Fracture characterization of bone under mode II loading using the end loaded split test

Fracture energy release rate under mode II loading of bovine cortical bone is determined using a miniaturized testing device of the end loaded split test. The energy release rate is evaluated by means of a data reduction scheme based on specimen compliance, beam theory and crack equivalent concept. Experimental tests were carried out to evaluate the Resistance curve which provides a successful method to characterize fracture behavior of quasi-brittle materials like bone. A numerical analysis including a cohesive damage model was used to validate the procedure. It was demonstrated that the end loaded split test and proposed data reduction scheme provide a valuable solution for mode II fracture characterization of bone.     

The double cantilever beam test applied to mode I fracture characterization of cortical bone tissue

The primary objective of this work was to analyse the adequacy of the Double Cantilever Beam (DCB) test in determining fracture toughness under pure mode I loading of cortical bone tissue. A new data reduction scheme based on specimen compliance and the crack equivalent concept was used to overcome the difficulties inherent in crack monitoring during its growth. It provides a complete resistance curve, which is fundamental in estimating the fracture energy. A cohesive zone model was used to simulate damage initiation and propagation, thus assessing the efficacy of the proposed testing method and data reduction scheme. Subsequently, the DCB test was applied to evaluate the mode I fracture energy of hydrated and thermally dehydrated cortical bone tissue from young bovine femur, in the tangential–longitudinal propagation system. The results obtained demonstrate the efficacy of the DCB test and the proposed data reduction scheme on the bone fracture characterization under mode I loading.     

皮持骨在拉伸型,剪切型和撕裂型加载条件下的断裂韧性——纵向断裂…

对以质骨在拉伸、剪切和撕裂型载荷下的裂纹启裂韧性进行了研究。总数为１３０个紧凑拉式样,紧凑剪切试样和三腿型试样分别用于测量骨的拉伸型、剪切型和撕裂型启裂韧性。多试样柔度法用来测定当ａ／Ｗ＝０．５５（１,裂纹长度,Ｗ,试样宽度）时的临界能量释放率。临界应力强度因子由ａ／Ｗ＝０．５５的试样在试验中得到的临界载荷来计算。为了考察骨力学 各是性对于它的剪切型和撕裂裂纹启裂韧性的影响,对骨试样的裂纹扩展方向     

Mixed-mode fracture of human cortical bone

Although the mode I (tensile opening) fracture toughness has been the focus of most fracture mechanics studies of human cortical bone, bones in vivo are invariably loaded multiaxially. Consequently, an understanding of mixed-mode fracture is necessary to determine whether a mode I fracture toughness test provides the appropriate information to accurately quantify fracture risk. In this study, we examine the mixed-mode fracture of human cortical bone by characterizing the crack-initiation fracture toughness in the transverse (breaking) orientation under combined mode I (tensile opening) plus mode II (shear) loading using samples loaded in symmetric and asymmetric four-point bending. Whereas in most structural materials, the fracture toughness is increased with increasing mode-mixity (i.e., where the shear loading component gets larger), in the transverse orientation of bone the situation is quite different. Indeed, the competition between the maximum applied mechanical mixed-mode driving force and the weakest microstructural paths in bone results in a behavior that is distinctly different to most homogeneous brittle materials. Specifically, in this orientation, the fracture toughness of bone is markedly decreased with increasing mode-mixity.     

Fracture length scales in human cortical bone: the necessity of nonlinear fracture models

Recently published data for fracture in human humeral cortical bone are analyzed using cohesive-zone models to deal with the nonlinear processes of material failure. Such models represent the nonlinear deformation processes involved in fracture by cohesive tractions exerted by the failing material along a fracture process zone, rather than attributing all damage to a process occurring at a single point, as in conventional linear-elastic fracture mechanics (LEFM). The relationship between the tractions and the net displacement discontinuity across the process zone is hypothesized to be a material property for bone. To test this hypothesis, the cohesive law was evaluated by analyzing published load vs. load-point displacement data from one laboratory; the calibrated law was then used to predict similar data taken for a different source of bone using a different specimen geometry in a different laboratory. Further model calculations are presented to illustrate more general characteristics of the nonlinear fracture of bone and to demonstrate in particular that LEFM is not internally consistent for all cases of interest. For example, the fracture toughness of bone deduced via LEFM from test data is not necessarily a material constant, but will take different values for different crack lengths and test configurations. LEFM is valid when the crack is much longer than a certain length scale, representative of the length of the process zone in the cohesive model, which for human cortical bone ranges from 3 to 10mm. Since naturally occurring bones and the specimens used to test them are not much larger than this dimension for most relevant orientations, it is apparent that only nonlinear fracture models can give an internally consistent account of their fracture. The cohesive law is thus a more complete representation of the mechanics of material failure than the single-parameter fracture toughness and may therefore provide a superior measure of bone quality. The analysis of fracture data also requires proper representation of the approximately orthotropic elasticity of the bone specimen; if the specimen is incorrectly assumed to be isotropic, the initial measured compliance cannot be reproduced to within a factor of four and the fracture toughness deduced from the measured work of fracture will be overestimated by approximately 30%.     

