combining with full-field deformation measurement, Brazilian disk can be used ..... CL also wants to thank Drs. Matthew W. Lewis and Philip Rae of Los Alamos ...
Proceedings of the XIth International Congress and Exposition June 2-5, 2008 Orlando, Florida USA ©2008 Society for Experimental Mechanics Inc.
Elastic Constants Determination and Deformation Observation Using Brazilian Disk Geometry C. Liu, M.L. Lovato Materials Science & Technology Division, Los Alamos National Laboratory Los Alamos, New Mexico 87545, USA Abstract Brazilian disk compression has been proposed as an alternative for measuring elastic constants of brittle solids with very low tensile strength [1]. Subsequently however, the Brazilian disk geometry was mainly used for measuring fracture toughness and tensile strength of brittle materials, like rocks and concretes. In this study, we revisit the Brazilian disk specimen as a tool for determining elastic constants and for observing the deformation process up to failure. We used the optical digital image correlation (DIC) technique to measure the displacement field on the specimen surface and proposed a scheme for obtaining the elastic constants from the measured displacement field and the applied load. Details of the elastic constant determination of a homogeneous material, epoxy resin, were presented. Comparison of the elastic constant measured using Brazilian disk with those obtained through more conventional means was carried out. We also present observation of the deformation evolution of the epoxy resin disk subjected to large nonlinear deformation up to failure and subjected to compressive loading and unloading.
1
Introduction
The determination of elastic constants of brittle materials with very low tensile strength remains a challenging task. Hondros [1] proposed the Brazilian disk compression test as an alternative for measuring elastic constants of brittle solids. The test configuration involves a circular disk with a pair of compressive forces acting across the diameter. Hondros [1] used a series of strain gages along the horizontal and vertical diameters of the disk to measure local strains. The local stress was obtained using solutions based on elasticity. Elastic constants were determined from these local measurements. Subsequently, however, this sample geometry was mainly used for determining the tensile strength and fracture toughness of brittle materials, e.g., concretes and rocks [2, 3]. There have been debates regarding the validity of the Brazilian test [4–6]. For example, Fairhurst [4] carried out a detailed study on the relationships among the half angle that characterizes the size of contact region between the disk sample and loading surface, the compression/tension strength ratio, and the measured tensile strength from the disk compression and the true tensile strength of the brittle material. As a result, he proposed several restrictions on how to implement the test configurations. Wang et al. [7] suggested using a Brazilian disk with flattened ends so that a crack would initiate at the center of the disk. All these investigations either attempted to ensure or just assumed that failure would initiate at the center of the disk specimen, so that the critical stress can be calculated by the elastic solution. They rely solely on monitoring the applied load for measuring the strength, while the fracture patterns are observed post-mortem. In this investigation, we revisit the Brazilian disk specimen as a tool both for determining elastic constants and for observing the deformation process up to failure. Rather than relying on the load measurement only, we use the optical digital image correlation (DIC) technique to monitor the deformation field on the disk specimen surface during the entire compression process. Consequently, the issues raised in those debates become insignificant. Meanwhile, by combining with full-field deformation measurement, Brazilian disk can be used to study not only brittle materials, but also quasi-brittle or even ductile materials. We organize the paper as follows. In Section 2, the Brazilian disk specimen and the associated elastic displacement and stress fields are described. The experimental setup, the full-field displacement measurement technique, and the overall response of the Brazilian disk, made of an epoxy resin and subjected to compressive load are presented in Section 3. The
scheme for determining elastic constants and the result for the epoxy resin material are given in Section 4, where the results based on the present scheme are also compared with those obtained through conventional techniques. Finally, observation of the full-field deformation associated with different loading conditions is presented in Section 5.
