The area moment of inertia of the tibia: A risk factor for stress fractures

In a prospective study of stress fractures among Israeli infantry recruits, the area moment of inertia of the tibia was found to have a statistically significant correlation with the incidence of tibial, femoral and total stress fractures. Recruits with "low" area moments of inertia of the tibia were found to have higher stress fracture morbidity than those with "high" area moments of inertia. The best correlation was obtained when the area moment of inertia was calculated about the AP axis of bending at a cross-sectional level corresponding to the narrowest tibial width on lateral X-rays, a point which is at the distal quarter of the tibia. This finding indicates that bending forces about the approximate AP axis are an important causal factor for tibial and many other stress fractures. The bone's bending strength, or ability to resist bending moments, as measured by the area moment of inertia, helps determine risk to stress fracture.     

A technique for quantifying the bending moment acting on the lumbar spine in vivo

The bending properties of cadaveric lumbar spines were measured and used to convert in vivo measurements of lumbar flexion into bending moments ('stresses'). Forty-two lumbar motion segments were subjected to complex physiological loading and graphs were obtained of bending moment vs flexion angle. Variability was reduced by expressing both variables as a percentage of their values at the elastic limit. Data were averaged for each lumbar level, and a composite bending curve was compiled for the lumbar spine, L1-S1. A linear relationship was established between lumbar flexion measured in vitro and in vivo. This enabled values of 'per cent lumbar flexion' measured in vivo to be converted into 'per cent maximum bending moment' with a maximum likely error of about +/- 8%, which is equivalent to about +/- 5 Nm at L5-S1 for an average person. The technique was applied to 28 subjects, using dynamic measurements of lumbar flexion obtained with the '3-Space Isotrack' system. The bending moment at L5-S1 was 12 Nm on average when picking a pen up off the floor. Highly significant increases in bending moment were observed when heavier and bulkier objects were lifted.     

The human proximal femur behaves linearly elastic up to failure under physiological loading conditions

It has not been demonstrated whether the human proximal femur behaves linearly elastic when loaded to failure. In the present study we tested to failure 12 cadaveric femurs. Strain was measured (at 5000Hz) on the bone surface with triaxial strain gages (up to 18 on each femur). High-speed videos (up to 18,000frames/s) were taken during the destructive test. To assess the effect of tissue preservation, both fresh-frozen and formalin-fixed specimens were tested. Tests were carried out at two strain-rates covering the physiological range experienced during daily motor tasks. All specimens were broken in only two pieces, with a single fracture surface. The high-speed videos showed that failure occurred as a single abrupt event in less than 0.25ms. In all specimens, fracture started on the lateral side of the neck (tensile stress). The fractured specimens did not show any sign of permanent deformation. The force-displacement curves were highly linear (R(2)>0.98) up to 99% of the fracture force. When the last 1% of the force-displacement curve was included, linearity slightly decreased (minimum R(2)=0.96). Similarly, all force-strain curves were highly linear (R(2)>0.98 up to 99% of the fracture force). The slope of the first part of the force-displacement curve (up to 70% fracture force) differed from the last part of the curve (from 70% to 100% of the fracture force) by less than 17%. Such a difference was comparable to the fluctuations observed between different parts of the curve. Therefore, it can be concluded that the proximal femur has a linear-elastic behavior up to fracture, for physiological strain-rates.     

The mechanical properties of cranial bone: The effect of loading rate and cranial sampling position

Linear and depressed skull fractures are frequent mechanisms of head injury and are often associated with traumatic brain injury. Accurate knowledge of the fracture of cranial bone can provide insight into the prevention of skull fracture injuries and help aid the design of energy absorbing head protection systems and safety helmets. Cranial bone is a complex material comprising of a three-layered structure: external layers consist of compact, high-density cortical bone and the central layer consists of a low-density, irregularly porous bone structure.In this study, cranial bone specimens were extracted from 8 fresh-frozen cadavers (F=4, M=4; 81±11 years old). 63 specimens were obtained from the parietal and frontal cranial bones. Prior to testing, all specimens were scanned using a μCT scanner at a resolution of 56.9 μm. The specimens were tested in a three-point bend set-up at different dynamic speeds (0.5, 1 and 2.5 m/s). The associated mechanical properties that were calculated for each specimen include the 2nd moment of inertia, the sectional elastic modulus, the maximum force at failure, the energy absorbed until failure and the maximum bending stress. Additionally, the morphological parameters of each specimen and their correlation with the resulting mechanical parameters were examined.It was found that testing speed, strain rate, cranial sampling position and intercranial variation all have a significant effect on some or all of the computed mechanical parameters. A modest correlation was also found between percent bone volume and both the elastic modulus and the maximum bending stress.     

