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Fracture Mechanics of Bone—The Effects of Density, Specimen Thickness and Crack Velocity on Longitudinal Fracture
Abstract
The fracture mechanics parameters of critical stress intensity factor (Kc) and critical strain energy release rate (Gc) for longitudinal fracture of bovine tibia cortical bone were determined by the compact tension method. It was demonstrated that, for a given bone density, Kc and Gc depended on the loading rate, and resultant crack velocity, with a maximum in fracture toughness (Kc approximately 6.3 MNm-3/2, Gc approximately 2900 Jm-2) at a crack velocity approximately 10(-3) ms-1. For a given loading rate, or crack velocity, an increase in bone density, in the range from 1.92 to 2.02 Mgm-3, produced increases in Kc and Gc, but a variation in specimen thickness (from 0.5 to 2 mm) had no effect on the measured fracture mechanics parameters.
... The study of fracture properties, particularly through tests on the initiation and propagation of cracks in human bone tissue, represents a vast area of study within the field of biomechanics. These tests serve, on one hand, to illustrate and understand the physical mechanisms [7,8,9,10,11], and on the other hand, to determine mechanical properties such as strength and toughness [7,12,13,14,15,16]. These data are essential for the development of cracking models. ...
... The study of fracture properties, particularly through tests on the initiation and propagation of cracks in human bone tissue, represents a vast area of study within the field of biomechanics. These tests serve, on one hand, to illustrate and understand the physical mechanisms [7,8,9,10,11], and on the other hand, to determine mechanical properties such as strength and toughness [7,12,13,14,15,16]. These data are essential for the development of cracking models. ...
... Subsequently, around 80 to 85% of the maximum force, these nonlinearities intensify more markedly, leading to the peak. These nonlinearities are indicative of physical processes corresponding to the formation of the Fracture Process Zone (FPZ) at the notch tip [5,7,12,18,19,20], where stresses are concentrated. The literature has extensively detailed their potential origins, all related to phenomena affecting each scale of the material's microstructure [3]. ...
A mechanical test of macrocrack propagation in a cortical bone shaft is developed to comprehend the failure mechanisms at this scale and identify fracture processes, which are paramount to a future mod-elisation. A first set of three-point bending tests was performed on notched segments of human femoral dyaphisis [2, 3]. However, macrocrack orientation is influenced by the material's anisotropy. In order to drive cracking processes in the transversal direction only, a modification of the testing protocol is submitted. From work found in the litterature [4, 5], notched shafts are previously grooved : a hemispherical groove is milled on the external surface of the bone to guide the crack in the transversal direction in the notch section. Special care is taken for the three-point bending test allowing a mastered procedure despite geometric variability of specimens : positioning and boundary conditions of the samples, notch displacement sensor's supports alignment regarding bone axis, positioning and height of the notch, constant groove's depth. The hydraulic jack is driven indirectly with the notch opening, measured with an in-ductive displacement sensor. A series of unloading/reloading cycles are done to plot residual quantities, pre-peak damage and macrocrack propagation in post-peak. Obtained results are expressed in terms of loading as a function of the notch opening displacement. Load-unloading cycles and residual openings analysis highlight the prominent role of cracking in failure mechanisms. Results show that the residual opening is directly linked to a structural weakening, consequently to the cracking processes, and that no plasticity is found at this scale. Cycles analysis allows us to develop a hypothesis about the role of internal residual stresses on permanent deformation. Furthermore, results point out that the groove guides the macrocrack in the transversal direction. Comparison between results on non-grooved samples exhibits the impact of the geometric feature. Future perspectives rely on the reverse analysis determination of the mechanical properties in the transversal direction.
English translation, January 2024 "Propagation de fissure sur tronçon d'os long rainuré" [1]
... In case of = 0 , experimental and numerical data show an almost collinear extension of the pre-existing main crack of the CT specimen. The theoretical peak load P can be then calculated by [63,141] where h is the thickness, ȃ is a geometrical parameter depicted in Fig. 28, whereas Y = 6.554 is a non-dimensional constant depending on the geometry of the specimen [141]. The peak load P = 98N is determined from Eq. 110 for K cř = 4.0MPa ⋅ m 1∕2 and thickness h = 1 mm. ...
... In case of = 0 , experimental and numerical data show an almost collinear extension of the pre-existing main crack of the CT specimen. The theoretical peak load P can be then calculated by [63,141] where h is the thickness, ȃ is a geometrical parameter depicted in Fig. 28, whereas Y = 6.554 is a non-dimensional constant depending on the geometry of the specimen [141]. The peak load P = 98N is determined from Eq. 110 for K cř = 4.0MPa ⋅ m 1∕2 and thickness h = 1 mm. ...
... The peak load P = 98N is determined from Eq. 110 for K cř = 4.0MPa ⋅ m 1∕2 and thickness h = 1 mm. This theoretical prediction, calculated analytically on the basis of (110) P = K cř hȃ 1∕2 Y explicitly given experimental data [141], is in very good agreement with the peak load value P CM = 96 N obtained by the CM model (Fig. 32). ...
This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive
the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors.
The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics.
... In general, these studies can be classified into three families. First, those focusing on illustrating and understanding physical mechanisms generally do so at microscopic or nanoscopic scales [1,2,3,4,5,6,7]. Second, experiments concentrating on the determination of mechanical properties such as strengths or toughnesses [3,8,9,10,11,12,13] are commonly performed on small specimens with controlled geometries machined from bones. ...
... First, those focusing on illustrating and understanding physical mechanisms generally do so at microscopic or nanoscopic scales [1,2,3,4,5,6,7]. Second, experiments concentrating on the determination of mechanical properties such as strengths or toughnesses [3,8,9,10,11,12,13] are commonly performed on small specimens with controlled geometries machined from bones. ...
... It reveals physical processes that develop at the notch tip, where stresses are localized, and are assumed to correspond to the Fracture Process Zone formation phase (FPZ). For studies related to the behavior of notched structures, the literature [3,9,13,18,19,17,20,21] has abundantly attempted to describe the physical mechanisms potentially at the origin of developing an FPZ at the notch tip (which concerns the pre-peak part of the behavior essentially). At a nano-or sub-nanoscale (according to scales defined in [22]), they are often assumed to be related to plastic deformation of the collagen fibrils, slippage/friction between the hydroxyapatite crystals and the fibrils or possible ruptures of these fibres and/or hydroxyapatite crystals [23]. ...
Experimenting with crack propagation in human cortical tissue is a necessary prerequisite for developing a cracking model. A three-point bending test on a shaft section of a notched human long bone is presented. A procedure for carrying out the experimental test, including unloading/reloading cycles, is implemented. The results obtained are analyzed regarding the physical mechanisms which occur in the different phases of the test, and during the cycles. The prominent role of cracking is highlighted. In addition a hypoth-esis is proposed concerning the potential effect of initial internal residual stresses, due to bone remodelling, on the significant residual notch openings after unloading and on the cycles’ shape.
... A wide range of specimen thickness (1-10 mm) might be critical to get plane strain conditions in the fracture analysis of human cortical bone, depending on location, age, and specimen orientation. 5,25,26 One study stated that fracture toughness of longitudinally fractured specimens is independent of thickness for 1.8-3.8 mm thick specimens under mode-I loading conditions. ...
... 26 Another study showed a similar conclusion; however, the thickness was ranged between 0.5 and 2 mm. 25 Conversely, one study reported that fracture toughness of cortical bone varies with the thickness from 2 to 6 mm and became constant after 6 mm thickness under mode-I loading conditions. 5 Therefore, it is difficult to compare the fracture toughness data of bone from different studies due to consideration of different specimen thicknesses. ...
... 75 Some researchers reported that mineral density and crack velocity affected the K C of bovine cortical bone on longitudinal fracture and found that the specimen thickness did not affect the measured fracture parameters. 25,26 However, tensile fracture stress in a longitudinally fractured specimen of the bovine tibia is a function of the curvature radius and the length of the machined edge. 77 Robertson et al. 78 studied the K C for v-notched cracked specimens of the adult bovine femur, using a three-point bending test at a strain rate from 0.03 mm/s, and found that the correlation between K IC and strain rate. ...
This paper comprehensively reviews the various experimental and numerical techniques, which were considered to determine the fracture characteristics of the cortical bone. This study also provides some recommendations along with the critical review, which would be beneficial for future research of fracture analysis of cortical bone. Cortical bone fractures due to sports activities, climbing, running, and engagement in transport or industrial accidents. Individuals having different diseases are also at high risk of cortical bone fracture. It has been observed that osteon orientation influences cortical bone fracture toughness and fracture mechanisms. Apart from this, recent studies indicate that fracture parameters of cortical bone also depend on many factors such as age, sex, temperature, osteoporosis, orientation, location, loading condition, strain rate, and storage facility, etc. The cortical bone regains its fracture toughness due to various toughening mechanisms. Owing to these factors, several experimental, clinical, and numerical investigations have been carried out to determine the fracture parameters of the cortical bone. Cortical bone is the dense outer surface of the bone and contributes to 80%–82% of the skeleton mass. Cortical bone experiences load far exceeding body weight due to muscle contraction and the dynamics of motion. It is very important to know the fracture pattern, direction of fracture, location of the fracture, and toughening mechanism of cortical bone. A basic understanding of the different factors that affect the fracture parameters and fracture mechanisms of the cortical bone is necessary to prevent the failure and fracture of cortical bone. This review has summarized the advancement considered in the various experimental techniques and numerical methods to get complete information about the fracture mechanisms of cortical bone.
