Contributions to the Biomechanics of Plants.; II. Stability Against Local Buckling in Hollow Plant Stems*

Hollow plant stems are, compared to solid ones, endangered by a particular kind of mechanical damage: local buckling. In upright stems centric forces due to the weight of the plant itself are not critical. Bending, however, as caused by wind forces causes transverse stresses that lead to local buckling, long before the critical tension or compression stresses are reached. A new numerical method is proposed to treat local buckling of hollow plant stems caused by bending forces. The results emphasize the importance of transverse reinforcements in the nodes and/or the nodal thickenings for the stability of hollow structures.     

A three-dimensional mechanical analysis of a stentless fibre-reinforced aortic valve prosthesis

Failure of bioprosthetic and synthetic three-leaflet valves has been shown to occur as a consequence of high tensile and bending stresses, acting on the leaflets during opening and closing. Moreover, in the stented prostheses, whether synthetic or biological, the absence of contraction of the aortic base, due to the rigid stent, causes the leaflets to be subjected to an unphysiological degree of flexure, which is related to calcification. It is shown that the absence of the stent, which gives a flexible aortic base and leaflet attachment, and leaflet fibre-reinforcement result in reduced stresses in the weaker parts of the leaflets in their closed configuration. It is postulated that this leads to a decrease of tears and perforations, which may result in a improved long-term behaviour. The effect of a flexible leaflet attachment and aortic base of a synthetic valve is investigated with a finite element model. Different fibre-reinforced structures are analysed with respect to the stresses that are likely to contribute to the failure of fibre-reinforced prostheses and compared with the results obtained for a stented prosthesis. Results show that for the stentless models a reduction of stresses up to 75% is obtained with respect to stented models with the same type of reinforcement.     

Three-dimensional mechanics of eukaryotic flagella.

Equations are derived that account for the contribution of internal structure of cilia and flagella to motion in three dimensions according to a sliding filament model of the motile system. It is shown that for reasonable amounts of bending and twisting, the bending properties of an axoneme can be described by a linear elastic bending resistance, and approximate values for the bending and twisting resistances are computed. Expressions for the shear moments contributed by purely elastic or pinned links between filaments are also derived. It is shown that within the confines of a strict sliding filament model such internal structures cannot by themselves produce twist. Thus planar bending will occur if the internal shear force lies in a plane. Application of an external force, however, will in general produce twisting. Computer simulations of flagellar shape in response to a constant external force applied to the distal end of the axoneme are presented. It is shown that a small amount of twist may arise because of acylindrical bend resistance. Large twists, however, result when the external force is applied to an axoneme with internal shear resistant links.     

On the Strength of Herbaceous Vascular Plant Stems

During the past 120 years researchers have tried to providean understanding of the relationship between the arrangementof stiffening tissue in the cross-section of plant stems andtheir mechanical integrity. The mechanical analysis of verticalstems of self-supporting plants has traditionally been concernedwith issues involving global and local stability of the stem,and with stresses developed due to wind loads. Plant stem tissue,considered as a material, is both heterogeneous and highly anisotropic,and this must be reflected in any characterization of its mechanicalbehaviour. This fact strongly influences the type of failurecriterion which should be applied for compressive failure ofthe stem subjected to loads causing bending. It is shown, here,that applying modern ideas as to the appropriate criteria forcompressive failure of fibre-reinforced composite materialscan influence how we assess the efficacy of various stelar arrangementsconcerning their ability to fulfill their mechanical function.Specifically, it is demonstrated that peripheral arrangementsof supporting tissue are, in some circumstances, less advantageousthan more uniform distributions of this tissue. Plant stems; stem bending; compressive strength; stelar types     

Interaction between bending and tension forces in bilayer membranes.

A theoretical analysis is presented of the bending mechanics of a membrane consisting of two tightly-coupled leaflets, each of which shears and bends readily but strongly resists area changes. Structures of this type have been proposed to model biological membranes such as red blood cell membrane. It is shown that when such a membrane is bent, anisotropic components of resultant membrane tension (shear stresses) are induced, even when the tension in each leaflet is isotropic. The induced shear stresses increase as the square of the membrane curvature, and become significant for moderate curvatures (when the radius of curvature is much larger than the distance between the leaflets). This effect has implications for the analysis of shape and deformation of freely suspended and flowing red blood cells.     

Functional significance of the mandibular symphysis.