2
Brazilian disk specimen
The Brazilian disk specimen is a circular disk with a pair of compressive forces acting across the diameter. The test configuration is simple without complicated loading fixtures and thus, is suitable for studying failure and fracture behavior of brittle or quasi-brittle materials. The original premise of using the disk specimen is that for an elastic circular disk, closed-form solution exists for plane strain or plane stress deformations. Detailed expressions of the elastic solution of both displacement and stress of a linearly elastic circular disk subjected to a pair of concentrated forces across its diameter, are listed in the Appendix. Here we only present some numerical results relevant to our testing situations. Figure 1(a) presents the contour plots of the displacement field of the circular disk, where a pair of concentrated forces is applied across the vertical diameter. The deformation is assumed to be plane stress and the Poisson’s ratio ν to be 0.3. In the plot, F is the magnitude of the applied compressive load per unit thickness of the circular disk, and both of the displacement components, u(x, y) and v(x, y), are normalized by the factor F/µR, where µ is the shear modulus and R the radius of the disk. In the same figure, we also present the profiles of the displacement u(x) along the horizontal diameter and the displacement v(y) along the vertical diameter. The contour plots of all three in-plane stress components are shown in Fig. 1(b). In these plots, the stresses are normalized by F/R. Also note that for traction-prescribed boundary conditions, the stress field within the circular disk is the same for both plane stress and plane strain, and the stress field does not depend on the elastic constants of the material.
(a)
(b)
Figure 1: (a) Contour plots of the displacement components u(x, y) and v(x, y). The yellow lines represent the displacement profiles, u(x) and v(y), along the horizontal and vertical diameters, respectively. (b) Contour plots of the three stress components σx (x, y), σy (x, y), and τxy (x, y). During the Brazilian disk compression test, a contact zone develops near each of the loading points, and the applied load F is distributed over a finite region rather than a singular point. Whether or not the formation of such contact regions will change the distribution of displacement and stress, and thus change the nature of failure and fracture, is worth investigating. Here we assume that the contact surface remains flat since compared to the testing material the material made of the loading fixture is much stiffer and can be treated as rigid during the compression. Meanwhile, we neglect the friction along the contact surface. The contact zone spans over a region −b 6 x 6 b along the top and bottom boundaries of the disk, where b is the half length of the contact zone, as shown in Fig. 2. We may also use the half angle, β, formed by the end of the contact region with respect to the disk center to characterize the contact zone. Figure 2(b) presents the displacement profiles along both the horizontal diameter and the vertical diameter for several different half angles β. Here, only plane stress is considered, and ν = 0.3. The displacement profile u(x) along the horizontal diameter changes very little as the half angle β varies from 0◦ (point-loading case) to 15◦ . The half angle β = 15◦ corresponds to the half contact zone size b = 0.26R. Along the vertical diameter, the displacement profile v(y) changes very little in the center section of the disk, but changes noticeably near the loading regions. Also as the half angle β increases, the magnitude of v(y) decreases due to the reduction of singularity at the loading points. The
0.15
1.
β = 0° β = 5° β = 10° β = 15°
0.1
β = 0° β = 5° β = 10° β = 15°
0.5
y
y/R
u/(FR/μ)
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Plane stress Poisson’s ratio ν = 0.3
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Plane stress Poisson’s ratio ν = 0.3 −0.6
1.
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β
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x 1.
1.
σx
0.5
β = 0° β = 5° β = 10° β = 15°
−1.
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σ/(F/πR)
0.
(a)
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(b) R
F
0.
v/(FR/μ)
β = 0° β = 5° β = 10° β = 15°
0.
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σy
σy
σx
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−3. −1.
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x/R
(c) Figure 2: (a) Contact of the disk boundary with the flat loading surface. (b) Displacement profiles along the horizontal diameter and the vertical diameter for uniformly distributed boundary tractions. (c) Normal stress profiles along the horizontal and the vertical diameters, respectively, for uniformly distributed boundary tractions. variations of the two normal stress components, σx and σy , along the horizontal and vertical diameters are shown in Fig. 2(c) for several different half angles β. Along the horizontal diameter, both stress components only reduce their magnitude slightly in the center section of the disk as the angle β increases. However, along the vertical diameter, both stress components change significantly when the half angle β becomes larger. Meanwhile, close to the loading region, the normal stress component σx changes from tensile to compressive for the distributed compressive load. Because of this drastic change of normal stress profile, some studies have proposed to use circular anvils with larger radii than that of the disk, for applying compressive load [8], in order to better control the fracture initiation location within the brittle sample.
