University of Nevada Reno
Knoop Microhardness Anisotropy on the Cleavage Plane of Single Crystals with the Calcite Structure
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Metallurgical Engineering
By Thomas Wong May, 1^91 MINES LIBRARY
Thesis Approval Tfcsi $ ilb3
The M.S. thesis of Thomas Wong is approved by:
Dr. Richard C. Bradt (Thesis Advisor, Dean, Mackay School of Mines)
Dr. Denny A. Jones (Chairperson, Chemical/Metallurgical Engineering Department)
Dr. Kenneth W. Hunter (Dean, Graduate School)
University of Nevada Reno
May, 1991
ii Copyright Note
In presenting this thesis in partial fulfillment of the Master of Science degree reguirements, I agree that the library may make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowed for all who are interested in this research.
Please send all inquires to the following addresses:
Current Address 69 Galen Place Reno, NV 89503 U.S.A.
Permanent Address 28-D Jalan Bukit Assek 96000 Sibu, Sarawak Malaysia
Signature: Thomas Wong
May, 1991
in Acknowledgements
Every worthwhile accomplishment of my life has been achieved with the help of many benevolent individuals. The completion of this thesis is no exception.
I owe my deepest gratitude to Dr. Richard C. Bradt for his valuable guidance throughout the research. His wit and insightful suggestions have made my experience most enjoyable. Besides him, I owe my sincere appreciation to Dr. Dhanesh Chandra and Dr. Malcolm J. Hibbard for their advice and objective criticism as members of my committee.
I also wish to thank my colleagues: Debashis Dey, Young Han and Hong Li for sharing their expertise in computer programming, instruction in the use of lab equipment and experimental techniques. Mr. Martin Jensen also deserves special acknowledgement for his assistance to obtain the carbonate single crystals for measurement, and with the scanning electron microscopy.
My desire to pursue a Master's degree is largely inspired by my previous instructors. Dr. John W. Patterson at Iowa State University, Dr. Peter J. Hansen at Northwestern College, and Mrs. Sarada Menon at the Sacred Heart Secondary School are the three persons I am especially indebted to.
Finally, I thank my caring parents, Mr. and Mrs. Wong Kee Chiek, for their encouragement and financial support throughout all of my educational endeavors.
IV Table of Contents
List of Figures ...... vii
List of Tables ...... x
Abstract ...... xi
I. Introduction and Statement of the Problem ...... 1
II. Review of the Literature ...... 4
(1) Physical Properties of the Rhombohedral Carbonate Single Crystals ...... 4 A. Crystal Structure ...... 5 B. Physical Appearance and Mohs' Hardness ...... 5 C. Cleavage ...... 7 D. Lattice Dimensions ...... 11 E. Dislocations and Slip Systems ...... 11 F. Twinning ...... 14
(2) Microhardness Indentation Measurements ...... 20 A. Fundamentals of Hardness Testing ...... 20
(3) Knoop Microhardness Anisotropy ...... 22 A. Effective-Resolved-Shear-StressModel ...... 22
(4) Indentation Load/Size Effect ...... 28 A. The Classical Meyer's Law ...... 28 B. The Normalized Meyer's Law ...... 3 0
v III. Experimental Procedures ...... 33
(1) Samples and Sample Preparations ...... 33
(2) Crystallographic Orientation Determinations ..... 35
(3) Knoop Indentation Measurements ...... 35
(4) Scanning Electron Microscopy ...... 37
IV. Results and Discussion ...... 39
(1) Experimental Observations ...... 39 A. Knoop Microhardness Profiles ...... 40 B. Anisotropy of Microhardness ...... 46 C. Scanning Electron Microscopy ...... 55
(2) Deformation Mechanisms ...... 55
(3) Effective-Resolved-Shear-Stress Diagrams ...... 59
(4) Indentation Load/Size Effect ...... 64 A. The Classical Meyer's Law ...... 64 B. The Normalized Meyer's Law ...... 68
V. Summary and Conclusions ...... 82
Appendix A: Knoop Indentation and Microhardness Data ..... 85
Appendix B: Effective-Resolved-Shear-Stress Values ...... 91
Appendix C: Knoop Software ...... 94
References ...... 95
vi List of Figures
II-l Crystal Structure of Calcite ...... 6
II-2 Relationship of the Cleavage Rhombohearon to the True Rhombohedral Unit Cell ...... 10
II-3 Critical-Resolved-Shear-Stress Deformation Model ... 13
II-4 Transformation of the Twinned Crystal Lattice ...... 16
II-5 Geometry of Twinning ...... 17
II-6 Enlarged View of a Knoop Indenter and Impression on a Test Block ...... 23
II—7 ERSS Model Showing the Knoop Indentation and the Cylinder of Deformation ...... 2 5
III- l Geometric Relation between the (1011) Cleavage Plane and the Morphological Unit Cell ...... 3 6
III- 2 Assignment of the Angles on the (1011) Cleavage Plane ...... 38
IV-1 Knoop Microhardness Profiles on the (1011) Cleavage Planes for Test Loads of (a) 25 g, (b) 50 g, (c) 100 g, and (d) 200 g ...... 41
IV—2 Knoop Microhardness Profiles on the (1011) Cleavage Planes of (a) Magnesite (b) Smithsonite, (c) Siderite, (d) Dolomite, (e) Rhodochrosite, (f) Calcite 1 ...... 47
vii IV-3 A Plot of the Anisotropy of Microhardness versus the Indentation Test Load, P ...... 54
IV—4 SEM Photomicrographs of (a) Calcite and (b) Repre- sentative of the Other Five Carbonate Single Crystals at 400X Magnification and for the 200 g Test Load ...... 56
IV-5 A Plot of the M-0 Bond Length versus the Lattice Volume of the Calcite-Structure Carbonates ...... 58
IV-6 (a) ERSS Diagrams of_(a)_the {2021}<0112> Slip Systems, (b) the_jf 1011}
IV—7 ERSS Diagrams of the {Ioi2}<1012> Twin Systems .... 65
IV-8 Logarithmic Plots of P versus d to Illustrate the Linear Relationship between log (P) and log (d) ...... 66
IV-9 A Plot of the Exponents or n-values versus the A Coefficients of the Classical Meyer's Law ...... 69
IV-10 Plots of P% versus d to Illustrate the Linear Relationship between P^ and d ...... 72
IV-11 The True Microhardness Profiles on the (1011) Cleavage planes ...... 73
IV-12 Logarithmic Plots of P versus (d/dQ) to Illustrate the Linear Relationship between log (P) and log (d/dQ ) ...... 74
Vlll IV-13 A Plot of the Meyer's Law Exponents or n-values versus the A0 Coefficients of the Normalized Meyer's Law ...... 77
IV-14 A Plot of the Critical Indentation Load, Pc , versus the Meyer's Law Exponents or n-values ...... 79
IV-15 A Plot of the Critical Indentation Load, Pc , versus the Characteristic Indentation Size, dQ .... 80
IV-16 A Plot of the Meyer's Law Exponents or n-values versus the Characteristic Indentation Size, dQ .... 81
IX List of Tables
II-l The Rhombohedral Calcite and Dolomite Groups ...... 4
II-2 Physical Appearances and Mohs' Hardnesses of the Carbonate Minerals ...... 7
II-3 Hexagonal Morphological and Standard Unit Cell Indices ...... 9
II-4 Lattice Dimensions ...... 11
II-5 Reported Slip Systems for Calcite and Dolomite .... 14
II- 6 Reported Twin Systems for Calcite and Dolomite .... 19
III- l Sources of the Carbonate Single Crystals ...... 34
IV-1 The A and n Parameters of the Classical Meyer's Law and the Linear Regression Coefficients, R 2 .... 67
IV-2 The True Hardness, H0, the Difference between the Diagonal Lengths due to Elastic Recovery, de, and the Linear Regression Coefficients; R 2 ...... 70
IV-3 The A0 and nQ Parameters of the Normalized Meyer's Law and the Linear Regression Coefficients, R 2 .... 75
IV-4 The Critical Indentation Loads, Pc, in grams for (a) Magnesite, (b) Smithsonite, (c) Siderite, (d) Dolomite, (e) Rhodochrosite, and (f) Calcite 1 ...... 76
x Abstract
Knoop microhardness profiles were determined on the (1011) cleavage planes for a series of natural rhombohedral carbonate single crystals, including calcite, magnesite, rhodochrosite, siderite, smithsonite, and the structurally related dolomite. Indentation test loads of 25 g, 50 g, 100 g, and 200 g were utilized. The resulting deformation features were examined using scanning electron microscopy (SEM).
Twinning was observed in the calcite sample only. The physical properties of the other carbonate samples were considered to account for their deformation. The primary slip systems, in particular, were applied through the effective- resolved-shear-stress (ERSS) concept:
F (cos S + sin a) ERSS = --- cos 6 cos
The indentation load/size effect, or the increase in hardness at low test loads, was addressed through the normalized form of Meyer's law:
2 Pc n P n where the indentation load/size effect was related to two physical parameters: the critical indentation load, Pc, and the characteristic indentation size, dQ.
xi I. Introduction and Statement of the Problem
The hardness of single crystals varies with the crystallographic plane and the crystallographic directions on the plane of measurement [1-5]. This variation of hardness is termed hardness anisotropy. The hardness anisotropy of single crystals originates from the crystallographic nature of the deformation processes which occur beneath an indenter.
Researchers have related the hardness anisotropy profile to the primary slip systems in different crystal structures [3- 11]. Brookes and co-workers [3] have suggested that the Knoop microhardness anisotropy is inversely related to the effective resolved shear stress (ERSS) on the primary slip system. From the reported studies on the microhardness anisotropy of single crystals, the Brookes's ERSS concept has been successfully confirmed. These studies have mostly been for ceramics with cubic crystal structures including MgO, CaF2 , FeS2 , Tie, ZrC, TaC, NbC, cubic Zr02 , and MgAl204. Several non-cubic exceptions are Sic [4], AI2O3 [7], SnC>2 , and Ti02 [11]. It is, therefore, of considerable interest to investigate the microhardness anisotropy of other single crystals that have noncubic crystal structures.
Hardness measurements at lower indentation test loads result in smaller indentations and yield higher apparent hardnesses. This phenomenon is known as the indentation load/size effect and has been described by Meyer's law:
P = A dn
where P is the indentation test load, d is the indentation size, A is the coefficient, and n is the exponent. Attempts to
1 correlate the A and n parameters to the deformation processes beneath the Knoop indenter have not been very successful and no general theory has evolved.
For Knoop microhardness testing, the following are now well accepted: (1) Both elastic and plastic deformation occur during the loading process, while elastic recovery also occurs during the unloading process. (2) As explained by dislocation theory, the plastic deformation is directly related to those slip systems which are activated by the resolved shear stress beneath and in the vicinity of the indenter. (3) The indentation load/size effect is significant at low test loads for both polycrystalline and single-crystal samples.