The effect of isolated valgus moments on ACL strain during single-leg landing: A simulation study

Valgus moments on the knee joint during single-leg landing have been suggested as a risk factor for anterior cruciate ligament (ACL) injury. The purpose of this study was to test the influence of isolated valgus moment on ACL strain during single-leg landing. Physiologic levels of valgus moments from an in vivo study of single-leg landing were applied to a three-dimensional dynamic knee model, previously developed and tested for ACL strain measurement during simulated landing. The ACL strain, knee valgus angle, tibial rotation, and medial collateral ligament (MCL) strain were calculated and analyzed. The study shows that the peak ACL strain increased nonlinearly with increasing peak valgus moment. Subjects with naturally high valgus moments showed greater sensitivity for increased ACL strain with increased valgus moment, but ACL strain plateaus below reported ACL failure levels when the applied isolated valgus moment rises above the maximum values observed during normal cutting activities. In addition, the tibia was observed to rotate externally as the peak valgus moment increased due to bony and soft-tissue constraints. In conclusion, knee valgus moment increases peak ACL strain during single-leg landing. However, valgus moment alone may not be sufficient to induce an isolated ACL tear without concomitant damage to the MCL, because coupled tibial external rotation and increasing strain in the MCL prevent proportional increases in ACL strain at higher levels of valgus moment. Training that reduces the external valgus moment, however, can reduce the ACL strain and thus may help athletes reduce their overall ACL injury risk.     

Structural Mechanics of DNA Wrapping in the Nucleosome

Experimental X-ray crystal structures and a database of calculated structural parameters of DNA octamers were used in combination to analyse the mechanics of DNA bending in the nucleosome core complex. The 1kx5 X-ray crystal structure of the nucleosome core complex was used to determine the relationship between local structure at the base-step level and the global superhelical conformation observed for nucleosome-bound DNA. The superhelix is characterised by a large curvature (597°) in one plane and very little curvature (10°) in the orthogonal plane. Analysis of the curvature at the level of 10-step segments shows that there is a uniform curvature of 30° per helical turn throughout most of the structure but that there are two sharper kinks of 50° at ± 2 helical turns from the central dyad base pair. The curvature is due almost entirely to the base-step parameter roll. There are large periodic variations in roll, which are in phase with the helical twist and account for 500° of the total curvature. Although variations in the other base-step parameters perturb the local path of the DNA, they make minimal contributions to the total curvature. This implies that DNA bending in the nucleosome is achieved using the roll-slide-twist degree of freedom previously identified as the major degree of freedom in naked DNA oligomers. The energetics of bending into a nucleosome-bound conformation were therefore analysed using a database of structural parameters that we have previously developed for naked DNA oligomers. The minimum energy roll, the roll flexibility force constant and the maximum and minimum accessible roll values were obtained for each base step in the relevant octanucleotide context to account for the effects of conformational coupling that vary with sequence context. The distribution of base-step roll values and corresponding strain energy required to bend DNA into the nucleosome-bound conformation defined by the 1kx5 structure were obtained by applying a constant bending moment. When a single bending moment was applied to the entire sequence, the local details of the calculated structure did not match the experiment. However, when local 10-step bending moments were applied separately, the calculated structure showed excellent agreement with experiment. This implies that the protein applies variable bending forces along the DNA to maintain the superhelical path required for nucleosome wrapping. In particular, the 50° kinks are constraints imposed by the protein rather than a feature of the 1kx5 DNA sequence. The kinks coincide with a relatively flexible region of the sequence, and this is probably a prerequisite for high-affinity nucleosome binding, but the bending strain energy is significantly higher at these points than for the rest of the sequence. In the most rigid regions of the sequence, a higher strain energy is also required to achieve the standard 30° curvature per helical turn. We conclude that matching of the DNA sequence to the local roll periodicity required to achieve bending, together with the increased flexibility required at the kinks, determines the sequence selectivity of DNA wrapping in the nucleosome.     