3
Experimental observation
The Brazilian disk specimen we used in this study is made of an epoxy resin and has the nominal dimension of 16.84 mm in diameter and 6.75 mm in thickness. We placed the Brazilian disk specimen between two parallel flat surfaces for applying compressive load. In this investigation, we use the digital image correlation (DIC) technique to obtain the deformation field on the specimen surface. This technique relies on the computer vision approach to extract the whole-field displacement data, that is, by comparing the features in a pair of digital images of a specimen surface before and after deformation. To apply the DIC technique, on the surface of the disk specimen, a random speckle pattern was generated by first spraying a very thin layer of white paint and then spraying black paint on top of the white background. Detailed description of the underlying principle of the DIC technique can be found in literature [9–11]. We used a conventional Instron 1125 screw-driven loading frame, with the MTS ReNew/E System controller, to load the specimen at a constant crosshead moving velocity. We chose the crosshead moving velocity to be about 0.33 mm/minute. The change of disk diameter during the compression was measured using the Keyence EX-V02 mag-
netic displacement sensor. The applied compressive load, crosshead displacement, and the output from the magnetic displacement sensor were monitored and recorded at the sampling rate of 10 Hz. A CCD camera, with 8-bit depth and resolution of 640 × 480 pixels, was setup in front of the disk specimen. A series of images was captured during the test at the framing rate of 1 frame/second. The random speckle image has a spatial resolution of about 27 µm/pixel. Before each test, a small amount of pre-load (∼ 2 – 3 N) was applied to keep the disk specimen in place. All experiments were conducted in ambient temperature (∼ 21◦ C). Figure 3(a) presents the variation of applied compressive load as a function of the change of diameter of the specimen for the three Brazilian disk compression tests. In this figure, the applied compressive force P was normalized by the product of the initial thickness W and the initial radius R of the specimen, and the diameter change ∆ was normalized by the initial specimen diameter D. The disk sample in test No.1 was loaded monotonically till it was fractured. In the other two tests, the disk specimens were loaded to a certain load level and then unloaded. One observes that the loading part of the three tests is almost exactly the same which indicates good repeatability. No.1
150.
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125.
B
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P
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u (mm)
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No.2
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2R (D)
Applied Load, P/(WR) (MPa)
175.
No.3
r o
θ
x
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Displacement, ∆/D (%)
(a)
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0.036
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−0.24
−0.014
−0.28
−0.022
−0.32
−0.029
−0.36
−0.036
−0.40
(b)
Figure 3: (a) Variation of applied compressive load P as a function of displacement ∆. (b) Contour plots of displacement fields u and v of test No.1 at the moment where P/(W R) = 36.1 MPa. DIC calculation was carried out on the series of random speckle images to determine the displacement field during the deformation of the disk specimens. The subcell size that we used in the correlation calculation is 15 pixels, which is about 0.4 mm. The ratio of this size to the disk diameter is 0.024, which is much smaller than the ratio suggested by Hondros [1] regarding the size of strain gages used for local strain measurement. Figure 3(b) shows the contour plots of the two displacement components u and v of the test No.1 at the moment where the applied compressive load P/(W R) = 36.1 MPa. Since this moment is in the early stage of the deformation process and the material is deforming within the elastic range, the displacement field indeed resembles those shown in Fig. 1(a).
4
Determination of the elastic constants
Using the DIC technique, the displacement field on the surface of the disk specimen is determined at each moment during the compression process. In particular, the horizontal displacement along the x-axis and the vertical displacement along the y-axis are known at any given time. According to the elastic solution listed in the Appendix, the expressions for the two non-zero displacement components along the two axes, u(x, 0) and v(0, y), are F κ n −1 2Rx x o F n −1 2Rx x 3R2 − x2 o tan − + tan + , u(x, 0) = − 4πµ x2 − R2 R R2 + x2 4πµ x2 − R2 R (1) Fκ n R + y y o F n R + y y o ln + − ln − . v(0, y) = − 4πµ R−y R 4πµ R−y R Note that the two elastic constants µ and κ only appear in the coefficients F/4πµ and F κ/4πµ, where µ is the shear modulus, κ = (3 − ν)/(1 + ν) for plane stress and κ = 3 − 4ν for plane strain, respectively, and ν is the Poisson’s ratio.
