A new data reduction scheme for mode I wood fracture characterization using the double cantilever beam test

Characterisation of shear behaviour of bovine cortical bone1

by coupling the Arcan test with digital image correlation2

J. Xavier∗, B. Diaquino, J. Morais, F. Pereira3

CITAB, University of Tras-os-Montes e Alto Douro, Apartado 1013, 5001-801 Vila Real,4

Portugal.5

Abstract6

In this work, the characterisation of the shear behaviour of bovine cortical boneby the Arcan test was investigated. Both numerical and experimental analyses ofthe Arcan shear test were carried out. Specimens oriented in the longitudinal-tangential (LT ) plane were considered. Finite element analyses were performed inorder to assess the uniformity of the shear stress/shear strain states at the gaugesection with regard to geometry and boundary conditions. Experimentally, digitalimage correlation was coupled with the Arcan test for strain evaluation. A home-made Arcan fixture was built to transfer shear loading on small bone specimens.The access to full-field measurements provided a qualitative validation of predom-inant shear behaviour between V-notches of the Arcan specimen. Moreover, directevaluation of the shear modulus was obtained by integrating shear strain withinthe gauge section; thus, avoiding the need for numerical correction factors as itis the case in the classical data reduction scheme based on strain gauge measure-ments. The shear modulus of bovine cortical bone was found in good agreementwith references from literature. Besides, the shear stress at maximum load wasintended to give a suitable estimation of the shear strength.

Keywords: Arcan shear test, Cortical Bone, Digital image correlation, Finite7

element method8

1. Introduction9

Cortical (compact) bone is a composite material, consisting of a mineral rein-10

forcement embedded in an organic matrix, with a complex hierarchical, heteroge-11

neous and anisotropic microstructure (Rho et al., 1998). In order to quantify the12

behaviour of bone tissue when submitted to external mechanical loading, experi-13

mental studies can be carried out in an engineering approach. In the analysis and14

modelling of bone tissue at the macroscopic scale, it is convenient to consider three15

axis of material symmetry defined along longitudinal (harvesian system orianta-16

∗Corresponding author: E-mail: [email protected]; Tel.: +351 259 350 356Preprint submitted to JMBBM April 2, 2013

tion), radial and circumferential directions. Moreover, as a first approximation,17

continuity and homogeneity assumptions of the material can be assumed. The me-18

chanical and fracture properties of bone along its orthotropic directions are funda-19

mental properties that must be identified through suitable test methods. However,20

this characterisation posses several difficulties due to the inherent anisotropy and21

heterogeneity of the material. This is particularly the case for shear behaviour of22

bone. Several test methods have been proposed in the literature and applied over23

a spectrum of different anisotropic materials such as composites, wood and bone.24

Among them there is the Arcan shear test, which was first proposed for the shear25

characterisation of plastic materials (Goldenberg et al., 1958). In last decades,26

several achievements were carried out on the Arcan test, applied to the character-27

isation of both fracture and mechanical properties on composite materials (Hung28

and Liechi, 1997), wood (Xavier et al., 2009b), and bone (Turner et al., 2001).29

Experimental mechanics typically rely on surface measurements. Moreover,30

simplification assumptions are commonly introduced in the mechanical models31

yielding to closed-form solutions for material parameter identification, knowing32

specimen dimensions, loading conditions and some kinematic response (explicit33

solution for the inverse problem of material characterisation). Conventionally, a34

homogeneous state of stress/strain is assumed at the gauge section and, there-35

fore, punctual measurements are usually carried out using strain gauges or exten-36

someters. However, in the last decades, the progress on computer science, digital37

cameras and automatic image processing has allowed the development of novel38

optical methods (Rastogi (ed.), 2000). Both white-light (e.g., moire, grid and39

digital image correlation methods) and interferometric (e.g., speckle and moire in-40

terferometry, holography and shearography) techniques have been proposed among41

simplicity, cost and performance criteria. Contrasting with convencional devices,42

these techniques provide full-field measurements and are contact free. This type of43

information has progressively opened new perspectives in solid mechanics such as44

verification of experimental boundary conditions (parasitic effects) (Pierron et al.,45