... For elastic analysis, the validation of the proposed model is achieved by comparing the predicted displacement maps of a 3D transversely isotropic block subjected to uniform extension with those from FEM analyses. Subsequently, the fracture behavior of a cortical bone compact tension (CT) specimen is simulated by the proposed model and compared with numerical and experimental results from the works of literature [55,57,76,77] to validate the conceived formulation for crack problems. ...
... When the thickness is taken to be 1 mm, the predicted maximum damage propagation load is approximately 105.05 N, which is extremely close to the experimental value in [76] and the numerical value in [55]. This shows that the numerical results from the proposed model are reliable. ...
... Additionally, the predicted maximum damage propagation loads for various thicknesses lead to the back-calculation of the critical stress intensity factor K c (K Ic2 in the longitudinal fracture, see Fig. 23) in terms of the following equation [ These estimated critical stress intensity factors for various thicknesses are also presented in Fig. 25. The mean value of these estimated critical stress intensity factors is 4.19 MPa·m 1/2 , which agrees well with the value directly measured in [76]. In addition, Fig. 25 shows that there is no significant dependence between the critical stress intensity factors and thicknesses of the specimens. ...
In this work, linearized elastic isotropic continuum-kinematics-inspired-peridynamics (CPD) is further generalized to study the elastic, damage, and fracture behavior of anisotropic continua, focusing on 3D transversely isotropic and 2D orthotropic body. The anisotropy is generated by changing the material properties, including the micromodulus, critical stretch, and critical micropotential energy of basic finite kinematic elements with their direction in the principal material axes in terms of eight-order double Fourier expansion. It is critically proven that the volumetric micromodulus of finite volume elements in the 3D case is independent of their direction, which is similarly applied to the areal micromodulus in the 2D case. For this reason, the 2D orthotropic CPD formulation characterized by three independent elastic moduli and the 3D transversely isotropic CPD formulation characterized by four independent elastic moduli are distinguished. The accuracy of the computational models applied to elastic analysis is assessed by comparing the predicted displacement fields with the results from finite element analysis. Quantitative simulation of an orthotropic lamina with a central rectangular hole under a tensile load, eccentric three-point bending test for orthotropic lamina, and compact tension test for cortical bone are performed to verify the ability of the proposed model to describe the damage and fracture behavior of anisotropic materials.
... However, under dynamic fractures, bone could exhibit a remarkably different behavior, showing a general transition from ductility to brittleness when compared with that under quasi-static rates [5,[20][21][22][23][24]. Studies at macroscopic length scales [20,21,[25][26][27] revealed that, the fracture toughness of bone was lower and the fracture surface was less rough and more cleavage-like under dynamic loadings [20,21,[25][26][27] when comparing with quasi-static loadings. Shannahan et al. [26] measured the mode I and mode II fracture initiation toughness separately and found a transition from mode II dominant fracture to a mixed mode (mode I and mode II) fracture as loading rate increased from quasi-static to dynamic. ...
... However, under dynamic fractures, bone could exhibit a remarkably different behavior, showing a general transition from ductility to brittleness when compared with that under quasi-static rates [5,[20][21][22][23][24]. Studies at macroscopic length scales [20,21,[25][26][27] revealed that, the fracture toughness of bone was lower and the fracture surface was less rough and more cleavage-like under dynamic loadings [20,21,[25][26][27] when comparing with quasi-static loadings. Shannahan et al. [26] measured the mode I and mode II fracture initiation toughness separately and found a transition from mode II dominant fracture to a mixed mode (mode I and mode II) fracture as loading rate increased from quasi-static to dynamic. ...
... Shannahan et al. [26] measured the mode I and mode II fracture initiation toughness separately and found a transition from mode II dominant fracture to a mixed mode (mode I and mode II) fracture as loading rate increased from quasi-static to dynamic. More interestingly, Behiri and Bonfield [27] found a critical range of loading rates in cortical bone, above which unstable fractures would occur in a sudden manner while below which fractures were stable and controllable. ...
The initiation and propagation of physiological cracks in porcine cortical and cancellous bone under high rate loading were visualized using high-speed synchrotron X-ray phase-contrast imaging (PCI) to characterize their fracture behaviors under dynamic loading conditions. A modified Kolsky compression bar was used to apply dynamic three-point flexural loadings on notched specimens and images of the fracture processes were recorded using a synchronized high-speed synchrotron X-ray imaging set-up. Three-dimensional synchrotron X-ray tomography was conducted to examine the initial microstructure of the bone before high-rate experiments. The experimental results showed that the locations of fracture initiations were not significantly different between the two types of bone. However, the crack velocities in cortical bone were higher than in cancellous bone. Crack deflections at osteonal cement lines, a prime toughening mechanism in bone at low rates, were observed in the cortical bone under dynamic loading in this study. Fracture toughening mechanisms, such as uncracked ligament bridging and bridging in crack wake were also observed for the two types of bone. The results also revealed that the fracture toughness of cortical bone was higher than cancellous bone. The crack was deflected to some extent at osteon cement line in cortical bone instead of comparatively penetrating straight through the microstructures in cancellous bone. Statement of Significance: Fracture toughness is with great importance to study for crack risk prediction in bone. For those cracks in bone, most of them are associated with impact events, such as sport accidents. Consequently, we visualized, in real-time, the entire processes of dynamic fractures in notched cortical bone and cancellous bone specimens using synchrotron X-ray phase contrast imaging. The onset location of crack initiation was found independent on the bone type. We also found that, although the extent was diminished, crack deflections at osteon cement lines, a major toughening mechanism in transversely orientated cortical bone at quasi-static rate, were still played a role in resisting cracking in dynamically loaded specimen. These finding help researchers to understand the dynamic fracture behaviors in bone.
... Despite a large number of studies having been conducted on the fracture behavior of bovine long bones Fig. 7 An overview of the energy release rate for each heat treatment condition for a stress intensity factor rate ofK = 1.5× 10 5 M Pam 1/2 /s (e.g., Behiri and Bonfield 1989;Melvin 1973;Robertson et al. 1978;Bonfield and Datta 1976;Moyle and Gavens 1986;Behiri and Bonfield 1980;Norman et al. 1992;De Santis et al. 2000;Chen et al. 2008;Yan et al. 2006;Behiri and Bonfield 1984;Adharapurapu et al. 2006;Bonfield et al. 1978), comparing results obtained here to those in literature is challenging due to differences in loading rate, loading configuration, mode of fracture, specimen geometry, and the formulation used to determine fracture toughness. A common feature of the studies listed above is the determination of a single-value fracture toughness from the peak load at the onset of cracking under a prescribed loading mode (i.e., opening (mode-I), shearing (mode-II), or tearing (mode-III)). ...
... From Table 3 it can be observed that there is a trend of the mean energy release rate decreasing for heat treatment temperatures above 160 • C, as would be expected with the trend of fracture initiation toughness decreasing with increasing heat treatment temperatures. The relative magnitude of the mean and median energy release rates for the control group representing healthy cortical bone, reported in Table 3 and Fig. 7 respectively, are in general agreement with the energy release rates reported for bovine cortical bone by Behiri and Bonfield (1984) (2.9k J/m 2 ), and the range given by Ritchie et al. (2005) for generalized cortical bone (0.15-3.25 kJ/m 2 ). ...
The fracture resistance of bone has been attributed to a competition of sub-micron lengthscale intrinsic mechanisms, including plasticity conferred by collagen stretching and intermolecular sliding and much larger lengthscale extrinsic mechanisms such as crack deflection and bridging. In this study, the contribution of intrinsic toughening mechanisms on the dynamic fracture behavior of bovine cortical bone is investigated. Single edge notched cortical bone specimens were extracted from the mid-diaphysis of a bovine femur with dimensions in accordance with ASTM E399. Four specimen groups are studied, a control group, and groups subjected to two-hour heat treatments of 130 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C, 160 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C and 190 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C, respectively. Using a trypsin-hydroxyproline assay to determine the percent of denatured collagen achieved by each heat treatment, it is shown that the 160 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C and 190 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C groups have accumulated substantial collagen network damage compared to the 130 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C and control groups. Three-point bend drop tower experiments with impact velocities of 1.6m/s. The selected impact velocity results in a nominal stress intensity factor rate of K˙=1.5×105MPam1/2/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{K}}=1.5\times 10^5 MPa \, \, m^{1/2}/s$$\end{document}.Specimen’s speckled surfaces were imaged at 500,000 fps during deformation and post-processed using digital image correlation to determine the in-plane displacement fields. Using an orthotropic material linear elastic fracture mechanics formulation and over-deterministic least-squares analysis, the critical mode-I and mode-II stress intensity factors (i.e., fracture initiation toughness) were determined immediately proceeding fracture. As the heat treatment temperature increases (and the damaged collagen content increases), a weak but decreasing trend in fracture toughness was observed. Of particular note, for the 160 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C and 190 ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}C heat treatments, it was observed that the mode-II fracture initiation toughness is larger than the mode-I fracture initiation toughness. Regardless of the heat treatment condition, the mode-II fracture initiation toughness was comparatively less affected. For the specific case of Haversian bovine cortical bone whose collagen network has been denatured using heat treatment, a trend is observed pointing to collagen primarily conferring mode-I fracture initiation toughness, opposed to mode-II fracture initiation toughness, for the transverse fracture orientation.
