Fracture mechanics of bone with short cracks

Tensile fracture experiments were performed upon specimens of wet mature bovine Haversian bone, with short, controlled notches. Stress concentration factors were found to be significantly less than values predicted using a maximum stress criterion in the theory of elasticity. Results were also modeled with the aid of linear elastic fracture mechanics. Agreement of experiment with theory was better in this case, however deviations were seen for short notches. Two mechanisms were evaluated for the behavior: plasticity near the crack tip, and effects of the Haversian microstructure, modelled by Cosserat elasticity, a generalized continuum theory. Plastic zone effects were found to be insignificant. Cosserat elasticity, by contrast, predicted stress concentration factors which better approximated observed values. To explore strain redistribution processes, further experiments were conducted upon notched specimens in torsion at small strain. They disclosed a strain redistribution effect consistent with Cosserat elasticity. These microelastic effects were attributed to the Haversian architecture of bone. The implications of the results are that bone resists the effect of stress raisers such as fatigue microcracks and surgical sawcuts to a much greater extent than anticipated on the basis of its elastic or elastoplastic properties.     

Molecular dynamics simulation of crack growth in pure titanium under uniaxial tension

The analysis of crack growth in titanium was performed using molecular dynamics simulation with Embedded Atom Method potentials. The effect of temperature and strain rate on the mechanism of crack growth and the change of microstructure were discussed. After setting an initial crack, the specimen was subjected to uniaxial tension strain up to the total strain level of 0.2 with a constant strain rate. During the period, the shape and the microstructure of crack tip as well as the stress–strain curves were monitored. In the simulation, the gather of voids and stress concentration leading to the crack growth occurred, which are in agreement with experimental results observed by transmission electron microscopy. The transformation from HCP to BCC also occurred at crack tip. The remarkable effect of temperature and strain rate on the growth direction and rate of stacking fault of crack tip was observed. Moreover, initial crack greatly lowered the tension yield point of pure titanium. In the stage of deformation, simulation results showed that loading strain rate and temperature strongly influenced peak stress point, which was increased by the low temperature and high strain, whereas the initial slope of the stress strain curve was independent of loading strain rate.     

Machine vision photogrammetry: a technique for measurement of microstructural strain in cortical bone.

Understanding local microstructural deformations and strains in cortical bone may lead to a better understanding of cortical bone damage development, fracture, and remodeling. Traditional experimental techniques for measuring deformation and strain do not allow characterization of these quantities at the microstructural level in cortical bone. This study describes a technique based on digital stereoimaging used to measure the microstructural strain fields in cortical bone. The technique allows the measurement of material surface displacements and strains by comparing images acquired from a specimen at two distinct stress states. The accuracy of the system is investigated by analyzing an undeformed image set; the test image is identical to the reference image but translated by a known pixel amount. An increase in the correlation sub-image train parameter results in an increase in displacement measurement accuracy from 0.049 to 0.012 pixels. Errors in strain calculated from the measured displacement field were between 39 and 564 microstrain depending upon the sub-image train size and applied image displacement. The presence of a microcrack in cortical bone results in local strain at the crack tip reaching 0.030 (30,000 microstrain) and 0.010 (10,000 microstrain) near osteocyte lacunae. It is expected that the use of this technique will allow a greater understanding of bone strength and fracture as well as bone mechanotransduction.     

The importance of the elastic and plastic components of strain in tensile and compressive fatigue of human cortical bone in relation to orthopaedic biomechanics

The longevity, success, or failure of an orthopaedic implant is dependent on its osseointegration especially within the initial six months of the initial surgery. The development of strains plays a crucial role in both bone modelling and remodelling. For remodelling, in particular, strains of substantial values are required to activate the osteoblastic and osteoclastic activity for the osseointegration of the implant. Bone, however, is subject to "damage" when strain levels exceed a certain threshold level. Damage is manifested in the form of microcracks; it is linked to increased elastic strain amplitudes and is accompanied by the development of "plastic" (irrecoverable, residual) strains. Such strains increase the likelihood for the implant to subside or loosen. The present study examines the rates (per cycle) by which these two components of strain (elastic and "plastic") develop during fatigue cycling in two loading modes, tension and compression. The results of this study show that these strain rates depend on the applied stress in both loading modes. It also shows that elastic and plastic strain rates can be linked to each other through simple power law relationships so that one can calculate or predict the latter from the former and vice versa. We anticipate that such basic bone biomechanics data would be of great benefit to both clinicians and bioengineers working in the field of FEA modelling applications and orthopaedic implant surgery.     

