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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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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 5
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