... It has several components that work together to provide humans with a large range of motion [1][2][3][4][5][6][7][8] in the neck and protects the spinal cord. Therefore, the recent literature shows how to ensure the health, strength and mechanical properties of the vertebrae [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. The objective of this research is to expand fracture toughness considerations in cortical bone. ...
... Several studies have investigated the limit of the fracture toughness of cortical bone [9][10][11][12][13][14][15][16]. The stress intensity factor of bone can be described in terms of longitudinal and transverse fracture toughness. ...
... GPa and C 66 =6.65 GPa. Regarding failure properties, Behiri and Bonfield measured average critical stress intensity factors in transversely oriented bovine cortical bone (ζ = 0 in Fig. 14), ranging from K Ic2 = 3.2 to 4.0 MPa.m 1/2 [62,59,60], whereas by grooving cortical bone compact tension specimens, they determined average values of K Ic1 from 6.3 to 6.5 MPa.m 1/2 . It should be noted that no direct measures of the fracture energy of the longitudinal oriented material was performed by the authors in their experimental studies. ...
... K Ic2 = 4.0 MPa m 1/2 [62,27] and thickness h = 1 mm. It should be highlighted that in this study we adopted a two parameter function to describe the anisotropic resistance of cortical bone, however the proposed energetic criterion can be 210 extended to other generalized multi-parameter directional dependent functions. ...
An anisotropic model for two-dimensional electrical conduction, elasticity and fracture is proposed in the peridynamic theoretical framework. Material particles interact through elastic non-central pair potentials and inelastic pair potential functions of pairwise elastic and inelastic deformation measure, allowing to obtain a bond-based type model for conductive Cauchy orthotropic media without restrictions in the number of independent material constants. The elasticity of pair interactions can be described mechanistically by equivalent normal and shear springs, whose stiffness varies continuously with the spatial orientation of the ligament and, preserving the elastic symmetries of the material, depends on four elastic material parameters defining in-plane orthotropic classical elasticity. The macroscopic anisotropic conductivity is described instead by continuous functions of the micro-conductive properties of the interparticles interactions. Moreover, non-uniform material toughness is modeled adopting an anisotropic energetic failure criterion related to direction-dependent fracture energy functionals.
The accuracy of the proposed model has been assessed by several problems including the anisotropic electrical conduction in multi-phases laminae with a central hole and evolving cracks, and the fracture and damage sensing in cortical bone considering different orientation of the material reference system.
... The fracture behavior of cracked materials like bone or other biomaterials is investigated by means of multiple test methods. For example, the compact tension specimen [1][2][3][4], single edge notch beam [4][5][6][7],edge cracked beam subjected to four point bend loading [8,9] and double cantilever beam specimen [10] are some of the test geometries used for conducting either mode I (tensile/opening type) or mixed mode I/II (opening-shearing mode) fracture toughness experiments on bone materials. Different influencing parameters such as loading type, environmental condition, age of bone source, direction of applied loading, etc. have been studied to characterize the fracture behavior of bone materials using the aforementioned test specimens. ...
... Fracture toughness values were calculated by replacing the fracture loads into Eqs. (1) and (2). Corresponding values of KIc and KIIc were obtained as 6 √ and 4 √ , respectively that are in good agreement with the reported fracture toughness data of bovine femur [5,7]. ...
Understanding the fracture behavior and associated crack growth mechanism in bone material is an important issue for biomechanics and biomaterial researches. Fracture of bone often takes place due to complex loading conditions which result in combined tensile-shear (i.e. mixed mode) fracture mechanism. Several parameters such as loading type, applied loading direction relative to the bone axis, loading rate, age and etc., may affect the mixed mode fracture resistance and damage mechanism in such materials. In this research, a number of mixed mode I/II fracture experiments are conducted on bovine femur bone using a sub-sized test configuration called “compact beam bend (CBB)” specimen to investigate the fracture toughness of bone under different mode mixities. The specimen is rectangular beam containing a mid-edge crack that is loaded by a conventional three-point bend fixture. The results showed the dependency of bone fracture toughness on the state of mode mixity. The fracture surfaces of broken CBB specimens under different loading conditions were studied via scanning electron microscopy (SEM) observations. Fracture surface of all investigated cases (i.e. pure mode I, pure mode II and mixed mode I/II) exhibited smooth patterns demonstrating brittle fracture of bovine femur. The higher density of vascular channels and micro-cracks initiated in the weakened area surrounded by secondary osteons were found to be the main cause of the decreased bone resistance against crack growth and brittle fracture.
... [44] The details of the orientation designation for the specimens are shown in Figure 2. The first letter in the designation refers to the direction normal to the crack plane, whereas the second letter refers to the direction of crack propagation. Behiri and Bonfield [7] also observed a similar conclusion for mode I fracture of bovine tibia, also in the longitudinal direction, where no thickness dependence was seen between 0.5 and 2 mm. Similar to most of the engineering materials, cortical bone shows the least resistance to fracture under mode I loading. ...
... Brookfield et al. [95] employed the small punch test and bulge test to determine the material properties and also used the finite element analysis for simulation purpose. Finite element model was used to establish a relationship between the yield stress and the punch force (F) for elastic-perfectly plastic situation as: 49.20 6 2.35 10 F y σ + = − × (7) where: σ y is in MPa and F is in N. ...
... mm thick specimens; 32 a similar conclusion was reached for mode I fracture of bovine tibia, also in the longitudinal direction, where no thickness dependence was seen between 0.5 and 2 mm. 33 Conversely, more recent studies by Norman et al. report that the mode I toughness varied significantly with thickness from 2 to 6 mm, becoming essentially constant after a thickness of 6 mm was achieved. 34 Limited exper- iments on human tibia also showed little change in mode I toughness for 2 to 3 mm thick specimens. ...
... 114 Bone does show subcritical cracking under sustained (non-cyclic) loads. Early results on bovine tibia demonstrated that cracks grew in the longitudinal direction at higher velocities (∼10 −5 -10 −3 ms −1 ) 33,115,116 as the driving force (K or G) was increased. Attempts to grow cracks at faster rates resulted in catastrophic failure, along with a change in fracture morphology and a lower toughness. ...
... First, mechanical tests on hard biological tissue are commonly performed on small 30 specimens. Controlled geometries improve the managing of the boundary conditions and allow the realization 31 of more standardized experiments [1][4]. However, taking specimens from the cortical thickness of a long bone 32 remains a delicate operation. ...
The determination of bone mechanical properties remains crucial, especially to feed up numerical models. An original methodology of inverse analysis has been developed to determine femoral cortical bone longitudinal elastic modulus. The method is based on a numerical twin of a specific three-point bending test. It has been designed to be reproducible on each test result. In addition, the biofidelity of the geometric acquisition method has been quantified. As the assessment is performed at the scale of a bone shaft segment, the Young modulus values obtained (between 9518.29 GPa and 14181.15 GPa) are considered average values for the whole tissue, highlighting some inter-subjects variability.
The material microstructure has also been studied through histological analysis, and bone-to-bone comparisons highlighted discrepancies in quadrants microstructures. Furthermore, significant intra-subjects variability exists since differences between the bone's medial-lateral and anterior-posterior quadrants have been observed. Thus, the study of microstructures can largely explain the differences between the elastic modulus values obtained. However, a more in-depth study of bone mineral density would also be necessary and would provide some additional information. This study is currently being set up.
... Mechanical behaviors of Sing-0 and Sint-1 (Table S1, Supporting Data) were 0.56 mm (Sint-0) and 0.55 mm (Sint-1), the honeycomb height was reduced by only 6% (Sint-0) or 5% (Sint-1) of the original size. After reaching the maximum stress, the SS curve of Sint-0 showed a gradual decrease in stress similar to that observed for cortical bone deflection, reported by Behiri and Bonfield [57]. In contrast, Sint-1 showed a profile characteristic of catastrophic failure. ...
Calcite particles (Source; CaCO3, ∼0.4 μm in size) were 3D-printed into cube honeycombs (honeycombs) with ∼0.7 mm thick struts and ∼0.6 mm cellular window opening. The sintered honeycombs, regardless of the presence of a sintering aid in the printing slurry, gave ∼3 MPa or more in the compressive stress, which exceeded that of trabecular bone. The conversion of calcite to hydroxyapatite (HAp) was assessed in 0.1 M K2HPO4 (pH: 7.0 or 7.4) at 20 °C ∼ 80 °C due to the X-ray diffraction intensity of the calcite and HAp peaks, surface microstructure, and P(V) accumulation on the honeycomb strut surface. Although a premature HAp layer yielded on all samples within 1 h soaking, further conversion depended on the samples. The conversion continued on Source until the volume fraction of the HAp shell layer reached ∼35% (37 °C). It was hardly detected on the grains of sintered honeycombs (from the slurry with the sintering aid) within the whole soaking period (≤24 h). In contrast, the conversion on the sintered honeycombs (from the slurry without the aid) became vigorous when soaked for 5 h, and, it reached at 24 h the highest level that the as-printed honeycombs achieved. The critical factors controlling the conversion were calcite dissolution, the equilibria among the carbonate and phosphate ions, and the rates of migration of Ca(II) and P(V) through the inter-granular channels of pores within the struts. A particle stacking model was proposed for the most plausible interpretation of the present results.
