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

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Characterisation of shear behaviour of bovine cortical bone
1
by coupling the Arcan test with digital image correlation
2
J. Xavier
∗
, B. Diaquino, J. Morais, F. Pereira
3
CITAB, University of Tr´ as-os-Montes e Alto Douro, Apartado 1013, 5001-801 Vila Real,
4
Portugal.
5
Abstract
6
In this work, the characterisation of the shear behaviour of bovine cortical boneby the Arcan test was investigated. Both numerical and experimental analyses of the 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, Finite
7
element method
8
1. Introduction
9
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 the
12
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 and
14
modelling of bone tissue at the macroscopic scale, it is convenient to consider three
15
axis of material symmetry deﬁned along longitudinal (harvesian system orianta-
16
∗
Corresponding author: E-mail: jmcx@utad.pt; Tel.: +351 259 350 356
Preprint 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 and
21
heterogeneity of the material. This is particularly the case for shear behaviour of
22
bone. Several test methods have been proposed in the literature and applied over
23
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 shear
25
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 (Hung
28
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 models
31
yielding to closed-form solutions for material parameter identiﬁcation, knowing
32
specimen dimensions, loading conditions and some kinematic response (explicit
33
solution for the inverse problem of material characterisation). Conventionally, a
34
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, digital
37
cameras and automatic image processing has allowed the development of novel
38
optical methods (Rastogi (ed.), 2000). Both white-light (
e.g.
, moir´e, grid and
39
digital image correlation methods) and interferometric (
e.g.
, speckle and moir´e in-
40
terferometry, holography and shearography) techniques have been proposed among
41
simplicity, cost and performance criteria. Contrasting with convencional devices,
42
these techniques provide full-ﬁeld measurements and are contact free. This type of
43
information has progressively opened new perspectives in solid mechanics such as
44
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/strain
47
ﬁelds (Avril et al., 2008; Xavier et al., 2007, 2009a).
48
The characterisation of bone mechanical properties is of major concern because
49
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) material
51
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 Arcan
53
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 at
55
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 of
57
the material until failure could be determined directly by integrating shear strain
58
along the vertical V-notch line. Thus, a direct evaluation of the shear modulus of
59
bovine cortical bone could be determined, in contrast with an apparent evaluation
60
when punctual measurements are carried out (in this case, the shear modulus is
61
overestimated or undestimated as a function of the anisotropic ratio in the plane
62
of analysis). Moreover, the shear stress at maximum load can be provided as an
63
estimation of the shear strength.
64
The ﬁrst part of the paper presents a new data reduction scheme of the Arcan
65
test coupled with DIC. Finite element analyses are then presented addressing the
66
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 approach
70
2.1. Digital image correlation
71
DIC provides full-ﬁeld displacements of a target object by correlating images
72
recorded before and after a given deformation. It is assumed herein that images
73
are grabbed by a monovision camera-lens optical system (DIC-2D). To solve the
74
correspondance problem in image processing, DIC requires that the surface of
75
interest has a random, textured pattern uniquely characterising the material sur-
76
face. The reference (undeformed) image is typically divided into subsets with size
77
Ω
≡
(2
M
+ 1)
×
(2
N
+ 1) pixels, where
M
and
N
represent the number of pixels
78
in the
x
and
y
directions, respectively. Subsets can slightly overlap by sharing
79
some pixels. In this case, the subset step (
f
d
) will be smaller than the subset size
80
(
f
s
). Adjacent (
f
s
=
f
d
) or spaced (
f
s
< f
d
) subsets can also be selected depend-
81
ing on the purpose. These are fundamental parameters since they will contribute
82
to the deﬁnition of spatial resolution (∆
u
) and resolution (
σ
u
) associated to DIC
83
measurements. Therefore, they must be carefully chosen with regard to the ap-
84
plication, in a compromise between correlation (small subsets) and interpolation
85
(large subsets) errors.
