Introduction to Contact Mechanics Part 9 pot

8.2 Pointed Indenters 143 unloading, the median cracks extend outward and upward and may join with the radial cracks to form a system of half-penny cracks, which are then referred to as “median/radial” cracks In glass, the observed cracks at the corners of the residual impression on the specimen surface are usually fully formed median/radial cracks However, in other brittle materials, with higher values of E/Y, radial cracks are usually quite distinct from the median cracks and form upon loading Experimental observations suggest that there is no one general sequence of indentation cracking Chiang, Marshall, and Evans4,5, and also Yoffe6, have proposed analytical models that attempt to describe the cracking sequence in terms of a superposition of the Boussinesq elastic stress field and that associated with the expanding cavity model of Johnson The Yoffe6 model describes the evolution of stresses for both loading and unloading in terms of material and E/Y and appears to account for the appearance of radial and lateral cracking for loading and unloading sequences for a wide range of test materials However, the model, as with that of Chiang, Marshall, and Evans4,5, deals with axis-symmetric, pointed indenters Clearly, the corners of a pyramidal indenter play an important role, since both median and radial cracks align themselves with the corners of the indentation However, development of a three-dimensional analytical model that deals with this type of geometry would be an extremely difficult undertaking and would be best left to numerical finite-element analysis The value of the axis-symmetric models is to provide physical insight, and that they quite well It is the radial and lateral cracks that are of particular importance, since their proximity to the surface has a significant influence on the fracture strength of the specimen Fracture mechanics treatments of these types of cracks seek to provide a measure of fracture toughness based on the length of the radial surface cracks (a) (b) (c) (d) Fig 8.2.4 Crack systems for Vickers indenter: (a) radial cracks, (b) lateral cracks, (c) median cracks, (d) half-penny cracks 8.2.3 Fracture toughness One of the main features of an indentation crack is that it is stable with increasing load.9 While straight cracks in beam-type specimens are routinely used for 144 Elastic-Plastic Indentation Stress Fields fracture toughness testing of ductile materials, this type of test is very difficult to undertake with brittle materials Attempts to so usually end with catastrophic failure of the specimen Further, indentation cracks require only a small surface test area, and multiple indentations can usually be made on the face of a single specimen For these reasons, it is desirable to be able to measure fracture toughness using crack length data available from indentation tests Attention is usually given to the length of the radial cracks as measured from the corner of the indentation and then radially outward along the specimen surface as shown in Fig 8.2.5 Palmqvist8 noted that the crack length l varied as a linear function of the indentation load Lawn, Evans, and Marshall10 formulated a different relationship, where they treated the fully formed median/radial crack and found the ratio P/c3/2 (where c is measured from the center of contact to the end of the corner radial crack) is a constant, the value of which depends on the specimen material Fracture toughness is found from: n ⎛E⎞ P K c = k ⎜ ⎟ 3/2 ⎝H⎠ c (8.2.3a) where k is a calibration constant equal to 0.016 and n = ½ with c = l+a (a) (b) l c l c Fig 8.2.5 Crack parameters for Vickers and Berkovich indenters Crack length c is measured from the center of contact to end of crack at the specimen surface Various other studies have since been performed, and Anstis, Chantikul, Lawn, and Marshall11 determined n = 3/2 and k = 0.0098 Laugier12-14 undertook an extensive review of previously reported experimental results and determined: 2/3 12⎛ E ⎞ K c = xv ( a l ) ⎜ ⎟ ⎝H⎠ P 3/2 c (8.2.3b) 8.3 Spherical Indenter 145 With xv = 0.015, Laugier showed that the radial and half-penny models make almost identical predictions of the dependence of crack length on load (note the similarity between Eqs 8.2.3a and 8.2.3b Experiments show that the term (a/l)1/2 shows little variation between glasses (median/radial) and ceramics (radial) The significance of this result is that it is thus generally not possible