1998), local damage characterisation (Kim et al., 2007), and multi-parameter iden-46

tification from single test configurations giving size to heterogeneous stress/strain47

fields (Avril et al., 2008; Xavier et al., 2007, 2009a).48

The characterisation of bone mechanical properties is of major concern because49

of its socio-economic impact. In this work the characterisation of the shear be-50

haviour of bovine cortical bone in the longitudinal-tangential (LT or 12) material51

axes was investigated by coupling the Arcan test method with full-field measure-52

ment provided by digital image correlation. Finite element analyses of the Arcan53

test were performed in order to verify the uniformity of the shear stress/strain re-54

sponse with regard to geometry and boundary conditions. Full-field strain fields at55

the centre of the specimen were determined from displacements measured by dig-56

2

ital image correlation (DIC). Therefore, the shear stress/shear strain response of57

the material until failure could be determined directly by integrating shear strain58

along the vertical V-notch line. Thus, a direct evaluation of the shear modulus of59

bovine cortical bone could be determined, in contrast with an apparent evaluation60

when punctual measurements are carried out (in this case, the shear modulus is61

overestimated or undestimated as a function of the anisotropic ratio in the plane62

of analysis). Moreover, the shear stress at maximum load can be provided as an63

estimation of the shear strength.64

The first part of the paper presents a new data reduction scheme of the Arcan65

test coupled with DIC. Finite element analyses are then presented addressing the66

pure and homogeneous shear stress/strain assumption at the gauge section be-67

tween V-notches of the Arcan specimen. The experimental work is then described.68

Results and discussion are then presented and finally main conclusions are drawn.69

2. Photo-mechanical data reduction approach70

2.1. Digital image correlation71

DIC provides full-field displacements of a target object by correlating images72

recorded before and after a given deformation. It is assumed herein that images73

are grabbed by a monovision camera-lens optical system (DIC-2D). To solve the74

correspondance problem in image processing, DIC requires that the surface of75

interest has a random, textured pattern uniquely characterising the material sur-76

face. The reference (undeformed) image is typically divided into subsets with size77

Ω ≡ (2M + 1)× (2N + 1) pixels, where M and N represent the number of pixels78

in the x and y directions, respectively. Subsets can slightly overlap by sharing79

some pixels. In this case, the subset step (fd) will be smaller than the subset size80

(fs). Adjacent (fs = fd) or spaced (fs < fd) subsets can also be selected depend-81

ing on the purpose. These are fundamental parameters since they will contribute82

to the definition of spatial resolution (∆u) and resolution (σu) associated to DIC83

measurements. Therefore, they must be carefully chosen with regard to the ap-84

plication, in a compromise between correlation (small subsets) and interpolation85

(large subsets) errors.86

Several mathematical correlation criteria have been proposed for estimation of87

the displacement fields in the subset matching process. It has been shown that the88

zero-normalized sum of squared differences (ZNSSD) is a robust algorithm since89

it take into account offset and linear scale variations of light intensity and is most90

efficient when using iterative procedure for the minimisation problem (Pan et al.,91

2009b)92

3

CZNSSD(p) =∑

Ω

f(xi, yi)− fm√∑Ω [f(xi, yi)− fm]2

−g(x′i, y

′j)− gm√∑

Ω

[g(x′i, y

′j)− gm

]22

(1)

where Ω is the subset domain, f(xi, yj) is the pixel grey level at (xi, yj) in the93

reference image, g(x′i, y′j) is the pixel grey level at (x′i, y

′j) in the deformed image,94

and fm and gm are the mean gray-level values over the subset in reference and95

deformed image, respectively, given by96

fm =1

(2M + 1)2

i=M∑i=−M

i=N∑i=−N

f(xi, yj) (2a)

gm =1

(2N + 1)2

i=M∑i=−M

i=N∑i=−N

g(x′i, y′j) (2b)

Eq. (1) has to be solved with regard to deformation parameter (p) which will char-97

acterise the mapping function. The first-order shape functions for the parameters98

p = u0, v0,u1,v1T write as99 x′i − xi = u0 + u1

Tdy′j − yj = v0 + v1

Td(3)

with u1 =

∂u∂x, ∂u∂y

T

, v1 =

∂v∂x, ∂v∂y

T

, d = xi − x0, yj − y0T . An iterative100

algorithm, such as Newton-Raphson or Levenberg-Marquardt, can then be used101

for finding the optimal set of deformation parameters for the correlation coefficient102

