Continuum Rod Research Articles

New Analytical Solution to Predict the Vertical Impedance of a Large-Diameter Pipe Pile in Soil Considering Wave Propagation in Visco-Elastic Continuum

According to the theories of wave propagation in visco-elastic continuum and Rayleigh-Love rod, a simplified model in axisymmetric conditions for the longitudinal vibration of large-diameter pipe pile (LDPP) in radially heterogeneous surrounding soil with viscous damping is presented. The relevant analytical solution for dynamic impedance at pile head is derived by using complex stiffness transfer method, which is also validated via independent comparisons with previous solutions. Besides, parametric analyses are carried out to reveal both the radial heterogeneity of surrounding soil and the lateral inertia effect of pile shaft on the dynamic impedance of LDPP. The obtained analytical solutions are suitable for the longitudinal vibration issues of floating LDPP in visco-elastic surrounding soil with radial heterogeneity, which can be conveniently degenerated to describe the longitudinal vibration of a floating solid pile in soil with radial heterogeneity as well as a floating LDPP or solid pile in radially homogenous soil.

Analytical Modeling and Design of Generalized Pneu-Net Soft Actuators with Three-Dimensional Deformations.

Pneu-net soft actuators, consisting of pneumatic networks of small chambers embedded in elastomeric structures, are particularly promising candidates in the society of soft robotics. However, there are few studies on the analytical modeling of pneu-net soft actuators, especially in the three-dimensional space. In this article, based on the minimum potential energy method and the continuum rod theory, we propose an analytical model and corresponding design approach for a class of generalized pneu-net soft actuators (gPNSAs) with both bending and twisting deformations by combining the geometric complexity and material elasticity. We experimentally verify our modeling approach and finally investigate the effects of geometric parameters, material properties, and external force on the deformations of gPNSAs, which can be used as a tool for the design of gPNSAs. We further demonstrate that our developed model can predict the deformations of gPNSAs made of multiple materials.

Analyzing the Bond Behavior of Fiber-Reinforced Polymer (FRP) Bars Embedded in Engineered Cementitious Composites (ECCs) with the Nonlocal Continuum Rod Model

In this study, a nonlocal elastic rod model is applied to analytically evaluate the bond behavior between fiber-reinforced polymer (FRP) bars and engineered cementitious composites (ECCs). The second-order differential equation, which is based on nonlocal elasticity theory, governs the bond behavior of the FRP bars along the bond length. The classical elasticity model is a special case of the nonlocal model. The solution of the second-order differential equation can be obtained by substituting three-stage linear bond stress-slip relationship of the FRP bars. The slip values (solution of the second-order differential equation) within the bond length calculated by the nonlocal continuum rod model are affected by the nonlocal parameter e0a. The results from a case study show that the maximum pullout force decreases when the nonlocal size effect is considered, thereby providing a closer approximation of the experimental data than the existing local model.

Unified nonlocal rational continuum models developed from discrete atomistic equations

In this paper, a unified nonlocal rational continuum enrichment technique is presented for improving the dispersive characteristics of some well known classical continuum equations on the basis of atomistic dispersion relations. This type of enrichment can be useful in a wide range of mechanical problems such as localization of strain and damage in many quasibrittle structures, size effects in microscale elastoplasticity, and multiscale modeling of materials. A novel technique of transforming a discrete differential expression into an exact equivalent rational continuum derivative form is developed considering the Taylor’s series transformation of the continuous field variables and traveling wave type of solutions for both the discrete and continuum field variables. An exact equivalent continuum rod representation of the 1D harmonic lattice with the non-nearest neighbor interactions is developed considering the lattice details. Using similar enrichment technique in the variational framework, other useful higher-order equations, namely nonlocal rational Mindlin–Herrmann rod and nonlocal rational Timoshenko beam equations, are developed to explore their nonlocal properties in general. Some analytical and numerical studies on the high frequency dynamic behavior of these novel nonlocal rational continuum models are presented with their comparison with the atomistic solutions for the respective physical systems. These enriched rational continuum equations have crucial use in studying high-frequency dynamics of many nano-electro-mechanical sensors and devices, dynamics of phononic metamaterials, and wave propagation in composite structures. These new models can help to circumvent the biggest problem regarding size and time restrictions in many atomistic simulations.

