A simple model for elastic wave propagation in hard sphere-filled random composites.

A simple model is proposed to study wave propagation in hard sphere-reinforced elastic random composites. Compared to existing related models, the proposed model is featured by a modified form of classical elastodynamic equations in which the inertia term is substituted by the acceleration field of the mass centre of a representative unit cell, supplied with a derived simple differential relation between the displacement field of the composite and the displacement field of the mass centre of a representative unit cell. The present model enjoys conceptual and mathematical simplicity although it is restricted to hard sphere-filled elastic composites in which the elastic moduli of embedded spheres are much (at least 4-5 times) stiffer than those of a softer matrix. Explicit formulas are derived for the attenuation coefficient and the effective phase velocity of plane longitudinal P-waves and transverse S-waves. The efficiency and reasonable accuracy of the present model are demonstrated by reasonably good agreement between the predicted results and some established known data. The proposed model could offer a potential general method to study various three-dimensional dynamic problems of hard sphere-filled elastic random composites.

Post-buckling of a pressured biopolymer spherical shell with the mode interaction.

Imperfection sensitivity is essential for mechanical behaviour of biopolymer shells characterized by high geometric heterogeneity. The present work studies initial post-buckling and imperfection sensitivity of a pressured biopolymer spherical shell based on non-axisymmetric buckling modes and associated mode interaction. Our results indicate that for biopolymer spherical shells with moderate radius-to-thickness ratio (say, less than 30) and smaller effective bending thickness (say, less than 0.2 times average shell thickness), the imperfection sensitivity predicted based on the axisymmetric mode without the mode interaction is close to the present results based on non-axisymmetric modes with the mode interaction with a small (typically, less than 10%) relative errors. However, for biopolymer spherical shells with larger effective bending thickness, the maximum load an imperfect shell can sustain predicted by the present non-axisymmetric analysis can be significantly (typically, around 30%) lower than those predicted based on the axisymmetric mode without the mode interaction. In such cases, a more accurate non-axisymmetric analysis with the mode interaction, as given in the present work, is required for imperfection sensitivity of pressured buckling of biopolymer spherical shells. Finally, the implications of the present study to two specific types of biopolymer spherical shells (viral capsids and ultrasound contrast agents) are discussed.

Imperfection sensitivity of pressured buckling of biopolymer spherical shells.

Imperfection sensitivity is essential for mechanical behavior of biopolymer shells [such as ultrasound contrast agents (UCAs) and spherical viruses] characterized by high geometric heterogeneity. In this work, an imperfection sensitivity analysis is conducted based on a refined shell model recently developed for spherical biopolymer shells of high structural heterogeneity and thickness nonuniformity. The influence of related parameters (including the ratio of radius to average shell thickness, the ratio of transverse shear modulus to in-plane shear modulus, and the ratio of effective bending thickness to average shell thickness) on imperfection sensitivity is examined for pressured buckling. Our results show that the ratio of effective bending thickness to average shell thickness has a major effect on the imperfection sensitivity, while the effect of the ratio of transverse shear modulus to in-plane shear modulus is usually negligible. For example, with physically realistic parameters for typical imperfect spherical biopolymer shells, the present model predicts that actual maximum external pressure could be reduced to as low as 60% of that of a perfect UCA spherical shell or 55%-65% of that of a perfect spherical virus shell, respectively. The moderate imperfection sensitivity of spherical biopolymer shells with physically realistic imperfection is largely attributed to the fact that biopolymer shells are relatively thicker (defined by smaller radius-to-thickness ratio) and therefore practically realistic imperfection amplitude normalized by thickness is very small as compared to that of classical elastic thin shells which have much larger radius-to-thickness ratio.

Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation.

Multiple elastic inclusions with uniform internal stress fields in an infinite elastic matrix are constructed under given uniform remote in-plane loadings. The method is based on the sufficient and necessary condition imposed on the boundary value of a holomorphic function that guarantees the existence of the holomorphic function in a multiply connected region. The unknown shape of each of the multiple inclusions is characterized by a conformal mapping. This work focuses on a major large class of multiple inclusions characterized by a simple condition that covers and is much beyond the known related results reported in previous works. Extensive examples of multiple inclusions with or without geometrical symmetry are shown. Our results showed that the inclusion shapes obtained for the uniformity of internal stress fields are independent of the remote loading only when all of the multiple inclusions have the same shear modulus as that of the matrix. Moreover, specific conditions are derived on remote loading, elastic constants of the inclusions and uniform internal stress fields, which guarantee the existence of multiple symmetric inclusions or multiple rotationally symmetrical inclusions with uniform internal stress fields.