From the DIC measurement, the horizontal displacement along the x-axis and the vertical displacement along the y-axis are obtained. The expressions in Eq. (1) can be fitted to the displacement data via a linear regression, and the two coefficients F/4πµ and F κ/4πµ are obtained. Then by using the compressive load measured at the corresponding moment, the two elastic constants µ and κ are solved. This fitting process is illustrated in Fig. 4(a) for the moment shown in Fig. 3(b). In Fig. 4(a), the symbols are the displacement from DIC and the solid lines are the fitted curves from Eq. (1). The fitting parameters, µ and κ, are also shown in the figure. This scheme is repeated for all the moments during the loading process, and the variations of the shear modulus µ and the Poisson’s ratio ν, as function of the overall deformation ∆/D are shown in Figs. 4(b) and (c).
0.5
x/R or y/R −0.75
−0.5
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Fitting parameters:
−1.
μ = 1.344GPa κ = 1.7866
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135.0
μ = 1.349GPa
90.0
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Displacement, ∆/D (%)
(b)
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P 150.0
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ν (plane stress) 0.5 0.4
100.0 ν = 0.364
75.0
ν (plane strain)
0.3
50.0
0.2
25.0
0.1
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Poisson’s Ratio, ν
u/R
175.0
2.0
P
Shear Modulus, μ (GPa)
1.
Applied Load, P/(WR) (MPa)
180.0
1.5
Applied Load, P/(WR) (MPa)
u/R or v/R (%)
0.0 0.
2.5
5.
7.5
10.
12.5
15.
17.5
Displacement, ∆/D (%)
(c)
Figure 4: (a) Scheme for determining elastic constants from load and displacement measurement. (b) Variation of shear modulus µ as a function of overall deformation. (c) Variation of Poisson’s ratio ν as a function of overall deformation for both plane strain and plane stress. At the very beginning of the deformation, the measured shear modulus µ, according to the scheme outlined above, oscillates markedly. This is due to the fact that deformation on the disk surface is so small that uncertainty of the two coefficients, F/4πµ and F κ/4πµ obtained by fitting Eq. (1) to the displacement data, is expected to be quite large. However, as the overall deformation increases, i.e., ∆/D > 0.25%, the measured shear modulus µ converges. The shear modulus remains relatively constant until ∆/D ∼ 3%, after which it monotonically decreases. Note that according to our proposed scheme, the shear modulus µ and the constant κ are uniquely determined at each moment during the compression. However, the value of Poisson’s ratio ν will depend on the assumption regarding the planar deformation, either plane strain or plane stress. Figure 4(c) presents the variation of the Poisson’s ratio ν for both plane strain and plane stress as function of the overall deformation. Like the shear modulus µ, at the very beginning of the deformation, Poisson’s ratio ν will oscillate and then becomes constant. One observes that the range of the overall deformation, over which the Poisson’s ratio remains constant, is much larger than that of the shear modulus as shown in Fig. 4(b). Eventually, the Poisson’s ratio ν will increase when the overall deformation continues to proceed. At one point, the Poisson’s ratio corresponding to plane stress starts to exceed the value of 0.5, which is outside the valid range of Poisson’s ratio, while the Poisson’s ratio corresponding to plane strain remains within the range of −1 < ν < 1/2 during the entire process of compression. The elastic constants of the epoxy resin material have been measured using more conventional test configurations, i.e., uniaxial tension, uniaxial compression, and simple shear, and using strain gages for local strain measurement. The results, as well as the associated measuring technique, are listed in Table 1. In uniaxial tension, Young’s modulus E and Poisson’s ratio ν are directly measured, and the shear modulus µ is derived from the relation µ = E/2(1 + ν). Young’s modulus E is also directly measured from uniaxial compression. In a simple shear experiment, shear modulus µ is directly measured. The measurement listed in Table 1 demonstrates the consistency of these measurements. The shear modulus µ (simple shear measurement) and the Poisson’s ratio ν (uniaxial tension measurement) from Table 1 are also shown in Figs. 4(b) and (c) as straight dashed lines. One can see that for small deformation the measured shear modulus using the Brazilian disk scheme gives the same result as those obtained using more conventional techniques. As we have mentioned before, although the constant κ is uniquely determined at each moment using the Brazilian disk
