Nonlinear Force Law Research Articles

Spatial parameterization of attachment processes in molecular motor-cargo systems

Intracellular transport is conducted largely by molecular motor proteins which process along cytoskeletal filaments, from which they can attach or detach. We describe an analytical framework to characterize motor attachment or reattachment rates to microtubules as a function of the physical and geometric properties of the motor, the cargo to which it is attached, and possibly a second motor attached to the same cargo and a microtubule. The biophysical model is coarse-grained at the level of the macromolecular motors and formulated in terms of stochastic differential equations, allowing for rotation of the cargo and nonlinear force laws for the motor-cargo tether.

From individual-based mechanical models of multicellular systems to free-boundary problems

In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson–Kendalli–Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.

Curvature effects on circular feed end-milling, part 2: stability analysis

The non-periodic delay differential equation (DDE) derived in Ozoegwu et al. (in press) to govern chatter of circular milling is averaged over a single discrete delay to get a periodic DDE that approximates chatter of circular end-milling. Chatter stability analysis of the averaged periodic DDE respectively using both the second order least squares approximated full-discretisation map and the modified full-discretisation map on systems governed by linear force law and nonlinear force law gives that curvature diminishes chatter stability because of its reducing effect on damping. This result is in conformity with the simulation-derived results in Ozoegwu et al. (in press). The destabilising effect of curvature gets stronger with lowering of the spindle speed. It is recommended that experimental validation of effects of curvature on stability of end-milling should be done at low spindle speed range. The observed destabilising effect of curvature on chatter stability increases with fall in natural frequency of the tool thus stiffer tools with more strengthening profiles should be selected for circular milling.

Classifying general nonlinear force laws in cell-based models via the continuum limit

Although discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

Effect of purely elastic flow instability on molecular conformation and drag

Linear stability analysis has shown that viscoelastic creeping flow of an Oldroyd-B liquid through a sinusoidal channel is unstable to stationary, wall-localized and short wavelength perturbations [B. Sadanandan, R. Sureshkumar, Global linear stability analysis of non-separated viscoelastic flow through a periodically constricted channel, J. Non-Newtonian Fluid Mech. 122 (2004) 55]. In this work, time-dependent simulations are performed to determine the nonlinear evolution of finite amplitude disturbances in the post-critical flow regime. It is shown that a nonlinear transition, which is facilitated by a supercritical pitchfork bifurcation, establishes a finite amplitude state (FAS) in which the average polymer stretch is highly modulated. The maximum normal stress, observed at the channel nip, can increase by up to approximately 100% when the Weissenberg number, defined as the ratio of the fluid relaxation time to an inverse characteristic shear rate, is increased by only 10% beyond its critical value. This is attributed to the amplification of configurational perturbations by the base flow shear rate, which attains its maximum at the channel nip. The effect of finite chain extensibility on the critical condition and nonlinear instability is investigated using the FENE-CR model. The stabilizing effect of finite extensibility can be expressed through a renormalization of the Weissenberg number by accounting for the screening effect of the nonlinear force law on the transmission of configurational perturbations to polymeric stress. The principal features of the FAS are qualitatively model-independent. The FAS exhibits a small, but numerically perceptible increase in the friction factor as compared to the base flow. The implication of the findings on the experimentally observed flow resistance enhancement phenomenon in viscoelastic creeping flows through converging/diverging geometries is discussed.

Mode-I fracture in a nonlinear lattice with viscoelastic forces.

We study mode-I fracture in a viscoelastic lattice model with a nonlinear force law, with a focus on the velocity and linear stability of the steady-state propagating solution. This study is a continuation both of the study of the piecewise-linear model in mode I, and the study of more general nonlinear force laws in mode-III fracture. At small driving, there is a strong dependency of the velocity curve on the dissipation and a strong sensitivity to the smoothness of the force law at large dissipation. At large driving we calculate, via a linear stability analysis, the critical velocity for the onset of instability as a function of the smoothness, the dissipation and the ratio of lattice spacing to critical extension. This critical velocity is seen to be very sensitive to these parameters. We confirm our calculations via direct numerical simulations of the initial value problem.

Nonlinear lattice model of viscoelastic mode III fracture.

We study the effect of general nonlinear force laws in viscoelastic lattice models of fracture, focusing on the existence and stability of steady-state mode III cracks. We show that the hysteretic behavior at small driving is very sensitive to the smoothness of the force law. At large driving, we find a Hopf bifurcation to a straight crack whose velocity is periodic in time. The frequency of the unstable bifurcating mode depends on the smoothness of the potential, but is very close to an exact period-doubling instability. Slightly above the onset of the instability, the system settles into a exactly period-doubled state, presumably connected to the aforementioned bifurcation structure. We explicitly solve for this new state and map out its velocity-driving relation.