Localized vibration of a microtubule surrounded by randomly distributed cross linkers.

Based on finite element simulation, the present work studies free vibration of a microtubule surrounded by 3D randomly distributed cross linkers in living cells. A basic result of the present work is that transverse vibration modes associated with the lowest frequencies are highly localized, in sharp contrast to the through-length modes predicted by the commonly used classic elastic foundation model. Our simulations show that the deflected length of localized modes increases with increasing frequency and approaches the entire length of microtubule when frequency approaches the minimum classic frequency given by the elastic foundation model. In particular, unlike the length-sensitive classic frequencies predicted by the elastic foundation model, the lowest frequencies of localized modes predicted by the present model are insensitive to the length of microtubules and are at least 50% lower than the minimum classic frequency for infinitely long microtubules and could be one order of magnitude lower than the minimum classic frequency for shorter microtubules (only a few microns in length). These results suggest that the existing elastic foundation model may have overestimated the lowest frequencies of microtubules in vivo. Finally, based on our simulation results, some empirical relations are proposed for the critical (lowest) frequency of localized modes and the associated wave length. Compared to the classic elastic foundation model, the localized vibration modes and the associated wave lengths predicted by the present model are in better agreement with some known experimental observations.

Localized buckling of a microtubule surrounded by randomly distributed cross linkers.

Microtubules supported by surrounding cross linkers in eukaryotic cells can bear a much higher compressive force than free-standing microtubules. Different from some previous studies, which treated the surroundings as a continuum elastic foundation or elastic medium, the present paper develops a micromechanics numerical model to examine the role of randomly distributed discrete cross linkers in the buckling of compressed microtubules. First, the proposed numerical approach is validated by reproducing the uniform multiwave buckling mode predicted by the existing elastic-foundation model. For more realistic buckling of microtubules surrounded by randomly distributed cross linkers, the present numerical model predicts that the buckling mode is localized at one end in agreement with some known experimental observations. In particular, the critical force for localized buckling, predicted by the present model, is insensitive to microtubule length and can be about 1 order of magnitude lower than those given by the elastic-foundation model, which suggests that the elastic-foundation model may have overestimated the critical force for buckling of microtubules in vivo. In addition, unlike the elastic-foundation model, the present model can capture the effect of end conditions on the critical force and wavelength of localized buckling. Based on the known data of spacing and elastic constants of cross linkers available in literature, the critical force and wavelength of the localized buckling mode, predicted by the present model, are compared to some experimental data with reasonable agreement. Finally, two empirical formulas are proposed for the critical force and wavelength of the localized buckling of microtubules surrounded by cross linkers.

Surface deflection of a microtubule loaded by a concentrated radial force.

Microtubules are hollow cylindrical filaments of a eukaryotic cytoskeleton which are sensitive to externally applied radial forces due to their low circumferential elastic modulus. In this work, an orthotropic elastic shell model for microtubules is used to study the surface radial deflection of a microtubule loaded by a concentrated radial force generated by either a single molecular motor or a radial indentation tip. Our results show that the maximum surface radial deflection of a microtubule generated by a concentrated radial force of a few pN can be as large as a few nanometers (a significant fraction of the radius of microtubules), which could cause significant surface morphological non-uniformity of the microtubule. In contrast, radial indentation under a much larger compressive force, which can be as large as a few hundreds of pN, will cause hardening of the circumferential elastic modulus almost equal to the longitudinal modulus of microtubules. In this case, our results show that a microtubule can withstand a concentrated radial compressive force as large as a few hundreds of pN, with a maximum radial deflection not more than a few nanometers, in good agreement with recent experiments on radial indentation of microtubules. These results offer useful data and new insights into the basic understanding of elastic interaction between microtubules and molecular motors and radial indentation of microtubules.