One goal of this research is to supplement the current knowledge of the Knoop microhardness anisotropy of noncubic single crystals by measuring a series of isostructural crystals. This goal was achieved by selecting carbonate single crystals including calcite (CaC0 3 ), magnesite (MgC03), rhodochrosite (MnC03), siderite (FeC03), smithsonite (ZnC03), and the structurally related mineral dolomite (CaMg(C03)2) for study. Other members in this structural series were excluded, because they could not be readily obtained.
This series of carbonate minerals has a rhombohedral crystal structure, although both the hexagonal morphological and standard unit cell indices are used to index them. The former readily describes the cleavage rhombohedron, while the later describes the true rhombohedral unit cell. The morphological unit cell indices are used in this thesis because of their direct description of the cleavage plane of the structure.
In addition to the cleavage-plane microhardness anisotropy, this thesis will also address the following 3
subjects: (1 ) the deformation mechanisms of these carbonate single crystals, (2) the ERSS model to consider the primary slip and twinning systems responsible for the microhardness anisotropy, and (3) the indentation load/size effect of the Knoop microhardness in terms of the classical and the recently proposed normalized forms of Meyer's law.
Theoretical complications and practical requirements demand the imposition of certain assumptions and limitations on this thesis problem. First, the series of carbonate single crystals is selected for their similar rhombohedral crystal structures. Although dolomite is not exactly isostructural with the other five carbonates in the series, its difference is considered incidental to the general study for it may be considered to be intermediate between calcite and magnesite, and thus, provides an interesting contrast. Second, the Knoop microhardness indentation measurements will be made only on the (1 0 1 1 ) cleavage planes of these fine carbonate structures. This limitation has been chosen to restrict the scope of this study to that of a M.S. thesis. II. Review of the Literature
This chapter, which summarizes the literature, consists of four main sections including: (1) the Physical Properties of the Rhombohedral Carbonate Single Crystals, (2) Microhardness Indentation Measurements, (3) Knoop Microhardness Anisotropy, and (4) the Indentation Load/Size Effect.
(1) Physical Properties of the Rhombohedral Carbonate Single Crystals
The group of divalent cation carbonates includes some very common and widespread minerals. The most common of these minerals fall into the rhombohedral calcite and dolomite groups, and the orthorhombic aragonite group. The rhombohedral calcite and dolomite groups [12-14] are listed in Table II-l.
Table II-l The Rhombohedral Calcite and Dolomite Groups
Calcite (CaC0 3 ) Dolomite (CaMg(C03)2)
Magnesite (MgC03) Ankerite (CaFe(C03)2)
Rhodochrosite (MnC03) Kutnahorite (CaMn(C03)2)
Siderite (FeC03)
Smithsonite (ZnC03)
4 5
A. Crystal Structure
The above five structures of the calcite group belong to the rhombohedral space group, R3c. This structure can be visualized as a derivative of the familiar cubic NaCl structure in which triangular (CC>3 )2“ groups replace the spherical Cl” anions. The triangular shape of the (CO3 )2- groups causes the resulting structure to become a rhombohedral one instead of being isometric. See Figure II-l and note the resemblance of the [0001] of the calcite structure to a [111] for the NaCl. The (CO3 )2- groups are planar and lie at right angles to the 3- fold c-axis. The Ca2+ cations, in alternate planes, are in 6- fold coordination with the oxygen ions of the (CaC03) groups. Each oxygen is coordinated to two Ca2+ cations as well as to a carbon ion at the center of the (CaC03) group.
The three structures of the dolomite group have the space group R3. The structure of dolomite is very similar to that of calcite, except that it contains alternating Ca2+ and Mg2+ layers along the c-axis. The large difference in the sizes of the Ca2+ and Mg2+ cations (33%) causes cation ordering with the two cations in specific and separate levels in the structure. Consequently, the dolomite structure differs from the calcite structure in that it does not possess the 2-fold rotation axes.
B. Physical Appearance and Mohs' Hardness
The physical appearance of the carbonate minerals varies considerably and can be distinguished quite easily. Table II-2 lists their physical appearances and Mohs' hardnesses. The corresponding microhardnesses range from 50 to 350 kg/mm2. 6
c' I I
group
Figure II-l Crystal structure of calcite [13]. 7
Table II-2 Physical Appearances and Mohs' Hardnesses of the Carbonate Minerals
Carbonate Physical Mohs 1 Minerals Appearances H
Calcite Luster vitreous to earthy. (CaC03) White to colorless; maybe tinted. 3 Transparent to translucent.
Rhodochrosite Luster vitreous. (MnC03) Rose-red, pink, brown. 3^-4 Transparent to translucent.
Siderite Luster vitreous. (FeC03) Light to dark brown. 3%-4 Transparent to translucent.
Smithsonite Luster vitreous. (ZnC03) White,colorless,brown,green,blue. 4-4% Translucent.
Magnesite Luster vitreous. (MgC03) White, gray, yellow, brown. 3-5 Transparent to translucent.
Dolomite Luster vitreous. (CaMg(C03)2) White,colorless,pink,green,brown. 3%—4 Transparent to translucent.
C. Cleavage
Cleavage occurs when a crystalline material fractures along planar surfaces controlled directly by its crystal structure. If it breaks along surfaces not totally dominated directly by the crystal planes, it is said to show parting; and if it fails irregularly, it is said to show random fracture. A material possesses a tendency to cleave because the bonding 8
strength within the crystal structure is selectively weak across particular crystal planes. This is often a dominant effect in materials that have layered structures in which the bonding is very strong within the layers, but relatively weak between the layers. Such a structure often shows perfect cleavage parallel to the layers [15]. Graphite and mica are excellent examples of this type of cleavage.
Cleavage is a function of the crystal structure. This fact is supported by the observation that within an isostructural group, the cleavage tends to be the same for all of the group members, although its guality may differ from one member to another. Cleavage is always parallel to a crystal plane, and it is generally one with low or simple Miller indices. Cleavage can occur almost an infinite number of times, down to the thickness of atomic layers. The theory of cleavage is usually presented in terms of classical elasticity and does not introduce the notion of plastic flow. However, describing cleavage in terms of dislocation theory leads to useful analogies with plastic deformation phenomena [16].
Materials may exhibit cleavage on more than one crystal plane but usually on closely-packed planes. Generally, the guality of cleavage can be distinguished quite easily, and descriptive terms such as perfect, good, distinct, and indistinct are used. Perfect cleavage is present when a material breaks to give a smooth lustrous surface on a specific plane and has difficulty breaking on other planes. Good cleavage is present if the material breaks easily along the cleavage plane, but can also be broken in other orientations. Distinct cleavage is present if the material breaks most readily along the cleavage plane, but also fractures easily on other planes. Indistinct cleavage is present if the material fractures as readily as it cleaves, and careful inspection may be necessary even to recognize the presence of a cleavage 9
plane. Cleavage is easily recognized in thin-section cuts for optical studies of minerals because cutting and grinding of the sections cause the minerals to fracture along their cleavage planes. Cleavage fractures appear as straight cracks in thin sections.
Both the calcite and dolomite structural groups exhibit near-perfect rhombohedral cleavage, to which the hexagonal morphological unit cell indices are usually assigned [12- 14,17]. The morphological indexing of the calcite and dolomite groups has been preserved in this thesis. Some of these indices together with their letter symbols are listed in Table II—3. X-ray structure determinations, however, have demonstrated that the cleavage rhombohedron does not correspond to the true unit cell and that the true unit cell is a much more extended rhombohedron. See Figure II-2. The cleavage rhombohedron is one half of the height of the true unit cell, along the c-axis.
Table II-3 Hexagonal Morphological and Standard Unit Cell Indices
Hexagonal Unit Cell Indices Mineralogical Morphological Standard Letter Symbols
(0001) (0004) c 2 110 2 110 ( ) ( ) al (12 10 ) (1210 ) a 2 (112 0 ) (1120 ) 1 0 1 1 a3 ( )* (1014)* r l (1 1 0 1 ) (1104) r 2 (0 1 1 1 ) (0114) 10 12 r3 ( ) (1018) el (110 2 ) (1108) e2 (0 112 ) (0118) 20 21 10 12 e3 ( ) ( ) fl (220 1) (110 2 ) 2 0 221 0 112 f ( ) ( ) f 3
*Cleavage plane 10
Hexagonal unit cell
Figure II-2 Relationship of the cleavage rhombohedron to the true rhombohedral unit cell [13]. 11
D. Lattice Dimensions
The lattice dimensions of the carbonate minerals are summarized in Table II-4. Calcite has the largest unit cell, while magnesite has the smallest. Dolomite is intermediate to these two carbonates, as might be expected.
Table II-4 Lattice Dimensions
Carbonate Minerals a (A) c (A)
Calcite (CaC03) 4.99 17.06 Rhodochrosite (MnC03) 4.78 15.67 Siderite (FeC03) 4.72 15.45 Smithsonite (ZnC03) 4.66 15.02 Magnesite (MgC03) 4.63 15.02
Dolomite (CaMg(C0 3 >2 ) 4.84 15.95
E. Dislocations and Slip Systems
Dislocation motion or slip occurs when a material deforms plastically under an applied load. It is described by slip systems which consist of slip planes and slip directions. The extent of slip depends on the magnitude of the shear stress produced on the slip plane by the applied load, the geometry of the crystal structure, and the orientation of the active slip planes with respect to the shear stress. Slip begins when the shear stress on the slip plane in the slip direction reaches a threshold value known as the critical resolved shear stress. This value is the single-crystal equivalent of the yield stress of an ordinary stress-strain curve. The level of the critical resolved shear stress of any material depends on the composition and temperature. 12
The concept of a critical resolved shear stress explains the effects of varying loads on single crystals of different crystallographic orientations [18]. The critical resolved shear stress can be calculated if one knows the orientation with respect to the tensile axis of the plane on which slip first appears and also the slip direction. See Figure II-3.