A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

We present a novel scheme to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.

A Study of the Dynamic Characteristics of the Cyclone-4 Launch Vehicle on the Basis of a Continuum Rod Model

A technology for the creation of dynamic simulation models of liquid-propellant launch vehicles on the basis of a combined continuum rod scheme is considered. The procedure has been successfully implemented and evaluated for the Cyclone-4 launch vehicle as an example. Amplitude-frequency characteristics of the launch vehicle in different flight segments have been determined, and the obtained results have been analyzed.

Models for twistable elastic polymers in Brownian dynamics, and their implementation for LAMMPS

An elastic rod model for semi-flexible polymers is presented. Theory for a continuum rod is reviewed, and it is shown that a popular discretised model used in numerical simulations gives the correct continuum limit. Correlation functions relating to both bending and twisting of the rod are derived for both continuous and discrete cases, and results are compared with numerical simulations. Finally, two possible implementations of the discretised model in the multi-purpose molecular dynamics software package LAMMPS are described.

Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models

In this paper, the size-effects in the torsional and axial response of microtubules by using the nonlocal continuum rod model is investigated. To this end, continuous and discrete rod models are performed for modeling of microtubules. A simple finite element procedure is used for modeling and solution of nonlocal discrete system equation for microtubules. The influence of the small length scale on the vibration frequencies is examined both torsional and axial vibration cases. Some parametric results are also presented for examination of the accuracy and performances of discrete and continuous models.

Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity

The axial vibration of single walled carbon nanotube embedded in an elastic medium is studied using nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are used in the formulations. The effect of various parameters like stiffness of elastic medium, boundary conditions and nonlocal parameters on the axial vibration of nanorods is discussed. It is obtained that, the axial vibration frequencies of the embedded nanorods are highly over estimated by the classical continuum rod model which ignores the effect of small length scale.

A Multiscale Dynamic Model of DNA Supercoil Relaxation by Topoisomerase IB

In this study, we report what we believe to be the first multiscale simulation of the dynamic relaxation of DNA supercoils by human topoisomerase IB (topo IB). We leverage our previous molecular dynamics calculations of the free energy landscape describing the interaction between a short DNA fragment and topo IB. Herein, this landscape is used to prescribe boundary conditions for a computational, elastodynamic continuum rod model of a long length of supercoiled DNA. The rod model, which accounts for the nonlinear bending, twisting, and electrostatic interaction of the (negatively charged) DNA backbone, is extended to include the hydrodynamic drag induced by the surrounding physiological buffer. Simulations for a 200-bp-long DNA supercoil in complex with topo IB reveal a relaxation timescale of ∼0.1–1.0 μs. The relaxation follows a sequence of cascading reductions in the supercoil linking number ( Lk), twist ( Tw), and writhe ( Wr) that follow companion cascading reductions in the supercoil elastic and electrostatic energies. The novel (to our knowledge) multiscale modeling method may enable simulations of the entire experimental setup that measures DNA supercoiling and relaxation via single molecule magnetic trapping.

Terahertz wave propagation in uniform nanorods: A nonlocal continuum mechanics formulation including the effect of lateral inertia

The dynamic testing of materials and components often involves predicting the propagation of stress waves in slender rods. The present work deals with the analysis of the wave propagation characteristics of nanorods. The nonlocal elasticity theory and also the lateral inertia are incorporated into classical/local rod model to capture unique features of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behaviors of nanorods from those of macroscopic rods. Nonlocal rod/bar model is developed for nanorods including the lateral inertia effects. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The wave propagation properties of the nanorod obtained from the present formulations are compared with the continuum rod model, nonlocal second and fourth order strain gradient models, Born-K a ´ rm a ´ n model and the nonlocal stress gradient model. It has also been shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in the present manuscript. The results provided in this article are useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Regional Variations in Flexibilities Limit Continuum Rod Description of Long Coiled Coils

