Plectoneme formation in twisted fluctuating rods

The mechanics of DNA supercoiling is a subject of crucial importance to uncover the mechanism and kinetics of several enzymes. It is therefore being investigated using several biochemical and biophysical methods including single molecule experimental techniques. An interesting problem within this realm is that of torsional buckling and plectoneme formation in DNA as it is simultaneously put under tensile and torsional stress. Analytical solutions to this problem are difficult to find since it involves nonlinear kinematics and thermal fluctuations. In this paper we use ideas from the Kirchhoff theory of filaments to find semi-analytical solutions for the average shape of the fluctuating DNA under the assumption that there is no self-contact. The basic step in our method consists of combining a helical solution of the rod with a non-planar localizing solution in such a way that the force, moment, position and slope remain continuous everywhere along the rod. Our solutions allow us to predict the extension vs. linking number behavior of long pieces of DNA for various values of the tension and temperature. An interesting outcome of our calculations is the prediction of a sudden change in extension at buckling which does not seem to have been emphasized in earlier theoretical models or experiments. Our predictions are amenable to falsification by recently developed single molecule techniques which can simultaneously track the force-extension as well as the torque-rotation behavior of DNA.     

The mechanics of short rod-like molecules in tension

The rapid development of single molecule experimental techniques in the last two decades has made it possible to accurately measure the force–extension response as well as the transverse fluctuations of individual rod-like macromolecules. This information is used in conjunction with a statistical mechanical model based on the treatment of the molecule as a fluctuating elastic rod to extract its bending and extension moduli. The models most commonly used to interpret the experimental data assume that the magnitude of the Brownian fluctuations are independent of the length of the macromolecule, an assumption that holds only in the asymptotic limit of infinitely long rods, and is violated in most experiments. As an alternative, we present a theoretical treatment of a finite length, fluctuating rod and determine its mechanical behavior by measuring the transverse Brownian fluctuations under the action of large stretching forces. To validate our theory, we have applied our methods to an experiment on short actin filaments whose force–extension relation is difficult to measure, but whose transverse deflections can be captured by current microscopy techniques. An important consequence of the short contour lengths is that the boundary conditions applied in the experiment affect the fluctuations and can no longer be neglected as is commonly done when interpreting data from force–extension measurements. Our theoretical methods account for boundary conditons and can therefore be deployed in conjunction with force–extension measurements to obtain detailed information about the mechanical response of rod-like macromolecules.     

A Catalogue of Stable Equilibria of Planar Extensible or Inextensible Elastic Rods for All Possible Dirichlet Boundary Conditions

We catalogue configurations that locally minimize energy for a planar elastic rod (extensible-shearable or inextensible-unshearable) subject to arbitrary Dirichlet boundary conditions in position and orientation. Via a combination of analysis and computation, we determine several bifurcation surfaces in the 3-parameter space of boundary conditions and explore how they depend on the rod material parameters, including in the inextensible limit. For each possible boundary condition, we find all stable equilibria with sufficiently low energy that they might be competitive within a Boltzmann distribution if the rod were used to model DNA with tens or hundreds of base pairs, the length-scale relevant for DNA looping. Depending on the boundary conditions, there are as many as three such equilibria.     

A non-linear viscoelastic model for ceramics at high temperatures

High-temperature mechanical behavior of ceramics is characterized by non-linear rate dependent responses, asymmetric behavior in tension and compression, and nucleation and coalescence of voids leading to rupture. Moreover, rupture experiments show considerable scatter or randomness in fatigue lives of nominally equal specimens. To capture the non-linear, asymmetric time-dependent behavior, a new non-linear viscoelastic model is proposed. Non-linearity and asymmetry are introduced in the volumetric component. To model the random formation and coalescence of voids, each element is assigned a failure strain sampled from a lognormal distribution. An element is deleted when its volumetric strain exceeds its failure strain. Temporal increases in strains produce a sequential loss of elements (a model for void nucleation and growth), which in turn leads to failure. Non-linear viscoelastic model parameters are determined from uniaxial tensile and compressive creep experiments on silicon nitride. The model is then used to predict the deformation of four-point bending and ball-on-ring specimens. Simulation is used to predict statistical moments of rupture lives. Numerical simulation results compare well with results of four-point bending experiments.     

Stability analysis of helical rod based on exact Cosserat model

Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff's kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler's angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov's stability of the helical equilibrium is discussed in static category. Euler's critical values of axial force and torque are obtained. Lyapunov's and Euler's stability conditions in the space domain are the necessary conditions of Lyapunov's stability of the helical rod in the time domain.     