The shear area, As, is not usually equal to the cross sectional area, A, perpendicular to the tensile or compressive forces. The forces must be resolved from the force, F, in the axial direction to the force, Fs, in the shear direction. The area increases as the plane is rotated away from the axis of the applied stress; thus, As = A / cos Fs F rs = ------cos 9 cos This is Schmid's law, which relates the resolved shear stress, rs, to the axial stress, (F/A) . It is simply a specific case of the general second-order tensor transformation for stress. Slip occurs with the minimum axial force when both 9 and cp are 45°. Under these conditions, rs is equal to one half the axial stress, (F/A). If the tension axis is normal to the slip plane or if it is parallel to the slip plane, the resolved shear stress is zero. Slip will not occur for these extreme orientations because there is no shear stress on the slip plane. Crystals oriented closely to these orientations tend to fracture rather than slip. 13 F Figure II-3 Critical-resolved-shear-stress (CRSS) deformation model. 14 The slip systems for the hexagonally-indexed carbonate single crystals includes the slip plane (hkil) and the slip direction [uvtw]. Generally, the slip plane is a plane of high atomic density and wide interplanar spacing. The slip direction is a direction of close packing within the slip plane. The reported slip systems for calcite and dolomite are listed in Table II-5. Table II-5 Reported Slip Systems for Calcite and Dolomite Slip Systems Temperature References Calcite {2021}<0112> 25-800 °C [19] {1 0 1 1 }<1012> 300-500 °C [19] Dolomite {0001}<2110> 25-700 °C [2 0 ] {2021}<1102> 25-700 °C [2 0 ] F. Twinning In addition to slip, twinning is also a mode of permanent deformation. It results when a crystalline material undergoes sudden localized shear processes. Generally, there are three modes of twinning: deformation twinning, phase- transition twinning, and growth twinning. Deformation twinning prevails in the calcite and may be produced as a result of mechanical deformation. It is common in metals and is freguently present in metamorphosed limestones, as shown by the presence of polysynthetically twinned calcite. Twinning frequently occurs in crystalline materials where the possible slip systems are severely limited, as in some of the close-packed hexagonal metals and many complex minerals. The process of twinning is a co-operative movement IS of atoms in which individual atoms move a fraction of the interatomic spacing relative to each other. The crystalline result is a transformation of the lattice such that the twinned lattice is a mirror image of the untwinned lattice. Figure II-4 shows that the resulting twin lattice may be exact (w = 0) or it may show a slight deviation (w > 0) . This possible deviation or the twin obliquity (w) is the angle between the normal to the twin plane and the deviated lattice row. The untwinned lattice can be related to the twinned lattice by either: (1 ) reflection by a mirror across the plane of symmetry (twin plane), (2 ) rotation about a line (twin axis), usually through an angle of 180°, or (3) inversion about a point (twin center) [13]. The best way to understand the lattice transformation in twinning is to consider a spherical construction in which the twinning plane K± is shown as an equatorial plane and is the direction of shear [21] as shown in Figure II-5. The magnitude of the shear, S, is defined as the distance moved by a point at a unit distance from the twinning plane. If the upper hemisphere is allowed to twin, it is distorted to an ellipsoid in which only the two planes, K]_ and K2, remain undistorted. The shear plane contains both the direction and the normal to the twinning plane while n2 is the line of intersection of the shear plane with the second undistorted plane K2 . A further criterion of the twinning transformation is that the second undistorted plane, K2, has the same angle with the twinning plane, K±, before and after the transformation. Assuming that the sphere is of a unit radius, then the magnitude of the shear S is represented by the distance BB' which can be expressed in terms of the angle 20 between the two undistorted planes as: S = 2 cot 20 (II—2) Consequently, if the two undistorted planes are known, the Figure II-4 Transformation of the twinned crystal lattice. 17 Figure II-5 Geometry of twinning. shear strain of the twin can be easily determined by measuring the interplanar angle. The indices of the above five parameters {K^, K2 , n^, n.2 , and S) define the twinning process. Twinning differs from slip in four specific aspects. First, the twinning process results in a crystallographic orientation difference across the twin plane, while for slip, the orientation of the crystal above and below the slip plane remains the same. Second, twinning occurs by atomic movements of less than a discrete atomic distance, while slip usually occurs in distinct multiples of the atomic spacing. Third, every atomic plane of the twinned region of a crystal is involved in the deformation, while slip usually occurs on relatively widespread planes. Fourth, twinning involves a small but well-defined volume within the crystal and is limited to one crystal plane, while slip involves a whole family of planes. The early literature of twinning in calcite is well presented by Klassen-Neyudova [22]. This includes the effects of pressure on twinning and crystallographic "channels" due to multiple twinning. Subsequent investigations by Turner [19] and his colleagues established the crystallography of the twin deformation elements in calcite. They interpreted their macroscopic observations in terms of glide and twinning and they were able to predict tension and compression axes from the orientations of twins after the deformation. In addition, Garber [23], Startsev [24], Kaga and Gilman [25], and Jourdan and Sauvage [26] have studied twinning of the regular twin systems in calcite. Twins can be introduced by applying pressure with a sharp point on the cleavage plane of a calcite single crystal. When the pressure is removed, these twins are often completely or partially expelled. These types of twins are sometimes referred to as elastic twins. 19 Compared with the wealth of publications concerning twinning in calcite, the literature addressing twinning in dolomite is rather limited. There is even less information about twinning in the structural analogues of calcite (magnesite, rhodochrosite, siderite and smithsonite) or the analogues of dolomite (ankerite and kutnahorite). Crystallographers have, however, clearly established that deformation twinning in the dolomite group is different from that of the calcite group [27]. Rogers [28] was first to report that the different crystallography of the twins in dolomite and calcite could be used to distinguish between the two minerals. Later, Turner [19] reported both the twinning and glide elements in dolomite, a result which has been supported by Higgs and Hardin [29]. The calcite and dolomite structures also differ markedly in that deformation twinning in calcite predominates at low temperatures, while twinning in dolomite occurs preferentially at higher temperatures. The reported twin systems of calcite and dolomite are listed in Table II-6. Barber and Wenk [30] observed that deformation twinning existed in one of their two softest carbonate samples (calcite and rhodochrosite) whose Knoop microhardness are below 200 kg/mm2 . For a 50 g test load, the soft calcite deformed by twinning, while the soft rhodochrosite deformed by only dislocation slip. Their other carbonate samples showed no twinning. Table II-6 Reported Twin Systems for Calcite and Dolomite Twin Systems Temperature References Calcite {1012}<1012> 25-500 °C [19] {1011}<1012> 500 °C [31] Dolomite {2021}<1014> 250-600 °C [20] 20 (2) Microhardness Indentation Measurements Microhardness indentation measurements of ceramic materials are of interest in mechanical testing because of their simplicity. Large quantities of data can be easily generated for the characterization of a material. Despite the uncertainties as to the exact stress patterns induced in different materials by various indenters [3,32-35], microhardness tests are widely used to explore the yielding and deformation behavior [35-38], near surface effects [39-42], indentation plasticity [43], and indentation fracture behavior [44-45]. Microhardness, as one form of hardness, is generally expressed as the resistance of a material to localized deformation. A simple way of quantifying this hardness is to impress an indenter with a specific geometry into the surface of the material and then divide the applied load (kg) by the area of the resulting indentation (mm2). For a plastic material, ductility exists and the material tends to flow irreversibly under the influence of the compressive stresses beneath the indenter. For a brittle material, the intensive stress field created during the indentation process often initiates cracks, especially from the sharp corners of the indenter at high test loads. A. Fundamentals of Hardness Testing In addition to the testing operator, four other elements are involved in microhardness indentation measurements: (a) the indenter, (b) the loading device, (c) the test sample, and (d) the measurement of the indentation size [46]. 21 (a) The indenter. An appropriate material for a hardness indenter must have a high hardness, a high elastic modulus, smooth surfaces, little plastic deformation, and low friction. Also, the indenter must have a perfect geometrical configuration. These reguirements are satisfied by using a highly polished diamond, while the perfect geometrical configuration can be achieved by a skillful diamond polisher. For microhardness indentation measurements, the unavoidable chisel edge of the Knoop or Vickers indenter is relatively smaller than the indentation size, and thus, satisfactory. (b) The loading device. Whatever loading is used— dead loading, electric or magnetic loading, or spring loading— the most important criterion is the minimization of incremental loading. Vibration of all sorts, including acoustic, should be kept at the lowest possible level. (c) The sample. For best results, a test specimen should possess near-ideal characteristics such as extreme smoothness, the absence of a work-hardened layer, and no contaminant film. In addition, the thickness of the sample should be many times larger than the indentation depth. (d) Measurement of the indentation size. One of the most difficult factors is the determination of the load-bearing area of the indentation. With microhardness testing, the impression should be measured immediately after the load is removed. The precise location of the edges of the indentation seems to depend on the technique of observation, even if elastic recovery is negligible. For example, with optical systems the angle of illumination affects the apparent position of the indentation edge, and thus, the measurements may differ from those obtained using an electron microscope. It is usually necessary to evaluate the indentation size statistically by averaging a number of measurements. 