to infer the existence of a fully formed median/radial crack from the observable crack length, and for opaque materials it is necessary to undertake sectioning of the specimen to obtain a full knowledge of the crack in any particular material 8.2.4 Berkovich indenter The vast majority of toughness determinations using indentation techniques are performed with a Vickers diamond pyramid indenter This indenter takes the form of a square pyramid with opposite faces at an angle of 136° (edges at 148°) However, the advantages of a Berkovich indenter have become increasingly important, especially in ultra-micro-indentation work, where the faces of the pyramid are more likely to meet at a single point rather than a line However, despite this advantage, the loss of symmetry presents some problems in determining specimen toughness because half-penny cracks can no longer join two corners of the indentation Ouchterlony15 investigated the nature of the radial cracking emanating from a centrally loaded expansion star crack and determined a modification factor for stress intensity factor to account for the number of radial cracks formed k1 = n2 2π n 1+ sin n 2π (8.2.4a) As proposed by Dukino and Swain16, this modification has relevance to the crack pattern observed from indentations with a Berkovich indenter The ratio of k1 values for n = (Vickers) and n = (Berkovich) is 1.073 and thus the length of a radial crack (as measured from the center of the indentation to the crack tip) from a Berkovich indenter should be 1.0732/3 = 1.05 that from a Vickers indenter for the same value of K116 The Laugier expression can thus be written: 2/3 12⎛ E ⎞ K c = 1.073 xv ( a l ) ⎜ ⎟ ⎝H⎠ P c3/2 (8.2.4b) 8.3 Spherical Indenter As mentioned previously, analytical treatments of the indentation stress field for elastic-plastic contact are made complex by the presence of plasticity beneath 146 Elastic-Plastic Indentation Stress Fields the indenter However, these difficulties can be circumvented by making use of the finite-element method17-21 Such modeling is of considerable importance because it provides numerical data for indentations involving complex geometries and material properties (see Chapter 12) Fig 8.3.1 shows a comparison between the elastic and elastic-plastic stress distributions for a spherical indenter −1 30 −0 00 −0.2 00 −0.150 −0.025 −0005 20 0.0 −1 −2 −3 −3 −4 −4 01 00 0 −0 −0 −0 −0 04 06 20 0 00 −2 00 200 −0.150 −0 00 50 −0 −0.0 20 0.0 0.01 −2 −4 00 −1 0 −1 −2 −3 −4 50 −0.0 −3 −0 10 50 −0 25 −0 −0.050 −2 −1 z/ao 75 −0 0 −0.5 25 0 −0.0 0.05 10 − −0 0 −2 −0.2 0.00 20 0.0 −0 0.004 005 08 −3 −4 −3 −3 0.0 0.0 0.0 10 (c) −4 10 02 −0.0 −0 −1 0.020 −0.004 −1 01 −0.0 10 −0.00 −0.006 0 −1 −0 10 −0 75 −0 02 −2 −4 −0.10 75 −0.0 −0 50 −3 z/ao −4 0.00 (b) 0 −2 0.01 r/ao 0.01 10 0.0 0.005 0 50 25 -0 -0 0.005 0.010 0.005 0.005 −1 02 r/ao −2 −3 −4 z/ao (a) Fig 8.3.1 Contours of principal stress for glass-ceramic material with spherical indenter Left side of figure shows elastic solutions from Eqs 5.4.2i to 5.4.2o in Chapter Right side of figure shows finite-element results for elastic-plastic response Magnitudes are shown for (a) σ1, (b) σ2, (c) σ3 For both elastic and elastic-plastic results, distances are expressed relative to the contact radius ao = 0.326 mm and stresses in terms of the mean contact pressure pm = 3.0 GPa for the elastic case at P = 1000 N (with kind permission of Springer Science and Business Media, Reference 22) 8.3 Spherical Indenter 147 The presence of the “plastic” or damage zone significantly alters the nearfield indentation stresses In general, the magnitudes of the maximum principal stresses appear to shift outward, away from the center of contact when compared to the elastic case The far-field stresses, however, appear to be little changed from the elastic case The shift in magnitudes of stresses away from the center of contact, indicates an outward shift in the distribution of upward pressure which serves to support the indenter This is reflected in the contact pressure distributions shown in Fig 8.3.2, where the contact pressure for the elastic-plastic case, appears to be more uniformly distributed (except for the rise in contact pressure near the edge of the contact circle) compared to the elastic case As will be seen in Chapter 9, the actual shape and size of the plastic zone depend on the mechanical properties of the specimen material, particularly the ratio E/Y, where E is the elastic modulus and Y is the yield