(Pan et al., 2009b; Bing et al., 2006).103

DIC provides displacements at a large set of discrete data points across a re-104

gion of interest. However, continuous strain fields are usually required in material105

parameter characterisation. Therefore, a suitable technique is needed to calcu-106

late the strain field from the measured displacement field, assuming the following107

relationships108

εij =1

2(uj,i + ui,j). (4)

It is worth noticing that the numerical differentiation of the measured displace-109

ment fields (Eqs. 4) is not straightforward since this procedure can amplify noise,110

inherently present in the measurements. For instance, direct differentiation us-111

ing finite differences can lead to a strain resolution in the range of 10−3 (e.g., for112

4

a displacement resolution of about 10−2 and a strain step of 5 subsets, a strain113

resolution of 2×10−3 is obtained using central finite differences), which is nor-114

mally to high for practical use in mechanical tests. Several strategies can be then115

used to solve Eqs. (4). Most of them consists in approximating the data points116

using smooth basis functions. The differentiation of the data is then based on117

the differentiation of the approximated basis functions in the least-square sense.118

Generically these methods can be sorted on global and local strategies. In this119

work, point-wise local least-squares fitting was used (Wattrisse et al., 2001; Pan120

et al., 2009a). This approach used a first order shape function to locally approxi-121

mate the displacement field measured by DIC. In this approach the regularisation122

parameters is the size of the strain window: (2m+ 1)× (2m+ 1). The value of m123

must be chosen in a compromise between level of low-pass filtering and accuracy124

of the representativeness of the strain field.125

2.2. Arcan shear test126

The Arcan shear test method is schematically shown in Fig. 1. The Arcan127

specimen has a rectangular configuration with two symmetrical V-notches at the128

centre. The geometry of the V-notches accounts both for weakling the specimen129

and enhancing uniform shear stress along the minimum cross-section. This spec-130

imen is mounted into an ad hoc fixture, which has two anti-symmetrical parts as131

shown in Fig. 1. This fixture allows transferring the vertical cross-head movement132

of a testing machine into a predominant shear stress between V-notches. The fix-133

ture can be adjusted to align the direction of applied load (P) with the specimen134

transverse axis (α = 0 in Fig. 1). Thus, an evaluation of the shear modulus can135

be determined from the shear stress-shear strain curve, according to the Hooke’s136

law137

Ga12 =

σ6

ε6

(5)

where σ6 is the nominal shear stress applied to the specimen and ε6 is the engi-138

neering shear strain. The shear stress can be estimated directly from the applied139

load (P ) and initial cross-section between V-notches (A) as140

σ6 =

∫ d/2

−d/2

σy dy 'P

A(6)

On the other hand, the shear strain can been measured at the specimen centre141

using two element rosette at ±45 according to the following transformation142

εgauge6 = ε+45 − ε−45 (7)

By using Eqs. (6) and (7) an apparent shear modulus is typically measured which143

5

can be corrected according to the following expression (Pierron et al., 1998; Xavier144

et al., 2004)145

G12 = CSGa12 (8a)

with146

C = σO6 /(P/A) and S = εgauge6 /εO6 (8b)

where σO6 and εO6 are the shear stress and strain at the central point O of the147

specimen (Fig. 1), respectively.148

This numerical correction factor (Eqs. 8) can be however discharged if the shear149

strain is evaluation based on full-field measurements. In this case, the shear strain150

along the V-notches of the Arcan specimen can be directly evaluated as151

ε6 =

∫ d/2

−d/2

εy(x, y) dy. (9)

It is worth noticing that a special strain gauge has been developed for the same152

purpose for the standard Iosipescu shear test, in which an average shear strain153

along the V-notch of the specimen (Eq. 9) was measured experimentally [-]. Finally,154

the shear strength can be evaluated from the maximum load at specimen failure155

X6 w σult6 =

Pmax

A(10)