22 (3 ) Knoop Microhardness Anisotropy For microhardness-anisotropy studies, the Knoop indenter is the one which is most commonly used [47]. It is a pyramidal indenter with 2-fold symmetry of the pyramidal facets. The edge angle along the long diagonal is 170.5° and the edge angle along the short diagonal is 130°. The ratio of the long diagonal to the short diagonal is 7.11:1, and the ratio of the long diagonal to the depth is 61:2. The Knoop indenter yields a long, shallow indentation as depicted in Figure II-6 . The Knoop microhardness derived from this indenter is expressed as: P Hk = 14.229 ------(kg/mm2) (II-3) d 2 where P is the indentation test load, and d is the indentation size or the length of the long diagonal. The main advantage of using a Knoop indenter is its long, shallow indentation which helps: (1 ) to avoid the "spring-back" effects during microhardness indentation measurements, (2 ) to reveal the intrinsic microhardness by minimizing elastic recovery and pileup near the vicinity of the indentation, and (3) to permit an unambiguous orientation identification during the study of microhardness anisotropy. A. Effective-Resolved-Shear-Stress Model A number of models have been proposed to explain the microhardness anisotropy of single crystals. Any such model must consider the effects of the indentation orientation on the resolved shear stress on each of the potentially active slip systems. If the active slip systems have low resolved shear 23 Operating position Ficrure n -6 Enlarged view of a Knoop indenter and impression y on a test block. 24 stress, a high hardness is predicted. Unfortunately, a complete description of the stress field beneath an indenter is far from simple. Various approximations have been made and some of the models are as below: 1. Effective Resolved Shear Stress [1,3] 2. Radial Displacement [33] 3. Long Flat Punch [48,49] Among these three models, the most consistently successful for explaining and predicting the nature of microhardness anisotropy has been based on adaptions of the Schmid [18] critical resolved shear stress (CRSS) criterion. When the CRSS is reached for the most favorably oriented slip systems, slip is initiated. The Schmid equation can serve as the foundation for microhardness-anisotropy evaluations based on the following two points. First, the angle between the axis of the stress responsible for deforming the single crystal and the surface of the indentation must be determined. Second, the available slip planes, unlike those submitted to a uniaxial stress, cannot rotate about the axes governed only by the slip directions in those planes. The indenter facets and the hinterland of the single crystal, which is only elastically deformed, will influence the choice of active slip systems. However, the material between these two regions will tend to be displaced from within the bulk onto the surface during the indentation process. A slip system which allows rotation about an axis parallel to an indenter facet will therefore be more favorably oriented for slip than one whose axis of rotation is normal to that facet. Daniels and Dunn [1] first proposed a model developed for the Knoop indenter along these principles in 1949. They 25 Figure II-7 ERSS model showing the Knoop indentation and the cylinder of deformation. 26 chose a tensile stress axis, FF' in Figure II-7, parallel to the steepest slope of each indenter facet, and then modified the equation for CRSS by including a constraint term, cos 6, to allow for the restricted rotation of slip planes during indentation. Figure II-7 shows the angles 8 , 0 , and F ERSS = -- cos 8 cos cp cos 6 (II-4) A Brookes, O'Neill, and Redfern [3,47] improved the model relating hardness values and the ERSS by introducing another constraint term with respect to the displacement of material in the slip direction. They noted that the orientation of the real HH' varies with that of the Knoop indenter's long diagonal and may also deviate from that of the slip direction by an angle of a. The latter deviation will affect the magnitude and effectiveness of the slip-plane rotation around the rotation axis, SR, in Figure II-7. They proposed that the minimum constraint is obtained only when the slip direction and HH' in Figure II-7 are coincident, and 6 is automatically 90°. If the angle between the slip direction and HH', designated as a, exceeds zero degrees, then the slip plane can rotate even though 6 is 90°. On the other hand, the minimum constraint is 27 obtained when the axis of rotation, SR, and HH' are coincident— i.e., when a is 90°. Thus the modifying function which reduces the ERSS, due to rotational constraint, is fe(cos 6 + sin a). The complete form of the ERSS equation is then: F (cos 5 + sin a) ERSS = --- cos 9 cos All four angles must be determined for every orientation of the indenter, for each of the four indenter facets, and all possible slip systems of the crystal. Thus, a considerable number of calculations are involved and a computer program is needed to reduce the repetitive aspect of the calculations. The computer program calculates each value of ERSS by first determining the unit vectors in the directions of the axes: F, H, SD, SN, and SR. Cosines of the required angles are then obtained by taking scalar products of the relevant vectors. All the possible slip planes and slip directions in a particular slip system are used. This is done by generating all possible unique permutations of the Miller indices, while excluding sets that are the negative of one another. The mean ERSS curve is obtained from the mean values of the maximum ERSS values at each indentation orientation. Those directions having the lowest ERSS will be the hardest. However, because F and A cannot be clearly specified, the product of the cosine factors and the rotational constraint term is used to demonstrate the anticipated microhardness anisotropy. The extent of anisotropy, or the anisotropy factor, is not directly approachable in this model. Furthermore, work-hardening effects and the contributions of alternate, or secondary slip systems are ignored. 28 (4) Indentation Load/Size Effect Conventional microhardness equipment [50,51] usually operates over a load range of 0.1 to 10 N (10 to 1000 g weight) . The indentation size may range from 10 to 100 nm, while the indentation depths are between 1 and 10 /im. At lower test loads, less than about 200 g, the measured microhardness appears to increase. Several explanations have been proposed to address this indentation load/size effect, although none appear adequate. These explanations include: (1) The resolution limit of the optical microscope reduces the accuracy of indentation measurements at low loads as the indentation appears to be smaller than its true value by a constant absolute amount [51]. (2) The surface layer may be hardened by the polishing process and a low load reveals more of this hardening effect for it measures proportionally more of the work-hardened layer. (3) The elastic recovery of the indentation is proportionately greater for lower test loads [52]. Due to the geometric similarity of indentations, however, this explanation is not tenable. (4) Buckle [50] and Mott [51] attributed the increase in microhardness at low loads to a limited number of dislocations in the low deformed volumes. This explanation also seems rather unlikely because dislocations can be easily generated at the surface during indentation. Moreover, the deformation processes involved in indentation plastic flow are much more complicated than those described by the basic dislocation theory. A. The Classical Meyer's Law The classical Meyer's law [32,51] describes the relationship between the indentation test load, P, and the 29 corresponding indentation size or the long diagonal length, d, for the Knoop microhardness test. It is P = A dn (II-6 ) where P is in grams, A is the Meyer's law coefficient with varying units of g*jLtm-n, d is in /im, and n is the dimensionless Meyer's law exponent. The n-value is greater than 1.0 and less than 3.0 for most crystalline materials. A n-value of 2.0 indicates the absence of any load/size effect as evident by substituting Equation II-6 into II-3. Sargent [53] has defined an indentation-size-effect index, n, in terms of the force (F) measured directly on the indenter and the diagonal (d) of the impression as: dF / F n = ------(H - 7 ) dd / d The indentation-size-effect index described by Sargent is just an alternate form of the classical Meyer's law: dd dF 8 n or n ln(d) = ln(F) + ln(A) (II— ) where In(A) expresses the constant of integration, or F = A dn . Sargent further assumed a standard microhardness, Hs, based on a standard indentation impression size, s. His choice of a standard size appears to be solely for convenience. This assumption allowed him to obtain the following expression: r d in-2 P = 0.5 Hs s2 (II-9) s 30 where s is the standard indentation size in micrometers. Sargent chose s to be 70 nm for a Knoop indentation and 10 lira. for a Vickers indentation. B. Normalized Meyer's Law The classical Meyer's law presents a serious problem in dimensions. The A coefficient of the Meyer's law has varying units, g ’/Ltm-n, which are dependent on the associated n values. This problem is similar to one encountered in fracture mechanics as demonstrated in the following equation: V = B (Kj )n (11-10) where V is the crack velocity, and Kj is the stress intensity. B has a dimensionality problem because the exponent, N, varies in value. Minnear and Bradt [54] first identified this problem, while Sines [55] provided an alternate expression by normalizing the V-K relationship: V = B' (1 1 -1 1 ) This normalized equation solved the dimensionality problem successfully. The same approach has been used to normalize the classical Meyer's law [56]. Before the normalization, it is necessary to determine the critical indentation load, Pc, because the measured Knoop microhardness involves the load/size effect and does not represent the true microhardness. The true microhardness, H0, can be measured only when the critical indentation load has been attained by eliminating the load/size effect as follows: Since the microhardness is a function of the indentation test load, the general expression of this function can be obtained by differentiating Equations II-3 and II-6 and combining the results to yield: dH r d - 2 P (n A dn _ 1 ) -1 --- = 14.229 ------0 (H-13) dP L d3 where P is now the critical indentation load, Pc , where the indentation load/size effect diminishes, and d is the corresponding characteristic indentation