Fracture toughening mechanism of cortical bone: an experimental and numerical approach

In this investigation, the crack propagation mechanisms contributing to the toughness of cortical bone were studied using a combination of experimental and numerical approaches. Compact tension (CT) specimens were prepared from bovine cortical bones to achieve crack propagation in the longitudinal and transverse directions. Stable crack extension experiments were conducted to distinguish the crack growth resistance curves, and virtual multidimensional internal bond (VMIB) modeling was adopted to simulate the fracture responses. Results from experiments indicated that cortical bone exhibited rising resistance curves (R-curves) for crack extension parallel and perpendicular to the bone axis; the transverse fracture toughness was significantly larger, indicating that the fracture properties of cortical bone are substantially anisotropic. Microscopic observations showed that the toughening mechanisms in the longitudinal and transverse directions were different. When the crack grew in the transverse direction, the crack deflected significantly, and crack bifurcations were found at the crack wake, while, in the longitudinal direction, the crack was straight and uncracked ligaments were observed. Numerical simulations also revealed that the fracture resistance in the transverse direction was greater than that in the longitudinal direction.     

Experimental study of damage and fracture of cancellous bone using a digital speckle correlation method

Cancellous bone is a widespread structure in a creatural body, for instance, in the femoral head and spondyle. The damage evolution and crack growth of cattle cancellous bone were studied under three-point-bending load conditions. A series of speckle images with deformation information surrounding the crack tip were recorded, and the full-field displacement distributions were obtained at different loading levels by means of digital speckle correlation method (DSCM). Characterizations of the damage deformation and fracture of cancellous bone were analyzed. These results provide some useful information for studying the fracture behavior of cancellous bone.     

Bone–cement interfacial behaviour under mixed mode loading conditions

Interfacial behaviour of the bone–cement interface has been studied under tensile, shear and mixed mode loading conditions. Bovine cancellous bone was used to bond with acrylic bone cement to form bone–cement interface samples, which were mechanically tested under selected tensile, shear and mixed mode loading conditions. The influence of the loading angle and the extent of the cement penetration on the interfacial behaviour were examined. The failure mechanisms with regard to loading mode were examined using micro-focus computed tomography. The measured tensile and shear responses were utilized in a cohesive zone constitutive model, from which the pre-yield linear and the post-yield exponential strain softening behaviour under mixed mode loading conditions was predicted. The implications of the work on the studies of cemented joint replacements are also discussed.     

Ultrasonic propagation in cortical bone mimics

Understanding the velocity and attenuation characteristics of ultrasonic waves in cortical bone and bone mimics is important for studies of osteoporosis and fractures. Three complementary approaches have been used to help understand the ultrasound propagation in cortical bone and bone mimics immersed in water, which is used to simulate the surrounding tissue in vivo. The approaches used were Lamb wave propagation analysis, experimental measurement and two-dimensional (2D) finite difference modelling. First, the water loading effects on the free plate Lamb modes in acrylic and human cortical bone plates were examined. This theoretical study revealed that both the S0 and S1 mode velocity curves are significantly changed in acrylic: mode jumping occurs between the S0 and S1 dispersion curves. However, in human cortical bone plates, only the S1 mode curve is significantly altered by water loading, with the S0 mode exhibiting a small deviation from the unloaded curve. The Lamb wave theory predictions for velocity and attenuation were then tested experimentally on acrylic plates using an axial transmission technique. Finally, 2D finite difference numerical simulations of the experimental measurements were performed. The predictions from Lamb wave theory do not correspond to the measured and simulated first arrival signal (FAS) velocity and attenuation results for acrylic and human cortical bone plates obtained using the axial transmission technique, except in very thin plates.     