... Therefore, it becomes crucial to understand the fracture mechanism of the cortical bone, which helps treat bone disorders and improves the design of artificial bone grafts and implants [8][9][10][11][12][13]. Fracture resistance of cortical bone is affected by bone quantity (bone mineral density) and bone quality, i.e., the hierarchical structure, variation in material properties, and the accumulation of microcracks [10,[14][15][16]. Cortical bone consists of a hierarchical composite structure of hydroxyapatite crystals and collagen fibres. ...
Bone fracture is a severe health concern; therefore, understanding the causes of bone fracture are crucial. This paper investigates the microstructure and fracture behaviour of cadaveric cortical bone of two different groups (Young, n= 6; Aged, n=7). The microstructure is obtained from µ-CT images, and the material parameters are measured with nanoindentation. Fracture behaviour in transverse and longitudinal orientations is investigated experimentally and numerically. The results show that the Haversian canal (HC) size increases and the osteon wall thickness (OWT) decreases significantly in the aged group, whereas a nonsignificant difference is found in tissue properties. The crack initiation (Jic) and crack growth (Jgrow) toughness of the aged group are found to be significantly lower (p<0.01) than the young group in the transverse orientation; however, for the longitudinal orientation, only the value of Jic in the aged group is found significantly lower. Further, a 4-phase XFEM (based on micro-CT image) model is developed to investigate the crack propagation behaviour in both orientations. For the transverse orientation, results show that in the aged group, the crack initially follows the cementline and then penetrates the osteon, whereas, in the young group, it propagates along the cementline. These results are in agreement with experimental results where the decrease in Jgrow is more significant than the Jic in the aged group. This study suggests that ageing leads to a larger HC and reduced OWT, which weakens the crack deflection ability and causes fragility fracture. Further, the XFEM results indicate that the presence of a small microcrack in the vicinity of a major crack tip causes an increase in the critical stress intensity factor.
... A strong relationship between fracture energy and bone density has been shown in a number of experimental studies (Behiri and Bonfield, 1984;Cook and Zioupos, 2009;Granke et al., 2016;Yeni et al., 1998). However, it is still unknown whether such relationships measured at the micro-or mesoscale would also be valid at the organ scale. ...
Hip fractures are a major health problem with high socio-economic costs. Subject-specific finite element (FE) models have been suggested to improve the fracture risk assessment, as compared to clinical tools based on areal bone mineral density, by adding an estimate of bone strength. Typically, such FE models are limited to estimate bone strength and possibly the fracture onset, but do not model the fracture process itself. The aim of this study was to use a discrete damage approach to simulate the full fracture process in subject-specific femur models under stance loading conditions. A framework based on the partition of unity finite element method (PUFEM), also known as XFEM, was used. An existing PUFEM framework previously used on a homogeneous generic femur model was extended to include a heterogeneous material description together with a strain-based criterion for crack initiation. The model was tested on two femurs, previously mechanically tested in vitro. Our results illustrate the importance of implementing a subject-specific material distribution to capture the experimental fracture pattern under stance loading. Our models accurately predicted the fracture pattern and bone strength (1% and 5% error) in both investigated femurs. This is the first study to simulate complete fracture paths in subject-spe- cific FE femur models and it demonstrated how discrete damage models can provide a more complete picture of fracture risk by considering both bone strength and fracture toughness in a subject-specific fashion.
... Fracture toughness experiments at physiological strain rates have shown that the resistance of cortical bone to fracture clearly diminishes at higher strain rates, indicating a change in the bone's extrinsic resistance to fracture (Figure 2.7a) [64][65][66][67][68]. At the microstructural scale (~10-100 μm), the lower toughness is associated with a distinct change in the path of the crack through the bone-matrix structure. ...
Bioinspired Structures and Design - edited by Wole Soboyejo September 2020
... Fracture toughness experiments at physiological strain rates have shown that the resistance of cortical bone to fracture clearly diminishes at higher strain rates, indicating a change in the bone's extrinsic resistance to fracture (Figure 2.7a) [64][65][66][67][68]. At the microstructural scale (~10-100 μm), the lower toughness is associated with a distinct change in the path of the crack through the bone-matrix structure. ...
Cambridge Core - Materials Science - Bioinspired Structures and Design - edited by Wole Soboyejo
... Because the 'cartilage bridge' is rarely injured and is presumably subjected to large shear and tension forces from the pull on the patellar tendon, we would expect this region to have greater fracture toughness than the horizontal growth cartilage, which is subjected primarily to compression. In fact, the fracture toughness is comparable to or greater than the 1700 -2800 N/m quoted for cortical bone [6]. The 'cartilage bridge' is an unusual growth plate in that it is bipolar; i.e. bone is formed on both sides of the plate, both on the tibial tuberosity side and on the epiphyseal side. ...
In the developing proximal tibial epiphysis the anterior ossifying tibial tuberosity is separated from the secondary ossification center of the tibial epiphysis by a bipolar growth plate known as the 'cartilage bridge.' We tested the fracture toughness of the central part of this growth cartilage in 18-week old calves in the direction perpendicular to the plate (mean 4962 N/m, SD 1846) and found it to be greater (p = 0.0004) than in the parallel direction (mean 2909 N/mm, SD 1122). Part of the reason for this anisotropy is the presence of vascular channels which cross the bridge from the epiphysis into the tuberosity. In addition, we hypothesize that the anisotropy reflects an arrangement of collagen primarily along the length of the 'cartilage bridge.'
... Early studies on fracture mechanics of cortical bone used experimental techniques [20,21] in conjunction with the concepts of Linear elastic fracture mechanics (LEFM). Behiri and Bonfield [22,23] obtained the values of critical stress intensity factor (Kc) and critical strain energy release rate (Gc) for different values of density, specimen thickness, crack velocity and orientation of bovine bone. Further, Norman et al. [24] used fracture toughness testing and principles of linear elastic fracture mechanics to compute the critical stress intensity factor and critical energy release rate. ...
Bone tissues are heterogeneous composites that consist of various microstructural constituents at different length scales. These microstructural constituents and their heterogeneous distribution significantly affect the fracture behavior of bones. Thus, the effect of parametric uncertainties on the fracture analysis of a cortical bone is presented in this study. A 2D model of the cortical bone is generated with the help of micro-CT image of a cortical bone and the fracture analysis is performed for the developed model with the help of an Extended Isogeometric Analysis (XIGA) using linear elastic fracture mechanics. The values of stress intensity factor are calculated with the help of interaction integral approach and the direction of crack propagation after each step is evaluated using the maximum principle stress criterion. The obtained results are compared with the finite element results using Abaqus software and found in good agreement. Further, uncertainties in the values of osteon Young’s modulus, cement line thickness and porosity percentage are taken into consideration for stochastic analysis. The effect of variation in the values of input parameters on the stress intensity factor values and the crack path trajectories is illustrated, and observed that the variation in osteon Young’s modulus and porosity percentage is more pronounced than the cement line thickness.
... Bone is not only anisotropic but also a viscoelastic material: its mechanical behaviors depend on loading rate significantly. Previous studies in macroscopic scales showed that, when loading rate increased from quasi-static to dynamic, a transition from quasi-ductile to brittle fracture took place in bone ( Zimmermann et al., 2014 ;Kulin et al., 2011a,b ;Evans et al., 1992 ;Kirchner, 2006 ;Zioupos et al., 2008 ), as the fracture toughness was largely reduced and a typical less tortuous crack surface was observed in postmortem specimen Ural et al., 2011 ;Behiri and Bonfield, 1984 ;Shannahan et al., 2015 ). A numerical study found that the initiation fracture toughness of bone decreased gradually as the strain rate increased from quasi-static to dynamic, while the propagation fracture toughness was only significantly sensitive to quasi-static strain rate (up to 1/s) ( Ural et al., 2011 ). ...
We visualized, in real time, the dynamic fracture behaviors in porcine cortical bone from humerus and porcine trabecular bone from nasal bone at a high loading rate using high-speed synchrotron X-ray phase-contrast imaging (PCI). Dynamic three-point bending loading was applied on notched bone specimens by a modified Kolsky compression bar and images of the entire fracture events were recorded with an ultra-high-speed camera. Experiments at a quasi-static loading rate on material testing system (MTS) were also performed to identify the loading-rate effects on the fracture toughness of the two types of bone. Three-dimensional synchrotron X-ray computed tomography was conducted to examine the initial microstructures in the bone specimens before mechanical loading. At the dynamic loading rate, the onset locations of crack initiation were found to be independent from the bone types. The deleterious effect of dynamic loading rate on bone's fracture toughness was verified in this study and the crack was found to propagate at higher speeds in cortical bone than in trabecular bone. In a comparison of the observed more torturous crack paths at the quasi-static loading rate, cracks in dynamically loaded bone specimens generally followed the paths with less in-plane deflections and out-of-plane twists. However, our experimental results also indicated that, although the extent was diminished at dynamic loading rate, the crack deflections at osteon cement lines still played a role as a major toughening mechanism to dynamic fractures in transversely orientated cortical bone.