The purpose of this investigation was to relate the morphology of connective tissues in the mandibular symphysis to the behavioral and experimental evidence for mobility and mechanical stress at the symphysis. The anatomy of the symphysis was examined histologically in 6 mammalian orders encompassing 22 species. Behavioral and experimental evidence of stress during the power stroke of the chewing cycle correspond with stresses at the symphysis implied by the location and orientation of symphyseal connective tissues. These stresses are: (1) dorsoventral shear of the symphysis due to the transfer of force from balancing to chewing sides, (2) bending of the symphysis causing tension along the inferior and compression along superior borders due to torsion on the dentaries from the jaw closing muscles, and (3) antero-posterior shear of the symphysis due to an anteriorly directed stress on the chewing side. Interspecific comparisons suggest that leaf eaters can resist greater dorsoventral shear than fruit or insect eaters, but no correlations exist between diet and bending or antero-posterior shear. This suggests that chewing leaves requires larger biting forces.     

Affinity of red blood cell membrane for particle surfaces measured by the extent of particle encapsulation.

An experimental technique and a simple analysis are presented that can be used to quantitate the affinity of red blood cell membrane for surfaces of small beads or microsomal particles up to 3 micrometers Diam. The technique is demonstrated with an example of dextran-mediated adhesion of small spherical red cell fragments to normal red blood cells. Cells and particles are positioned for contact by manipulation with glass micropipets. The mechanical equilibrium of the adhesive contact is represented by the variational expression that the decrease in interfacial free energy due to a virtual increase in contact area is balanced by the increase in elastic energy of the membrane due to virtual deformation. The surface affinity is the reduction in free energy per unit area of the interface associated with the formation of adhesive contact. From numerical computations of equilibrium configurations, the surface affinity is derived as a function of the fractional extent of particle encapsulation. The range of surface affinities for which the results are applicable is increased over previous techniques to several times the value of the elastic shear modulus. It is shown that bending rigidity of the membrane has little effect on the analytical results for particles 1--3 micrometers Diam and that results are essentially the same for both cup- and disk-shaped red cells. A simple analytical model is shown to give a good approximation for surface affinity (normalized by the elastic shear modulus) as a function of the fractional extent of particle encapsulation. The model predicts that a particle would be almost completely vacuolized for surface affinities greater than or equal to 10 times the elastic shear modulus. Based on an elastic shear modulus of 6.6 x 10(-3) dyn/cm, the range for the red cell-particle surface affinity as measured by this technique is from approximately 7 x 10(-4) to 7 x 10(-2) erg/cm2. Also, an approximate relation is derived for the level of surface affinity necessary to produce particle vacuolization by a phospholipid bilayer surface which possesses bending rigidity and a fixed tension.     

Characterization of in vivo strain in the rat tibia during external application of a four-point bending load.

This paper describes a technique for characterizing strains and stresses induced in vivo in the rat tibia during application of an external four-point bending load. An external load was applied through the muscle and soft tissue with a four-point bending device, to induce strain in a 11 mm section of the right tibiae of ten adult female Sprague-Dawley rats. Induced strains were measured in vivo on the lateral surface of the tibia. Inter-rat difference, leg positioning and strain gage placement were evaluated as sources of variability of applied strains. Beam bending theory was used to predict externally induced in vivo strains. Finite element analysis was used to quantify the magnitude of shear stresses induced by this type of loading. There was a linear relationship between applied load and induced in vivo strains. In vivo strains induced by external loading were linearly correlated (R2 = 0.87) with the strains calculated using beam bending theory. The finite element analysis predicted shear stresses at less than 10% of the longitudinal stresses resulting from four-point bending. Strains predicted along the tibia by finite element analysis and beam bending theory were well-correlated. Inter-rat variability due to tibia size and shape difference was the most important source of variation in induced strain (CV = 21.6%). Leg positioning was less important (CV = 9.5%).     

Further studies on the evolution of land and freshwater nemerteans: generic relationships among the paramonostiliferous taxa

The costs and benefits of fighting in bovids are high in terms of injury and reproductive success, respectively. The breakage of a horn would curtail reproductive success permanently. Therefore, the horns of bovids should include sufficient material so that they are strong enough to be unlikely to break in fighting but without being too heavy to carry around. The forces developed during fighting were measured in a computerized analysis of film of blackbuck and bighorn sheep. All possible modes of failure were investigated using a mechanical analysis to see how the horns are most likely to fail.
The maximum possible force developed during fighting is 3400 N for the bighorn sheep and 456 N for the blackbuck. Bending stress, shear stress, deflection, strain energy and critical crack lengths were calculated for the horns of these two bovids. Horns are most likely to fail in bending as indicated by safety factors. Most of the force is taken in compression due to the curvature of horns. Shear stresses are insignificant and deflections are negligible during the most forceful encounters.
The safety factor in bending of the horn of the bighorn sheep is greater (10) than that of the horn of the blackbuck (3.4) because the forces are probably more variable and unpredictable in the fighting of the former.
All of the energy of fighting is absorbed by the body musculature because horns store less than 1% of the energy produced in fighting as strain energy when they bend. Cracks and scratches mustbe more than 60% of the transverse basal dimensions of horns in order for there to be catastrophic failure at the maximum stresses developed during fighting.
The horns of the blackbuck and the bighorn sheep appear to be minimum weight structures given the variability of the forces acting on them and are unlikely to break in fighting given the forces calculated from the analyses of films.     