Finite element analysis of the influence of loading rate on a model of the full lumbar spine under dynamic loading conditions

Despite an increase in the number of experimental and numerical studies dedicated to spinal trauma, the influence of the rate of loading or displacement on lumbar spine injuries remains unclear. In the present work, we developed a bio-realistic finite element model (FEM) of the lumbar spine using a comprehensive geometrical representation of spinal components and material laws that include strain rate dependency, bone fracture, and ligament failure. The FEM was validated against published experimental data and used to compare the initiation sites of spinal injuries under low (LD) and high (HD) dynamic compression, flexion, extension, anterior shear, and posterior shear. Simulations resulted in force-displacement and moment-angular rotation curves well within experimental corridors, with the exception of LD flexion where angular stiffness was higher than experimental values. Such a discrepancy is attributed to the initial toe-region of the ligaments not being included in the material law used in the study. Spinal injuries were observed at different initiation sites under LD and HD loading conditions, except under shear loads. These findings suggest that the strain rate dependent behavior of spinal components plays a significant role in load-sharing and failure mechanisms of the spine under different loading conditions.     

Aspects of in vitro fatigue in human cortical bone: time and cycle dependent crack growth

Although fatigue damage in bone induced by cyclic loading has been recognized as a problem of clinical significance, few fracture mechanics based studies have investigated how incipient cracks grow by fatigue in this material. In the present study, in vitro cyclic fatigue experiments were performed in order to quantify fatigue-crack growth behavior in human cortical bone. Crack-growth rates spanning five orders of magnitude were obtained for the extension of macroscopic cracks in the proximal-distal direction; growth-rate data could be well characterized by the linear-elastic stress-intensity range, using a simple (Paris) power law with exponents ranging from 4.4 to 9.5. Mechanistically, to discern whether such behavior results from "true" cyclic fatigue damage or is simply associated with a succession of quasi-static fracture events, cyclic crack-growth rates were compared to those measured under sustained (non-cyclic) loading. Measured fatigue-crack growth rates were found to exceed those "predicted" from the sustained load data at low growth rates ( approximately 3 x 10(-10) to 5 x 10(-7) m/cycle), suggesting that a "true" cyclic fatigue mechanism, such as alternating blunting and re-sharpening of the crack tip, is active in bone. Conversely, at higher growth rates ( approximately 5 x 10(-7) to 3 x 10(-5) m/cycle), the crack-growth data under sustained loads integrated over the loading cycle reasonably predicts the cyclic fatigue data, indicating that quasi-static fracture mechanisms predominate. The results are discussed in light of the occurrence of fatigue-related stress fractures in cortical bone.     

Fracture toughness is dependent on bone location--a study of the femoral neck, femoral shaft, and the tibial shaft

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.     

Analysis of miniature single- and double-notch bending specimens for estimating the fracture toughness of cortical bone

Studies of the fracture behavior of cortical bone have determined multiple toughening mechanisms that are active during propagation of a crack. Common methods for measuring bone fracture toughness use single-notched specimens often in four-point (SN4PB) or three-point bending (SN3PB). A double-notch four-point bending (DN4PB) specimen is useful to study prefailure damage at the crack tip. Total failure occurs at one notch and only partial failure at the other allowing study of prefailure damage in the unbroken notch. There is no widely known method for calculating the fracture toughness of bone using a DN4PB specimen. A method for calculating the fracture toughness of cortical bone using a DN4PB is developed here and compared with results for a common SN3PB specimen. The new double-notch method permits using a single specimen to measure apparent fracture toughness and to study both pre- and postfailure microdamage in the bone matrix. When and how to use the new and the established test specimens for understanding bone mechanics is discussed.     

The effect of strain rate on fracture toughness of human cortical bone: a finite element study

Evaluating the mechanical response of bone under high loading rates is crucial to understanding fractures in traumatic accidents or falls. In the current study, a computational approach based on cohesive finite element modeling was employed to evaluate the effect of strain rate on fracture toughness of human cortical bone. Two-dimensional compact tension specimen models were simulated to evaluate the change in initiation and propagation fracture toughness with increasing strain rate (range: 0.08–18 s⁻¹). In addition, the effect of porosity in combination with strain rate was assessed using three-dimensional models of micro-computed tomography-based compact tension specimens. The simulation results showed that bone’s resistance against the propagation of a crack decreased sharply with increase in strain rates up to 1 s⁻¹ and attained an almost constant value for strain rates larger than 1 s⁻¹. On the other hand, initiation fracture toughness exhibited a more gradual decrease throughout the strain rates. There was a significant positive correlation between the experimentally measured number of microcracks and the fracture toughness found in the simulations. Furthermore, the simulation results showed that the amount of porosity did not affect the way initiation fracture toughness decreased with increasing strain rates, whereas it exacerbated the same strain rate effect when propagation fracture toughness was considered. These results suggest that strain rates associated with falls lead to a dramatic reduction in bone’s resistance against crack propagation. The compromised fracture resistance of bone at loads exceeding normal activities indicates a sharp reduction and/or absence of toughening mechanisms in bone during high strain conditions associated with traumatic fracture.     