Enhanced critical pressure for buckling of carbon nanotubes due to an inserted linear carbon chain.

Recent findings of linear carbon-atom chains (C-chains) inside carbon nanotubes have stimulated considerable interest. In this work, molecular dynamics (MD) simulation and an elastic string-elastic shell model is adopted to study radial pressure-induced buckling of single-walled carbon nanotubes (SWCNT) filled with a C-chain. The continuum model predicts that the C-chain increases critical buckling pressure considerably (about 40%-160%) for SWCNTs of diameters ranging from 0.68 to 0.72 nm, in reasonable quantitative agreement with the prediction of MD simulation. In particular, the MD simulation confirms that the originally circular cross section of filled SWNTs becomes elliptical after buckling, as predicted by the continuum model.

Torsion of the central pair microtubules in eukaryotic flagella due to bending-driven lateral buckling.

Inspired by recent interest in torsion of the central pair microtubules in eukaryotic flagella, a novel thin-walled elastic beam model is suggested to study critical condition under which uniform bending of a flagellum will cause lateral/torsional buckling of the central pair. The model is directed to the central pair itself and the role of all surrounding cross-linkings inside the flagellum is modeled as an equivalent surrounding elastic medium. The model predicts that bending-driven torsion of the central pair does occur when the radius of curvature of the bent flagellum reduces to a moderate critical value typically of tens of microns. In particular, this critical value is almost independent of the flagellum length, and more sensitive to the parameters defining the surrounding elastic medium than the shear modulus of microtubules. The predicted wavelengths of the torsional buckling mode are insensitive to the flagellum length and comparable to some known related experimental data. These results indicate that torsion of the central pair microtubules in flagella is inevitable as a result of bending-driven lateral buckling. This offers an entirely new insight into the ongoing research on the mechanism of the central pair torsion.

Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules.

Unexplained length-dependence of flexural rigidity and Young's modulus of microtubules is studied using an orthotropic elastic shell model. It is showed that vibration frequencies and buckling load predicted by the accurate orthotropic shell model are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is this inaccuracy of the isotropic beam model used by all previous researchers that leads to reported lower flexural rigidity and Young's modulus for shorter microtubules. In particular, much lower shear modulus and circumferential Young's modulus, which only weaken flexural rigidity of shorter microtubules, are responsible for the observed length-dependence of the flexural rigidity. These results confirm that longitudinal Young's modulus of microtubules is length-independent, and the observed length-dependence of the flexural rigidity and Young's modulus is a result of strongly anisotropic elastic properties of microtubules which have a length-dependent weakening effect on flexural rigidity of shorter microtubules.

Orthotropic elastic shell model for buckling of microtubules.

In view of the fact that microtubules exhibit strong anisotropic elastic properties, an orthotropic elastic shell model for microtubules is developed to study buckling behavior of microtubules. The predicted critical pressure is found to agree well with recent unexplained experimental data on pressure-induced buckling of microtubules [Needleman, Phys. Rev. Lett. 93, 198104 (2004); Biophys. J. 89, 3410 (2005)] which are lower than that predicted by the isotropic shell model by four orders of magnitude. General buckling behavior of microtubules under axial compression or radial pressure is studied. The results show that the isotropic shell model greatly overestimates the bucking loads of microtubules, except columnlike axially compressed buckling of long microtubules (of length-to-diameter ratio larger than, say, 150). In particular, the present results also offer a plausible explanation for the length dependency of flexibility of microtubules reported in the literature.

Elastic buckling of multiwall carbon nanotubes under high pressure.

This paper studies elastic buckling of individual multiwall carbon nanotubes under radial pressure. The analysis is based on a multiple-elastic-shell model in which each of the concentric tubes of a multiwall carbon nanotube is described as an individual elastic shell. According to their radius-to-thickness ratios, the multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical pressure for elastic buckling is calculated for examples of all three types. It is found that a thin N-wall nanotube (defined by a radius-to-thickness ratio larger than 4) is approximately equivalent to a single-layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of single-wall carbon nanotubes. Based on this result, an approximate method is suggested for replacing the problematic multiwall nanotube of many layers with a multilayer elastic shell of fewer layers. In particular, the critical pressure predicted by the present model is in good agreement with known experimental results.