Local and global measurements show that damage initiation in articular cartilage is inhibited by the surface layer and has significant rate dependence

Cracks in articular cartilage are a common sign of joint damage, but failure properties of cartilage are poorly understood, especially for damage initiation. Cartilage failure may be further complicated by rate-dependent and depth-dependent properties, including the compliant surface layer. Existing blunt impact methods do not resolve local cartilage inhomogeneities and traditional fracture mechanics tests induce crack blunting and may violate underlying assumptions of linear elasticity. To address this knowledge gap, we developed and applied a method to indent cartilage explants with a sharp blade and initiate damage across a range of loading rates (strain rates 0.5%/s–500%/s), while recording local sample deformation and strain energy fields using confocal elastography. To investigate the importance of cartilage’s compliant surface, we repeated the experiment for samples with the surface removed. Bulk data suggest a critical force at which the tissue cuts, but local strains reveals that the deformation the sample can sustain before reaching this force is significantly higher in the surface layer. Bulk and local results also showed significant rate dependence, such that samples were easier to cut at faster speeds. This result highlights the importance of rate for understanding cracks in cartilage and parallels recent studies of rate-dependent failure in hydrogels. Notably, local sample deformation fields were well fit by classical Hookean elasticity. Overall, this study illustrates how local and global measurements surrounding the initiation of damage in articular cartilage can be combined to reveal the importance of cartilage’s zonal structure in protecting against failure across physiologically relevant loading rates.     

A theoretical framework to analyze bend testing of soft tissue

It has been hypothesized that repetitive flexural stresses contribute to the fatigue-induced failure of bioprosthetic heart valves. Although experimental apparatuses capable of measuring the bending properties of biomaterials have been described, a theoretical framework to analyze the resulting data is lacking. Given the large displacements present in these bending experiments and the nonlinear constitutive behavior of most biomaterials, such a formulation must be based on finite elasticity theory. We present such a theory in this work, which is capable of fitting bending moment versus radius of curvature experimental data to an arbitrary strain energy function. A simple finite element model was constructed to study the validity of the proposed method. To demonstrate the application of the proposed approach, bend testing data from the literature for gluteraldehyde-fixed bovine pericardium were fit to a nonlinear strain energy function, which showed good agreement to the data. This method may be used to integrate bending behavior in constitutive models for soft tissue.     

3D micro-scale deformations of wood in bending: synchrotron radiation muCT data analyzed with digital volume correlation

A micro-scale three-point-bending experiment with a wood specimen was carried out and monitored by synchrotron radiation micro-computed tomography. The full three-dimensional wood structure of the 1.57 × 3.42 × 0.75 mm³ specimen was reconstructed at cellular level in different loading states. Furthermore, the full three-dimensional deformation field of the loaded wood specimen was determined by digital volume correlation, applied to the reconstructed data at successive loading states. Results from two selected regions within the wood specimen are presented as continuous displacement and strain fields in both 2D and 3D. The applied combination of synchrotron radiation micro-computed tomography and digital volume correlation for the deformation analysis of wood under bending stress is a novel application in wood material science. The method offers the potential for the simultaneous observation of structural changes and quantified deformations during in situ micro-mechanical experiments. Moreover, the high spatial resolution allows studying the influence of anatomical features on the fracture behaviour of wood. Possible applications of this method range from bio-mechanical observations in fresh plant tissue to fracture mechanics aspects in structural timber.     