size, dQ. Equation 11-13 can then be rearranged to yield: n Pc = A dDn (11-14) 2 The dQ can be related to the true microhardness, H0, by applying Equation II-3: Pc dQ2 = 14.229 ---- (11-15) Ho An alternate expression of dQ can be obtained by substituting Equation 11-14 into Equation 11-15: d 2-n = 14.229 ao (11-16) Hq and H0 can be obtained by using the following expression [57]: 32 r H0 -I u. s _ 0 .5 = H 4- - 14.229 - ae + r- 14.229 H ° -i (11-17) where de is the difference between the original and elastically recovered diagonal lengths. Both the H0 and de values can be evaluated by linear regression of a plot of P^ versus d. With d0, the classical Meyer's law can then be normalized as follows: (11-18) where A0, dQ, and nQ are parameters for the normalized Meyer's law. A0 has the units of force and dQ has the units of length. n0 is dimensionless and is equal to the original n. Equation 11-18 can also be expressed in another form. The following expression, which is obtained by combining Equations II-6 and 11-18, is needed. A0 = A d0n (11-19) Combining Equations 11-14, 11-18 and 11-19, the final form of the normalized Meyer's law results: 2 Pc P = --- (11- 20) n This normalization of the classical Meyer's law has successfully removed the awkward dimensionality of A. It also introduces two new parameters: Pc and dQ, which have real physical significance. III. Experimental Procedures The Experimental Procedures portion of this thesis is divided into four sections: (1) Samples and Sample Preparations, (2) Crystallographic Orientation Determinations, (3) Knoop Indentation Measurements, and (4) Scanning Electron Microscopy. (1) Samples and Sample Preparations The six different carbonate single crystals were provided by Dr. Bradt and Mr. Jensen who had previously obtained them from numerous sources, as listed in Table III-l. The physical appearances and sizes of these single crystals are as follows. The calcite was a transparent, clear white cleavage rhombohedron about 5x5x10 cm in size. The magnesite was also a transparent cleavage rhombohedron about 3x3x7 cm in size, but sightly off-white in color. The rhodochrosite was a translucent, reddish-pink cleavage rhombohedron about lxlxl cm in size, and the siderite was a translucent, brownish cleavage rhombohedron about 1x2x2 cm in size. The smithsonite was a translucent, light-blue cleavage rhombohedron about 2x1x3 cm in size. Finally, the dolomite was a translucent, yellowish cleavage rhombohedron about 4x4x6 cm in size. The quality of these single crystals was checked using x-ray microanalysis. All, except the siderite, contained only minor impurities such as aluminum, calcium, iron, manganese, or magnesium. The siderite contained slightly more manganese and magnesium. 33 34 Table III-l Sources of the Carbonate Single Crystals Carbonate Sources Single Crystals Calcite Wards Company, Montana Magnesite Brumado Mine, Brazil Rhodochrosite Grizzly Bear Mine, Colorado Siderite Eagle Mine, Colorado Smithsonite Tsomeb Mine, Namibia Dolomite Eagle Mine, Colorado Four of the six single crystals, namely the calcite, magnesite, dolomite, and rhodochrosite exhibited excellent large flat (1011) cleavage planes. This made identification of the (1011) cleavage planes relatively easy. The angles between the edges on the cleavage planes were measured and confirmed to agree with those documented in the literature [13]. For the other two minerals, siderite and smithsonite, cleavage planes were also easily identified as the smooth, mirror-like flat surfaces on the crystals for this kind of surfaces exists only in crystal structures that have near-perfect cleavage. The single-crystal samples were properly mounted using epoxy resin, and then finely polished using both AI 2O 3 and diamond polishing compounds. The sizes of the polishing compounds varied from 600 to 0.05 An automatic Vibromet polisher was used in the final stages of the polishing to ensure a mirror-like surface finish. Generally, the softer 35 the sample, the shorter the polishing time that was required. Polishing times varied from two days for the calcite to about ten days for the rhodochrosite. (2) Crystallographic Orientation Determinations After polishing and prior to the indentation measurements, each single-crystal sample was mounted in a Shimadzu (Type-m) microhardness tester. The crystallographic orientations, which are parallel to the long diagonal of the Knoop indenter on the cleavage plane as shown in Figure II-2, were determined by the following method. A hexagonal morphological unit cell, which is one quarter of the height of a standard unit cell, was used. By definition, the x^ direction in Figure III-l is the [1011]. Similarly, the X2 and X3 are the [2111] and [1121], respectively. Other directions are found by considering the geometrical relationships between the (1 0 1 1 ) cleavage plane and the unit cell. (3) Knoop Indentation Measurements Knoop microhardness indentations were made at room temperature for test loads of 25 g, 50 g, 100 g, and 200 g at an indentation rate of 0.017 mm/sec for a 15-sec dwell time. Indentations made at higher test loads caused severe cracking in some of the test samples, and thus, could not be used for reliable microhardness measurements. For this reason, test loads greater than 200 g were avoided. 36 / c Figure IIl-i Geometric relation between the (1011) cleavage plane and the morphological unit cell. 37 The angles on the (1011) cleavage plane were assigned to be positive if they were counterclockwise to the [1 0 1 1 ] direction, and negative if they were clockwise to the [1 0 1 1 ] direction. See Figure III-2. An interval of 16° was chosen for the assigned angles ranging from -48 to 48°. For the calcite, magnesite, dolomite, and rhodochrosite samples, these assigned angles were readily achieved because of the large test sample's geometry. For the siderite and smithsonite samples, the assigned angles were identified by alignment with the induced indentation cleavage cracks from an indentation test at a higher test load. At each assigned angle and for each indentation test load, ten individual Knoop indentations were made. The long diagonal of the Knoop impression of each indentation was measured immediately after the indentation. The average of the ten indentation measurements together with their standard deviations were calculated. The ± values are expressed as the 95% confidence intervals using the t-distribution. (4) Scanning Electron Microscopy Optical observations of the indented surfaces of the carbonate single crystals revealed interesting deformation features. Detailed examination of these features required the use of a scanning electron microscope (SEM). Before the SEM examination, however, a thin gold film was applied onto the surface of each sample so that the images could be obtained. Photomicrographs of these features were taken at the 0 ° assigned angle and for the 200 g test load. A magnification of 400X was chosen. The scale was 4 cm to 100 ^m. 38 (1 0 1 1 ) cleavage plane Figure III-2 Assignment of the angles on the (1011) cleavage plane. IV. Results and Discussion The Results and Discussion portion of this thesis consists of four sections: (1) Experimental Observations, (2) Deformation Mechanisms, (3) Effective-Resolved-Shear-Stress Diagrams, and (4) the Indentation Load/Size Effect. The results are presented in the form of graphs and tables; some of the numerical data are also tabulated in Appendices A and B. (1) Experimental Observations For all of the test loads (25 g, 50 g, 100 g, and 200 g), the indentations for calcite were the largest among the six rhombohedral carbonates, followed by rhodochrosite, dolomite, siderite, smithsonite, and magnesite. Calcite is thus the softest of these six carbonates. Calcite also appeared unique for several other aspects of the results; conseguently, two different calcite samples were used to double check the observations. The standard deviations of the microhardnesses vary from sample to sample, which are partially affected by the quality of polishing and the optical measurement errors. For most of the indentation measurements, however, the standard deviations are small, which imply that the measurements are very accurate. The majority of the 9 5% confidence intervals, as calculated using the t-distribution, have been omitted from the graphs for the sake of clarity. Moreover, the largest of these confidence intervals for each carbonate sample is found at the 25 g test load and in the [1 0 1 1 ] direction. 39 40 A. Knoop Microhardness Profiles The cleavage—plane microhardnesses (H]^) were determined using the following expression: P Hfc = 14.229 ------(kg/mm2) (IV-1) d 2 Figures IV-1 (a)-(d) illustrate the microhardnesses for the seven assigned angles of -48°, -32°, -16°, 0°, 16°, 32°, and 4 8 and for the indentation test loads of 25 g , 50 g, 100 g, and 200 g, respectively. These microhardness profiles are reduced and presented on a single page in Figure IV-1 (e) for ease of direct comparison. The microhardnesses range from about 100 to 800 kg/mm2. These values are comparable to the Mohs' hardness values ranging from 3 to 5 which have been documented in the literature and were previously summarized in Table II-2. The order ranking of these microhardnesses also matches favorably with their ranking by the Mohs' hardnesses. Magnesite has the highest microhardness at each of the test loads, followed by smithsonite, siderite, rhodochrosite, and calcite. The microhardnesses of dolomite are intermediate to those of magnesite and calcite. The prevailing feature of all of these microhardness profiles is a maximum in the microhardness for the 0 ° assigned angle or the [1011] direction. For example, the microhardness for the 0° assigned angle and for the 25 g test load on the cleavage plane of magnesite is nearly 3 0% more than that for the -48° assigned angle at the same test load. Figure IV-1 (e) shows that this microhardness anisotropy is more prominent for magnesite than it is for calcite. 