Fracture properties of bovine tibial bone

Linear elastic fracture mechanics predicts that the fracture stress of precracked materials is dependent on the length of the initial crack tip radius of curvature, as supported by the Griffith and Inglis equations. In order to determine the applicability of these equations and the effects of other variables, tensile specimens of bovine tibia were produced with edge cracks of known dimensions and tested to fracture. Longitudinal sheet tensile specimens were taken from the midposterior diaphysis of bovine tibiae that had been kept frozen in saline soaked towels. Each specimen had a milled gauge length of 25 mm, 16 mm width and 2 mm thickness. All specimen preparation was performed under a saline drip. An edge crack, centered along the gauge length, was milled in the specimen perpendicular to its long axis. The crack lengths used were 4, 6, 8, 10 and 12 mm. The crack tip was formed with drill bits having nominal diameters of 1/32, 1/16, and 3/32 in. All the combinations of crack length and crack tip radius were repeated five times for a total of 75 specimens. The testing order was randomly selected. Each specimen was tested in tension to fracture at a constant deformation rate of 7.5 X 10(-3) mm s-1, on an Instron mechanical testing device, and the fracture stress was measured. A linear load-deflection curve to fracture was exhibited by all of the specimens. The weight percent calcium of each specimen was determined by atomic absorption spectrophotometry. Microradiographs were used to determine the fractional void area and to histologically evaluate each bone sample.(ABSTRACT TRUNCATED AT 250 WORDS)     

Are plasticity models required to predict relative risk of lag screw cut-out in finite element models of trochanteric fracture fixation?

Using a finite element model of unstable trochanteric fracture stabilized with a sliding hip screw, the benefits of two plasticity-based formulations, Drucker–Prager and crushable foam, were evaluated and compared to the commonly used linear elastic model of trabecular bone in order to predict the relative risk of lag screw cut-out for five distinct load cases. The crushable foam plasticity formulation leads to a much greater strain localization, in comparison to the other two models, with large plastic strains in a localized region. The plastic zone predicted with Drucker–Prager is relatively more diffuse. Linear elasticity associated with a minimum principal strain criterion provides the smallest volume of elements susceptible to yielding for all loading modes. The region likely to undergo plastic deformation, as predicted by the linear elastic model, is similar to that obtained from plasticity-based formulations, which indicates that this simple criterion provides an adequate estimate of the risk of cut-out.     

Fracture analysis for biological materials with an expanded cohesive zone model

In this study, a theoretical framework for simulation of fracture of bone and bone-like materials is provided. An expanded cohesive zone model with thermodynamically consistent framework has been proposed and used to investigate the crack growth resistance of bone and bone-like materials. The reversible elastic deformation, irreversible plastic deformation caused by large deformation of soft protein matrix, and damage evidenced by the material separation and crack nucleation in the cohesive zone, were all taken into account in the model. Furthermore, the key mechanisms in deformation of biocomposites consisting of mineral platelets and protein interfacial layers were incorporated in the fracture process zone in this model, thereby overcoming the limitations of previous cohesive zone modeling of bone fracture. Finally, applications to fracture of cortical bone and human dentin were presented, which showed good agreement between numerical simulation and reported experiments and substantiated the effectiveness of the model in investigating the fracture behavior of bone-like materials.     

A model of tension and compression cracks with cohesive zone at a bone-cement interface

This paper gives an insight about compression and tension cracks as encountered at a bone-cement interface. Within the context of continuum theory of fracture, an analytical solution is presented for the problem of a bimaterial interface edge crack under uniaxial tension or compression, assuming no tangential slip along the crack faces since cement pedicles penetrate into the cancellous bone several millimeters. Also essential to the solution are cohesive zone effects that account for a strengthening mechanism over the crack faces. The solution provides a methodological framework for quantifying the influence of the cohesive zone on the magnitude of the stress singularity. Mode I crack tip stress intensity factors are calculated at different stages of the loading and unloading phases under uniaxial tension or compression. Finally, an inelastic mechanism is presented that gives theoretical support to explain the formation of interfacial compression cracks, a phenomenon that was not previously appreciated and that arises from the rigid cement being forced into the more compliant cancellous bone.     