... 7,8 To standardize the fracture properties, the American Society for Testing and Materials (ASTM E399:1997) 9 has been followed by Moyle and Bowden 10 and Behiri and Bonfield. 11 Different specimen and loading mechanism such as single-edge notched bend (SENB) specimen, 4 compact tension (CT) specimen, 12 double cantilever beam (DCB) specimen, 13 and corresponding loading mechanism have been used by many researchers to evaluate the SIF and energy release rate of cortical bone. 4,12,13 FEM has been successfully used over the past few decades in the field of biomechanics. ...
Stress intensity factor and energy release rate are important parameters to understand the fracture behaviour of bone. The objective of this study is to predict stress intensity factor and energy release rate using finite element method, element-free Galerkin method, and extended finite element method and compare these results with the experimentally determined values. For experimental purpose, 20 longitudinally and transversely fractured single-edge notched bend specimens were prepared and tested according to ASTM standard. All specimens were tested using the universal testing machine. For numerical simulations (finite element method, element-free Galerkin method, and extended finite element method), two-dimensional model of cortical bone was developed by assuming plane strain condition. Material properties of the cortical bone were considered as anisotropic and homogeneous. The values obtained through finite element method, element-free Galerkin method, and extended finite element method are well corroborated to experimentally determined values and earlier published data. However, element-free Galerkin method and extended finite element method predict more accurate results as compared to finite element method. In the case of the transversely fractured specimen, the values of stress intensity factor and energy release rate were found to be higher as compared to the longitudinally fractured specimen, which shows consistency with earlier published data. This study also indicates element-free Galerkin method and extended finite element method predicted stress intensity factor and energy release rate results are more close to experimental results as compared to finite element method, and therefore, these methods can be used in the different field of biomechanics, particularly to predict bone fracture.
... The fracture properties of cortical bone were evaluated with CT testing. Bone is considered to have a higher fracture resistance to transverse cracking compared to longitudinal cracking [48][49][50][51]. Various toughening mechanisms active at various hierarchical levels are considered responsible for the greater toughness in the transverse direction [52]. ...
The fracture properties of cortical bone are directly coupled to its complex hierarchical structure. The limited availability of bone material from many anatomic locations creates challenges for assessing the effect of bone heterogeneity and anisotropy on fracture properties. The small punch technique (SPT) was employed to examine the fracture behavior of cortical bone in terms of area under the curve values obtained from load-load point displacement behavior. Fracture toughness of cortical bone was also determined in terms of J-toughness values obtained using a compact tension (CT) test. Area under the curve values obtained from the small punch test were correlated with the J-toughness values of cortical bone. The effects of bone density and compositional parameters on area under the curve and J-toughness values were also analyzed using linear and multiple regression analysis. Area under the curve and J-toughness values are strongly and positively correlated. Bone density and %mineral content are positively correlated with both area under the curve and J-toughness values. The multiple regression analysis outcomes support these results. Overall, the findings support the hypothesis that area under the curve values obtained from small punch tests can be used to assess the fracture behavior of cortical bone.
... The fracture mechanics model has been widely used in solid materials [Hutchinson and Evans (2000); Sahimi and Arbabi (1993)], mechanical engineering [Metkar et al. (2013); Chen et al. (2017); Yazdanipour et al. (2015)] and human injury [Motomura et al. (2011); Gao (2006)] due to its advantage in simulating the process of object fracture loss. There have been many studies on the fracture mechanics of bones [Gao (2006); Behiri and Bonfield (1984); Malik et al. (2003); Yan et al. (2007); Yeni and Fyhrie (2002); Melvin (1993); Lakes et al. (1990)] showing that the cortical bone of the femoral head, normal cancellous bone and necrotic cancellous bone have different fracture properties. ...
... En ce qui concerne la densité, le fait d'expérimenter sur une espèce différente peut toutefois gêner la comparaison, puisqu'il est important de travailler sur des espèces et des éléments anatomiques structurellement proches, notamment pour éviter les différences de densité. Pour la fracture, cela a toute son importance puisque la densité osseuse joue un grand rôle dans la propagation de la ligne de fracture, grâce à la vélocité de l'impact (Behiri et Bonfield, 1984). La densité osseuse est toutefois délicate à intégrer à la réflexion car elle est sujette à de nombreuses variations qui dépendent de l'état de santé de l'animal, de son âge et de la portion de l'os mesurée. ...
Le cas de la bipartition des métapodes au Mésolithique à Zamostje 2 (région de Moscou, Russie) Résumé : À la fin du Mésolithique, dans la Volga supérieure, les « techniques de fracture » (sensu Christensen, 2015) sont très large-ment utilisées pour débiter les métapodes d'élan par bipartition et produire de longues baguettes que les Mésolithiques utilisaient pour confectionner certaines pointes de projectile et pointes barbelées. Cette manière de faire, typique de la période, n'en demeurait pas moins mal comprise. En vue de mieux caractériser les étapes de ces débitages, nous avons réalisé plusieurs expérimentations. En nous basant sur les résultats de notre analyse du matériel archéologique du site de Zamostje 2, nous avons débité une trentaine de métapodes de cerf en percussion diffuse directe (ou éclatement : Christensen, 2015) et en percussion linéaire indirecte (ou fendage : Christensen, 2015). Les résultats de ces expériences ont permis, sur la base d'une caractérisation fine des stigmates de percussion, d'identifier de nouveaux critères distinctifs entre les fractures obtenues par percussion diffuse directe, de celles obtenues par percussion linéaire indirecte. Nous avons ainsi pu mettre en évidence une évolution tout à fait inédite des techniques mises en oeuvre à la fin du Mésolithique dans la région de la Volga supérieure.
Mots-clés : Mésolithique, technologie osseuse, archéologie expérimentale, stigmates, éclatement par percussion directe, éclatement par percussion indirecte, débitage par bipartition, métapode.
... All CT specimens showed typical P-v records designated as Type I in the ASTM standards (Fig.5). Therefore, the critical load (P cr ) associated with the onset of crack growth in Eq.(1) should be determined as the load at the intersection of a 5% secant line with the test record, but the maximum load (P max ) recognised as a peak in the test record (Fig.5) was used in this study by the same way as reported in the other previous study [8]. ...
... The results of the present study clearly demonstrated what is well known in the recent literature. No one material is ideal on replacing the anatomical features of the tooth [15,17,19,26]. ...
The aim of this paper is to underline the mechanical properties of dental single crown prosthodontics materials in order to differentiate the possibility of using each material for typical clinical condition and masticatory load. Objective of the investigation is to highlight the stress distribution over different common dental crowns by using computer-aided design software and a three-dimensional virtual model. By using engineering systems of analyses like FEM and Von Mises investigations it has been highlighted the strength over simulated lower first premolar crowns made by chrome cobalt alloy, golden alloy, dental resin, and zirconia. The prosthodontics crown models have been created and put on simulated chewing stresses. The three-dimensional models were subjected to axial and oblique forces and both guaranteed expected results over simulated masticatory cycle. Dental resin presented the low value of fracture while high values have been recorded for the metal alloy and zirconia. Clinicians should choose the better prosthetic solution for the teeth they want to restore and replace. Both prosthetic dental crowns offer long-term success if applied following the manufacture guide limitations and suggestions.
... To-date most fracture assessment methods for compact bone rely on pure mode I testing and very few mixed-mode investigations [79,80,53] have been reported. Early studies focused on linear elastic fracture mechanics [54,58], based on traditional fracture tests such as the compact tension test [14,18,15,54,71] or the three point bending test on single-edge notched specimens [63,73,43]. However, bone exhibits a rising R-curve behavior [44,48,50,64] due to multiple toughening mechanisms [77] such as microcracking [65], diffuse damage [24,56], fiber bridging [51], crack deflection [36] or osteon pull-out [21]. ...
Bone is a composite material with five distinct structural levels: collagen molecules, mineralized collagen fibrils, lamellae, osteon and whole bone. However, most fracture testing methods have been limited to the macroscopic scale and there is a need for advanced characterization methods to assess toughness at the osteon level and below. The goal of this investigation is to present a novel framework to measure the fracture properties of bone at the microscopic scale using scratch testing. A rigorous experimental protocol is articulated and applied to examine cortical bone specimens from porcine femurs. The observed fracture behavior is very complex: we observe a strong anisotropy of the response with toughening mechanisms and a competition between plastic flow and brittle fracture. The challenge consists then in applying nonlinear fracture mechanics methods such as the J-integral or the energetic Size Effect Law to quantify the fracture toughness in a rigorous fashion. Our result suggests that mixed-mode fracture is instrumental in determining the fracture resistance. There is also a pronounced coupling between fracture and elasticity. Our methodology opens the door to fracture assessment at multiple structural levels, microscopic and potentially nanometer length scale, due to the scalability of scratch tests.