scheme, the Poisson’s ratio ν, however, depends on the assumption of the planar deformation, either plane strain or plane stress. Nevertheless, the Poisson’s ratios calculated from plane strain and plane stress do seem to provide the upper and lower bounds of the actual value when the Poisson’s ratios obtained from the Brazilian disk scheme remain constant during the compression, as shown in Fig. 4(c). Table 1: Elastic constants of the epoxy resin material Tension Compression Simple shear
E (GPa) 3.767 ± 0.041 3.329 ± 0.384 –
ν 0.364 ± 0.018 – –
µ (GPa) 1.367 ± 0.159 – 1.349 ± 0.031
In Fig 4, the total applied load is used for the determination of elastic constants and we have assumed that the disk deformation remains elastic and the expression Eq. (1) can be used for any load. The modulus obtained in such a way may be called secant. Using the Brazilian disk and the scheme proposed above, we can determine the so-called tangential or incremental modulus as well. This is illustrated in Fig. 5. First, we divide the entire loading range into a set of equal-spacing subsections as shown in Fig. 5(a). For each load increment δP , the displacement increments δu and δv can be obtained from the DIC measurement. Meanwhile for each load increment δP , Eq. (1) still holds and can be used to determine the elastic constants for the corresponding load increment. The shear modulus µ obtained in such a way is presented in Fig. 5(b). Not surprisingly, for the load versus deformation curve with a convex shape, as shown in Fig. 3(a), the tangential or incremental modulus would always be equal to or less than the secant modulus.
Applied Load, P/(WR) (MPa)
δP
125. 100. 75. 50. 25.
2.0
P 135.0
μ = 1.349GPa μ (secant)
90.0
0.
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5.
7.5
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Displacement, Δ/D (%)
(a)
15.
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μ (incremental)
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5.
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10.
Shear Modulus, μ (GPa)
Applied Load, P/(WR) (MPa)
180.0
150.
0.0 12.5
15.
17.5
Displacement, Δ/D (%)
(b)
Figure 5: (a) Subdivide the entire loading range into equal-spacing subsection. (b) Variation of shear modulus µ as a function of overall deformation for both incremental and secant schemes.
5
Full-field deformation observation
One advantage of using the optical DIC technique for measuring deformation is be able to monitor the whole deformation field up to failure without a prior knowledge about the location and the manner of failure. In Fig. 6, both the displacement and the strain fields of the Brazilian disk test No.1, which is subjected to monotonic compression load, are shown. The moments shown in Fig. 6 correspond to those indicated by the open symbols in Fig. 3(a), and the strain is the so-called Green-Lagrangian strain, which is appropriate for characterizing large deformation but coincides with the engineering strain for infinitesimal deformation. At moment A, or the early stage of the deformation, the deformation field behaves as that shown in Fig. 1(a). At moment B, contact between the deforming sample and loading surface becomes apparent. However, the elastic solution that takes into account of contact, can still describe the deformation field quite well. Beyond moment B, the deformation is no longer infinitesimal and large deformation analysis is required. Moment E is the moment just before the sample failed in a brittle and dynamic fashion. At this moment, the Brazilian disk sample has been severely distorted. At the center of the sample, where the deformation is most intense, the strain
A
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E
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0.40 0.32 0.24 0.16 0.08 0.00 −0.08 −0.16 −0.24 −0.32 −0.40
u-field 0.00 −0.30 −0.60 −0.90 −1.20 −1.50 −1.80 −2.10 −2.40 −2.70 −3.00
v-field
A
B
A
B
18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 −2.0
x -field 2.0 0.0 −2.0 −4.0 −6.0 −8.0 −10.0 −12.0 −14.0 −16.0 −18.0
y -field
A
B
10.0 8.0 6.0 4.0 2.0 0.0 −2.0 −4.0 −6.0 −8.0 −10.0
xy -field
Figure 6: Displacement and strain fields of the Brazilian disk subjected to monotonic compression at the selected moments A–E indicated in Fig. 3(a).