A continuous dynamic beam model for swimming fish

When a fish swims in water, muscle contraction, controlled by the nervous system, interacts with the body tissues and the surrounding fluid to yield the observed movement pattern of the body. A continuous dynamic beam model describing the bending moment balance on the body for such an interaction during swimming has been established. In the model a linear visco-elastic assumption is made for the passive behaviour of internal tissues, skin and backbone, and the unsteady fluid force acting on the swimming body is calculated by the 3D waving plate theory. The body bending moment distribution due to the various components, in isolation and acting together, is analysed. The analysis is based on the saithe (Pollachius virens), a carangiform swimmer. The fluid reaction needs a bending moment of increasing amplitude towards the tail and near-standing wave behaviour on the rear-half of the body. The inertial movement of the fish results from a wave of bending moment with increasing amplitude along the body and a higher propagation speed than that of body bending. In particular, the fluid reaction, mainly designed for propulsion, can provide a considerable force to balance the local momentum change of the body and thereby reduce the power required from the muscle. The wave of passive visco-elastic bending moment, with an amplitude distribution peaking a little before the mid-point of the fish, travels with a speed close to that of body bending. The calculated muscle bending moment from the whole dynamic system has a wave speed almost the same as that observed for EMG-onset and a starting instant close to that of muscle activation, suggesting a consistent matching between the muscle activation pattern and the dynamic response of the system in steady swimming. A faster wave of muscle activation, with a variable phase relation between the strain and activation cycle, appears to be designed to fit the fluid reaction and, to a lesser extent, the body inertia, and is limited by the passive internal tissues. Higher active stress is required from caudal muscle, as predicted from experimental studies on fish muscle. In general, the active force development by muscle does not coincide with the propulsive force generation on the tail. The stiffer backbone may play a role in transmitting force and deformation to maintain and adjust the movement of the body and tail in water.     

Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified.

Computer simulation is used to examine a simple flagellar model that will initiate and propagate bending waves in the absence of viscous resistances. The model contains only an elastic bending resistance and an active sliding mechanism that generates reduced active shear moment with increasing sliding velocity. Oscillation results from a distributed control mechanism that reverses the direction of operation of the active sliding mechanism when the curvature reaches critical magnitudes in either direction. Bend propagation by curvature-controlled flagellar models therefore does not require interaction with the viscous resistance of an external fluid. An analytical examination of moment balance during bend propagation by this model yields a solution curve giving values of frequency and wavelength that satisfy the moment balance equation and give uniform bend propagation, suggesting that the model is underdetermined. At 0 viscosity, the boundary condition of 0 shear rate at the basal end of the flagellum during the development of new bends selects the particular solution that is obtained by computer simulations. Therefore, the details of the pattern of bend initiation at the basal end of a flagellum can be of major significance in determining the properties of propagated bending waves in the distal portion of a flagellum. At high values of external viscosity, the model oscillates at frequencies and wavelengths that give approximately integral numbers of waves on the flagellum. These operating points are selected because they facilitate the balance of bending moments at the ends of the model, where the external viscous moment approaches 0. These mode preferences can be overridden by forcing the model to operate at a predetermined frequency. The strong mode preferences shown by curvature-controlled flagellar models, in contrast to the weak or absent mode preferences shown by real flagella, therefore do not demonstrate the inapplicability of the moment-balance approach to real flagella. Instead, they indicate a need to specify additional properties of real flagella that are responsible for selecting particular operating points.     

Three-dimensional load measurements in an external fixator

On the basis of a six-degree-of-freedom adjustable fracture reduction hexapod external fixator, a system which can be used for measuring axial and shear forces as well as torsion and bending moments in the fixator in vivo was developed. In a pilot study on 9 patients (7 fresh fractures and 2 osteotomies of the tibia), the load in the fixator during the healing process was measured after 2, 4, 8 and 12 weeks and at fixator removal. The measured values enabled both the type of fracture to be determined as well as the monitoring of the healing process. In well-reduced type A3 fractures small axial (direction of the bone axis) forces were found in the fixator. A2, B2 and C3 fractures showed distinct axial forces, which decreased during the healing process, according to an increasing load transfer over the bone. Bending moments in the fixator showed good correspondence with the clinical healing process, except in the case of a C3 fracture. A combination of bending moment and axial force proved to be particularly suitable to assess fracture healing. In transverse fractures, the well-known resorption phenomenon of bone in the fracture gap at approximately 4 weeks was detected by the system. Compared with other external fixator load measurements in vivo, the hexapod offers the advantage of being able to measure all forces and moments in the fixator separately and with a relatively simple mechanical arrangement. In our opinion, it will be possible to control fracture healing using this system, thereby minimizing radiation exposure from radiographs. Furthermore, the measurement system is a step towards the development of external fixator systems that enable automatic adjustments of the callus mechanical situation ("automatic dynamization") and inform the patients about the optimal weight bearing of their extremity ("intelligent fixator").     