Convection and overstability in a viscoelastic fluid layer

In this paper, a study is made of the stability of a plane layer of a simple fluid heated from below. A differential (second-order fluid) and a single integral model are applied under various boundary conditions. The conditions for which the principle of the exchange of stabilities is valid are presented for both models and selected boundary conditions. It is shown that the second-order-fluid model proposed by Dunn and Fosdick [10], without surface tension, will not exhibit overstability.     

The strongest rotating rod

By using the Rayleigh quotient, we present the variational formulation for the strongest rotating rod stable against buckling. This variational formulation is converted to fifth-order singular non-linear boundary value problem. The optimal shape and the critical rotating speed are determined with special numerical-analytical integration procedure. We found the explicit linear relation between the volume and the squared critical speed. Although, in general, the linear stability problem for the optimal rotating rod does not have purely discrete spectra, we show that in the present case, the critical speed is determined with lowest eigenvalue. This fact verifies our optimization strategy based on a linear spectral problem.     

An isotropic-plasticity-based constitutive model for martensitic reorientation and shape-memory effect in shape-memory alloys

In this work, we develop an isotropic-plasticity-based constitutive model for initially martensitic shape-memory alloys (SMA) which exhibit martensitic reorientation and the shape-memory effect. The constitutive model is then implemented in the [Abaqus reference manuals. 2006. Providence, R.I.] finite-element program by writing a user-material subroutine. The results from the constitutive model and numerical procedure are then compared to representative physical experiments conducted on polycrystalline rod and sheet Ti–Ni. The constitutive model and the numerical simulations are able to reproduce the stress–strain responses from these physical experiments to good accuracy. Finally, two different boundary value problems utilizing the one-way shape-memory effect are studied: (a) the deformation of an arterial stent, and (b) a micro-clamper. We show that our constitutive model can be used to model the response of the aforementioned boundary value examples.     

On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.     

A fluctuating elastic plate and a cell model for lipid membranes

The thermal fluctuations of lipid bi-layer membranes are key to their interaction with cellular components as well as the measurement of their mechanical properties. Typically, membrane fluctuations are analyzed by decomposing into normal modes or by molecular simulations. Here we propose two new approaches to calculate the partition function of a membrane. In the first approach we view the membrane as a fluctuating von Karman plate and discretize it into triangular elements. We express its energy as a function of nodal displacements, and then compute the partition function and co-variance matrix using Gaussian integrals. We recover well-known results for the dependence of the projected area of the membrane on the applied tension and recent simulation results on the dependence of membrane free energy on geometry, spontaneous curvature and tension. As new applications we compute the fluctuations of the membrane of a malaria infected cell and analyze the effects of boundary conditions on fluctuations. Our second approach is based on the cell model of Lennard-Jones and Devonshire. This model, which was developed for liquids, assumes that each molecule fluctuates within a cell on which a potential is imposed by all the surrounding molecules. We adapt the cell model to a lipid membrane by recognizing that it is a 2D liquid with the ability to deform out of plane whose energetic penalty must be factored into the partition function of a cell. We show, once again, that some results on membrane fluctuations can be recovered using this new cell model. However, unlike some well established results, our cell model gives an entropy that scales with the number of molecules in a membrane. Our model makes predictions about the heat capacity of the membrane that can be tested in experiments.     

Helical Birods: An Elastic Model of Helically Wound Double-Stranded Rods

We consider a geometrically accurate model for a helically wound rope constructed from two intertwined elastic rods. The line of contact has an arbitrary smooth shape which is obtained under the action of an arbitrary set of applied forces and moments. We discuss the general form the theory should take along with an insight into the necessary geometric or constitutive laws which must be detailed in order for the system to be complete. This includes a number of contact laws for the interaction of the two rods, in order to fit various relevant physical scenarios. This discussion also extends to the boundary and how this composite system can be acted upon by a single moment and force pair. A second strand of inquiry concerns the linear response of an initially helical rope to an arbitrary set of forces and moments. In particular we show that if the rope has the dimensions assumed of a rod in the Kirchhoff rod theory then it can be accurately treated as an isotropic inextensible elastic rod. An important consideration in this demonstration is the possible effect of varying the geometric boundary constraints; it is shown the effect of this choice becomes negligible in this limit in which the rope has dimensions similar to those of a Kirchhoff rod. Finally we derive the bending and twisting coefficients of this effective rod.     

Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

Writhing instabilities of twisted rods: from infinite to finite length

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into account as a second approach, reveal some possible experiments in which the configuration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real finite length and we show that another type of instability may occur, leading to possible hysteresis behavior. As we go from infinite to finite length, we compare the different information given by the three approaches on the possible equilibrium configurations of the rod and their stability. These finite size effects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the infinite length approach is understood.     

Optimal shape of a twisted and compressed rod

We formulate and solve the problem of determining the shape of an elastic rod stable against buckling and having minimal volume. The rod is loaded by a concentrated force and a couple at its ends. The equilibrium equations are reduced to a single nonlinear second-order equation. The eigenvalues of the linearized version of this equation determine the stability boundary. By using Pontryagin's maximum principle we determine the optimal shape of the rod.     

Modelling of fluidelastic instability in a square inline tube array including the boundary layer effect

Flow-induced vibration (FIV) is a design concern in many engineering applications such as tube bundles in heat exchangers. When FIV materializes, it often results in fatigue and/or fretting wear of the tubes, leading to their failure. Three cross-flow excitation mechanisms are responsible for such failures: random turbulence excitation, Strouhal periodicity, and fluidelastic instability. Of these three mechanisms, fluidelastic instability has the greatest potential for destruction. Because of this, a large amount of research has been conducted to understand and predict this mechanism. This paper presents a time domain model to predict the fluidelastic instability forces in a tube array. The proposed model accounts for temporal variations in the flow separation. The unsteady boundary layer is solved numerically and coupled with the structure model and the far field flow model. It is found that including the boundary layer effect results in a lower stability threshold. This is primarily due to a larger fluidelastic force effect on the tube. The increase in the fluidelastic effect is attributed to the phase difference between the boundary layer separation point motion and the tube motion. It is also observed that a non-linear limit cycle is predicted by the proposed model.     

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.     

Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys

A crystal-inelasticity-based constitutive model for martensitic reorientation and detwinning in shape-memory alloys (SMAs) has been developed from basic thermodynamics principles. The model has been implemented in a finite-element program by writing a user-material subroutine. We perform two sets of finite-element simulations to model the behavior of polycrystalline SMAs: (1) The full finite-element model where each finite element represents a collection of martensitic microstructures which originated from within an austenite single crystal, chosen from a set of crystal orientations that approximates the initial austentic crystallographic texture. The macroscopic stress-strain responses are calculated as volume averages over the entire aggregate: (2) The Taylor model (J. Inst. Metals 62 (1938) 32) where an integration point in a finite element represents a material point which consist of sets of martensitic microstructures which originated from within respective austenite single-crystals. Here the macroscopic stress-strain responses are calculated through a homogenization scheme.Experiments in tension and compression were conducted on textured polycrystalline Ti-Ni rod initially in the martensitic phase by Xie et al (Acta Mater. 46 (1998) 1989). The material parameters for the constitutive model were calibrated by fitting the tensile stress-strain response from a full finite-element calculation of a polycrystalline aggregate to the simple tension experiment. With the material parameters calibrated the predicted stress-strain curve for simple compression is in very good accord with the corresponding experiment. By comparing the simulated stress-strain response in simple tension and simple compression it is shown that the constitutive model is able to predict the observed tension-compression asymmetry exhibited by polycrystalline Ti-Ni to good accuracy. Furthermore, our calculations also show that the macroscopic stress-strain response depends strongly on the initial martensitic microstructure and crystallographic texture of the material.We also show that the Taylor model predicts the macroscopic stress-strain curves in simple tension and simple compression reasonably well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory materials.     

Material symmetry group of the non-linear polar-elastic continuum

We extend the material symmetry group of the non-linear polar-elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors. The group consists of an ordered triple of tensors which makes the strain energy density of polar-elastic continuum invariant under change of reference placement. An analog of the Noll rule is established. Four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed. Definitions of polar-elastic fluids, solids, liquid crystals and subfluids are given in terms of members of the symmetry group. Within polar-elastic solids we discuss in more detail isotropic, hemitropic, cubic-symmetric, transversely isotropic, and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density. For physically linear polar-elastic solids, when the density becomes a quadratic function of strain measures, reduced representations of the density are established for monoclinic, orthotropic, cubic-symmetric, hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch, wryness and undeformed structure curvature tensors.