state is approximately x = −y , xy = 0 and the magnitude of x and y is close to 20%. However, elastic solution, according to Eq. (1), would give x =
F (5 − κ) , 4πµR
y = −
F (3 + κ) , 4πµR
xy = 0,
which is obviously different from the observation shown in Fig. 6. In the tests, when the disk specimen was loaded to a certain load level and then gradually unloaded at the same speed as that of loading, a hysteresis loop was observed, see Fig. 3(a). The area within the hysteresis loop is directly proportional to energy dissipated during that loading/unloading cycle. For test No.2, we selected three moments shown in Fig. 3(a) as solid circles and designated them as a, b, and c, where moment a is on the loading portion of the curve, moment b is when the loading stopped and deformation started to reverse, and moment c is on the unloading portion. Moments a and c are chosen so that the overall deformation of the disk specimen is the same for those two moments while the difference of the applied compressive force is the largest for any given overall displacement during the loading/unloading cycle. Both the displacement and the strain fields of the disk specimen at those three selected moments are shown in Fig. 7. From moment a to moment b, the disk specimen was further deformed, as the contour plots of the two displacement components u and v indicated. Due to such a deformation increase, strain localization near the two contact regions became apparent. As the overall displacement returned from moment b to moment c, it becomes clear that part of the additional deformation from moment a to moment b is permanent, see the u-field contour plot. From the contour plots of the strain components, one observes that although the magnitude of strain was reduced from moment b to moment c due to unloading, the strain pattern remained the same as that of moment b, and this strain state pattern is different from that at moment a. It is this deformation pattern change that causes the apparent hysteresis behavior of the disk specimen when subjected to cyclic loading/unloading.
6
Summary
In this investigation, we revisited the Brazilian disk specimen for measuring elastic constants of a homogeneous material and for observing deformation evolution up to failure and subjected to complex loading histories. The major component of our proposed scheme is to use the optical technique DIC for measuring the full-field deformation. Based on the elastic solution of a circular disk, the shear modulus of the testing material can be uniquely determined, but the value of Poisson’s ratio depends on the assumption regarding the deformation of being either plane strain or plane stress. As an example, we conducted experiments on disk specimens made of a homogeneous epoxy resin. Our result of the shear modulus agreed well with the measurements using conventional testing specimens, i.e., uniaxial tension, uniaxial compression, and simple shear. With DIC, we also studied the deformation field of the disk samples subjected to large compressive load up to the point of failure and subjected to cyclic loading/unloading. The deformation state at the moment of failure was quantitatively determined and this information is valuable for establishing the failure criterion of the material. The quantitative description of the deformation fields associated with the hysteresis loop will also help us to understand and model the viscoelastic behavior of the material. Finally, the detailed deformation field measurement will play an important role for material model validations.
Acknowledgments This study was supported by the Joint DoD/DOE Munitions Technology Development Program and by the Enhanced Surveillance Campaign. CL also wants to thank Drs. Matthew W. Lewis and Philip Rae of Los Alamos National Laboratory for the valuable comments.
References [1] Hondros, G., “The evaluation of Poisson’s ratio and the modulus of materials of a low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete,” Australian Journal of Applied Science, 10, pp.243–268, 1959.
u (mm) 0.20 0.16 0.12 0.08 0.04 0.00 -0.04 -0.08 -0.12 -0.16 -0.20
v (mm) 0.00 -0.15 -0.30 -0.45 -0.60 -0.75 -0.90 -1.05 -1.20 -1.35 -1.50 -1.65 -1.80
ex (%)
14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0
-2.0
ey (%)
1.0 -0.5 -2.0 -3.5 -5.0 -6.5 -8.0 -9.5 -11.0 -12.5 -14.0
exy (%)
7.0 5.6 4.2 2.8 1.4 0.0 -1.4 -2.8 -4.2 -5.6 -7.0
(a)
(b)
(c)
Figure 7: Displacement and strain fields of the Brazilian disk subjected to loading then unloading at the selected moments a, b, and c indicated in Fig. 3(a).