Prediction of microdamage formation using a mineral-collagen composite model of bone

Age-related changes in bone quality are mainly manifested in the reduced toughness. Since the post-yield deformation of bone is realized through microdamage formation (e.g., microcracking and diffuse damage), it is necessary to understand the mechanism of microdamage formation in bone in order to elucidate underlying mechanisms of age-related bone fractures. In this study, a two-dimensional shear lag model was developed to predict stress concentration fields around an initial crack in a mineral-collagen composite. In this model, non-linear elasticity was assumed for the collagen phase, and linear elasticity for the mineral. Based on the pattern of the stress concentration fields, the condition for microdamage formation was discussed. The results of our analyses indicate that: (1) an initial crack formed in mineral phase may cause stress concentration in the adjacent mineral layers; (2) the pattern of stress concentration fields depends not only on the spatial but also mechanical properties of the collagen and mineral phases; (3) the pattern of the stress concentration fields could determine either coalescence or scattering of nano cracks around the initial crack.     

Fracture-toughening mechanisms responsible for differences in work to fracture of hydrated and dehydrated dentine

This study investigates the nature of deformation and differences in the mechanisms of fracture and properties of dentine where there has been a loss of moisture, as may occur with removal of the pulp in the endodontic treatment of teeth. Controlled fracture toughness testing was conducted on bovine teeth to determine the influence of hydration on the work of fracture of dentine. Significant differences (p<0.01) were observed between the fracture toughness of hydrated (554+/-27.7J/m2) and dehydrated (113+/-17.8J/m2) dentine. Observations of the crack tip region during crack extension revealed extensive ligament formation occurred behind the crack tip. These ligaments provide considerable stability to the crack by significantly increasing the work of fracture, thereby acting as a fracture-toughening mechanism. Micro-cracking, reported as a fracture-toughening mechanism in bone, is also clearly seen. A zone of in-elastic deformation may occur as hydrated specimens revealed upon crack extension, a region about the tip that appeared to suck water into the structure and to exude water behind the crack tip. In dehydrated dentine, no in-elastic zone was observed. Micro-cracking is present though the cracks are smaller, straighter and with less opening than hydrated dentine. Only limited ligament formation just behind the crack tip was observed. These differences resulted in a significantly lower work of fracture with unstable brittle fracture characteristics. Based on these results, several fracture-toughening mechanisms were identified in dentine, with micro-cracking not considered the most important. These findings may be relevant for bone, a similar mineralised hydrated tissue.     

The cellular transducer in damage-stimulated bone remodelling: a theoretical investigation using fracture mechanics

This paper reports on some theoretical work which used fracture mechanics concepts to draw conclusions about the nature of the so-called 'cellular transducer': the means by which bone cells detect the presence of damage and thus initiate remodelling and adaptation activities. Using analytical and numerical methods, we estimated the strains and displacements around cracks of the typical size, shape and orientation that normally occur in compact bone. We predicted that it is not possible for osteocytes or their processes to be fractured as a result of direct tensile strains, because the strains generated are much less than the expected failure strains of cellular material. We proposed a new failure mechanism by which osteocyte processes spanning the crack are cut by shearing motions between the crack faces. We predicted that failures of this type can occur. Failures begin to occur if crack lengths become greater than normal (100 microm), so this could act as a signal to initiate repair processes for individual cracks. Very large numbers of cell processes (greater than 1000) will fail if the crack length and/or applied stress reach dangerous levels (300 microm and 60 Mpa, respectively) at which point bone deposition may be required to prevent stress fractures. Similar results also occurred if we proposed a different mechanism of damage detection, involving cells' ability to detect the high levels of strain that occur near crack tips. This work, though based on theoretical mechanics considerations, suggests some biological experiments which might confirm our findings.     

Non-interacting modes for stress, strain and energy in anisotropic hard tissue.

The six non-interacting modes for stress, strain and energy in an orthotropic elastic model of human femoral cortical bone tissue are discussed and illustrated. The stress and strain modes are illustrated using the representation of the stress and strain fields around a circular hole in a flat plate of cortical bone subjected to a uniaxial field of tension as the example. The six modes play a role in the stress analysis of orthotropic elastic materials similar to the roles played by the hydrostatic and deviatoric non-interacting stress, strain and energy modes in isotropic elasticity. The biomechanical significance of the six non-interacting modes for stress, strain and energy in hard tissue is both practical and suggestive. The modes suggest a practical scheme for the representation of stress and strain fields in hard tissue. The existence of the modes suggests physical insights, for example, possible failure mechanisms or adaptation strategies possessed by the hard tissues.     