Continuum rod description of coiled coils yielded a persistence length that range between 150–300 nm (Hvidt, S. et al., 1982. Biochem. 21, 4064, Adamovic, I. Et al. 2008. Biophys. J. 94, 3779). However, recent studies show that long coiled coils such as the stalk of a Rad50 complex exhibit regional variations in flexibilities (van Noort, J. et al., 2003. PNAS, 100, 7581). Apart from a length dependence due to non-bonded interactions (Lakkaraju, S. et al., 2008. BioPhys J. 94, 2404-Pos) our investigations reveal that stiffness of coiled coils is also strongly sequence dependent, hence limiting the region within which a coiled coil maybe described as a continuum rod. Bending stiffness (Kb) of a 161 A long coiled coil with a leucine zipper periodicity calculated using normal mode analysis, forced bending dynamics and thermal fluctuation analysis (TFA) with locally built triad systems is about 1.86X10−27 Nm2. Replacing the apolar residues at the a and d positions of the heptad repeat that are involved in the knob-into hole packing, with a hydrophilic (Glu and Lys) or polar charged residues (Ser), decreased Kb to about 0.864X10−27 Nm2. Importantly, TFA of tropomyosin (PDB: 1C1G) mapped regions of varied flexibilities whose Kb varied between 0.33X10−27 Nm2 (Ala 211 to Lys 221) to about 1.83X10−27 Nm2 (Ser 36 to Leu 46), which confirms that stiffness of the structure decreases towards the C-terminus. Four of the seven actin binding sites (Brown, J.H. et al., 1986. Adv. Prot. Chem. 71, 121) lie within the regions of increased flexibility (Kb ∼ (0.75–1.36)X10−27 Nm2), which possibly has a mechanical role during the on state of muscle contraction.

Axial vibration of the nanorods with the nonlocal continuum rod model

Nonlocal elastic rod model is developed and applied to investigate the small-scale effect on axial vibration of nanorods. Explicit expressions are derived for frequencies for clamped–clamped and clamped–free boundary conditions. It is concluded that the axial vibration frequencies are highly over estimated by the classical (local) rod model, which ignores the effect of small-length scale. Present results can be used for axial vibration of single-walled carbon nanotubes.

Radial breathing-like mode of wide carbon nanoribbon

The radial breathing-like mode (RBLM), an interesting Raman mode, of the carbon nanoribbons (CNRs) with different widths have been calculated by the empirical Brenner potential. It is found that the RBLM frequencies of CNRs whose widths are larger than 25 Å follow a new 1 / w rule, which is different from the 1 / w rule, obtained for the narrow CNRs [J. Zhou, J. Dong, Appl. Phys. Lett. 91 (2007) 173108]. A continuum rod model is proposed to explain the new 1 / w rule, which is also different from the previous simple one-dimensional oscillator model, suitable only for the narrow CNRs. Based upon the CNR's RBLMs, we have proposed a practical experimental method to determine the widths of CNRs by their Raman spectra.

Model for polymorphic transitions in bacterial flagella

Many bacteria use rotating helical flagellar filaments to swim. The filaments undergo polymorphic transformations in which the helical pitch and radius change abruptly. These transformations arise in response to mechanical loading, changes in solution temperature and ionic strength, and point substitutions in the amino acid sequence of the protein subunits that make up the filament. To explain polymorphism, we propose a coarse-grained continuum rod theory based on the quaternary structure of the filament. The model has two molecular switches. The first is a double-well potential for the extension of a protofilament, which is one of the 11 almost longitudinal columns of subunits. Curved filament shapes occur in the model when there is a mismatch strain, i.e., when intersubunit bonds in the inner core of the filament prefer a subunit spacing which is intermediate between the two spacings favored by the double-well potential. The second switch is a double-well potential for twist, due to lateral interactions between neighboring protofilaments. Cooperative interactions between neighboring subunits within a protofilament are necessary to ensure the uniqueness of helical ground states. We calculate a phase diagram for filament shapes and the response of a filament to external moment and force.

The parametrically excited upside-down rod: an elastic jointed pendulum model

A model is studied which consists of a chain of N identical pendulums coupled by damped elastic joints subject to vertical sinusoidal forcing of its base. Particular attention is paid to the stability of the upright equilibrium configuration with a view to understanding recent experimental results on the stabilization of an unstable stiff column under parametric excitation. It is shown via an appropriate scaling argument how the continuum rod model arises by taking the limit N→∞. The effect of the inclusion of bending stiffness is first studied via asymptotics and numerics for the case N=1, showing how the static bifurcation of the pendulum varies with the four dimensionless parameters of the system; damping, bending stiffness and amplitude and frequency of excitation. For the multiple pendulum system, the bifurcation behaviour of the upright position as a function of the same four parameters is studied via numerical methods applied to the linearized equations. The damping term is found to be crucial in destroying many of the resonant instabilities that occur in the limit as N→∞. At realistic damping levels only a few instabilities remain, which are shown to be largely independent of N. These instabilities agree qualitatively with the experiments.