stress22 In Fig 8.3.2, results are plotted normalized to the mean contact pressure pm = 3.0 GPa and contact radius ao = 0.326 mm for the elastic case Stresses have been calculated for a yield stress Y = 770 MPa with the load P = 1000 N and R = 3.18 mm Absolute values may be obtained by multiplying by these factors -1.5 Elastic -1.0 σz/pm Elastic-plastic -0.5 0.0 0.0 1.0 r/ao 2.0 Fig 8.3.2 Contact pressure distribution for elastic (equation), and elastic-plastic (finiteelement) contact for P = 1000 N, R = 3.18 mm Results are normalized to ao and pm as in Fig 8.3.1 The bar at the bottom of the horizontal axis indicates the radius of circle of contact a = 0.437 mm for the elastic-plastic condition (with kind permission of Springer Science and Business Media, Reference 22) Figure 8.3.3 shows the variation in stress along the surface and downward along the axis of symmetry As for Fig 8.2.3, results are plotted normalized to the elastic contact pressure (pm = 3.0 GPa) and contact radius (ao = 0.326 mm) Also shown in this figure are the results for the elastic solution for the maximum principal stress σ1 As shown in Fig 8.3.3 (a), along the surface, the maximum value of σ1 is very much the same as for the elastic case, although the tact pressure is correspondingly lower due to the larger area of contact in the Elastic-Plastic Indentation Stress Fields 148 elastic-plastic case (with a = 0.437 mm) There is an interesting change in magnitude for all stresses within the contact zone near the edge of the circle of contact Later we shall see that this arises due to the plastic zone being contained within the area of the circle of contact, and this discontinuity flattens out for contacts on materials, such as metals, where the plastic zone extends beyond the radius of circle of contact In Fig 8.3.3 (b), the variation in stresses along the axis of symmetry downward into the specimen material is shown Note that the maximum tensile stress occurs at the elastic-plastic boundary and is larger by a factor of ≈3.6 than that calculated for an elastic contact The significance of these results is particularly important when we wish to use indentation stress fields for structural reliability analysis The failure of brittle materials is often attributed to the action of tensile stresses on surface flaws For example, the procedure for determining the conditions for initiation of a Hertzian crack in soda-lime glass (presented in Chapter 7) applies to a purely elastic contact For the type of material described here—a nominally brittle ceramic which undergoes shear-driven failure within a “plastic” zone—an analysis involving Weibull statistics may not be appropriate or may need to be applied to flaws within the interior of the specimen rather than on the surface Whatever analysis is selected, a detailed knowledge of the state of stress within the material is required (a) (b) 0.4 σ1E τmax σ/pm 0.2 σ1 0.0 0.0 σ/pm -0.2 σ2 σ3 σΗ -0.4 τmax σ1E σ3 σ1,2 -0.4 -0.8 σH plastic elastic -0.6 -0.8 -1.2 0.0 1.0 2.0 r/ao 3.0 0.0 -2.0 -4.0 -6.0 -8.0 z/ao Fig 8.3.3 Variation in stresses σ1, σ2, σ3, the hydrostatic component σH, and maximum shear stress τmax along (a) the surface of the specimen at z = and (b) downward along the axis of symmetry at r = In both (a) and (b), the dashed curve σ1E shows the variation in σ1 as calculated from elastic formulas for comparison with the elastic-plastic finite-element result The horizontal bar in (a) indicates the radius of circle of contact a = 0.437 mm for the elastic-plastic condition The shaded area in (b) indicates the region in which plastic strains have occurred (with kind permission of Springer Science and Business Media, Reference 22) References 149 References D.M Marsh, “Plastic flow in glass,” Proc R Soc London, Ser A279, 1964, pp 420–435 R Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950 K.L Johnson, “The correlation of indentation experiments,” J Mech Phys Solids 18, 1970, pp 115–126 S.S Chiang, D.B Marshall, and A.G Evans, “The response of solids to elastic/plastic indentation Stresses and residual stresses,” J Appl Phys 53 1, 1982, pp 298–311 S.S Chiang, D.B Marshall, and A.G Evans, “The response of solids to elastic/plastic indentation Fracture initiation,” J Appl Phys 53 1, 1982, pp 312–317 E.H Yoffe, “Elastic stress fields caused by indenting brittle materials,” Philos Mag A, 46, 1982, pp 617–628 R.F Cook and G.M Pharr, “Direct observation and analysis of indentation cracking in glasses and ceramics,” J