Internal strain gradients quantified in bone under load using high-energy X-ray scattering

High-energy synchrotron X-ray scattering (>60 keV) allows noninvasive quantification of internal strains within bone. In this proof-of-principle study, wide angle X-ray scattering maps internal strain vs position in cortical bone (murine tibia, bovine femur) under compression, specifically using the response of the mineral phase of carbonated hydroxyapatite. The technique relies on the response of the carbonated hydroxyapatite unit cells and their Debye cones (from nanocrystals correctly oriented for diffraction) to applied stress. Unstressed, the Debye cones produce circular rings on the two-dimensional X-ray detector while applied stress deforms the rings to ellipses centered on the transmitted beam. Ring ellipticity is then converted to strain via standard methods. Strain is measured repeatedly, at each specimen location for each applied stress. Experimental strains from wide angle X-ray scattering and an attached strain gage show bending of the rat tibia and agree qualitatively with results of a simplified finite element model. At their greatest, the apatite-derived strains approach 2500 με on one side of the tibia and are near zero on the other. Strains maps around a hole in the femoral bone block demonstrate the effect of the stress concentrator as loading increased and agree qualitatively with the finite element model. Experimentally, residual strains of approximately 2000 με are present initially, and strain rises to approximately 4500 με at 95 MPa applied stress (about 1000 με above the strain in the surrounding material). The experimental data suggest uneven loading which is reproduced qualitatively with finite element modeling.     

Nonlinear viscoelasticity and the thrombelastograph: 1. Studies on bovine plasma clots

To study the possible role of nonlinear viscoelastic effects in the thrombelastograph (TEG), clotting of bovine plasma was studied by both thrombelastography and with a controlled strain rheometer. Clot rheology is dominated by elastic effects at frequencies of interest. There is a well-defined regime of linear elasticity for strains less than around 2%, while at larger strains the clots show significant strain hardening. Oscillatory shear applied during clotting has little effect on the resulting clot provided that the strain is less than 2%, but leads to substantial weakening of clots formed at larger strains. The TEG operates within the regime of nonlinear elasticity, significantly obscuring the interpretation of TEG amplitude in terms of an elastic modulus. Comparing the results of standard TEG experiments with those conducted with a modified TEG, having no oscillation during clotting, shows that deformation during standard thrombelastography leads to weaker clots than are produced under quiescent conditions.     

Fracture analysis of single-layer graphene sheets with edge crack under tension

In this study, the fracture of single-layered graphene sheets (SLGSs) with edge crack under simple tension is investigated using molecular dynamics simulations, and the variations in fracture strength of SLGSs with crack length, strain rate and temperature are analysed. It is found that the existing edge crack weakens mechanical properties of SLGSs. Fracture strength and strain decrease with the increase in crack length and temperature, but increase with the increase in strain rate. It is also shown that shorter initial cracks propagate faster than longer initial cracks, but shorter initial cracks begin propagating at higher axial strain at a certain temperature and strain rate. And cracks are found to propagate faster in higher strain rates.     

Mechanical properties of pristine and nanoporous graphene

We present molecular dynamics simulations of monolayer graphene under uniaxial tensile loading. The Morse, bending angle, torsion and Lennard-Jones potential functions are adopted within the mdFOAM library in the OpenFOAM software, to describe the molecular interactions in graphene. A well-validated graphene model using these set of potentials is not yet available. In this work, we investigate the accuracy of the mechanical properties of graphene when derived using these simpler potentials, compared to the more commonly used complex potentials such as the Tersoff-Brenner and AIREBO potentials. The computational speed up of our approach, which scales O(1.5N), where N is the number of carbon atoms, enabled us to vary a larger number of system parameters, including graphene sheet orientation, size, temperature and concentration of nanopores. The resultant effect on the elastic modulus, fracture stress and fracture strain is investigated. Our simulations show that graphene is anisotropic, and its mechanical properties are dependant on the sheet size. An increase in system temperature results in a significant reduction in the fracture stress and strain. Simulations of nanoporous graphene were created by distributing vacancy defects, both randomly and uniformly, across the lattice. We find that the fracture stress decreases substantially with increasing defect density. The elastic modulus was found to be constant up to around 5% vacancy defects, and decreases for higher defect densities.     