[2] Mellor, M. and Hawkes, I., “Measurement of tensile strength by diametral compression of discs and annuli,” Engineering Geology, 5, pp.173–225, 1971. [3] Hudson, J. A., Brown, E. T., and Rummel, F., “The controlled failure of rock discs and rings loaded in diametral compression,” International Journal of Rock Mechanics and Mining Technology, 9, pp.241–248, 1972. [4] Fairhurst, C., “On the validity of the ‘Brazilian’ test for brittle materials,” International Journal of Rock Mechanics and Mining Science, 1, pp.535–546, 1964. [5] Andreev, G. E., “A review of the Brazilian test for rock tensile strength determination. Part I: calculation formula,” Mining Science and Technology, 13, pp.445–456, 1991. [6] Andreev, G. E., “A review of the Brazilian test for rock tensile strength determination. Part II: contact conditions,” Mining Science and Technology, 13, pp.457–465, 1991. [7] Wang, Q. Z., Jia, X. M., Kou, S. Q., Zhang, Z. X., and Lindqvist, P.-A., “The flattened Brazilian disc specimen used for testing elastic modulus, tensile strength and fracture toughness of brittle rocks: analytical and numerical results,” International Journal of Rock Mechanics and Mining Sciences, 41, pp.245–253, 2004. [8] Awaji, H. and Sato, S., “Combined mode fracture toughness measurement by the disk test,” Engineering Materials and Technology, 100, pp.175–182, 1978. [9] Chu, T. C., Ranson, W. F., Sutton, M. A., and Peters, W. H., “Applications of digital image correlation techniques to experimental mechanics,” Experimental Mechanics, 25, pp.232–244, 1985. [10] Bruck, H. A., McNeil, S. R., Sutton, M. A., and Peters, W. H., “Digital image correlation using Newton-Raphson method of partial differential correction,” Experimental Mechanics, 29, pp.261–267, 1989. [11] Vendroux, G. and Knauss, W. G., “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Experimental Mechanics, 38, pp.86–92, 1998.
Appendix: Elastic solution of a circular disk For planar deformation, either plane strain or plane stress, and in the Cartesian system (x, y) with origin located at the center of the disk, consider the case where a pair of concentrated compressive forces with the magnitude F , acting across the diameter of the circular disk along the y direction. Here F represents the compressive force per unit thickness of the two-dimensional disk. The stresses and the displacements can be expressed as o x2 (R + y) x2 (R − y) 2F n + + π (R2 + x2 − 2Ry + y 2 )2 (R2 + x2 + 2Ry + y 2 )2 o 2F n (R + y)3 (R − y)3 σy = − + + π (R2 + x2 − 2Ry + y 2 )2 (R2 + x2 + 2Ry + y 2 )2 o 2F n 4Rxy(R2 − y 2 − x2 )(R2 − y 2 + x2 ) τxy = , π (R2 + x2 − 2Ry + y 2 )2 (R2 + x2 + 2Ry + y 2 )2 σx = −
F , πR F , πR
(2)
and (1 − κ)x o 4Rx(R2 − y 2 + x2 ) F n 2Rx − (1 − κ) tan−1 2 + , 4πµ x + y 2 − R2 R (R2 + x2 − 2Ry + y 2 )(R2 + x2 + 2Ry + y 2 ) o 8Rx2 y F n 1 + κ (R + y)2 + x2 (1 − κ)y − 2 v=− ln + , 2 2 2 2 2 2 2 4πµ 2 (R − y) + x R (R + x − 2Ry + y )(R + x + 2Ry + y )
u=−
(3)
where R is the radius of the circular disk, constant κ = (3 − ν)/(1 + ν) for plane stress and κ = 3 − 4ν for plane strain, respectively, and µ is the shear modulus, ν the Poisson’s ratio.