Am Ceram Soc 73 4, 1990, pp 787–817 S Palmqvist, “A method to determine the toughness of brittle materials, especially hard materials,” Jernkontorets Ann 141, 1957, pp 303–307 F.C Roesler, “Brittle fractures near equilbrium,” Proc R Soc London, Ser B69 1956, pp 981–992 10 B.R Lawn, A.G Evans, and D.B Marshall, “Elastic/plastic indentation damage in ceramics: the median/radial crack system,” J Am Ceram Soc 63, 1980, pp 574– 581 11 G.R Anstis, P Chantikul, B.R Lawn, and D.B Marshall, “A critical evaluation of indentation techniques for measuring fracture toughness: I Direct crack measurements,” J Am Ceram Soc 64 9, 1981, pp 533–538 12 M.T Laugier, “Palmqvist indentation toughness in WC-Co composites,” J Mater Sci Lett 6, 1987, pp 897–900 13 M.T Laugier, “Palmqvist toughness in WC-Co composites viewed as a ductile/brittle transition,” J Mater Sci Lett 6, 1987, pp 768–770 14 M.T Laugier, “New formula for indentation toughness in ceramics,” J Mater Sci Lett 6, 1987, pp 355–356 15 F Ouchterlony, “Stress intensity factors for the expansion loaded star crack,” Eng Frac Mechs 8, 1976, pp 447–448 16 R Dukino and M.V Swain, “Comparative measurement of indentation fracture toughness with Berkovich and Vickers indenters,” J Am Ceram Soc 75 12, 1992, pp 3299–3304 17 C Hardy, C.N Baronet, and G.V Tordion, “The elastic-plastic indentation of a halfspace by a rigid sphere,” Int J Numer Methods Eng 3, 1971, pp 451–462 18 P.S Follansbee and G.B Sinclair, “Quasi-static normal indentation of an elastoplastic half-space by a rigid sphere-I,” Int J Solids Struct 20, 1981, pp 81–91 19 R Hill, B Storakers and A.B Zdunek, “A theoretical study of the Brinell hardness test,” Proc R Soc London, Ser A423, 1989, pp 301–330 20 K Komvopoulos, “Finite element analysis of a layered elastic solid in normal contact with a rigid surface,” J Tribology, Trans ASME, 111, 1988, pp 477–485 150 Elastic-Plastic Indentation Stress Fields 21 K Komvopoulos, “Elastic-plastic finite element analysis of indented layered media,” J Tribology, Trans ASME, 111, 1989, pp 430–439 22 G Caré and A.C Fischer-Cripps, “Elastic-plastic indentation stress fields using the finite element method,” J Mater Sci 32, 1997, pp 5653–5659 Chapter Hardness 9.1 Introduction Indentation tests involving hard, spherical indenters have been the basis of hardness testing since the time of Hertz in 18811,2 Conventional indentation hardness tests involve the measurement of the size of a residual plastic impression in the specimen as a function of the indenter load Theoretical approaches to hardness can generally be categorized according to the properties of the indenter and the assumed response of the specimen material For sharp indenters, the specimen is usually approximated by a rigid-plastic material in which plastic flow is assumed to be governed by flow velocity considerations For blunt indenters, the specimen responds in an elastic-plastic manner, and plastic flow is usually described in terms of the elastic constraint offered by the surrounding material The hardness of brittle materials is conveniently measured using a diamond pyramid indenter, since a residual impression is readily obtained at relatively low values of indenter load The use of spherical indenters in hardness measurements is usually restricted to tests involving ductile materials but has recently been applied to certain brittle ceramics that exhibit a shear-driven “plasticity” in the indentation stress field 9.2 Indentation Hardness Measurements 9.2.1 Brinell hardness number The Brinell test involves the indentation of the specimen with a hard spherical indenter The method was developed by J.A Brinell in 19003 For an indenter diameter D, and diameter of the residual impression d, the hardness number is calculated from: BHN = 2P ⎛ D − D2 − d ⎞ πD⎜ ⎟ ⎝ ⎠ (9.2.1a) 152 Hardness D P d Fig 9.2.1 Geometry of a Brinell hardness test This formula gives the hardness as a function of the load and the actual surface area of the residual impression The diameter of the impression d is measured in the plane of the original surface Brinell hardness measurements are usually made such that the ratio d/D is between 0.25 and 0.5 and the load applied for at least 30 seconds For iron and steel, a ball diameter of 10 mm with a load of 30 kN is commonly used Figure 9.2.1 shows the relevant features of the test geometry The Brinell hardness number is favored by some engineers because of the existence of an empirical relationship between it