The research on the crack propagation mechanism of bone has important research significance and clinical medical value for the selection of cutting parameters and the development of new surgical tools. In this paper, an extended finite element method (X-FEM) model of ultrasonic bone cutting considering microstructure was developed to further study the ultrasonic bone cutting mechanism and to quantitatively analyze the effects of cutting direction, ultrasonic parameters, and cutting parameters on the mechanism of ultrasonic bone cutting crack propagation. The results show that ultrasonic bone cutting is essentially a controlled crack propagation process, in which brittle crack and fatigue crack are the main crack propagation mechanisms. In order to improve the efficiency of ultrasonic bone cutting, large amplitude and high-frequency ultrasonic vibration are preferred. Compared with the other two cutting directions, the crack propagation deflection angle in the transverse cutting direction is the largest, resulting in the worst cutting surface. Therefore, in the path planning of orthopedic surgical robots, the transverse cutting direction should be avoided as much as possible. Frequency only has a significant effect on the crack propagation rate and has a positive correlation. There is a positive correlation between the deflection angle, propagation length, propagation rate, and amplitude, which provides the possibility to control the direction and length of crack propagation by controlling the amplitude of ultrasonic. The feed speed is much lower than the ultrasonic vibration speed, which makes the influence of ultrasonic vibration speed on the crack propagation characteristics dominant. The X-FEM model of ultrasonic bone cutting provides an effective method for selecting reasonable machining parameters of orthopedic robot and optimize the design of ultrasonic osteotome.
Continuum mechanics has been very effective in solving many mechanical problems and is widely used in the field. However, the classical continuum mechanics theory faces challenges when dealing with fracture problems, where the displacement field is discontinuous. Peridynamics is a new theory of continuum mechanics, which addressed this fundamental issue. In peridynamic theory in contrast to the classical continuum mechanics theory, the spatial derivative of the displacement field is replaced with an integral of forces exchanged between particles inside an influence domain, called horizon. In this article, we introduce three versions of the peridynamic theory, the bond-based, the ordinary state-based and the non-ordinary state-based. We will also explain the concept of dual horizon, which extends the capabilities of the computational implementation of the theory. We will then present some challenging fracture problems and their solutions with the peridynamic theory. These include predicting fracture in isotropic and anisotropic materials under static and dynamic loading conditions. We show that the peridynamic theory can accurately predict the fracture force and patterns in a wide range of materials, but we also show some limitations of the theory, which need to be addressed in future work.
The variety of forces that can result in failure of bone infrastructure are described. Their effects on and response induced in bone are discussed. Monotonic and fatigue fractures are differentiated. The mechanical concepts of fatigue, toughness, modelling and remodelling and their implications for predisposition or resistance to fractures are discussed. Methods of fracture classification and topographic description are provided.
This Perspectives provides a back-to-basics rationale for the ideal exercise prescription for osteoporosis. The relevance of fundamental principles of mechanical loading and bone adaptation determined from early animal studies is revisited. The application to human trials is presented, including recent advances. A model of broad-scale implementation is described and areas for further investigation identified.
Elastic and elastic–plastic theoretical and numerical analyses are performed to evaluate the bone containing initial crack. The mode I + II of fracture is applied to identify the most critical incline crack and the analytical results are compared with experimental evaluation. The loading capacity in bones with smaller thickness decreases more significantly in the elastic–plastic analysis. Both plastic zone and the elastic–plastic stress are higher in smaller initial crack length. Simultaneous initiation of non-close multiple cracks has no significant influence on the ultimate tensile load and the effect of crack width on fracture decreases in higher crack width and longer length.
Crustose coralline algae (CCA), and in particular Porolithon onkodes, play an important
reef-building role in modern tropical coral reefs. CCA form thick crusts of Mg-calcite
and grow over corals and loose substrate to bind these together. This binding and
cementing process is fundamental to the development of structural reefs that are capable
of withstanding the high-energy waves in the shallow to inter-tidal areas of the reef. As
anthropogenic CO2 emissions continue to increase, the oceans absorb part of this extra
CO2 and become more acidic, a process known as Ocean Acidification (OA). There are
concerns that OA will have a negative affect on the reef-building capacity of coral reef
organisms, in particular on CCA. This is because Mg-calcite is meta-stable and more
susceptible to dissolution than aragonite, the mineral used by corals to build skeletons.
The goal of this thesis work was to firstly understand the physical and mechanical
properties that enable the CCA to cement the reef and withstand damage from highenergy waves, bioerosion and chemical dissolution. Secondly, to anticipate how OA may
interfere with these reef-building properties. These goals were pursued by setting clear
aims with associated specific objectives designed to elucidate information relevant to
these questions.
Methods were developed for X-ray diffraction to identify the mineral composition of
CCA. Nanoindentation was investigated as a tool for determining the mechanical
properties of CCA and the measurement of fracture toughness was found to return
physically meaningful information relevant to structural reef development.
Study of CCA calcification showed that cell wall Mg-calcite exhibited radial crystal
morphology in agreement with published studies on temperate species. However, highresolution imaging showed the radial crystals were made of banded stacked sub-micron
grains within an organic framework. Dolomite was found not only as cell lining by
submicron rhombs, but also as the primary calcification of hypothallial cell walls.
5
Dolomite is shown to be resistant to bacterial erosion. A model is developed whereby it is
proposed that dolomite formation is dependent on polysaccharide accumulation.
Using nanoindentation, P. onkodes are found to be extraordinarily tough, on par with the
measured fracture toughness for metamorphic minerals quartz and corundum. The
fracture toughness is enabled by the presence of dolomite cell lining. Contrary to the
literature, bacterial erosion is found to be a constructive, not destructive, process.
A survey of P. onkodes from Heron Island fore reef and reef flat showed that dolomite
was present in all the fore reef crusts but none of the reef flat crusts. The reef flat crusts
did not have fracture resistance except where remineralised. The presence of dolomite
cell lining was shown to decrease skeletal dissolution rates by an order of magnitude.
OA experiments showed that skeletal dissolution rates increased with elevated pCO2, but
dolomite continued to confer resistance to dissolution. pCO2 levels did not affect the
skeletal Mg content or dolomite formation in living CCA. Of concern, and in agreement
with the literature, bacterial erosion is accelerated under a combination of elevated pCO2
and temperatures, suggesting this may be the main threat to CCA reef-building in the
future. The experimental findings were corroborated by results of a field survey along a
natural pCO2 gradient.
In summary, dolomite was found to be an essential component of modern reef
development via its contribution to enabling CCA P. onkodes thick crust development
and persistence. Reef building by CCA P. onkodes is likely to continue as pCO2 rises up
until a tipping point is reached whereby bacterial erosion switches from constructive to
destructive.
Background:
Understanding the fracture mechanics of bone is very important in both the medical and bioengineering field. Bone is a hierarchical natural composite material of nanoscale collagen fibers and inorganic material.
Objective:
This study investigates and presents the fracture toughness of bovine cortical bone by using elastic plastic fracture mechanics.
Methods:
The J-integral was used as a parameter to calculate the energies utilized in both elastic deformation (Jel) and plastic deformation (Jpl) of the hipbone fracture. Twenty four different types of specimens, i.e. longitudinal compact tension (CT) specimens, transverse CT specimens, and also rectangular unnotched specimens for tension in longitudinal and transverse orientation, were cut from the bovine hip bone of the middle diaphysis. All CT specimens were prepared according to the American Society for Testing and Materials (ASTM) E1820 standard and were tested at room temperature.
Results:
The results showed that the average total J-integral in transverse CT fracture specimens is 26% greater than that of longitudinal CT fracture specimens. For longitudinal-fractured and transverse-fractured cortical specimens, the energy used in the elastic deformation was found to be 2.8-3 times less than the energy used in the plastic deformation.
Conclusion:
The findings indicate that the overall fracture toughness measured using the J-integral is significantly higher than the toughness calculated by the stress intensity factor. Therefore, J-integral should be employ to compute the fracture toughness of cortical bone.
In this study, the dynamic cracking processes in porcine cortical bone were visualized in real-time using the high-speed synchrotron X-ray phase-contrast imaging (PCI) technique in three osteon orientations: in-plane transverse, out-of-plane transverse and in-plane longitudinal. The dynamic flexural loading applied on the pre-notched bone specimens was introduced by a modified Kolsky compression bar. High-speed X-ray images of the entire loading events were documented with a high-speed camera. Three-dimensional X-ray micro-computed tomography was conducted to examine the intact microstructures and obtain the basic material properties of the bone material used for mechanical characterizations. The onset location, where crack initiated, and the subsequent direction, along which the incipient crack propagated, were measured quantitatively using the high-speed X-ray images and the latter was found dependent on the osteon direction significantly. The crack propagation velocities were dependent on crack extension over the entire crack path significantly for all the three directions while the initial velocity for in-plane longitudinal direction was lower than the other two directions. Straight-through crack paths were observed for in-plane longitudinal specimens while the cracks were deflected and twisted in the in-plane transverse direction. For out-of-plane transverse direction, the cracks follow paths with tortuosity fall in between the other two directions, showing a mixed mode of fractures of the former two extreme cases. The toughening mechanisms, visualized by the high-speed X-ray images, and the corresponding fracture toughness, evaluated in terms of fracture initiation toughness and crack growth resistance curve (R-curve), were also found significantly different among the three osteon directions, suggesting an overall transition from brittle to ductile-like fracture behaviors at the dynamic displacement rate (5.4 m/s) as the osteon orientation varies from in-plane longitudinal to out-of-plane transverse, and to in-plane transverse eventually.