3. Finite element analyses156

3.1. Finite element model157

[...]158

3.2. Numerical results159

[...]160

4. Experimental work161

4.1. Material and specimens162

The material used in this work was cortical bone taken from the femur mid-163

diaphysis of juvenile bovines with about eight months of age (male or female164

genus of the bovines was not known) (Fig. 2). In the machining process, the165

endosteal and periosteal tissues were removed. Arcan specimens of cortical bone166

where then manufactured, oriented along the longitudinal-transverse plane, with167

nominal dimensions of l = 60, w = 6, t = 2, d = 2, r = 2 mm and θ = 90168

(see Fig. 1). The bone tissue was carefully kept moist using physiological saline169

6

in order to prevent degradation (dehydration) from in vivo conditions. Before170

testing the specimens were kept freezing. Tests were carried out under hydrate171

and quasi-static (neglecting inertia effects) conditions.172

4.2. Photo-mechanical set-up173

4.2.1. Arcan shear test174

The photomechanical set-up of the Arcan test coupled with DIC is shown in175

Fig. 3(a). The Arcan shear test was carried out on an Instron 5848 MicroTester176

machine. A specific Arcan fixture was built to transfer shear loading to the Arcan177

bone specimen (Fig. 3(b)). The specimen was fixed and centred on the fixture178

by means of lateral grips. A torque wrench was used to tighten the specimen179

to a torque of 3.5 Nm. The free, central length between left and right parts of180

the Arcan fixture was set to 15 mm (visible length of the specimen). Adherent181

150-grit sandpaper was used between the specimen and grips in order to prevent182

slipping or premature failure. The applied load was measured using a load cell of183

2 kN (setting the gain to 200 N). The tests were performed under displacement184

control, with a cross-head displacement rate of 0.5 mm/min. In image acquisition,185

the vertical movement of the specimen observed during an exposure time of 0.4186

ms (Table 1) was then negligible with regard to the pixel size on the object plane187

(4.4 µm/pixel). The specimens were systematically pre-loaded before testing to a188

load of about 30 N. At this stage, the focus of the image was rechecked in order189

to prevent any parasitic out-of-plane displacement, with regard to the camera,190

due to the accommodation of the specimen. Although with this procedure the191

initial applied strain (and therefore the shear stress-shear strain curve) will be192

missing, image quality (constant magnification of the optical system) can be safely193

guarantee during image acquisition through the test (telecentric lens).194

4.2.2. DIC measurements195

The bone specimens preserved on physiological saline were unfrozen at room196

temperature before testing. The textured pattern required in DIC was created197

across the region of interest by means of an airbrush. At the scale of observation,198

a suitable random and isotropic pattern was created by spreading dark spots over199

the natural bright texture of the cortical bone. The final speckle pattern is shown in200

Fig. 4 together with the image histogram. The Aramis DIC-2D system was used in201

this work (Xavier et al., 2012; ARAMIS, 2009). An 8-bit Baumer Optronic FWX20202

camera coupled with a TC 23 09 telecentric lens were used for image grabbing at203

an acquisition frequency of 1 Hz (Table 1). The working distance was set to 63.3204

mm to have a focused image, with a magnification factor of 228.7 pixels/µm.205

A rigid-body translation test was performed before testing to guide in the se-206

lection of DIC measuring parameters (Table 1). The advantage of this simple207

test is the fact that both displacement and strain fields are theoretically known.208

7

Firstly, a mean value of displacement was calculated over the entire displacement209

field and subtracted to each displacement data point. In this way, the noise in210

the displacement measurements can be sorted out. The strain field in this case211

study is theatrically zero (no applied load), therefore a noise signal will be actually212

obtained. The resolution associated to the measurements can then be defined on213

a statistical basis as the standard deviation of a normal distribution function (the214

amont of data points is assumed suitable for a representative sampling). Fig. 5(a)215

shows both resolution (σu) and spatial resolution (∆u)(defined as the smallest dis-216

tance separating two independent measurements) on the displacement as a function217

of the subset size. As already pointed out in previous studies (Xavier et al., 2012;218