41 Figure IV-1 (a) Knoop microhardness profiles on the (1011) cleavage planes for the test load of 25 g. Fig ure IV-1 IV-1 ure (kg/m b Kopmcoadespoie o h (1011) the on profiles microhardness (b) Knoop laaepae frtets la f 0 g. 50 of load test the for planes cleavage 42 Figu k (kg/mm2) e IV-1 re c Kopmcoadespoie nte (1011) the on profiles microhardness (c) Knoop cleavage planes for the test load of 100 g. 100 of load test the for planes cleavage 44 Figure IV-1 (d) Knoop microhardness profiles on the (1011) cleavage planes for the test load of 200 g. (kg/m m 2) ^ Hh (kg/mm2) 46 The microhardnesses also decrease substantially as the test loads are increased from 25 to 200 g. See Figures XV-2 (a)~(f)/ and (g) for an overall comparison. This change in the microhardnesses is much greater for magnesite (57%) than it is for calcite (15-s) . The significance of this increase in the microhardnesses at the lower test loads will be further addressed in the Indentation Load/Size Effect section. B. Anisotropy of Microhardness The degree of microhardness anisotropy, or the anisotropy of microhardness (AOM), can be determined using the following equation [58]: Hmax ~ Hmin AOM = ------(IV-2) (Hmax ' Hmin)^ where Hmax and Hmin are the maximum and the minimum microhardnesses, respectively. Figure IV-3 illustrates the calculated anisotropy of microhardness for each carbonate at all of the four test loads. This anisotropy of microhardness is significantly affected by the indentation test load. It generally appears to decrease with an increase of the testing load, except in the case of calcite, where it increases with an increase of the testing load. The exceptional phenomenon in calcite is due to the increased amount of twinning, which reduces the microhardness of calcite dramatically, at higher test loads. In addition to the test load, the effect of the carbonate samples was also investigated. The anisotropy of microhardness, however, seems to be independent of the carbonate samples as they do not order as logically as their individual microhardness values. Figure IV-2 (a) Knoop microhardness profiles on the (1011) cleavage plane of magnesite. cleavage plane of snuorsonite. (b) Knoop(b) microhardness profiles on the (1011) re IV-2 ( uuuu/6>i) ^ C Figu 49 Figure IV-2 (c) Knoop microhardness profiles on the (1011) cleavage plane of siderite. Figure IV-2 (d) Knoop microhardness profiles on the (1011) cleavage plane of dolomite. Figure IV-2 (e) Knoop microhardness profiles on the (1011) cleavage plane of rhodochrosite. 52 Figure IV-2 (f) Knoop microhardness profiles on the (1011) cleavage plane of calcite 1. iue IV-2 Figure (kg/mrr,1) _ H, (kg/m m 1) 0 6 - - 0 * sind nl (deg) AngleAssigned 0 2 - (g) An overall comparison of the Knoop microhardness microhardness Knoop the of comparison overall An profiles for (i) magnesite, (ii) smithsonite, smithsonite, (ii) (i) magnesite, for profiles n (i clie 1. (vi) calcite and 0 (iii) siderite, (iv) dolomite, (v) rhodochrosite, (v) rhodochrosite, (iv) dolomite, siderite, (iii) 20 *0 60 60 400-4 ^ -1 i ■ - ' 1 O' * 2 0 ° -i -i ° 0 2 * 600 -< 600 80 0 -5 0 o- 60 -6 t \ 4 ; 40 - 6 0 -2 0 -4 “ T T Assigned (deg) Angle As ind nl (deg) signed Angle v ( . — - - 2 T 2 ) i 0 0 6 40 20 *5 T ...... «« O« * ” JOC« . « O W « 53 iue IV-3Figure Anisotropy of Hardness 0.40 0.50 0.6 0.30 0.20 0.00 0.10 0 A plot of the anisotropy of microhardness versus versus microhardness of anisotropy the of plot A h idnaints od P.load, test indentation the Tl II | ' r 50 | | I I | II I I I II I I | I IN M I I 0 10 200 150 100 (g) P COCGO Magnesite Magnesite COCGO /W W \ Sidente \ W /W 1 m x x x x x Calcite Calcite x x x x x ***** Calcite Calcite ***** 1 1111 Smithsonite Smithsonite Dolomite Rhodochrosite Rhodochrosite 1 | 2 1 1 1 1 54 C. Scanning Electron Microscopy Among the six rhombohedral carbonate samples, only the calcite showed twinning during the Knoop indentation process. See Figure IV-4 (a) . The lines in the lower half of the photomicrograph and perpendicular to the long diagonal of the 200 g indentation are (1012) twin striations. These twins intersect the (1011) cleavage cracks which are at about a 45° angle to them. The two lines extending from the sides of the indentation are cleavage cracks on the (1011) cleavage planes. The angle between the two large cleavage cracks is about 90°. The smooth surface reflects the quality of polishing, while the white spots in the background are accumulated dust particles. A similarly oriented indentation which is in the [1011] direction and is representative of the other five carbonates (magnesite, smithsonite, siderite, dolomite, and rhodochrosite) is shown in Figure IV-4 (b) . A few small cracks are occasionally observed at some of the indentation corners in all of these carbonate samples. The crack sizes seem to vary, but their intensity at the same test loads appears to be independent of the carbonate samples. Unlike calcite, however, no twinning was observed in any of the other carbonate samples. (2) Deformation Mechanisms Scanning electron microscopy suggests that both twinning and dislocation slip occur during the indentation of calcite; while for the other carbonate samples, there is no evidence of twinning and dislocation slip dominates. The uniqueness of calcite can be justified by considering the effects of substituting different metal cations (M^+) in the crystal structures of the rhombohedral carbonates. 56 (a) (b) Figure IV-4 SEM photomicrographs of (a) calcite and (b) repre sentative of the other five carbonate single crystals at 400X magnification and for the 200 g test load. 57 The most obvious effect is the change in lattice volume. This change is directly related to the cation size. The larger the cation size, the greater the lattice volume. The other effect is the change in the bond length between the metal cation and the oxygen anion (M-O) . The M-0 bond length increases with the cation size. Figure IV-5 illustrates these two effects. Note that calcite is very distinctly detached from the other four carbonates as it has the largest lattice volume and the longest M-0 bond length. The corresponding results for calcite are low critical resolved shear stresses due to much larger slip-plane areas and weak bond strength. Thus, the opportunity for twinning is promoted. Twinning in calcite may generate dislocations and initiate cleavage cracks which appear to reduce the work- hardening effect. The cleavage cracks seem to be initiated because large deformations are confined to a limited number of twins. The net result is a reduction in the hardness of calcite. In fact, calcite has the lowest microhardness of all these carbonate samples by a substantial amount, as observed in Figures IV-1 (a)-(e) and IV-2 (a)-(g). The microhardness ranking of the other four carbonates may be attributed to the ease of dislocation motion. The weaker the bond strength, the easier it is for the dislocations to move during indentation and the lower the microhardness. The bond strength for these carbonates is governed by the both the cation size and the M-0 bond length. Therefore, the microhardness ranking of these carbonates can also be correlated to Figure IV-5. A comparison of Figure IV 5 with Figure IV-2 (g) indicates that the correlation is excellent. 2.5 2.4 - C a lc it e (CaC03 ) '2.3 2.2 2.1 1 1—1 1 1 1 1 1 1 1 1 1 -i 2.0 t— i— I— i— |— i— i— r n—|—!—i—r T------I I I I r — i—i i i i 250 270 290 310 330 350 370 390 Lattice Volume (A) A plot of the M-0 bond length versus the lattice Figure IV-5 calcite-structure carbonates [12]. volume of the 59 (3) Effective-Resolved-Shear-Stress njaarams The shapes of the Knoop microhardness profiles can be explained by using the effective-resolved-shear-stress (ERSS) model: F (cos S + sin a) ERSS = --- cos 0 cos cp ------(IV-3) A 2 The primary slip systems responsible for the dislocation slip yield net ERSS diagrams that are inversely related to the Knoop microhardness profiles. This inverse relationship was explored for all of the reported slip systems of calcite and dolomite. The applicability of the ERSS concept to the twin systems was also investigated. Figures IV-6 (a) and (b) illustrate the ERSS diagrams for the reported {2021}<0112> and {1011} For the reported {0001}<2110> and {2021}<1102> slip systems of dolomite, the net ERSS profiles are also inversely related to the Knoop microhardness profiles. See Figures IV-6 (c) and (d) . These results together with those of calcite confirm that the reported slip systems are the primary slip systems for these carbonates and that those slip systems dominate the deformation processes. 0.40 - Net ERSS profile 00 0.20 - 00 QC Ld 0.00 - lr-4 l«HIrH IfHo i i i i i i i i i | i i n i i i i | ii 1 1 1 ' 1 1 1 M ! I II I I I | I I I T i m i | m i i ii m rf •60 -40 -2 0 0 20 40 60 Assigned Angle (deg) Figure IV-6 (a) ERSS diagrams of the {2021}<0112> slip systems, Fig ERSS ure IV-6 IV-6 ure (b) ERSS diagrams of diagrams ERSS (b) h { the 1 1 0 1 }< 1012 si systems. slip > 61 Fig ERSS ure IV-6 IV-6 ure (c) ERSS diagrams ERSS (c) fte { the of 1 0 0 0 }< 2110 > slip systems. slip > 62 iue IV-6 Figure ERSS 60 -0 2 U -20 -40 0 -6 r i- i i i n i ~ i i i i i i i ~i m y r r i n i i i i i r r r i -i i i rr ~ (d) ERSS diagrams of the of diagrams ERSS (d) gned Angle A d e n ig s s A I I I I I I I ' l I | I I I T { de ) eg (d 1 2 0 2 20 }< 1102 i i i i | i i i i i 0 60 40 > slip systems. slip > m i r 64 Similar ERSS calculations were performed for the reported {1012}<1012> twin systems of calcite, as shown in Figure IV-7. These calculations were based on the assumption that twinning was replacing dislocation slip in the deformation of calcite. Unfortunately, the resulting ERSS diagrams do not yield an inverse relationship to the Knoop microhardness profiles. This deviation implies that the ERSS model may not be applicable to twin systems, at least not for calcite. (4) Indentation Load/Size Effect The indentation load/size effect, also known as the ISE, was addressed by evaluating the results obtained from the classical and normalized Meyer's laws. A. The Classical Meyer's Law The relationship between the indentation test load, P, and the indentation size or the long-diagonal length, d, is described by: P = A dn (IV-4) The A coefficient and the n-value are obtained by linear regression of the logarithmic form of the classical Meyer s law as follows: log P = log A + n log d (IV -5) The intercepts yield the A coefficients, while the slopes yield the n-values. Figure IV-8 illustrate this linear relationship ERSS iue IV-8 Figure '°9 (p) log (p ) log (H) Logarithmic plots of P versus d versus P of plots Logarithmic ierrltosi ewe o () n lg (d). log (P) and log between relationship linear for all of the carbonate samples. All the linear regression coefficients, R 2 , are greater than 0.99 which indicate that the regressions are very good. A summary of the data is included in Table IV-1. Table IV-1 The A and n Parameters of the Classical Meyer's Law and the Linear Regression Coefficients, R2 2 Angles A n R A n R2 Magnesite Smithsonite -48° 0.104 1.714 1.000 0.067 1.803 0.999 -32° 0.110 1.705 1.000 0.072 1.792 1.000 -16° 0.124 1.684 0.999 0.094 1.738 0.999 0° 0.157 1.643 1.000 0.127 1.692 1.000 16° 0.120 1.692 0.999 0.088 1.756 1.000 32° 0.109 1.707 1.000 0.075 1.781 0.999 48° 0.102 1.718 1.000 0.068 1.801 0.999 Siderite Dolomite -48° 0.067 1.772 1.000 0.047 1.865 1.000 -32° 0.071 1.768 1.000 0.048 1.857 1.000 -16° 0.087 1.735 1.000 0.054 1.835 1.000 0° 0.114 1.686 1.000 0.072 1.780 1.000 16° 0.084 1.738 1.000 0.053 1.837 1.000 32° 0.069 1.776 1.000 0.051 1.843 1.000 48° 0.069 1.765 1.000 0.048 1.857 1.000 Rhodochrosite Calcite 1 -48° 0.028 1.906 1.000 0.036 1.700 1.000 -32° 0.032 1.884 1.000 0.036 1.719 1.000 -16° 