Impulse propagation in granular media

Grains can be described as elastic objects. When two grains are pushed against one another, they repel via Hertz’ nonlinear force law, F∝δn, n>2, δ≥0, being the overlap between the grains. We show that the propagation of an impulse of any magnitude in a 1D chain of grains at zero loading can be described as a solitary wave. The width of the solitary wave, L(n)→1 as n→∞, L∼5 grain diameters for typical granular contacts and L(n)→∞ when n→2. The condition δ≥0 leads to the formation of secondary solitary waves when two identical solitary waves propagating in opposite directions collide. Randomness in grain densities and sizes and restitutional losses lead to approximately exponential decays in distance of the energy of a propagating solitary wave. It turns out that impulses can be exploited to fingerprint a buried impurity mass in a 1D chain with and without gravity. In closing, we shall discussthe backscattering of an impulse from a buried object in 3D beds at shallow depths. Implications of this work with respect to humanitarian demining and related applications will be presented. [Work supported in part by U.S. Army Corps of Engineers and Sandia National Laboratories. Work done in collaboration with Marian Manciu (SUNY-Buffalo) and Alan J. Hurd (Sandia National Labs).]

Arrested cracks in nonlinear lattice models of brittle fracture.

We generalize lattice models of brittle fracture to arbitrary nonlinear force laws and study the existence of arrested semi-infinite cracks. Unlike what is seen in the discontinuous case studied to date, the range in driving displacement for which these arrested cracks exist is very small. Also, our results indicate that small changes in the vicinity of the crack tip can have an extremely large effect on arrested cracks. Finally, we briefly discuss the possible relevance of our findings to recent experiments.

Effective transverse response of fiber composites with nonlinear interface

This paper presents a framework for the analysis of the constitutive response of composites in which nonlinear behavior originates solely from the force-separation response of the interfaces. The direct method of composite materials theory is employed to pass from local nonlinear behavior of a solitary inclusion problem to nonlinear aggregate response. The resulting model, which involves no adjustable parameters, falls within the conceptual framework of continuum damage mechanics with “damage” variables that have geometrical meaning on the microscale. They are shown to be the expansion coefficients arising in the eigenfunction representation of the displacement jump at a representative inclusion—matrix interface. Detailed calculations of stress—strain response are carried out for the case of transverse shear and plane dilatation of unidirectional fiber composites in dilute concentration. Bifurcation of equilibrium separation in the solitary inclusion problem is shown to precipitate instability in composite response. Interfaces are assumed smooth and such that normal separations are governed by the nonlinear force law of Ferrante et al. (1982) . Comments on the extension of the formulation to include the effects of finite concentration are provided as well.

Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields

Numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model; the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbell model of Chilcott–Rallison (CR). In the Giesekus model, intramolecular forces are described by a Hookean spring, whereas a finitely extensible spring whose modulus is given by the Warner approximation is used in both the FENE-P and CR models. Hydro dynamic drag on the beads is taken to be anisotropic for the Giesekus model and isotropic for the other two models. The CR and FENE-P models differ subtly in their approximate treatment of the nonlinear force law. The three models exhibit very similar rheological behavior in viscometric flow and steady elongational flow, with the notable exception that the viscosity for the CR model is shear-rate independent. Finite element simulations are performed by using two different formulations: the elastic-viscous split-stress gradient (EVSS-G) method and a new variant of this formulation, the discrete EVSS-G (DEVSS-G) formulation, in which the elliptic stabilization term is added only to the discrete version of the momentum equation, and the constitutive equation is solved directly in terms of the polymer contribution to the stress tensor. Calculations are performed for all models up to a Weissenberg number We, where the configuration tensor 〈 QQ 〉 loses positive definiteness. However, by locally refining the mesh in the gap region, the positive definiteness of 〈 QQ 〉 is recovered. The flow and stress fields predicted by the three constitutive equations are qualitatively similar. A `birefringent strand' of highly stretched polymer molecules, which appears to emanate from the rear stagnation point in the cylinder, strengthens as We is increased. Not surprisingly, the molecular extension computed for the Giesekus model is considerably larger than that of the two FENE spring models. The drag force on the cylinders differs for the FENE-P and CR models, because of the difference in the shear-thinning viscosity resulting from the different approximations used in these models.