Effect of the endothelial surface layer on transmission of fluid shear stress to endothelial cells

Responses of vascular endothelial cells to mechanical shear stresses resulting from blood flow are involved in regulation of blood flow, in structural adaptation of vessels, and in vascular disease. Interior surfaces of blood vessels are lined with a layer of bound or adsorbed macromolecules, known as the endothelial surface layer (ESL). In vivo investigations have shown that this layer has a width of order 1 microm, that it substantially impedes plasma flow, and that it excludes flowing red blood cells. Here, the effect of the ESL on transmission of shear stress to endothelial cells is examined using a theoretical model. The layer is assumed to consist of a matrix of molecular chains extending from the surface, held in tension by a slight increase in colloid osmotic pressure relative to that in free-flowing plasma. It is shown that, under physiological conditions, shear stress is transmitted to the endothelial surface almost entirely by the matrix, and fluid shear stresses on endothelial cell membranes are very small. Rapid fluctuations in shear stress are strongly attenuated by the layer. The ESL may therefore play an important role in sensing of shear stress by endothelial cells.     

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.     

Stresses on joints of Opuntia laevis: forces necessary for mechanical failure and role of lignified xylem cells as resistance components

Species of Opuntia exhibit a wide range of morphologies. Understanding these morphologies may require knowledge of the mechanical stresses on joints of stem segments and as well as the internal components in joints that withstand joint failure (separation of the terminal cladode from the sub-terminal cladode after weights were applied perpendicularly to the long axis). Results of stress testing terminal cladodes of Opuntia laevis provided the following conclusions: (1) amounts of applied stress for joint failure were not related to the amounts of stress on joints before stress testing; (2) breaking strength (failure stress) was accurately determined for joints from linear plots of M (bending moment) versus I/c (section modulus) [breaking stress for O. laevis was 2.77 kPa]; (3) bending moments at failure were twice as high for tensile portions than for compressive portions of joints; and (4) bending moments at failure were positively correlated with amounts of lignified xylem cells in joints [for each mm² of lignified xylem cells in joints there was an increase of 0.06 N m of bending moment]. These data support the overall hypothesis that bending stresses are the main stresses at joints of Opuntia laevis and that lignified xylem cells are the main components that resist joint failure. Moreover, since tensile portions have more lignified xylem cells than other stem portions, tensile portions can resist more applied stress.     

Compressive/Tensile Stresses and Lignified Cells as Resistance Components in Joints between Cladodes of Opuntia laevis (Cactaceae)

The Cactaceae are a diverse group of plants with a wide variety of morphologies. Many species of Opuntia have segmented stems in which terminal cladodes may be separated from main-stem cladodes with varying amounts of resistance. From a geometric approach, derivations were used to calculate normal (axial and bending) and shear (transverse force and torque) stresses at joints due to the weight of the cladodes. Normal and shear stresses act perpendicular and parallel to (along) the cross sections of joints, respectively. Normal stress caused by bending was >10 times that of the mean value of any other stress. Analyses were performed to determine the relationship between maximum normal stress and the amount of lignified xylem cells. Such cells had thicker cell walls compared with the various other cells of stem joints that had thin cell walls and that thus would provide the most resistance to normal stresses. An analogy was made between cactus joints and a composite beam with reinforcing rods. In such joints, thin-walled parenchyma cells might be analogous to concrete that has little resistance to tensile stress, while the thick-walled, lignified xylem cells would be analogous to reinforcing rods. There were statistically significant relationships between normal stresses (from bending and axial loads) and mean percentage of lignified xylem cells (r=0.73) and between normal stresses and total areas of lignified xylem cells (r=0.65) (more stress, more reinforcing xylem cells). Tensile portions of cactus joints had 23% lignified xylem cells, while compressive portions had only 10% lignified xylem cells in joint areas (more tension, more reinforcing xylem cells). In addition, tensile joint tissues had two to three times more thick-walled, lignified xylem cells in the outer 30% of the radius compared with other joint tissues types (more reinforcing near the surface). To our knowledge, this is the first publication to present mechanical stresses at stem joints of cacti and the first to relate these stresses to characteristics of resisting tissues in the joints of a cactus.     