Effects of differences in mineralization on the mechanical properties of bone

There is a considerable variation in the mineralization of bone; normal, non-pathological compact bone has ash masses ranging from 45 to 85% by mass. This range of mineralization results in an even greater range of mechanical properties. The Young modulus of elasticity can range from 4 to 32 GPa, bending strength from 50 to 300 MPa, and the work of fracture from 200 to 7000 Jm-2. It is not possible for any one type of bone to have high values for all three properties. Very high values of mineralization produce high values of Young modulus but low values of work of fracture (which is a measure of fracture toughness). Rather low values of mineralization are associated with high values of work of fracture but low values of Young modulus and intermediate values of bending strength. The reason for the high value for the Young modulus associated with high mineralization is intuitively obvious, but has not yet been rigorously modelled. The low fracture toughness associated with high mineralization may be caused by the failure of various crack-stopping mechanisms that can act when the mineral crystals in bone have not coalesced, but which become ineffective when the volume fraction of mineral becomes too high. The adoption of different degrees of mineralization by different bones, leading to different sets of mechanical properties, is shown to be adaptive in most cases studied, but some puzzles still remain.     

Variability of tissue mineral density can determine physiological creep of human vertebral cancellous bone

Creep is a time-dependent viscoelastic deformation observed under a constant prolonged load. It has been indicated that progressive vertebral deformation due to creep may increase the risk of vertebral fracture in the long-term. The objective of this study was to examine the relationships of creep with trabecular architecture and tissue mineral density (TMD) parameters in human vertebral cancellous bone at a physiological static strain level. Architecture and TMD parameters of cancellous bone were analyzed using microcomputerized tomography (micro-CT) in specimens cored out of human vertebrae. Then, creep and residual strains of the specimens were measured after a two-hour physiological compressive constant static loading and unloading cycle. Creep developed (3877 ± 2158 με) resulting in substantial levels of non-recoverable post-creep residual strain (1797 ± 1391 με). A strong positive linear correlation was found between creep and residual strain (r = 0.94, p < 0.001). The current results showed that smaller thickness, larger surface area, greater connectivity of trabeculae, less mean tissue mineral density (TMD, represented by gray levels) and higher variability of TMD are associated with increasing logarithmic creep rate. The TMD variability (GL(COV)) was the strongest correlate of creep rate (r = 0.79, p < 0.001). This result suggests that TMD variability may be a useful parameter for estimating the long-term deformation of a whole vertebral body. The results further suggest that the changes in TMD variability resulting from bone remodeling are of importance and may provide an insight into the understanding of the mechanisms underlying progressive failure of vertebral bodies and development of a clinical fracture.     

Molecular dynamics study of deformation mechanisms of poly(vinyl alcohol) hydrogel

Molecular dynamics (MD) simulations have been performed on the physically crosslinking poly(vinyl alcohol) (PVA) hydrogel to study the deformation mechanisms under uniaxial tensile conditions. The distributions of hydroxyl oxygens and dihedral angle and the number of hydrogen bonds have been analysed to study the structure of the hydrogel. The water content and temperature dependency of mechanical properties have been investigated. The energy contributions from the partially united atom potential have been calculated as a function of strain. It is found that the stress–strain curve comprises toe region, linear region and yield and failure region which is close to most biomaterials. In the toe and yield region, all the contributions to the internal energy change a little. However, in the linear region, the bond stretching and angle bending energy increase rapidly and mainly dominate the region, and the energy increases more rapidly with the increasing water content but the decreasing temperature. The degree of crosslinking decreases with the increasing deformation. The polymer chains occur significant torsional activity in the toe region. Hydrogen bonds are stable in the toe and yield region, but the hydrogen bonds between hydroxyl groups and waters decrease in the linear region.     