Lecompte et al., 2006), the displacement resolution is improved by increasing sub-219

sets size. A power law was fitted to this set of points yielding a displacement220

resolution function of σu = 1.405 × ∆u−1.283 pixel (R2 = 0.9948). On the other221

hand, however, a decrease on spatial resolution is obtained by increasing subsets222

size, so a compromise must be found between correlation and interpolation errors.223

Fig. 5(b) shows both resolution (σε) and spatial resolution (∆ε) on strains as a224

function of gauge length for several subset sizes. By increasing ∆ε accuracy is also225

increased, but coupled with a degradation on spatial resolution. In this work, a226

subset size and a subset step of 15×15 pixel and 11×11 pixel were chosen, respec-227

tively (Table 1). A slightly overlapping of 4 pixels between adjacent subsets was228

chosen in order to enhance spatial resolution since the distance between V-notches229

is only 2 mm (see Fig. 1). Besides, a strain length of 7 subsets was chosen for230

strain computation. Fig. 6(a) shown the typical noisy displacement field obtained,231

together with the histogram of the displacement values. As it can be confirmed, a232

normal (Gaussian) distribution is obtained roughly centered at zero (i.e., no sig-233

nificant bias is introduced by the optical system). The standard deviation of this234

distribution function is around 4×10−2 pixel, defining the displacement resolution235

of the measurements with a spatial resolution of 48.5 µm. A similar analysis was236

performed directly on the strain fields. Typical noisy strain fields and its distribu-237

tion is shown in Fig. 6(b). The strain resolution was estimated about 0.03% with238

a spatial resolution of about 338.8 µm.239

5. Results and discussion240

The access to full-field measurement allows the experimental verification of241

the assumption of predominant shear behaviour at the central part of the Arcan242

specimen. Fig. 7 shows the in-plane components of the strain field at the V-notch243

region. As it can be seen, shear is the predominant response of the specimen.244

Eventually, a strain (stress) concentration can be identified at the root of the V-245

notches, suggesting failure of the specimen at these locations. The shear strain246

was then evaluated over a gauge area between V-notches of 0.19×1.44 mm (Eq. 9,247

8

see Fig. 7).248

The shear stress-shear strain curves were then evaluated as shown in Fig. 8.249

Some dispersion can be observed typical of a biological material as bone. Fig. 9250

shows the typical failure of the specimen at the centre between V-notches. From251

the constitutive curves both shear modulus (Eq. 5) and shear strength (Eq. 10)252

were evaluated. The shear modulus was obtained from regression analysis in the253

initial part of the shear stress-strain curve, whilst the shear strength was estimated254

from the ultimate loading at failure. This study is summarised in Table 2. Ac-255

cordingly, a shear modulus of 5.08 ± 0.43 GPa and a shear strength of 47.91 ±256

5.51 MPa were determined. These results are in agreement with reference value re-257

ported in the literature (Table 3) confirming the validity of the proposed approach258

for measuring the shear behaviour of bone tissue.259

6. Conclusions260

In this work, the Arcan test method was proposed for determining the shear261

behaviour of bovine cortical bone. This test was continently integrated with digital262

image correlation. Qualitative verification of predominant shear behaviour at the263

central region of the specimen was verified experimentally. Moreover, a procedure264

based on full-field measurement was proposed for the correct evaluation of the shear265

modulus, avoiding the need of numerical correction factors taking into account the266

degree of stress and strain heterogeneity at the gauge section, as it is typically267

the case when using a classical data reduction scheme. From the shear stress-268

shear strain curves, shear modulus and shear strength of bovine cortical bone were269

evaluated. These results are in good agreement with reference values reported in270