Sub-10-micrometer toughening and crack tip toughness of dental enamel

In previous studies, enamel showed indications to occlude small cracks in-vivo and exhibited R-curve behaviors for bigger cracks ex-vivo. This study quantifies the crack tip's toughness (K(I0),K(III0)), the crack's closure stress and the cohesive zone size at the crack tip of enamel and investigates the toughening mechanisms near the crack tip down to the length scale of a single enamel crystallite. The crack-opening-displacement (COD) profile of cracks induced by Vickers indents on mature bovine enamel was studied using atomic force microscopy (AFM). The mode I crack tip toughness K(I0) of cracks along enamel rod boundaries and across enamel rods exhibit a similar range of values: K(I0,Ir)=0.5-1.6MPa?m(0.5) (based on Irwin's 'near-field' solution) and K(I0,cz)=0.8-1.5MPa?m(0.5) (based on the cohesive zone solution of the Dugdale-Muskhelishvili (DM) crack model). The mode III crack tip toughness K(III0,Ir) was computed as 0.02-0.15MPa?m(0.5). The crack-closure stress at the crack tip was computed as 163-770?MPa with a cohesive zone length and width 1.6-10.1μm and 24-44?nm utilizing the cohesive zone solution. Toughening elements were observed under AFM and SEM: crack bridging due to protein ligament and hydroxyapatite fibres (micro- and nanometer scale) as well as microcracks were identified.     

Mixed-mode stress intensity factors for kink cracks with finite kink length loaded in tension and bending: Application to dentin and enamel

Fracture toughness resistance curves describe a material’s resistance against crack propagation. These curves are often used to characterize biomaterials like bone, nacre or dentin as these materials commonly exhibit a pronounced increase in fracture toughness with crack extension due to co-acting mechanisms such as crack bridging, crack deflection and microcracking. The knowledge of appropriate stress intensity factors which depend on the sample and crack geometry is essential for determining these curves. For the dental biomaterials enamel and dentin it was observed that, under bending and tensile loading, crack propagation occurs under certain constant angles to the initial notch direction during testing procedures used for fracture resistance curve determination. For this special crack geometry (a kink crack of finite length in a finite body) appropriate geometric function solutions are missing. Hence, we present in this study new mixed-mode stress intensity factors for kink cracks with finite kink length within samples of finite dimensions for two loading cases (tension and bending) which were derived from a combination of mixed-mode stress intensity factors of kink cracks with infinitely small kinks and of slant cracks. These results were further applied to determine the fracture resistance curves of enamel and dentin by testing single edge notched bending (SENB) specimens. It was found that kink cracks with finite kink length exhibit identical stress fields to slant cracks as soon as the kink length exceeds 0.15 times the initial straight crack or notch length. The use of stress intensity factor solutions for infinitely small kink cracks for the determination of dentin fracture resistance curves (as was done by other researchers) leads to an overestimation of dentin’s fracture resistance of up to 30%.     

A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion

In this paper, a simple and practical finite element (FE) model coupled to a quasi-brittle damage law to describe the initiation and progressive propagation of multiple cracks based on element deletion is developed to predict the complete force–displacement curve and the fracture pattern of a human proximal femur under quasi-static load. The motivation of this work was to propose a FE model for possible clinical use with a good compromise between complexity and capability of the simulation. The model considers a limited number of parameters that can predict proximal femur fracture in more adequate physical terms than criteria-based fracture models. Based on experimental results, different damage laws for cortical and trabecular bone are proposed to describe inelastic damage accumulation under excessive load. When the damage parameter reaches its critical value inside an element of the mesh, its stiffness matrix is set to zero, leading to the redistribution of the stress state in the vicinity of the damaged zone (crack initiation). Once a crack is initiated, the propagation direction is simulated by the propagation of the broken elements of the mesh. To illustrate the potential of the proposed approach, the left femur of a male (age 61) previously investigated by Keyak and Falkinstein [37] (Model B: male, age 61) was simulated till complete fracture under one-legged stance quasi-static load. The proposed finite element model leads to more physical results concerning the shape of the force–displacement curve (yielding and fracturing) and the profile of the fractured edge.     