Compliance calibration for fracture testing of equine cortical bone

An experimental compliance calibration method for measuring crack length in fracture toughness tests of cortical bone was developed. Calibration tests were conducted on twenty compact type fracture specimens machined from the mid-diaphysis of five pairs of equine third metacarpal bones. Specimens were oriented for crack propagation in a direction transverse to the longitudinal axis of the bone. Specimen compliance was determined from the load vs. crack opening displacement record over a range of crack lengths from 0.48 to 0.75 times the specimen width. The results demonstrate that the compliance calibration method developed for isotropic materials can be used to determine crack length in bone, which is transversely isotropic. However, specimens from lateral and dorsal regions exhibited significantly different compliance calibrations even after differences in elastic modulus were taken into account in the normalized compliance.     

A crack model of a bone cement interface

This paper is concerned with the fracture mechanics of a bone-cement interface that includes a cohesive zone effect on the crack faces. This accounts for the experimentally observed strengthening mechanism due to the mechanical interlock between the crack faces. Edge crack models are developed where the cohesive zone is simulated by a continuous or a discrete distribution of linear or nonlinear springs. It is shown that the solution obtained by assuming a homogeneous material is fairly close to the exact solution for the bimaterial interface edge crack problem. On the basis of that approximation, the analysis is conducted for the problem of two interacting edge cracks, one at the interface, and the other one in the cement. The small crack that was observed to initiate in the cement, close to the bone-cement interface, does not affect much the mode I stress-intensity factor at the tip of the interface crack. However it may grow, leading to a catastrophic breakdown of the cement. The analysis and following discussion point out an interdependency between bone-cement interface strength and cement strength not previously appreciated. The suggested crack models provide a framework for quantifying the fracture mechanisms at the bone-cement interface.     

Effect of temperature on the fracture toughness of compact bone

Bone is a composite composed mainly of organics, minerals, and water. Many researchers have studied effects such as crack velocity, density, orientation, storage media, porosity, and age on the fracture toughness (K(C), also called critical stress intensity factor) of compact bone. Most of these studies were conducted at room temperature. Considering that the body temperature of animals is greater than room temperature, and that bone has a large volumetric percentage of organics and water (generally, 55-65%), it is hypothesized that temperature has a significant effect on the fracture toughness of compact bone. Single-edge V-notched (SEVN) specimens were prepared to measure the fracture toughness of bovine femur and manatee rib in water at 0, 10, 23, 37, and 50 degrees C in four-point flexure. The fracture toughness values of bovine femur and manatee rib were found to decrease from 7.0 to 4.3MPam(1/2) and from 5.5 to 4.0MPam(1/2), respectively, as temperature increased over a temperature range of 50 degrees C. The results support the hypothesis that temperature has a significant effect on the fracture toughness of compact bone. Therefore, we suggest that study on fracture toughness of bone should be done at physiologically relevant temperatures.     

Self-healing fish gelatin/sodium montmorillonite biohybrid coacervates: structural and rheological characterization

Complex coacervation driven by associative electrostatic interactions was studied in mixtures of exfoliated sodium-montmorillonite (Na(+)-MMT) nanoplatelets and fish gelatin, at a specific mixing ratio and room temperature. Structural and viscoelastic properties of the coacervate phase were investigated as a function of pH by means of different complementary techniques. Independent of the technique used, the results consistently showed that there is an optimum pH value at which the coacervate phase shows the tightest structure with highest elasticity. The solid-like coacervates showed an obvious shear-thinning behavior and network fracture but immediately recovered back into their original elastic character upon removal of the shear strain. The nonlinear mechanical response characterized by single step stress relaxation experiments revealed the same trend for the yield stress and isochronal shear modulus of the coacervates as a function of pH with a maximum at pH 3.0 and lower values at 2.5 and 3.5 pHs, followed by a very sharp drop at pH 4.0. Finally, small-angle X-ray scattering (SAXS) data confirmed that at pHs lower than 4.0 the coacervate phases were dense and structured with a characteristic length scale (ξ(SAXS)) of ~7-9 nm. Comparing the ξ(SAXS) with rheological characteristic length (ξ(rheol)) estimated from low-frequency linear viscoelastic data and network theory, it was concluded that both the strength of the electrostatic interactions and the conformation of the gelatin chains before and during of the coacervation process are responsible for the structure and rigidity of the coacervates.