and the ultimate tensile strength of the specimen material* However, for physical reasons, it is preferable to use the projected contact area rather than the actual surface area of the specimen as the divisor since this gives the pressure beneath the indenter which opposes the applied force 9.2.2 Meyer hardness The Meyer hardness is similar to the Brinell hardness except that the projected area of contact rather than the actual curved surface area is used to determine the hardness In this case, the hardness number is equivalent to the mean contact pressure between the indenter and the surface of the specimen As we shall see, the mean contact pressure is a quantity of considerable physical significance The Meyer hardness is given by: H = pm = 4P πd (9.2.2a) where d is the diameter of the contact circle at full load (assumed to be equal to the diameter of the residual impression in the surface—see Chapter 10) * Ultimate tensile strength is defined as the maximum nominal stress measured using the original cross-sectional area of the test specimen 9.2 Indentation Hardness Measurements 153 9.2.3 Vickers diamond hardness The Vickers diamond indenter takes the form of a square pyramid with opposite faces at an angle of 136° (edges at 148°) The indenter was suggested by Smith and Sandland in 19244 The Vickers diamond hardness, VDH, is calculated using the indenter load and the actual surface area of the impression The area of the base of the pyramid, at a plane in line with the surface of the specimen, is equal to 0.927 times the surface area of the faces that contact the specimen The mean contact pressure pm is given by the load divided by the projected area of the impression† Thus, the Vickers hardness number is lower than the mean contact pressure by ≈ 7% The Vickers diamond hardness is found from: VDH = 2P 136° P sin = 1.86 2 d2 d (9.2.3a) with d equal to the length of the diagonal measured from corner to corner as shown in Fig 9.2.3 From geometry, it can be easily shown that the length d of the diagonal is precisely times the total penetration depth d α Fig 9.2.3 Geometry of Vickers hardness test 9.2.4 Knoop hardness The Knoop indenter is similar to the Vickers indenter except that the diamond pyramid has unequal length edges, resulting in an impression that has one diagonal with a length approximately seven times the shorter diagonal The angles for the opposite faces of a Knoop indenter are 172°30′ and 130° The Knoop indenter is particularly useful for the study of very hard materials since the length of the long diagonal of the residual impression is more easily measured compared to the dimensions of the impression made by Vickers or spherical indenters † Assuming frictionless contact between the indenter and the specimen 154 Hardness d = d' b' b 172.5Њ 130Њ Fig 9.2.4 Geometry of a Knoop indenter Due to the unequal lengths of the diagonals, this indenter is also very useful for investigating anisotropy of the surface of the specimen The indenter was invented in 1934 at the National Bureau of Standards in the United States by F Knoop, C.G Peters, and W.B Emerson5 As shown in Fig 9.2.4, the length d of the longer diagonal is used to determine the projected area of the impression The Knoop hardness number calculated from: KHN = 2P 172.5 130 ⎤ ⎡ tan a ⎢cot 2 ⎥ ⎣ ⎦ (9.2.4a) For indentations in highly elastic materials, there is observed a substantial difference in the length of the short axis diagonal for a condition of full load compared to full unload Marshall, Noma and Evans6 likened the elastic recovery along the short axis direction to that of a cone with major and minor axes and applied elasticity theory to arrive at an expression for the recovered indentation size in terms of the geometry of the indenter and the ratio H/E b′ b H = −α ′ a E a (9.2.4b) In Eq 9.2.4b, α is a geometry factor found from experiments on a wide range of materials to be equal to 0.45 The ratio of the dimension of the short diagonal b to the long diagonal a at full load is given by the indenter geometry and for a Knoop indenter, b/a = 1/7.11 The primed values of a and b are the lengths of the long and short diagonals after removal of load Since there is observed to be negligible recovery along the long diagonal, we can say that a′ ≈ a When H is small and E is large (e.g., metals), then b′ ≈ b indicating negligible elastic recovery along the short diagonal When H is large and E is small (e.g., glasses and ceramics), there we would expect b′