基于不同应变判定准则模拟皮质骨断裂的准确性

文题释义：基于等效应变断裂模拟：即在大鼠股骨皮质骨断裂模拟过程中，应用皮质骨有限元模型在外部载荷作用下所产生的等效应变数值，与皮质骨组织的失效应变进行对比，当等效应变数值大于皮质骨组织失效应变时，有限元模型内的单元便发生失效，直至失效单元达到一定数量，模型便发生整体失效，此过程为基于等效应变的断裂模拟。 基于主应变断裂模拟：即在大鼠股骨皮质骨断裂模拟过程中，应用皮质骨有限元模型在外部载荷作用下所产生的主应变数值，与皮质骨组织的失效应变进行对比，当主应变数值大于皮质骨组织失效应变时，有限元模型内的单元便发生失效，直至失效单元达到一定数量，模型便发生整体失效，此过程为基于主应变的断裂模拟。 背景：由于意外碰撞等外力因素所产生的皮质骨裂纹是引起骨折的重要原因之一，要防止此类骨折发生，首先需弄清不同载荷作用下皮质骨裂纹的产生与扩展机制。由于实验分析对样本具有破坏性，难以同时了解骨结构在断裂前后的内部力学状态，找到一种能够准确模拟皮质骨从裂纹产生、扩展，直至断裂过程的有限元方法就显得尤为重要。当前模拟方法主要应用主应变或等效应变判定模型单元力学状态，继而进行断裂模拟，却鲜有关于这2种应变进行模拟准确性的探究。 目的：验证应用主应变与等效应变进行皮质骨断裂模拟的准确程度。 方法：结合实验与仿真分析，应用主应变与等效应变进行皮质骨断裂模拟，将仿真与实验结果进行对比，确定应用哪种应变进行模拟更加准确。 结果与结论：①应用主应变模拟的皮质骨断裂时间要明显晚于应用等效应变；②通过与实验对比发现，相比主应变，应用等效应变进行仿真所得结果与实验值更为接近；③因此，应用等效应变进行皮质骨断裂模拟相对更加准确。 ORCID: 0000-0003-0313-1359(王伟军) 中国组织工程研究杂志出版内容重点：人工关节；骨植入物；脊柱；骨折；内固定；数字化骨科；组织工程     

Mechanisms of short crack growth at constant stress in bone

This paper describes an experimental study of the growth of small (i.e. sub-millimetre) cracks in samples of cortical bone subjected to a constant tensile stress. Slow, stable crack growth occurred at a rate and angle which were dependent on the orientation of the sample: tests were conducted with the loading axis both parallel and perpendicular to the longitudinal axis of the bone. All cracks showed intermittent growth in which periods of relatively rapid propagation alternated with periods of temporary crack arrest or relatively slow growth. In some cases crack arrest could be clearly linked to microstructural features such as osteons or Volkmann's canals, which acted as barriers to crack growth. Crack-opening displacement increased over time during the arrest periods. These observations suggest a mechanism for the growth of small cracks in bone at constant stress, involving microstructural barriers, time-dependent deformation of material near the crack tip and strain-controlled propagation.     

Study of the toughening mechanisms in bone and biomimetic hydroxyapatite materials using Raman microprobe spectroscopy

A Raman microprobe spectroscopy characterization of microscopic fracture mechanisms is presented for a natural hydroxyapatite material (cortical bovine femur) and two synthetic hydroxyapatite-based materials with biomimetic structures-a hydroxyapatite skeleton interpenetrated with a metallic (silver) or a polymeric (nylon-6) phase. In both the natural and synthetic materials, a conspicuous amount of toughening arose from a microscopic crack-bridging mechanism operated by elasto-plastic stretching of unbroken second-phase ligaments along the crack wake. This mechanism led to a rising R-curve behavior. An additional micromechanism, responsible for stress relaxation at the crack tip, was recognized in the natural bone material and was partly mimicked in the hydroxyapatite/silver composite. This crack-tip mechanism conspicuously enhanced the cortical bone material resistance to fracture initiation. A piezo-spectroscopic technique, based on a microprobe measurement of 980 cm(-1) Raman line of hydroxyapatite, enabled us to quantitatively assess in situ the microscopic stress fields developed during fracture both at the crack tip and along the crack wake. Using the Raman piezo-spectroscopy technique, toughening mechanisms were assessed quantitatively and rationally related to the macroscopic fracture characteristics of hydroxyapatite-based materials.