Deformation behaviour and damage accumulation of cortical bone specimens from the equine tibia under cyclic loading

Despite its clinical importance, the fatigue behaviour of cortical bone has not been examined as widely as its static behaviour. In the present study, specimens from the tibiae of horses have been subjected to load-controlled single step tests. The cyclic deformation behaviour was described by the development of stress-strain hysteresis parameters over the lifetime. The fatigue behaviour of bone is characterised by cyclic softening which is most distinctive towards the end of the lifetime. The microstructural damage accumulated during cyclic loading results in a loss of stiffness, asymmetrical deformation of the bone in tension and compression in cyclic creep. As shown by light and scanning electron microscopy, microcrack formation and growth is the main damage mechanism. The crack growth behaviour is strongly influenced by the microstructure, the stress components and the absolute value of the local stresses. Lower local stresses and/or compressive mean stresses lead to a dominant influence of the shear stress components with shear failure at inner interfaces. With increasing crack length, that is, higher local stress amplitudes, or tensile mean stresses, the microstructure is more and more ignored and failure occurs primarily under the influence of the normal stress components. This can be clearly seen on the fracture and specimen surfaces.     

Transverse masticatory movements, occlusal orientation, and symphyseal fusion in selenodont artiodactyls

Based on extensive experimental work on primates, two masticatory loading regimes have emerged as the likely determinants of mandibular symphyseal fusion-dorsoventral shear and lateral transverse bending (wishboning) (Ravosa and Hylander, 1994; Hylander et al., 1998, 2000). Recently, however, it has been argued that, rather than functioning to strengthen the symphysis during mastication, fusion serves to stiffen the symphyseal joint so as to facilitate increased transverse jaw movements during occlusion (Lieberman and Crompton, 2000). As part of this transverse stiffness model, it has been suggested that taxa with fused symphyses should also exhibit more horizontally oriented occlusal wear facets. Using a series of univariate and bivariate analyses, we test predictions of these three models in a sample of 44 species of selenodont artiodactyls. Consistent with the wishboning and transverse stiffness models, taxa with fused symphyses (camelids) have more horizontally oriented M(2) and M(2) occlusal wear facets, anteroposteriorly (AP) elongate symphyses, and relatively wider corpora. Contrary to the dorsoventral shear model, camelids do not have relatively deeper corpora (due to greater parasagittal bending). While taxa with ossified symphyses have relatively larger symphysis cross-sectional areas, this appears to be the byproduct of an increase in AP symphysis length due to greater lateral transverse bending of the mandible. Theoretical consideration of the biomechanics of mastication further suggests that strength, not stiffness, is the critical factor in determining symphyseal ossification. Thus, like anthropoid primates, fusion in selenodont artiodactyls appears to function in resisting increased wishboning stresses arising from an emphasis on transverse occlusal/mandibular movements and loads.     

Micromechanically based poroelastic modeling of fluid flow in Haversian bone

To explore the hypothesis that load-induced fluid flow in bone is a mechano-transduction mechanism in bone adaptation, unit cell micro-mechanical techniques are used to relate the microstructure of Haversian cortical bone to its effective poroelastic properties. Computational poroelastic models are then applied to compute in vitro Haversian fluid flows in a prismatic specimen of cortical bone during harmonic bending excitations over the frequency range of 10(0) to 10(6) Hz. At each frequency considered, the steady state harmonic response of the poroelastic bone specimen is computed using complex frequency-domain finite element analysis. At the higher frequencies considered, the breakdown of Poisueille flow in Haversian canals is modeled by introduction of a complex fluid viscosity. Peak bone fluid pressures are found to increase linearly with loading frequency in proportion to peak bone stress up to frequencies of approximately 10 kHz. Haversian fluid shear stresses are found to increase linearly with excitation frequency and loading magnitude up until the breakdown of Poisueille flow. Tan delta values associated with the energy dissipated by load-induced fluid flow are also compared with values measured experimentally in a concurrent broadband spectral analysis of bone. The computational models indicate that fluid shear stresses and fluid pressures in the Haversian system could, under physiologically realistic loading, easily reach the level of a few Pascals, which have been shown in other works to elicit cell responses in vitro.     