In this paper an angled edge cracked short beam specimen subjected to symmetric three-point bend loading was designed and examined for conducting mixed-mode I + II fracture toughness experiments. This specimen is suitable for being used in those experimental crack growth studies in which a very limited amount of raw material is available for the preparation of the test specimen. By conducting several finite element analyses, it was shown that the short bend beam specimen is able to produce full modes I and II mixities if the affecting parameters such as the crack length ratio, loading span ratio and crack inclination angle are set in suitable ranges. The practical ability of the short bend beam specimen was investigated for obtaining the mixed-mode I + II fracture toughness of bovine bone and the corresponding values of KIc and KIIc were determined for different mode mixities. While a significant discrepancy was observed between the experimental results and predictions of the conventional singular based fracture theories, it was demonstrated that the two-parameter (K & T) based fracture models such as GMTS and EMTSN criteria can provide significantly improved predictions for the fracture behavior of tested short bend beam specimen made of bone material.
A proximal humerus fracture is an injury to the shoulder joint that necessitates medical attention. While it is one of the most common fracture injuries impacting the elder community and those who suffer from traumatic falls or forceful collisions, there are almost no validated computational methods that can accurately model these fractures. This could be due to the complex, inhomogeneous bone microstructure, complex geometries and the limitations of current fracture mechanics methods.
In this paper, we develop and implement a novel phase field method to investigate the proximal humerus fracture. To model the fracture in the inhomogeneous domain, we propose a power law relationship between bone mineral density and critical energy release rate. The method is validated by an in vitro experiment, in which a human humerus is constrained on both ends while subjected to compressive loads on its head in the longitudinal direction that lead to fracture at the anatomical neck. CT‐scans are employed to acquire the bone geometry and material parameters, from which detailed finite element meshes with inhomogeneous Young modulus distribution in the bone are generated.
The numerical method, implemented in a high performance computing environment, is used to quantitatively predict the complex 3D brittle fracture of the bone, and is shown to be in good agreement with experimental observations. Furthermore, our findings show that the damage is initiated in the trabecular bone‐head and propagates outward towards the bone cortex. We conclude that the proposed phase field method is a promising approach to model bone fracture.
The structural constituents of tissues in organisms are composed primarily of minerals and proteins. Collagen is the most common protein used to construct such natural materials in vertebrates; among these structures, a wide variety of hierarchical architectures with structural and property gradients have evolved to induce desired combinations of stiffness, strength, ductility and toughness for a diverse range of mechanical functionalities. The soft collagen provides biological materials the ability to resist tensile tractions and to dissipate energy under mechanical deformation. Here we seek to understand the structure, deformation and toughening mechanisms of collagenous materials from the perspective of the hierarchical assembly of individual collagen molecules, fibrils, fibers, as well as the other nature-designed hierarchical structural elements. This review summarizes the structural designs of collagenous materials focusing on Type-I collagen, the most abundant extracellular protein that forms linear arrays, as well as examining its deformation and toughening mechanisms by illustrating how nature uses hierarchical structures and gradients, at nano-, micro- to macro-levels, to confer different functions to its organisms. The organization of collagen is discussed for different structures in order to illustrate the broad range of its functional and mechanical properties: specifically, skin, arteries, eye cornea, fish scales, bone, ligaments and tendons. We conclude by highlighting important developments in tissue engineering where synthetic and natural collagen has been incorporated into the architecture of the body. We trust that such insight may provide guidance for the design of the next-generation of synthetic structural materials with unprecedented functionality.
A fracture is estimated every three seconds in the world, leading to an increased risk of impairment or even mortality. The biomechanical knowledge of bone fracture mechanisms in a fall configuration of loading is of great interests for the development of clinical method for the prediction of the risk of fracture. Toughness seems to be a good candidate to investigate this fracture process as it corresponds to the energy needed to propagate a crack through cortical bone complex microstructure. The aim of this study was thus to evaluate human cortical bone toughness parameter under both quasi-static and fall-like loading conditions paired anatomical locations. Micro-computed tomography images using synchrotron radiation and collagen cross-links maturation measurements were performed to investigate the influence of the tissue architecture on crack propagation. Results found showed that under quasi-static condition, the different anatomical locations present different mechanical behavior. Radius significantly better resist crack propagation than the other studied location. Considering a fall-like loading condition, no more difference is observed between the locations but a significant decreased is measured compare to the first configuration. Human cortical bone has a better capacity to resist crack propagation under a standard quasi-static loading condition. By investigating the tissue morphometric and biochemical parameters, we observed different organization from a location to another that explains the mechanical differences. The architectural features appear to be determinant for crack propagation mechanisms under quasi-static condition, but they play a lesser role under fall-like condition. These results imply that the tissue microstructure is not a determinant when dealing with the prediction of the risk of fracture. Further work has to be done to reach out which parameters are more determinants under a specific fall-like loading condition
The innovative use of specially designed ceramics to repair and reconstruct diseased or damaged parts of the body has improved the quality of life, and in some cases the length of life, for thousands of people. Ceramics used for this purpose are termed “bioceramics” and can be single crystals (sapphire), polycrystalline (alumina or hydroxylapatite), glass (Bioglass®), glass-ceramics (Ceravital® or A/W glassceramic), or composites (stainless-steel fiber-reinforced Bioglass or polyethylene-hydroxylapatite).
Ceramics and glasses have long been used in the health care industry for eye glasses, diagnostic instruments, chemical ware, thermometers, tissue culture flasks, and fiber optics for endoscopy. Insoluble porous glasses have been used as carriers for enzymes, antibodies, and antigens. Ceramics are also widely used in dentistry as restorative materials, gold porcelain crowns, glass-filled ionomer cements, and dentures.
This review describes bioceramics used as implants to repair parts of the body, usually hard tissues such as bones or teeth, but also to replace heart valves. Dozens of ceramic compositions have been tested, but few have achieved human clinical application. Clinical success requires the simultaneous achievement of a stable interface with tissue and a match of the mechanical behavior of the implant with the tissue to be replaced.
The mechanism of tissue attachment depends on the type of tissue response at the implant interface (Table I).
In Chapter 4, we examined the mechanical properties of bone when it is fractured monotonically by a load sufficient to exceed the failure stress of the material. Structures may also fail more gradually. There are two principal ways in which this can happen: creep and fatigue. In both, a stress less than the ultimate stress is applied, and damage from this stress grows and accumulates until failure occurs. In creep, the stress is applied continuously; in fatigue, the stress is applied cyclically. Obviously, these phenomena may occur simultaneously in the components of an engineering structure like a bridge, where the weight of the structure provides creep loading and the periodic passage of vehicles produces fatigue. Similarly, in the skeleton, bones often support more or less constant loads for prolonged periods of time (such as the vertebral bodies in your spine as you sit reading this book), and equally often carry cyclic loads (such as when you walk to class). In this chapter we shall see that creep and fatigue are closely related and explore the ways in which bone limits and repairs the damage that they produce. We also discuss what seems to happen when the limits of this capacity for damage control are exceeded and the bone fails by what is called a stress fracture.
The anisotropic nature exhibited by compact bone has been well documented in the literature, with differences in mechanical behaviour measured between orientations parallel to (designated as longitudinal) or normal (designated as transverse) to the long axis of the bone (1–5). Several investigators have studied the influence of a variation in the direction of osteons in bone in relation to the associated tensile, compressive and torsional deformation properties (6–10). but only limited data has been previously reported on the fracture characteristics as evaluated by K
c (the critical stress intensity factor) (11) and G
c (the critical strain energy release rate) as a function of orientation.
Based on the microscopic analyses of cracks and correlational studies demonstrating evidence for a relationship between fracture toughness and microstructure of cortical bone, an equation was derived for bone fracture toughness in longitudinal crack growth, using debonding at osteonal cement lines and weakening effect of pores as main crack mechanisms. The correlation between the measured and predicted values of fracture toughness was highly significant but weak for a single optimal value of matrix to cement line fracture toughness ratio. Using fracture toughness values and histomorphometrical parameters from an available data set, matrix to cement Line fracture toughness ratio was calculated for human femoral bone. Based on these calculations it is suggested that the effect of an osteon on fracture toughness will depend on the cement line's ability to compensate for the pore in an osteon. Matrix to cement line fracture toughness ratio significantly increased with increasing age, suggesting that the effectiveness of osteons in energy absorption may be reduced in the elderly due to a change in cement line properties. (C) 2000 John Wiley & Sons, Inc.