0.037 1.864 1.000 0.025 1.839 1.000 0° 0.051 1.814 1.000 0.024 1.858 1.000 16° 0.039 1.853 1.000 0.024 1.848 1.000 32 ° 0.031 1.892 1.000 0.035 1.732 1.000 0.030 1.739 1.000 48° 0.028 1.906 1.000 The A and n parameters are anisotropic, that is, both are orientation dependent. The A coefficients, exc p 68 those of calcite, increase with an increase in the microhardness. The A coefficients of calcite decrease with an increase in the microhardnesses. The opposite trends are observed for the n-values. These n-values, except for those of calcite, decrease with an increase in the microhardness. The n-values of calcite increase with an increase in the microhardness. The peculiar trend for calcite can be explained by relating the A and n parameters to the microhardness. A substitution of Equation II-6 into Equation II-3 yields: Hk = 14.229 A dn_2 (IV-6) Careful examination of this equation suggests that the reverse trend of the A and n parameters for calcite may be due to the large indentation size or the softness of calcite. Although the dimensionality problem of the A coefficient (g*/im_n) must be solved before further analysis is meaningful, the A and n parameters of the classical Meyer's law are linearly correlated. Figure IV-9 shows the resulting trends for the A and n parameters. The indentation load/size effect is evident since none of the n-values are equal to two. The linear fit for the calcite differs from that for the other four carbonate samples and dolomite by its slope, which is three times greater for calcite. This unique result is a consequence of the deformation mechanisms of calcite. As discussed earlier, the calcite deforms by dislocation slip and twinning whereas the other carbonates do not readily twin. B. The Normalized Meyer's Law Using the method provided by Sines [55] and Li [56], the dimensionality problem of the A coefficient (g-Mm ) 69 ©no 4 n r n-values versus the A Figure IV-9 ^ p l o t ^ t ^ e x p e n classical Meyer,s la„. 70 by normalizing the classical Meyer's law. This method requires the determination of the critical indentation load, Pc, at which the indentation load/size effect diminishes. But first, it is necessary to evaluate the true microhardness using Equation 11-17. The results are summarized in Table IV-2. Table IV-2 The True Hardness, H0 , the Difference between the Diagonal Lengths due to Elastic Recovery, de, and the Linear Regression Coefficients, RJ 2 2 Angles Ho de R Ho de R Magnesite Smithsonite -48° 356 7.657 1.000 368 4.401 1.000 -32° 368 7.233 1.000 368 4.939 1.000 -16° 384 7.210 1.000 372 6.123 1.000 0° 394 8.514 1.000 389 8.175 0.999 16° 386 7.046 1.000 375 6.182 1.000 32° 370 7.246 1.000 370 4.976 1.000 1.000 48° 357 7.447 1.000 369 4.568 Siderite Dolomite 1.000 1.000 -48° 298 6.849 336 3.748 1.000 1.000 -32° 317 6.140 331 4.175 1.000 4.562 1.000 -16° 322 7.610 331 1.000 0 318 10.266 0.999 335 6.333 ° 1.000 1.000 319 7.492 329 4.543 16° 1.000 1.000 322 5.733 330 4.404 32° 1.000 1.000 48° 296 7.073 333 3.924 Rhodochrosite Calcite 1 2.971 1.000 94 16.136 0.999 -48° 243 1.000 1.000 -32° 249 3.783 105 14.462 1.000 6.614 1.000 -16° 266 3.953 145 1.000 0 5.772 1.000 158 5.348 ° 281 1.000 1.000 4.465 148 6.070 16° 262 1.000 110 1.000 250 3.511 13.094 32 ° 1.000 0.999 48° 242 2.989 96 14.172 71 to Basically, linear regression of the plots of p% versus d, as shown in Figure IV-10, was used. The true microhardness, H0 , was obtained from the slope as follows: H0 = 14.229 (slope) 2 (IV-7) The true microhardness profiles are illustrated in Figure IV-11 and resemble the original microhardness profiles with the exception that the dolomite appears to be slightly harder. The microhardness profiles of magnesite, smithsonite, siderite, dolomite, and rhodochrosite are more closely grouped together, but remain distinctly separated from the calcite. This result confirms the previous observation that the twinning in calcite r considerably lowers its microhardness. Using the H0, the characteristic indentation size, dQ, was evaluated using the following expression: 14.229 do2-n = ------(IV-8) Ho With the d0 values, the classical Meyer's law was normalized: P = A0 (IV-9) The logarithmic form is log (P) = log A0 + nQ log (d/dQ) (IV-10) The plots in Figure IV-12 confirm such a linear relationship. They resemble the log(P)-versus-log(d) plots of the classical Meyer's law. The intercepts are the A0 coefficients and the iue IV-10 Figure i'/J Plots of versus d to illustrate the linear linear the illustrate to d versus P between relationship of Plots d ( »m) 2 0 and d. and 50 ( ) m (u d 0 IO 200 ISO 100 72 73 Figure IV-11 The true microhardness profiles on the (1011) cleavage planes. Fi ue IV-12 gure log (P) log (P) log (P) 1000 -]— 1 / i ------s ______h lna eainhpbten log between relationship linear the of plots Logarithmic log ^d/do) ^d/do) log log .. ' Rhodochro i 1 \ (d/dQ). J 7 Si .. .. derite site - io-i sr o a* P 100 10 0.1 - - versus versus ------(d/dQ) to to (d/dQ) / - - ■■-—■------o (d/do) log v * ------(P) - --- X Do - ■ lomite and illustrate ------74 lO 75 slopes are the Hq —values of the normalized Meyer's law. Since the normalization does not affect the slopes, the nQ-values are equal to the classical Meyer's law exponents or the n-values. The A0 values, on the other hand, are 10,000 times larger than the classical Meyer's law A coefficients and have different units— grams. See Table IV-3 for more details. Table IV-3 The A0 and nQ Parameters of the Normalized Meyer's Law and the Linear Regression Coefficients, R 2 2 2 Angles Aq nO R Ao nO R Magnesite Smithsonite -48° 214 1.714 1.000 161 1.803 0.999 -32° 186 1.705 1.000 179 1.792 1.000 -16° 170 1.684 0.999 174 1.738 0.999 0° 187 1.643 1.000 229 1.692 1.000 16° 171 1.692 0.999 204 1.756 1.000 32° 191 1.707 1.000 166 1.781 0.999 48° 208 1.718 1.000 169 1.801 0.999 Siderite Dolomite 1.000 1.000 -48° 227 1.772 224 1.865 1.000 243 1.857 1.000 -32° 189 1.768 1.000 223 1.735 1.000 218 1.835 -16° 1.000 1.000 0 219 1.686 235 1.780 ° 1.000 1.000 219 1.738 219 1.837 16° 1.000 1.000 181 1.776 223 1.843 32° 1.000 1.000 48° 228 1.765 217 1.857 Rhodochrosite Calcite 1 1.000 210 1.906 1.000 226 1.700 -48° 1.000 1.000 -32° 229 1.884 233 1.719 1.000 211 1.839 1.000 -16° 195 1.864 1.000 0 1.000 1.858 ° 232 1.814 195 1.000 1.853 1.000 203 1.848 16° 209 1.000 220 1.000 1.892 1.732 32° 228 1.000 1.000 48° 212 1.906 239 1.739 76 Unlike the A coefficients which either increase or decrease with increasing microhardness, the A0 coefficients seem to vary rather randomly with the crystallographic orientation. The general trend as shown in Figure IV-13 indicates that the softer carbonates are represented by larger A0 and n parameters with the exception of calcite whose results are rather messy. As explained in Chapter II, the final form of the normalized Meyer's law _s as follows: (IV-11) The possibility of relationships between the various parameters of Equation IV-11 was explored. But first, the Pc values were obtained using the following expression: Ho Pc = ------(IV-12) 14.229 See Table IV-4 for a summary of the results. Table IV-4 The Critical Indentation Loads, Pc , in grams for (a) Magnesite, (b) Smithsonite, (c) Siderite, (d) Dolomite, (e) Rhodochrosite, and (f) Calcite 1 77 2.00 1.90 c 1.80 1.70 1.60 A plot of the Meyer's law exponents or ^values Figure IV-13 versus the A0 coefficients of the normalized Meyer's law. 78 rirst, the critical indentation loads, the pc , were plotted versus the n-values, as shown in Figure IV-14. Unfortunately, this plot does not yield very much information as no good correlation between the parameters appears to exist. The general trend is lower Pc values and lower n-values for the harder carbonates. For example, magnesite has the lowest Pc values and the lowest n-values, while calcite has the highest. Next, the critical indentation loads, the Pc, were plotted versus the characteristic indentation size, dQ, as shown in Figure IV-15. This plot is interesting for two reasons. First, it reveals a linear relationship between the Pc and the dQ. The indentation size diminishes as the critical indentation load approaches zero. Second, the combination of dislocation slip and twinning in the calcite, as observed in earlier plots, again distinctly separates calcite from the other carbonates. Lastly, the relationship between the n-values and the characteristic indentation size, dQ, was examined. Figure IV-16 indicates that the dQ at the same Pc and for the same n- value is considerably larger for calcite than for the other five carbonates. In fact, the dQ for calcite increases with decreasing n-values while those of the other carbonates decreases with decreasing n-values. This plot also suggests that two linear correlations exist and that the n-values are directly related to the characteristic indentation size. The relationships as found in Figure IV—14 through IV—16 conclude that the indentation load/size effect is associated with two physical parameters: the critical indentation load, Pc, and the characteristic indentation size, dQ. Both of which can be determined from the experimental data. The differences as observed in calcite are due to the twinning which has complicated the indentation load/size effect. 79 A plot of the critical indentation load, Pc, Figure IV-14 exponents or n-values. versus the Meyer's law 80 „ , i +. t-hP critical indentation load, Pc, Figure IT-1, ^plot indentation size, d0 . 2.00 C0 0 0 D Magnesite i i 11 11Smithsonite 1.90 1.80 c 1.70 1.60 twinning 1.50 ------1 1 I | I T~I 1 1 1 I I ! I 1 I I I I 1 I I I I I I I I I ! I I I I I V 1 I I | 'I "I I 'I 'IT T T T 0 50 100 150 200 250 d0 ( p m ) Figure IV-16 A plot of the Meyer's law exponents or n-values versus the characteristic indentation size, dQ . V. Summary and Conclusions Knoop indentation measurements were made on the (1011) cleavage planes for a series of rhombohedral carbonate single crystals for seven crystallographic directions as designated by the assigned angles of -48°, -32°, -16°, 0°, 16°, 32°, and 48°. This series of rhombohedral carbonates: calcite (CaC03), magnesite (MgC03), rhodochrosite (MnC03), siderite (FeC03), smithsonite (ZnC03), and dolomite (CaMg(C03)2) were chosen for their unique cleavage-rhombohedron structures. Indentation test loads of 25 g, 50 g, 100 g, and 200 g were used. The Knoop microhardness profiles were evaluated using the measured long diagonal lengths. The prevailing trend is an increase in the microhardness as one approaches the [1011] direction. This microhardness anisotropy is more pronounced for magnesite than it is for calcite. In addition to the microhardness anisotropy, the increase of the microhardness, or the indentation load/size effect, was also observed in all of these carbonate samples. The degree of anisotropy, or the anisotropy of microhardness, was determined. Calcite again is unique as its anisotropy of hardness increases with an increase of the testing load while that of the other carbonate samples decreases with an increase of the test loads. Calcite is the only one of the carbonate samples that showed twinning during the indentation process. The resulting indentation features were evaluated using scanning electron microscopy (SEM). The twin striations as observed in calcite are remarkable. In addition, large cleavage cracks, which appear to be induced by the twinning, initiated from the corners of the indentations. 