Mechanisms for mechanical damage in the intervertebral disc annulus fibrosus

Intervertebral disc degeneration results in disorganization of the laminate structure of the annulus that may arise from mechanical microfailure. Failure mechanisms in the annulus were investigated using composite lamination theory and other analyses to calculate stresses in annulus layers, interlaminar shear stress, and the region of stress concentration around a fiber break. Scanning electron microscopy (SEM) was used to evaluate failure patterns in the annulus and evaluate novel structural features of the disc tissue. Stress concentrations in the annulus due to an isolated fiber break were localized to approximately 5 microm away from the break, and only considered a likely cause of annulus fibrosus failure (i.e., radial tears in the annulus) under extreme loading conditions or when collagen damage occurs over a relatively large region. Interlaminar shear stresses were calculated to be relatively large, to increase with layer thickness (as reported with degeneration), and were considered to be associated with propagation of circumferential tears in the annulus. SEM analysis of intervertebral disc annulus fibrosus tissue demonstrated a clear laminate structure, delamination, matrix cracking, and fiber failure. Novel structural features noted with SEM also included the presence of small tubules that appear to run along the length of collagen fibers in the annulus and a distinct collagenous structure representative of a pericellular matrix in the nucleus region.     

Finite element assessment of block-augmented total knee arthroplasty

Loosening and migration of tibial prostheses have been identified as causes of early total knee replacement (TKR) failure. The problem is made more complex when defects occur in the proximal tibia compromising fixation and alignment. Clinical studies using metal augments have shown these to be an alternative to other means of defect treatment. Finite element (FE) analysis can be used to identify regions that may be prone to loosening and migration. In the current work, 3D FE models of TKR uncontained type-2 defects treated with block augments have been constructed and analysed. It has been shown that a metal augment is the most suitable. The use of bone cement (PMMA) to fill proximal defects is not considered suitable as stresses carried by the cement block exceed those of the fatigue limit of bone cement. It has been shown that the stresses in the proximal cancellous bone of block-augmented models are significantly below levels likely to cause damage due to overloading. Furthermore, the use of stem extensions has been shown to reduce the cancellous bone stresses in the proximal region thus increasing the likelihood of bone resorption. Given this, it is recommended that stem extensions are not required unless necessary to mitigate some other problem.     

An accurate finite element model of the cervical spine under quasi-static loading

Cervical disc injury due to impact has been observed in clinical and biomechanical investigations; however, there is a lack of data that helps to elucidate the mechanisms of disc injury during these collisions. Therefore, it is necessary to understand the behavior of the cervical spine under different types of loading situations. A three dimensional finite element (FE) model for the multi-level cervical spine segment (C0-C7) was developed using computed tomography (CT) data and applied to study the internal stresses and strains of the intervertebral discs under quasi-static loading conditions. The intervertebral discs were treated as nonlinear, anisotropic and incompressible subjected to large deformations. The model accuracy was validated by comparing it with previously published experimental and numerical results for different movements. It was shown that the use of a fiber reinforced model to describe the behavior of the annulus of the discs would predict higher maximum shear strains than an isotropic one, being therefore important the use of complex constitutive models in order to be able to detect the appearance of injured zones, since those strains and stresses are supposed to be related with damage to soft tissues. Several movements were analyzed: flexion, extension and axial rotation, obtaining that the maximum shear stresses in the disc were higher for a flexo-extension movement.     

Effect of heel height on in-shoe localized triaxial stresses

Abnormal and excessive plantar pressure and shear are potential risk factors for high-heeled related foot problems, such as forefoot pain, hallux valgus deformity and calluses. Plantar shear stresses could be of particular importance with an inclined supporting surface of high-heeled shoe. This study aimed to investigate the contact pressures and shear stresses simultaneously between plantar foot and high-heeled shoe over five major weightbearing regions: hallux, heel, first, second and fourth metatarsal heads, using in-shoe triaxial force transducers. During both standing and walking, peak pressure and shear stress shifted from the lateral to the medial forefoot as the heel height increased from 30 to 70mm. Heel height elevation had a greater influence on peak shear than peak pressure. The increase in peak shear was up to 119% during walking, which was about five times that of peak pressure. With increasing heel height, peak posterolateral shear over the hallux at midstance increased, whereas peak pressure at push-off decreased. The increased posterolateral shear could be a contributing factor to hallux deformity. It was found that there were differences in the location and time of occurrence between in-shoe peak pressure and peak shear. In addition, there were significant differences in time of occurrence for the double-peak loading pattern between the resultant horizontal ground reaction force peaks and in-shoe localized peak shears. The abnormal and drastic increase of in-shoe shear stresses might be a critical risk factor for shoe-related foot disorders. In-shoe triaxial stresses should therefore be considered to help in designing proper footwear.