum on [1¯1¯2]. Fig. 2 illustrates the Young’s modulus of pyrite (1 0 0), showing a minimum on [0 1 1] and a maximum on [0 0 1] [0 1 0]. Also, as for the (1 1 1) plane, though in the opposite sense, the shear modulus has a maximum on [0 1 1] and a minimum on [0 0 1] [0 1 0], respectively.

Chung and Buessen [2] explained that the above orientation-dependent process arises from a negative S value and shows the opposite behavior when the S value is positive. Chung and Buessen also found that negative S implies that the value of the anisotropy factor A has to be less than one, and positive S implies A greater than unity. Thus, one can classify the orientation-dependence of the technical elastic moduli in cubic crystals into three distinct kinds. The first kind includes those crystals for S = 0 and A = 1 and thus A = 0 and, as noted above, these are physically evidenced by the fact that the elastic properties are invariant with crystalline direction. Examples of this kind are tungsten at about 200 K and BaF2 at about 220 K. The second kind is represented by all those crystals with a negative S and A < 1. Examples of this kind include pyrite single crystals and most of the transition metals, alkali halides (except for the lithium halides) and fluorides. The third kind of cubic crystals is characterized by positive S and A > 1 and includes most face-centered cubic (fcc) and body-centered cubic (bcc) metals, semiconductor compounds, some simple oxides (e.g. MgO), diamond and spinel.

Indentation hardness testing is widely applied to describe numerous material parameters for a variety of research, development and industrial applications [9], [10], [11]. Two important phenomena reported are: (i) the hardness anisotropy [12] and (ii) the indentation size effect (ISE) [13]. Relative to anisotropy, indentation hardness has often been reported to be a function of both the crystal plane and the crystallographic direction of the plane [13], [14]. This anisotropy of crystalline hardness has frequently been explained in terms of the primary slip system for a particular crystal structure of interest. These include systems in bcc metals, fcc metals, Fe2O3 (hematite), Al2O3 (sapphire), NaCl (rock salt), C (diamond cubic) and CaF2 (fluorite) [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

The ISE is the increase in the apparent hardness with decrease in the applied indentation test force; in other words, the ISE is dependent on the hardness on the applied indentation test force. The ISE is usually represented by plotting the hardness as a function of the indentation test force, or resulting indentation size. Fig. 3 illustrates the two distinct regions, or regimes, that are of significance for determining hardness in general. The apparent hardness is a function of the applied test force for low-force indentation testing where there is no single value of hardness. This is the ISE. At high indentation forces, the hardness is constant with respect to the indentation test load and a single, well-defined value exists. That value is the force-independent hardness, HK-FIH, a quantity that has also been referred to as the “true” hardness in some hardness literature. This paper addresses this specific issue by analyzing the force-independence of the hardness and its correlation with the Young’s modulus of single crystal pyrite (FeS2). This is important because it will allow the reader to understand the fundamental mechanical and elastic properties of this compound, which is mainly used for steel fabrication, where it is essential from the early beneficiation stages including fracture and wear at room temperature. This study focusses on the (1 0 0) and (1 1 1) planes of single crystal pyrite by using the Knoop hardness test, which allows both the hardness anisotropy and the ISE to be investigated.

Section snippets

Experimental procedures

Natural pyrite single crystals with a purity of 99.96% were prepared for hardness measurements using a diamond saw to cut test specimens in the (1 1 1) plane and (1 0 0) planes. Fig. 4 illustrates the crystallographic orientation for each of the test specimens relative to the unit cell. Oriented and sawn samples were mounted and then manually polished to 400 grit SiC. Final polishing was completed with 0.5 μm diamond paste followed by 0.01 μm silica in an automatic vibratory polisher. This resulted in

Anisotropy and the indentation size effect

For the (1 1 1) and (1 0 0) planes, the Knoop hardness was measured as a function of the crystallographic orientation and the indentation test force (Fig. 5, Fig. 6). The hardness anisotropy is evident. For the (1 1 1) plane, hardness maxima are observed for low indentation test forces when the long axis of the Knoop indenter is aligned parallel to the [1¯1¯2] direction. Hardness minima occur in the [2¯11] directions. Surprisingly, these maxima and minima orientations are exactly the same as the

Summary and conclusions

Knoop hardnesses were measured on the (1 1 1) and (1 0 0) planes of single crystal pyrite for a range of indentation test forces and different indenter orientations of these planes. Hardness anisotropy and an ISE were observed. The hardness anisotropy at low indentation forces is attributed to the Young’s modulus of a specific crystallographic direction and the friction character of the experiment. This can be clearly seen by comparison of Fig. 1, Fig. 2 with Fig. 5, Fig. 6 as well as with Fig. 9.

Acknowledgments

The author would like to thank the CONACYT institution and the PIFI program of the National Polytechnic Institute for its sponsorship of his studies at the National Polytechnic Institute, Mexico DF, Mexico and at The University of Alabama, Tuscaloosa, AL, USA. The author would like to address special thanks and recognition to Prof. Richard C. Bradt, University of Alabama, whose mentoring, encouragement and teaching helped him to complete the present study and write his Ph.D. studies.

References (43)

  • M.E. Stevenson et al.

    J Eur Ceram Soc

    (2002)
  • H. Li et al.

    J Mater Sci

    (1996)
  • H. Li et al.

    Mater Sci Eng A

    (1991)
  • W.D. Nix et al.

    J Mech Phys Solids

    (1998)
  • P.-L. Larsson

    Int J Mech Sci

    (2001)
  • K.L. Johnson

    J Mech Phys Solids

    (1970)
  • J. Bystrzycki et al.

    Scripta Metall Mater

    (1993)
  • M. Mata et al.

    J Mech Phys Solids

    (2004)
  • K.L. Johnson

    Contact mechanics

    (1985)
  • Cheng DH, Buessen WR. The elastic anisotropy of crystals, mechanical properties, part 1. Materials research laboratory....
  • W. Voigt

    Lehrbuch der kristallphysik

    (1928)
  • G. Simmons et al.

    J Appl Phys

    (1963)
  • S. Bragavantam et al.

    Proc R Soc London A

    (1946)
  • Aleksandrov et al.

    Bull (Izvestiya) Acad Sci USSR

    (1961)
  • Aleksandrov et al.

    Bull (Izvestiya) Acad Sci USSR

    (1962)
  • S. Bragavantham

    Proc Ind Acad Sci A

    (1959)
  • I.J. McColm

    Ceramic hardness

    (1990)
  • J. Westbrook et al.

    The science of hardness testing and its research applications

    (1973)
  • Lawn BR, Blau PJ, Microindentation techniques in materials science and engineering. ASTM STP 889. West Conshohocken,...
  • F.W. Daniels et al.

    Trans ASTM

    (1949)
  • C.A. Brookes et al.

    Proc R Soc London A

    (1971)
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