86
Several mathematical correlation criteria have been proposed for estimation of
87
the displacement ﬁelds in the subset matching process. It has been shown that the
88
zero-normalized sum of squared diﬀerences (ZNSSD) is a robust algorithm since
89
it take into account oﬀset and linear scale variations of light intensity and is most
90
eﬃcient when using iterative procedure for the minimisation problem (Pan et al.,
91
2009b)
92
3
C
ZNSSD
(
p
) =
Ω
f
(
x
i
,y
i
)
−
f
m
Ω
[
f
(
x
i
,y
i
)
−
f
m
]
2
−
g
(
x
i
,y
j
)
−
g
m
Ω
g
(
x
i
,y
j
)
−
g
m
2
2
(1)where Ω is the subset domain,
f
(
x
i
,y
j
) is the pixel grey level at (
x
i
,y
j
) in the
93
reference image,
g
(
x
i
,y
j
) is the pixel grey level at (
x
i
,y
j
) in the deformed image,
94
and
f
m
and
g
m
are the mean gray-level values over the subset in reference and
95
deformed image, respectively, given by
96
f
m
= 1(2
M
+ 1)
2
i
=
M
i
=
−
M i
=
N
i
=
−
N
f
(
x
i
,y
j
) (2a)
g
m
= 1(2
N
+ 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 parameters
98
p
=
{
u
0
,v
0
,
u
1
,
v
1
}
T
write as
99
x
i
−
x
i
=
u
0
+
u
1
T
d
y
j
−
y
j
=
v
0
+
v
1
T
d
(3)with
u
1
=
∂u∂x
,
∂u∂y
T
,
v
1
=
∂v∂x
,
∂v∂y
T
,
d
=
{
x
i
−
x
0
,y
j
−
y
0
}
T
. An iterative
100
algorithm, such as Newton-Raphson or Levenberg-Marquardt, can then be used
101
for ﬁnding the optimal set of deformation parameters for the correlation coeﬃcient
102
(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 material
105
parameter characterisation. Therefore, a suitable technique is needed to calcu-
106
late the strain ﬁeld from the measured displacement ﬁeld, assuming the following
107
relationships
108
ε
ij
= 12(
u
j,i
+
u
i,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.
, for
112
4
a displacement resolution of about 10
−
2
and a strain step of 5 subsets, a strain
113
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 then
115
used to solve Eqs. (4). Most of them consists in approximating the data points
116
using smooth basis functions. The diﬀerentiation of the data is then based on
117
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 this
119
work, point-wise local least-squares ﬁtting was used (Wattrisse et al., 2001; Pan
120
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 regularisation
122
parameters is the size of the strain window: (2
m
+1)
×
(2
m
+1). The value of
m
123
must be chosen in a compromise between level of low-pass ﬁltering and accuracy
124
of the representativeness of the strain ﬁeld.
125
2.2. Arcan shear test
126
The Arcan shear test method is schematically shown in Fig. 1. The Arcan
127
specimen has a rectangular conﬁguration with two symmetrical V-notches at the
128
centre. The geometry of the V-notches accounts both for weakling the specimen
129
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 as
131
shown in Fig. 1. This ﬁxture allows transferring the vertical cross-head movement
132
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 specimen
134
transverse axis (
α
= 0
◦
in Fig. 1). Thus, an evaluation of the shear modulus can
135
be determined from the shear stress-shear strain curve, according to the Hooke’s
136
law
137
G
a
12
=
σ
6
ε
6
(5)where
σ
6
is the nominal shear stress applied to the specimen and
ε
6
is the engi-
138
neering shear strain. The shear stress can be estimated directly from the applied
139
load (
P
) and initial cross-section between V-notches (
A
) as
140
σ
6
=
d/
2
−
d/
2
σ
y
dy
P A
(6)On the other hand, the shear strain can been measured at the specimen centre
141
using two element rosette at
±
45
◦
according to the following transformation
142
ε
gauge
6
=
ε
+45
◦
−
ε
−
45
◦
(7)By using Eqs. (6) and (7) an apparent shear modulus is typically measured which
143
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