Elastic-Plastic Indentation Stress Fields 144 fracture toughness testing of ductile materials, this type of test is very difficult to undertake with brittle materials.. Further, indentat
Trang 18.2 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
me-dian/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 pro-posed 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
evolu-tion 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 undertak-ing 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 do 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
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 increas-ing load.9 While straight cracks in beam-type specimens are routinely used for
Trang 2Elastic-Plastic Indentation Stress Fields
144
fracture toughness testing of ductile materials, this type of test is very difficult to
undertake with brittle materials Attempts to do 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
tough-ness 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
sur-face as shown in Fig 8.2.5
Palmqvist8 noted that the crack length l varied as a linear function of the
in-dentation 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:
2 / 3
c
P H
E
k
⎠
⎞
⎜
⎝
⎛
where k is a calibration constant equal to 0.016 and n = ½ with c = l+a
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:
3/2
K x a l
c l
l c
Trang 38.3 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
(ra-dial) 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
increas-ingly 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
deter-mining 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
ra-dial cracks formed
n
n n
π
2 sin 2
1
2
1
+
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 = 4 (Vickers) and n = 3 (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:
3/2
8.3 Spherical Indenter
elastic-plastic contact are made complex by the presence of plasticity beneath
As mentioned previously, analytical treatments of the indentation stress field for
Trang 4Elastic-Plastic Indentation Stress Fields
146
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 geome-tries 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
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 5 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 a o = 0.326 mm and stresses in terms of the mean
contact pressure p m = 3.0 GPa for the elastic case at P = 1000 N (with kind permission of
Springer Science and Business Media, Reference 22)
− 4
− 3
− 2
− 1 0
− 4
− 3
− 2
− 1 0
r/ao
− 4
− 3
− 2
− 1 0
r/ao
− 4
− 3
− 2
− 1 0
− 4
− 3
− 2
− 1 0
− 4
− 3
− 2
− 1 0
(a)
(b)
(c)
0.005
0.005
0 01
0.010
.0
0.02 0 0.020 0.005
− 0.025
− − 0.050 0.100
0.010
0.0 05
0
−0.100
− 0.010
− 0.008
−0.006
− 0.004
− 0.002
0.0 0 0.004
0.006 0.008
− 0.300
− 0.200
− 0.150
− 0.100 − 0.050
10
− 0.500
− 0.250
− 0.100
00
−0.050
− 0.005 − 0.02 − 0.0550
− 0.250
−0
.500
− 0.075 − 0.050
− 0.
5
− 0.100
−
0.050
0.02 0
0.0 10
0.010
0.010
0.
0100.005-0.025-0.050
0.005
20
Trang 58 3 Spherical Indenter 147 The presence of the “plastic” or damage zone significantly alters the near-field 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 distribu-tions 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
me-chanical 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 p m = 3.0 GPa and contact radius a o =
Fig 8.3.2 Contact pressure distribution for elastic (equation), and elastic-plastic
(finite-element) contact for P = 1000 N, R = 3.18 mm Results are normalized to ao and p m 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 (p m = 3.0 GPa) and contact radius (a o = 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 con tact pressure is correspondingly lower due to the larger area of contact in the
Y = 770 MPa with the load P = 1000 N and R = 3.18 mm Absolute values may
be obtained by multiplying by these factors
0.326 mm for the elastic case Stresses have been calculated for a yield stress
r/ao
-1.5
-1.0
-0.5
0.0
σ z/p m
Elastic
Elastic-plastic
Trang 6Elastic-Plastic Indentation Stress Fields
148
elastic-plastic case (with a = 0.437 mm) There is an interesting change in
mag-nitude 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 maxi-mum 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 brit-tle 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 ce-ramic 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 mate-rial is required
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 = 0 and (b) downward along the axis of symmetry at r = 0 In both (a) and (b), the dashed curve σ1E shows the varia-tion 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 Busi-ness Media, Reference 22)
-1.2
-0.8
-0.4
0.0
0.4
σ1
σ2
σ3
σ1E
τmax
σΗ
-8.0 -6.0 -4.0 -2.0 0.0 -0.8 -0.6 -0.4 -0.2 0.0
0.2
τmax
σH
σ3
σ1E
σ1,2
plastic elastic
σ/p m
σ/p m
Trang 7References 149
References
1 D.M Marsh, “Plastic flow in glass,” Proc R Soc London, Ser A279, 1964, pp 420–435
2 R Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950
3 K.L Johnson, “The correlation of indentation experiments,” J Mech Phys Solids 18,
1970, pp 115–126
4 S.S Chiang, D.B Marshall, and A.G Evans, “The response of solids to elastic/plastic indentation 1 Stresses and residual stresses,” J Appl Phys 53 1, 1982, pp 298–311
5 S.S Chiang, D.B Marshall, and A.G Evans, “The response of solids to elastic/plastic indentation 2 Fracture initiation,” J Appl Phys 53 1, 1982, pp 312–317
6 E.H Yoffe, “Elastic stress fields caused by indenting brittle materials,” Philos Mag
A, 46, 1982, pp 617–628
7 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
8 S Palmqvist, “A method to determine the toughness of brittle materials, especially hard materials,” Jernkontorets Ann 141, 1957, pp 303–307
9 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 measure-ments,” 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 half-space 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 elasto-plastic 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
Trang 8Elastic-Plastic Indentation Stress Fields
150
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
Trang 9Chapter 9
Hardness
9.1 Introduction
Indentation tests involving hard, spherical indenters have been the basis of
hard-ness 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
as-sumed to be governed by flow velocity considerations For blunt indenters, the
specimen responds in an elastic-plastic manner, and plastic flow is usually
des-cribed 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:
⎟
=
2 2
2
d D D
D
P BHN
Trang 10Hardness
152
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
prefer-able 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:
2
4
d
P
p
π
=
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
D
d
P