Injury tolerance and moment response of the knee joint to combined valgus bending and shear loading

Valgus bending and shearing of the knee have been identified as primary mechanisms of injuries in a lateral loading environment applicable to pedestrian-car collisions. Previous studies have reported on the structural response of the knee joint to pure valgus bending and lateral shearing, as well as the estimated injury thresholds for the knee bending angle and shear displacement based on experimental tests. However, epidemiological studies indicate that most knee injuries are due to the combined effects of bending and shear loading. Therefore, characterization of knee stiffness for combined loading and the associated injury tolerances is necessary for developing vehicle countermeasures to mitigate pedestrian injuries. Isolated knee joint specimens (n=40) from postmortem human subjects were tested in valgus bending at a loading rate representative of a pedestrian-car impact. The effect of lateral shear force combined with the bending moment on the stiffness response and the injury tolerances of the knee was concurrently evaluated. In addition to the knee moment-angle response, the bending angle and shear displacement corresponding to the first instance of primary ligament failure were determined in each test. The failure displacements were subsequently used to estimate an injury threshold function based on a simplified analytical model of the knee. The validity of the determined injury threshold function was subsequently verified using a finite element model. Post-test necropsy of the knees indicated medial collateral ligament injury consistent with the clinical injuries observed in pedestrian victims. The moment-angle response in valgus bending was determined at quasistatic and dynamic loading rates and compared to previously published test data. The peak bending moment values scaled to an average adult male showed no significant change with variation in the superimposed shear load. An injury threshold function for the knee in terms of bending angle and shear displacement was determined by performing regression analysis on the experimental data. The threshold values of the bending angle (16.2 deg) and shear displacement (25.2 mm) estimated from the injury threshold function were in agreement with previously published knee injury threshold data. The continuous knee injury function expressed in terms of bending angle and shear displacement enabled injury prediction for combined loading conditions such as those observed in pedestrian-car collisions.     

Utilizing scanning electron microscopy to observe the initiation and development of slip bands and fracture in a nickel based metallic glass

To gain a better understanding of the elastic/plastic deformation and fracture of metallic glasses, in situ bending of a nickel based metallic glass ribbon was observed utilizing scanning electron microscopy. The results verified that the formation of slip bands was the first observable indication of plastic deformation. The slip band formation was followed by a veined fracture and a rapid unstable fracture. It was also observed that the rapid fracture was arrested by the formation of box-like hinges in the material. It is believed that the failure of this metallic glass occurred in progressive stages of slip, veined fracture and rapid fracture.     

A three-dimensional digital image correlation technique for strain measurements in microstructures

A three-dimensional digital image correlation technique is presented for strain measurements in open-cell structures such as trabecular bone. The technique uses high-resolution computed tomography images for displacement measurements in the solid structure. In order to determine the local strain-state within single trabeculae, a tetrahedronization method is used to fill the solid structure with tetrahedrae. Displacements are calculated at the nodes of the tetrahedrae. The displacement data is subsequently converted to a deformation tensor in each of the tetrahedral element centers with a least-squares estimation method. Because the trabeculae are represented by a mesh, it is possible to deform this mesh according to the deformation tensor and, at the same time, visualize the calculated local strain in the deformed mesh with a finite element post-processing tool. In this way, the deformation of a single trabecula from an aluminum foam sample was determined and validated with rendered images of the three-dimensional sample. A precision analysis showed that a rigid translation or rotation does not affect the accuracy. Typical values for the standard deviation in the displacement and strain components are 2.0 microm and 0.01, respectively. Presently, the precision limits the technique to strain measurements beyond the yield strain.     

Determining parameters related to the likelihood of failure of red oak (Quercus rubra L.) from winching tests

Key message

This study provides data necessary to develop mechanistic models of the failure of open-grown trees. The literature contains few such data. Some results contrast previous studies on conifers.

Abstract

In cities and towns, tree failure can cause damage and injury. Few studies have considered large, open-grown trees when measuring parameters related to tree failure. To measure elastic modulus and maximum bending moment and stress, we winched red oaks (Quercus rubra L.), including some with co-dominant stems and others with extant decay. To simulate decay in a subsample of trees, we cut voids in the trunk before pulling trees to failure. Maximum bending moment was greatest for uprooted trees, but maximum bending and shear stresses were greatest for trees that failed in the crown in the vicinity of branches. The likelihood of failure at a void or area of extant decay increased as the loss in area moment of inertia increased. The moduli of elasticity and rupture of specimens taken from trees were greater than values measured on the trees themselves. Failure at the union of co-dominant stems only occurred when we pulled them apart, loading them perpendicular to the plane bifurcating the union. Some of the results are inconsistent with previous work on conifers; more data on open-grown trees are necessary to develop mechanistic models to predict tree failure.