82 83 ano Twinning in calcite was explained by considering the l effects of substituting metal cations of different sizes. Both the lattice volume and the bond length increase with an increase in the cation size. Calcite has the largest cation size, and thus, the largest lattice volume and the longest bond length. The corresponding results are low critical resolved shear stresses and weak bond strength which make twinning possible. The effective-resolved-shear-stress (ERSS) model was applied to explain the Knoop microhardness profiles. The net ERSS profiles of the slip systems: {2021}<0ll2> and {10ll}<1012> for calcite, as well as the {0001}<2110> and {2021}<1102> for dolomite, as have been reported in the literature, are inversely related to the Knoop microhardness profiles. These results confirm that the reported slip systems are the primary slip systems. The ERSS model was also applied to the {1012}<1012> twin system of calcite, but no inverse relation was obtained. This result suggested that the ERSS model is impractical for the twin systems of calcite. The Meyer's law was evaluated by using a logarithmic plot of the indentation test load, P, versus the long-diagonal length, d, of the indentation. A linear relationship is confirmed. A plot of these two parameters yields two fitted lines which, in addition to the SEM, reveals the two deformation mechanisms: dislocation slip and twinning in calcite, and dislocation slip in the other five carbonate samples. The A and n parameters are orientation dependence and vary with the microhardness. The A coefficients of calcite decrease with an increase in the microhardness, while its n- values increase with an increase in the microhardness. The opposite trends are observed for the other carbonate samples. 34 Since the A coefficient has varying units as g-/m_n, normalization of the Meyer's law was used to address this dimensionality problem. The true microhardness profiles obtained during the normalization resemble the original microhardness profiles with the exception that dolomite appears to be harder. The microhardness profiles of magnesite, smithsonite, siderite, rhodochrosite, and dolomite become more similar and distinctly separate themselves from that of calcite. A logarithmic plot of P versus (d/dQ) for the Normalized Meyer's law yields a linear relationship as expected. The slopes are the nQ-values which are egual to the classical Meyer's law n-values. The A0 coefficients, on the other hand, are 10,000 times larger the A coefficients and have different units— grams. No correlations between the A 0 and n parameters are found because the A0 coefficients vary with the orientation rather randomly. The various parameters of the final form of the normalized Meyer's law were also investigated. Only the plots of Pc versus dQ and n versus dQ yield good correlations. The plot of Pc versus dQ reveals that the indentation load/size effect diminishes as the critical indentation load approaches zero. The plot of n versus dQ indicates that the dQ at the same critical load and for the same n-value is considerably larger for calcite than for the other five carbonates. In fact, the dQ for calcite increases with decreasing n-value, while those of the other carbonates decreases with decreasing n-value. This plot also suggests that two linear correlations exist and that the n-values are somehow directly related to the dQ. In conclusion, the relationships found among these plots confirm the association of the indentation load/size effect with two physical parameters, the Pc and the dQ . Appendix A Knoop Indentation and Microhardness Data I. Knoop indentation measurements of the carbonate single crystals for the four test loads and seven assigned angles. (1) Magnesite Diagonal Lengths and Standard Deviations, 25 g 50 g 100 g 200 g -48 24.4 ± 0.4 36.8 ± 0.4 55.1 ± 0.4 82.1 ± 0.4 -32 23.9 ± 0.4 36.4 ± 0.4 55.4 ± 0.4 80.5 ± 0.4 -16 22.9 ± 0.4 35.9 ± 0.5 54.2 ± 0.4 78.5 ± 0.4 0 21.6 ± 0.7 33.9 ± 0.5 51.6 ± 0.5 76.5 ± 0.3 16 23.0 ± 0.4 36.0 ± 0.4 54.1 ± 0.3 78.5 ± 0.4 32 23.8 ± 0.5 36.6 ± 0.7 54.7 ± 0.3 80.5 ± 0.5 48 24.5 ± 0.2 36.8 ± 0.3 55.5 ± 0.4 82.0 ± 0.4 (2) Smithsonite Diagonal Lengths and Standard Deviations, 200 25 g 50 g 100 g g -48 26.2 ± 0.5 40.1 ± 0.6 58.0 ± 0.6 83.3 ±1.1 26.0 ± 0.5 38.9 ± 0.7 57.7 ± 0.8 82.7 ± 0.9 -32 0.6 -16 24.6 ± 0.6 37.5 ± 0.5 56.4 ± 0.7 81.0 ± 0 22.7 ± 0.5 34.4 ± 0.6 51.2 ± 0.4 77.9 ± 0.9 0.6 16 24.8 ± 0.3 37.3 ± 0.6 55.2 ± 0.5 81.1 ± 32 25.7 ± 0.3 39.0 ± 0.7 57.7 ± 0.8 82.4 ± 0.5 48 26.1 ± 0.5 39.8 ± 1.0 57.6 ± 0.8 83.1 ± 0.9 85 86 (3) Siderite Diagonal Lengths and Standard Deviations, 25 g 50 g 100 g 200 g -48 28.2 ± 0.5 41.8 ± 0.5 61.6 ± 0.5 91.3 ± 0.7 -32 27.3 ± 0.5 41.2 ± 0.6 60.9 ± 0.6 88.5 ± 0.4 -16 26.3 ± 0.6 38.6 ± 0.6 58.7 ± 0.7 86.6 ± 0.6 0 24.2 ± 0.4 35.8 ± 0.7 56.6 ± 0.7 84.6 ± 0.5 16 26.6 ± 0.6 38.8 ± 0.5 59.5 ± 0.7 87.1 ± 0.4 32 27.4 ± 0.5 41.2 ± 0.6 61.2 ± 0.5 88.1 ± 0.4 48 28.2 ± 0.4 41.3 ± 0.5 61.9 ± 0.5 91.2 ± 0.7 (4) Dolomite Diagonal Lengths and Standard Deviations, /im 25 g 50 g 100 g 200 g -48 29.1 ± 0.6 41.9 ± 0.5 61.4 ± 0.5 88.4 ± 0.5 0.6 88.8 -32 29.0 ± 0.5 41.9 ± 0.4 61.0 ± ± 0.4 88.2 0.6 -16 28.5 ± 0.5 41.4 ± 0.6 61.0 ± 0.7 ± 0 26.9 ± 0.6 39.2 ± 0.4 58.5 ± 0.6 86.2 ± 0.6 16 28.6 ± 0.8 41.6 ± 0.4 61.1 ± 0.4 88.5 ± 0.6 0.6 32 28.7 ± 0.4 41.9 ± 0.6 61.0 ± 88.7 ± 0.5 88.6 48 29.0 ± 0.6 42.0 ± 0.5 61.5 ± 0.4 ± 0.7 87 (5) Rhodochrosite Diagonal Lengths and Standard Deviations, /m 25 g 50 g 100 g 200 g -48 35.5 ± 0.6 50.6 ± 0.6 74.0 ± 0.6 105.1 ± 0 . 6 -32 34.4 ± 0.6 48.9 ± 0.7 72.1 ± 0.6 103.0 ± 0 . 6 -16 32.7 ± 0.7 47.5 ± 0.6 69.6 ± 0.7 99.4 ± 0.5 0 30.3 ± 0.5 44.2 ± 0.5 65.0 ± 0.6 95.2 ± 0.7 16 32.6 ± 0.5 47.1 ± 0.8 69.5 ± 0.5 99.6 ± 0.7 32 34.6 ± 0.5 49.1 ± 0.6 72.3 ± 0.6 103.1 ± 0.9 48 35.5 ± 0.5 50.8 ± 0.6 73.9 ± 0.6 105.3 ± 0.7 (6) Calcite 1 Diagonal Lengths and Standard Deviations, /m 25 g 50 g 100 g 200 g -48 46.6 ± 0.7 71.1 ± 1.5 104.7 ± 1.4 159.4 ± 1.9 -32 45.1 ± 0.8 67.5 ± 1.0 100.4 ± 1.6 151.5 ± 0.9 -16 43.1 ± 0.5 63.2 ± 0.6 92.1 ± 1.2 133.5 ± 0 . 9 0 42.0 ± 0.5 62.0 ± 1.2 89.3 ± 0.8 128.9 ±0 . 8 16 43.0 ± 0.5 63.1 ± 0.7 91.8 ± 1.3 132.5 ± 1.0 100.2 1.6 32 44.8 ± 1.3 66.4 ± 1.1 ± 148.3 ± 2.3 48 47.7 ± 0.7 71.9 ± 1.3 104.6 ± 1.5 158.9 ± 2 . 0 88 II. Knoop Microhardness and Their 95% Confidence Intervals for Test Loads of 25 g, 50 g, 100 g, and 200 g (1) Magnesite Hjc and 95% Cl (kg/mm2) 25 g 50 g 100 g 200 g -48 597 ± 7.0 525 ± 4.1 469 ± 2.4 422 ± 1.5 -32 623 ± 7.5 537 ± 4.2 464 ± 2.4 439 ± 1.6 -16 678 ± 8.5 552 ± 5.5 484 ± 2.6 462 ± 1.7 0 762 ± 17.7 619 ± 6.5 534 ± 3.7 486 ± 1.4 16 672 ± 8.4 549 ± 4.4 486 ± 1.9 462 ± 1.7 32 628 ± 9.4 531 ± 7.3 476 ± 1.9 439 ± 2.0 48 593 ± 3.5 525 ± 3.1 462 ± 2.4 423 ± 1.5 (2) Smithsonite Hfc and 95% Cl (kg/mm2) 200 25 g 50 g 100 g g -48 518 ± 7.1 442 ± 4.7 423 ± 3.1 410 ± 3.9 -32 526 ± 7.2 470 ± 6.1 427 ± 4.2 416 ± 3.2 -16 588 ± 10.3 506 ± 4.8 447 ± 4.0 434 ± 2.3 0 690 ± 10.9 601 ± 7.5 543 ± 3 . 0 469 ± 3.9 16 578 ± 5.0 511 ± 5.9 467 ± 3.0 433 ± 2.3 32 539 ± 4.5 468 ± 6.0 427 ± 4.2 419 ± 1.8 48 522 ± 7.2 449 ± 8.1 429 ± 4.3 412 ± 3.2 o\ CO in CM CO CT> CO in o tp rH rH rH rH rH rH CM O + i + i + i + i + i + i + i o CM rH VO CO CO CM CO vo VO vo CO vo VO vo co CO CO CO CO CO CO CM t-' rH rH CO CO n • § tp CM CM CO CO rH CM rH -5. o +i + i + i + i +1 +1 + i tp O rH r- CM CM VO rH CM vo r- CO CO rH CO CO CO CO CO CO CO CO H U oY> ID Ch in 00 CO CO CM ■O4 TJ tp CO CM CO CM CO c o + i + i +1 fO in +i +1 +i +i in .y in in CO rH in CO o o rH vo rH o o X "M4 CM CM in CO r- CO CO CP VO in in r- 00 vo in + i + i + i + i +1 + i +1 CM o CO CO CM in CM CO CM CM CO Ch CO CO CM ■O4 M4 CO CM vo o vo CM CO CO rH rH CO l 1 i (4) Dolomite (4) V. o o\ r H o r - r - O in CO 00 VO VO H VO CO ID VO rH CM in VO CM t" N in tn rH rH CM CM rH rH rH c p rH rH rH rH rH rH rH o o +i +i +i +i +i +i +i +i +i +i +i +i +i +i tp o o X o rH O rH CO CO cn T) tp CM CO CM CM CO CM CM d 95% Cl (kg/mm2) tp CM rH rH CM rH rH rH c an o +1 +i +i + i + i +i +i o +i +i +i +i +i +i +i in in , y CO CO rH in VO rH VO CO in G\ rH CO inrH K r * o> VO CM o> r * in n* CO VO CO CM CM CO CO CO CM CM rH rH rH rH rH rH rH CO rH VO rH CO CO CM VO VO r - vo 0) CP CO CO in CO CO CM CP i—1 CM rH rH rH CO rH •H-p in +i +i +i m in +i +i +1 +i +i +i +i CM +i +i +i +1 o CM in CM CM r - vo CM rH CO in t" CM rH XIM CO O CO CO CO CT\ CO VO r - CTl O a \ r - in o CM CO CO CO CO CM CM rH rH rH CM rH rH rH o TJ x3o o CO fM VO o IX CM CO « CO CM VO VO CM CO n H H m "d" CO rH rH CO T 1 1 I I I in (6) Calcite (6) Appendix B Effeetive-Resolved-Shear-Stress Values Computed ERSS Values for the Reported Slip and Twin Systems of Calcite and Dolomite. (1) ERSS Values for the {2021}<0112> Slip System of Calcite -48 0.005 0.014 0.061 0.220 0.069 0.220 -32 0.009 0.021 0.092 0.119 0.040 0.1 3 -16 0.013 0.029 0.110 0.052 0.089 0.128 0 0.018 0.090 0.110 0.020 0.110 0.080 0.037 0.131 0.092 0.012 0.108 0.032 16 0.020 32 0.094 0.164 0.040 0.011 0.086 48 0.205 0.203 0.074 0.009 0.056 0.015 ERSS Values for the {1011}<1012> Slip System of Calcite 0.010 -48 0.221 0.120 0.198 0.005 0.033 0.021 -32 0.119 0.100 0.141 0.008 0.045 0.011 0.022 -16 0.052 0.062 0.087 0.047 0 0.021 0.060 0.037 0.014 0.027 0.048 16 0.012 0.053 0.008 0.030 0.074 0.060 32 0.011 0.051 0.001 0.081 0.130 0.080 48 0.009 0.042 0.003 0.195 0.191 0.119 91 92 (3) ERSS Values for the {0001}<2ll0> Slip system of Dolomite -48 0.000 0.001 0.156 -32 0.002 0.001 0.104 -16 0.014 0.001 0.059 0 0.031 0.001 0.030 16 0.058 0.000 0.005 32 0.099 0.000 0.000 48 0.160 0.000 0.000 (4) ERSS Values for the {2021}<1102> Slip System of Dolomite 0.020 0.200 -48 0.013 0.004 0.262 0.085 0.025 0.020 0.040 0.160 0.083 0.124 -32 0.110 -16 0.036 0.041 0.050 0.070 0.130 0 0.053 0.080 0.065 0.090 0.062 0.072 0.112 0.062 0.052 0.065 0.031 16 0.084 0.022 0.128 0.111 0.081 0.045 0.041 32 0.102 0.010 48 0.244 0.174 0.025 0.034 93 (5) ERSS Values for the {1012}<1012> Twin System of Calcite -48 0.176 0.030 0.074 0.189 0.035 0.015 -32 0.109 0.081 0.047 0.077 0.042 0.022 -16 0.046 0.125 0.020 0.023 0.030 0.023 0 0.019 0.151 0.004 0.006 0.036 0.020 16 0.011 0.152 0.002 0.004 0.072 0.003 32 0.010 0.133 0.009 0.006 0.027 0.039 48 0.009 0.100 0.026 0.008 0.164 0.113 Appendix C Knoop Software This appendix refers to the supplemental Knoop software, which was used to perform all the analytical calculations in this research. 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