Liapunov Stability of Force-Controlled Grasps with a Multi-Fingered Hand

Holding an object stably is a building block for dexterous ma nipulation with a multi-fingered hand. In recent years a rather large body of literature related to this topic has developed. These works isolate some desired property of a grasp and use this property as the definition of stable grasp. To varying degrees, these approaches ignore the system dynamics. The purpose of this article is to put grasp stability on a more basic and fundamental foundation by defining grasp stability in terms of the well-established stability theory of differential equations. This approach serves to unify the field and to bring a large body of knowledge to bear on the field. Some relationships be tween the stability concepts used here and the previously used grasp stability concepts are discussed. A hierarchy of three levels of approach to the problem is treated. We consider that the grasp force applied is a basic consideration, in terms of ensuring both that there is sufficient force to prevent dropping the object and that the forces are not too large to cause break age. As a result, we first investigate the use of constant force grasps. It is shown that with the proper combination of finger locations and grasp forces, such grasps can be Liapunov sta ble, and methods are presented that help find such grasps. It is also shown that such grasps cannot be asymptotically stable. To produce asymptotic stability, one must alter the forces ap plied to an object when the object deviates from equilibrium, and a linear feedback force law is given for this. It results in local asymptotic stability guaranteeing convergence to the desired grasp equilibrium from all states within a region of attraction in the state space. Some of the results are similar to results obtained previously, but this time they have a stronger meaning in terms of the dynamic response of the system. In the third level, a nonlinear force law is given that, to within certain limitations, produces global asymptotic grasp stability, so that all initial states are guaranteed to converge to the de sired grasp equilibrium. The method is robust to large classes of inaccuracies in the implementation.

Separation at a circular interface under biaxial load

T his p aper is concerned with the plane problem of the decohesion of a smooth circular elastic inclusion from an unbounded elastic matrix subject to a remote biaxial load. The phenomenon of brittle decohesion is treated as a problem of bifurcation of equilibrium separation at the inclusion-matrix interface. In accordance with other cohesive zone type models the interface is characterized by a constitutive equation relating tractions across the interface to the displacement discontinuity that develops there. The specific form considered in this work is the physically based nonlinear force law of Ferrante, Smith and Rose that couples the normal component of displacement jump to the normal component of interface traction and which requires a characteristic length for its prescription. Under decreasing values of characteristic length to inclusion radius ratio, three types of separation behavior (ductile decohesion, brittle decohesion, closure) can occur provided the remote load, interface strength and elastic moduli of inclusion and matrix are within the required bounds. Within this context of interfacial separation, the phenomenon of void nucleation is briefly examined and a connection is made between the classical nucleation criteria of critical interfacial stress and critical energy release and the criteria for ductile and brittle decohesion. Interaction effects are analysed in an approximate way by considering the effects of decohesion on a hypothetical, rigidly bonded inclusion with the same elastic moduli as the matrix.

Linearized Flexible Multibody Dynamics Including Geometric Stiffening Effects∗

Abstract In multibody system simulation for vehicle dynamics applications, it is often sufficient to consider a set of system equations in which the kinematics are linearized. On the other hand, care must be taken to model the flexibility of the system bodies correctly and to be able to introduce the highly nonlinear force laws appearing in multibody vehicle models into the simulation. A formalism meeting these objectives is presented here. Special care has been taken to incorporate geometric stiffening terms that result from nominal loads on the system. This formalism is the core of the vehicle dynamic simulation code MEDYNA.

Measuring piano hammers and modeling their interaction

Comparison of measured piano string vibration spectra with predictions indicates that modeling the hammer with a simple mass and Hooke's law spring is not very satisfactory [Hall and Askenfelt, J. Acoust. Soc. Am. 83, 1627 (1988)]. Measurements have been made on additional hammer sets to see if values of the exponent p in a nonlinear force law F = Kξp can be connected either with hammer quality or with other aspects of piano design and history. The corresponding theory of hammer/string interaction requires solution of nonlinear equations of motion, and has previously been done only for a few isolated examples. Steps toward a flexible general computer program that can explore the dependence of string spectra upon striking parameters with nonlinear hammers will be discussed.

Lattice theory of three-dimensional cracks

The problem of the stability of a three-dimensional crack will be analyzed within a lattice-statics approximation. Hsieh and Thomson employed a similar approach for a two-dimensional crack. In that work the force was taken to be linear up to an arbitrary displacement and set equal to zero for larger displacements. This paper will investigate the consequence of introducing a jog into the crack face as well as the effects of various nonlinear-force laws. The phenomenon of lattice trapping (upper and lower bounds on the applied stress for an equilibrium crack of given length) is again obtained. By looking at various force laws, it is possible to obtain some physical insight into which aspects of the force law are critical for crack stability. In particular, the inadequacy of a thermodynamic approach (which relates the critical stress to a surface energy corresponding to the area under the cohesive-force–vs–displacement curve) will be demonstrated. Surface energy is a global property of the cohesive-force law. Crack stability is sensitive to much more refined aspects of the cohesive-force law. Crack healing is sensitive to the long-range portion of the cohesive force. It occurs when an applied load leads to a pair of atoms, whose bond had been broken, acquiring a net displacement such that the cohesive force is non-negligible (about 5% of peak value) and increasing with decreasing displacment. Crack expansion normally occurs when an increment in a net atomic displacement leads to a change in slope in the cohesive-force law: a positive spring constant becoming negative. Hence crack expansion is sensitive to the position of the maximum in the cohesive-force relation.