Modeling of continuous time homogeneous Markov system with fixed size as elastic solid: the two-dimensional case

Every attainable structure of a continuous time homogeneous Markov system (HMS) with n states and fixed size is considered as a point-particle of R n . Thus, the motion of an attainable structure corresponds to the motion of the respective point-particle in R n . Under the assumption that “the motion of every particle at every time point is due to its interaction with its surroundings”, R n becomes a continuum [Appl. Stochastic Models Data Anal. 13 (1998) 177]. The evolution of the set of the attainable structures corresponds to the motion of the continuum. For the particular case of a two-dimensional HMS ( n=2), it is examined to what extend the HMS could be considered as a linear elastic continuum (rod). Hence, it is concluded that the general concept of multidimensional anisotropic elasticity can further interpret the evolution of HMSs with n>3.

A continuum rod model of sequence-dependent DNA structure

Experimentally motivated parameters from a base-pair-level discrete DNA model are averaged to yield parameters for a continuum elastic rod with a curved unstressed shape reflecting the local DNA geometry. The continuum model permits computations with discretization lengths longer than the intrinsic discretization of the base-pair model, and, for this and other reasons, yields an efficient computational formulation. Obtaining continuum stiffnesses is straightforward, but obtaining a continuum unstressed shape is hindered by the ‘‘noisy’’ small-scale structure and rapid helix twist of the discrete unstressed shape. Filtering of the discrete data and an analytic transformation from the true normal-vector field to a natural (untwisted) frame allows a stable continuum fit. Equilibrium energies of closed rings predicted by the continuum model are found to match the energies of the underlying discrete model to within 0.5%. The model is applied to a set of 11 short DNA molecules (≊ 150 bp) and properly distinguishes their cyclization probabilities (J factors) when compared both to experimental cyclization rates and to Monte Carlo simulations. The continuum model does not include entropic contributions to the free energy. However, because of its rapid and accurate computation of internal energy, the continuum model should, when combined with further work on entropic effects, be a useful method for computing experimental DNA free energies.

A continuum theory for the prediction of lateral and rotational positioning of alpha-helices in membrane proteins: bacteriorhodopsin.

We have developed a new method for the prediction of the lateral and the rotational positioning of transmembrane helices, based upon the present status of knowledge about the dominant interaction of the tertiary structure formation. The basic assumption about the interaction is that the interhelix binding is due to the polar interactions and that very short extramembrane loop segments restrict the relative position of the helices. Another assumption is made for the simplification of the prediction that a helix may be regarded as a continuum rod having polar interaction fields around it. The polar interaction field is calculated by a probe helix method, using a copolymer of serine and alanine as probe helices. The lateral position of helices is determined by the strength of the interhelix binding estimated from the polar interaction field together with the length of linking loop segments. The rotational positioning is determined by the polar interaction field, assuming the optimum lateral configuration. The structural change due to the binding of a prosthetic group is calculated, fixing the rotational freedom of a helix that is connected to the prosthetic group. Applying this method to bacteriorhodopsin, the optimum lateral and rotational positioning of transmembrane helices that are very similar to the experimental configuration was obtained. This method was implemented by a software system, which was developed for this work, and automatic calculation became possible for membrane proteins comprised of several transmembrane helices.

特集 生体膜と人工膜の接点 膜タンパク質の姿・形を予測しよう

Principles of the structure prediction of membrane proteins is discussed : (1) Dominant interaction for the binding between transmembrane helices is polar interactions. (2) A helix may be simplified to a continuum rod with surface charges. (3) The configuration of helices is determined by the interactions between transmembrane helices and the structure of linking segments. These principles are based upon various denaturation experiments and physical considerations. A software system for the strcture prediction of membrane proteins is being developed and applied to bacteriorhodopsin whose molecular structure has been already reported.