the literature for this type of material.271

7. Acknowledgements272

This work is supported by European Union Funds (FEDER/COMPETE - Op-273

erational Competitiveness Programme) and by national funds (FCT - Portuguese274

Foundation for Science and Technology) under the project FCOMP-01-0124-287275

FEDER-022692. The authors would like to thank FCT for supporting this work276

through the research project PTDC/EME-PME/119093/2010 and Ciencia 2008277

program.278

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11

List of Figures342

1 Schematic representations of the Arcan specimen and fixture (cor-343

tical bone with: l = 60, w = 6, t = 2, d = 2, r = 2 mm and344

θ = 90). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13345

2 Cortical bone cut from juvenile bovine femur. . . . . . . . . . . . . 14346

3 (a) photomechanical set-up of the Arcan test coupled with digital347

image correlation; (b) home-made Arcan fixture. . . . . . . . . . . . 15348

4 Speckle pattern typically used in digital image correlation. . . . . . 16349

5 (a) Displacement resolution (σu) as a function of subset size ∆u;350

(b) Strain resolution (σε) as a function of gauge length (∆ε) (5351

and 4 stand for the resolution and spatial resolution, respectively,352

corresponding to a subset size of 15×15 pixels, a subset step of353

11×11 pixels and a strain gauge length of 7 subsets). . . . . . . . . 17354

6 (a) displacement and (b) strain noisy fields and histograms obtained355

from rigid-body translation tests for resolution evaluation. . . . . . 18356

7 Strain fields over the V-notch central region of the Arcan specimen. 19357

8 Shear stress-shear strain curves of cortical bone measured from the358

Arcan test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20359

9 Typical failure of the Arcan specimens. . . . . . . . . . . . . . . . . 21360

12

Figure 1: Schematic representations of the Arcan specimen and fixture (cortical bone with:l = 60, w = 6, t = 2, d = 2, r = 2 mm and θ = 90).

13

Figure 2: Cortical bone cut from juvenile bovine femur.

14

Figure 3: (a) photomechanical set-up of the Arcan test coupled with digital image correlation;(b) home-made Arcan fixture.

15

Figure 4: Speckle pattern typically used in digital image correlation.

16

Figure 5: (a) Displacement resolution (σu) as a function of subset size ∆u; (b) Strain resolution(σε) as a function of gauge length (∆ε) (5 and 4 stand for the resolution and spatial resolution,respectively, corresponding to a subset size of 15×15 pixels, a subset step of 11×11 pixels and astrain gauge length of 7 subsets).

17

Figure 6: (a) displacement and (b) strain noisy fields and histograms obtained from rigid-bodytranslation tests for resolution evaluation.

18

Figure 7: Strain fields over the V-notch central region of the Arcan specimen.

19

Figure 8: Shear stress-shear strain curves of cortical bone measured from the Arcan test.

20

Figure 9: Typical failure of the Arcan specimens.

21

List of Tables361

1 Components of the optical system and measuring parameters. . . . 23362

2 Shear modulus and shear strength of bovine cortical bone deter-363

mined from the Arcan shear test. . . . . . . . . . . . . . . . . . . . 24364

3 Reference values of shear modulus and shear strength for bone tissue. 25365

22

Table 1: Components of the optical system and measuring parameters.

Camera-lens optical systemCCD camera Baumer Optronic FWX20

8 bit, 1624×1236 pixels2

lens TC 23 09 Telecentric lensField of view: 7.1 × 5.4 mm2

Working distance: 63.3 ± 2 mmWorking F-number: 11

Image recordingAcquisition frequency 1 HzExposure time 0.4 ms

DisplacementSubset size (∆u) 15×15 pixels2 (66×66 µm2)Subset step 11×11 pixels2 (48.4×48.4 µm2)Resolution (σu) 4×10−2 pixels

StrainStrain length (∆ε) 7×7 subsets2 (0.339×0.339 mm2)Resolution (σε) 0.03%

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Table 2: Shear modulus and shear strength of bovine cortical bone determined from the Arcanshear test.

SpecimensShear modulus Shear strength(GPa) (MPa)

1 4.92 43.702 4.97 32.903 4.45 47.924 4.69 37.795 4.26 47.176 5.07 46.717 5.73 52.128 4.93 64.419 6.50 53.6710 5.58 41.7811 4.21 48.0012 5.68 58.81Mean 5.08 47.91I.C.(a) 0.43 5.51C.V.(b) (%) 13.36 18.10

(a) Confidence intervals at 95% confidence level;(b) Coefficient of variation (%).

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Table 3: Reference values of shear modulus and shear strength for bone tissue.

Shear modulus (GPa)Reilly and Burstein (1975) 3.6-5.1Knets (1978) 4.9Dong and Guo (2004) 4.74

Shear strength (MPa)Wang et al (2002) 53-57Bando (1961) 43.1Truner et al. (2001) 51.6

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