The fracture toughness of the right femoral neck, femoral shaft, and tibial shaft of matched cadaveric bones, ages 50 to 90 years, was compared. Results of this study indicate that tensile (G(Ic)) and shear (G(IIc)) fracture toughness vary depending on bone location. The femoral neck has the greatest resistance to crack initiation for both tension and shear loading while the femoral shaft has the least. The relationship between age and the fracture toughness of the femoral neck and shaft was investigated. G(c) of the femoral shaft significantly decreased with age for mode I and was nearly significant for mode II. Fracture toughness of the femoral neck did not change with age for the later decades of life. Implications of these findings are discussed. (C) 2000 John Wiley & Sons, Inc.
"... In this work, we shall focus on the brittle fracture of elastic materials. For that we will perform a numerical analysis of a cracked plate in a plane stress situation. This requires three distinct problems to be solved. Firstly, numerical methods to determine the stress and displacement fields around the crack must be available. The second problem consists of the numerical computation of the fracture parameters, such as the stress intensity factors, the J integral, the energy release rate or another. Finally one needs to decide on criteria to determine under which conditions the crack will propagate, as well as the direction of propagation."
Full article available on: https://www.win.tue.nl/analysis/reports/rana07-23.pdf
Cortical bone is notch sensitive and the presence of surface cracks significantly reduces the energy absorbed during fracture for both longitudinal and transverse fracture directions [1]. A variety of fracture mechanics techniques have been utilized to assess the fracture toughness of bone specimens with a “characterized” crack, which have included measurements of the critical strain energy release rate (GC) (or the specific surface energy γ, = GC/2), and the critical stress intensity factor, KC [2–5]. The specimen geometries used in the earlier experiments all produced rapid crack propagation, with an unknown and variable crack velocity. More recently, in contrast, with the use of the compact tension method [6], it has been possible to propagate a crack in bone at a relatively slow and measurable rate and GC and KC values for transversely oriented bovine femur and tibia bone specimens were determined by this method for various crack velocities [7–8]. The results of these investigations are shown for comparison in Table 1.
The paper examines the behavior of anisotropic compact bone in tension at a range of strain rates. Specimens of fresh bovine
bone were loaded at strain rates between. 001 and 200 sec???1. This bone was shown to exhibit considerable plasticity throughout the range, except when tested in a direction normal to
the long axis. The modulus of elasticity, breaking stress and breaking strain were found to vary with strain rate. There is
a maximum energy absorbtion capability at a strain rate of .1 sec???1.
An evaluation of the fracture characteristics of bovine tibia compact tension specimens associated with controlled crack propagation in the longitudinal direction has been made. The fracture mechanics parameters of critical strain energy release rate (G
c) and critical stress intensity factor (K
c) were determined for a range of crack velocities. A comparative fracture energy (W) was also evaluated from the area under the load-deflection curve. It was found that an increase in the average crack velocity from 1.75 to 23.610–5 m sec–1 produced increases in G
c (from 1736 to 2796 J m–2), K
c (from 4.46 to 5.38 MN m–3/2) and W. At crack velocities >23.610–5 m sec–1, W decreased appreciably. Microstructural observations indicated that, for crack velocities –1, relatively rough fracture surfaces were produced by the passage of the crack around intersecting osteons (or lamellae), together with some osteon pull-out. In contrast, at a higher crack velocity, fracture was characterized by relatively smooth surfaces, as the crack moved indiscriminately through the microstructural constituents.
A shock tube technique was utilized to assess the impact fracture of bovine femur compact bone, in the form of cylindrical tube specimens. The effect of wall thickness on the fracture pressure was established and an unique hoop stress (= 18.0 MN m–2) for fracture obtained. In addition, the effect of crack length and radius of curvature of the crack tip on the fracture hoop stress was established, together with a determination of Young's modulus at the shock loading rate. The possible correlations of these results with fracture mechanics concepts are discussed.
The microscopic yield stress (the stress to produce a plastic strain of 2×10<sup>-6</sup> in./in.) and the variation of plastic strain with applied stress have been established for bone. A feature of the deformation is the large anelastic contraction which occurs after unloading plastically strained specimens. The fracture characteristics of bone, as determined in impact loading and tensile experiments, are a function of specimen orientation and are also notch sensitive. A study of the temperature dependence of fracture in the range from -190° to 900°C shows that specimens oriented parallel to the length of the bone exhibit a pronounced maximum in strength at 0°C.
Thesis (Ph. D.)--Dept. of Materials Science, Stanford University. Includes bibliographical references.
Book on plain strain crack toughness testing of high strength metallic materials
THE anisotropy in elastic deformation exhibited by compact bone sections is well known1, but has been démonstrated mainly for orientations parallel to, (designated as longitudinal) or, normal to, (designated as transverse or radial), the long axis of the bone. Various models based on the concept that bone is a hydroxyapatite-reinforced collagen composite have been proposed (for example ref. 2) to account for the elastic behaviour of compact bone, as defined by the Young's modulus (E), but lack of data for a range of orientations has precluded one critical test of their validity. We report here preliminary measurements on the Young's modulus of mature bovine femur cortical bone for specimens orientated at various angles from 0° to 90° to the bone long axis. These results were obtained on a series of rectangular-shaped, machined, specimens (4 mm × 3 mm) with two different lengths (15 mm and 20 mm), by a novel differential ultrasonic technique, which is illustrated in Figs 1 and 2.
The fracture mechanics parameters associated with the fracture of transversely oriented bovine femur compact tension specimens have been determined for a slowly propagating and stable crack, as a function of cross head speed. It was found that an increase in cross head speed from 1.7-33 × 10-6 m sec-1 produced an increase in the crack velocity from 2.1-27 × 10-5 m sec-1 and an associated increase in the critical strain energy release rate (Gc) from 920 to 2780 J m-2 and in the critical stress intensity factor (Kc) from 2.4 to 5.2 MN m- 3 2.
An investigation has been made of the compressive strength of the porcine mandible and its depedence upon microstructure and strain rate. The results are compared with the fracture behavior of bovine femoral bone. Regarding microstructural dependence, it was found that fracture behavior depends upon regularity of structure, morphology of subunits, orientation of lamellae with respect to the stress axis, amount of ground substance, density and mineral content. Fracture mode was found to be a strong function of strain rate. For both porcine mandibular and bovine femoral bone, there is a ductile-to-brittle transition which results in a change of strain rate sensitivity coefficient from 0.1 to 0.0 at the transition region. This is corroborated by a large change in work-to-fracture values at this region. Therefore, the existence of a critical velocity for bone is supported by the present data.
Linear elastic fracture mechanics was used to study longitudinal crack propagation in bovine compact bone. The resistance to crack initiation in slotted specimens was examined by measuring both the critical stress intensity factor, Kc, and the critical strain energy release rate, Gc. A change in specimen thickness (from 0.185 to 0.380 cm) did not affect the value of Kc (or Gc). A significant positive correlation (P < 0.01) was found between Kc (or Gc) and dry density. A comparison of the experimentally determined effective modulus (relating Kc and Gc) with an effective modulus calculated from the transversely isotropic model for bone showed good agreement.
The tensile fracture stress (σfr) of longitudinal bovine tibia compact bone specimens was measured as a function of the length (c) and radius of curvature (r) of machined edge cracks. It was established that, for a given value of r, then where A and B are constants. A fracture mechanics method was utilised to derive values of the fracture toughness (as defined by the critical stress intensity factor, KIC), the specific surface energy (γ) and the “intrinsic” flaw size (c0). The advantages and limitations of this approach are discussed.
AcCno~~ledgements-The provision of a Research Student-ship by the Science and Engineering Research Council for J. C. Behiri is gratefully acknowledged Fracture mechanics of bone Crack velocity depen-dence of longitudinal fracture in bone
- J C Behiri
- W Bonfield
0 mm), there was no significant effect of thick-ness on K, or G,. AcCno~~ledgements-The provision of a Research Student-ship by the Science and Engineering Research Council for J. C. Behiri is gratefully acknowledged. REFERENCES Behiri, J. C. (1982) Fracture mechanics of bone. Ph.D. Thesis, University of London. Behiri, J. C. and Bonfield, W. (1980) Crack velocity depen-dence of longitudinal fracture in bone. J. Mater. Sci. 15, 1841-1849.
Dynamic response of biological materials. A.S.M.E. publication 65-WA-HUF-9, ;;p$3. American Society of Mechanical Engineers Studies 9f f?acture in bone Compressive strength of mandibular bone as a function of microstruc-ture and strain rate
- F G Evans
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- Il Springfield
- J H Mcelhaney
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Evans, F. G. (1973) Mechanical Properties of Bone. Thomas, Springfield, IL. McElhaney, J. H. and Byers, E. F. (1965) Dynamic response of biological materials. A.S.M.E. publication 65-WA-HUF-9, ;;p$3. American Society of Mechanical Engineers. New Margel-Robertson, D. (1973) Studies 9f f?acture in bone. Ph.D. thesis, Stanford University. Margel-Robertson, D. and Smith, D. C. (1978) Compressive strength of mandibular bone as a function of microstruc-ture and strain rate. J. Biomechanics 11, 455-471.