Effective elasticity tensor of a periodic composite

The effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of Nemat-Nasseret al., also agree well with the numerical results for cubic lattices.     

Numerical study of two-dimensional problems of nonsymmetric elasticity

We consider the algorithm of the finite element method for solving two-dimensional problems of nonsymmetric elasticity. We discuss the possibilities of the algorithm and its efficiency by comparing the numerical results with the well-known analytic solutions. We present the results obtained by solving the problem of tension of a plate weakened by a series of holes and the problem of tension for a plate with a central crack. The numerical results thus obtained are considered as an addition to the analytic solutions in the context of experimental justification of couple-stress effects arising under deformation of elastic materials and in the context of solving the identification problem for mechanical constants in nonsymmetric elasticity.     

Lie symmetries of geodesic equations and projective collineations

We prove a theorem which relates the Lie symmetries of the geodesic equations in a Riemannian space with the collineations of the metric. We apply the results to Einstein spaces and spaces of constant curvature. Finally with examples we show the use of the results.     

An anti-plane shear problem

We consider an anti-plane shear of an elastic cylinder with a non-convex stored energy function. So, we look for solutions of a non-convex problem of the calculus of variations with Dirichlet boundary conditions. We give sufficient conditions on the boundary data to get existence or non-existence results for this non-convex problem. We also prove some uniqueness results for the relaxed problem associated with the initial problem.     

On the dynamics of swimming linked bodies

In this paper we study the motion of three linked ellipses moving through a viscous fluid in two dimensions. The angles between the ellipses change with time in a specified manner (the gait) and the resulting time varying configuration is similar to the appearance of a swimming leech. We simulate the motion using the particle method Smoothed Particle Hydrodynamics (SPH) which we test by convergence studies and by comparison with the inviscid results of Kanso et al. (2005) [2] and the viscous results of Eldredge (2006, 2007, 2008) [4], [5], [6]. We determine how the average speed and power output depends on the amplitude and oscillation frequency of the gait. We find that the results fit simple scaling rules which can be related to the analytical results of G.I. Taylor for the swimming of long narrow animals (1952). We apply our results to estimate the speed of a swimming leech with reasonable accuracy, and we determine the minimum power required to propel the bodies at a specified average speed.     

Thermovisco-hyperelastic behavior of elastomeric materials modified by filler nanoparticles

We present the results of experimental studies of hyperelastic and relaxation properties of polymer composites with elastomeric matrix made of hydrogenated nitrile butadiene rubber filled with nanoparticles of technical carbon in the temperature range 19–150° C. We present typical experimental diagrams of deformation of the material with constant strain rate and the stress relaxation curves at different strain levels under tension and compression conditions. We consider a possible version of constitutive relations for describing some singularities of the behavior of the material under study. We developed a method for determining all the parameters of the accepted relations on the basis of the results of uniaxial tests. We found a nonmonotone dependence of the relaxation modulus on the temperature and proposed a formula for describing this dependence in the temperature range under study. To justify the possible use of the considered constitutive relations to perform calculations under conditions of arbitrary compound stress state, we performed numerical modeling of the compression experiment for cylindrical samples. A rather satisfactory agreement between the computational results and experimental data was obtained.     

Convective oscillations in interconnected containers

We study an oscillating Instability of a nonuniformly heated liquid in interconnected containers. We examine the influence of the conditions of heat transfer and the geometric parameters on the critical Rayleigh number and the frequency of the stationary oscillations. We make a comparison of the theoretical results obtained with experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 4–9, July–August, 1974.     

Anticipatory, complete and lag synchronization of chaos and hyperchaos in a nonlinear delay-coupled time-delayed system

We study the synchronization of chaos and hyperchaos in first-order time-delayed systems that are coupled using the nonlinear time-delay excitatory coupling. We assign two characteristic time delays: the system delay that is same for both the systems, and the coupling delay associated with the coupling path. We show that depending upon the relative values of the system delay and the coupling delay the coupled systems show anticipatory, complete, and lag synchronization. We derive a general stability condition for all the synchronization processes using the Krasovskii–Lyapunov theory. Numerical simulations are carried out to corroborate the analytical results. We compute a quantitative measure to ensure the occurrence of different synchronization phenomena. Finally, we set up an experiment in electronic circuit to verify all the synchronization scenario. It is observed that the experimental results are in good agreement with our analytical results and numerical observations.     

Attractivity in non-autonomous systems

We discuss some sufficient conditions for attractivity of a closed set by using Liapunov functions and show how to apply our results to the study of asymptotic behaviors of solutions in nonautonomous systems. We also discuss the convergence of solutions by the same idea. These results are extendable to functional differential equations.     

Numerical Experiments in Scalar Diffusion under Stochastic Convection

We study scalar diffusion, both from Eulerian and Lagrangian perspectives, advected by two dimensional flows. Emphasis is devoted to the problem of scalar diffusion under a synthetic turbulent flow. We present numerical and analytical results for the turbulent diffusion coefficient either from the influence of the turbulent synthetic field and a periodic array of eddies. Preliminary results concerning Lagrangian dispersion are also reported.     

Mathematical modeling of dynamic fracture processes in concrete

We solve the problem on the collision of steel cylindrical hammers in the velocity range up to 800 m/s with rectilinear concrete slabs. We consider the following two approaches to calculating concrete fracture under impact loading: the phenomenological approach in which the strength criteria are expressed in terms of invariant constraints imposed on the critical macrocharacteristics of the process, i.e., stresses and strains, and the approach in which the fracture is considered as the process of formation, growth, and confluence of microdefects under the action of applied stresses. We compare the results of mathematical modeling with the experimental data concerning the penetration depth and the exterior flushing value. In the framework of the proposed model of dynamic fracture in concrete, we calculate the strength of a concrete tetrahedral prism under the action of longitudinal loads. We obtain a satisfactory agreement of the results of mathematical modeling with experimental results.     

Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations

We investigated theoretically and experimentally the nonlinear responseof a clamped-clamped buckled beam to a subharmonic resonance of orderone-half of its first vibration mode. We used a multi-mode Galerkindiscretization to reduce the governing nonlinear partial-differentialequation in space and time into a set of nonlinearly coupledordinary-differential equations in time only. We solved the discretizedequations using the method of multiple scales to obtain a second-orderapproximate solution, including the modulation equations governing itsamplitude and phase, the effective nonlinearity, and the effectiveforcing. To investigate the large-amplitude dynamics, we numericallyintegrated the discretized equations using a shooting method to computeperiodic orbits and used Floquet theory to investigate their stabilityand bifurcations. We obtained interesting dynamics, such as phase-lockedand quasiperiodic motions, resulting from a Hopf bifurcation,snapthrough motions, and a sequence of period-doubling bifurcationsleading to chaos. Some of these nonlinear phenomena, such as Hopfbifurcation, cannot be predicted using a single-mode Galerkindiscretization. We carried out an experiment and obtained results ingood qualitative agreement with the theoretical results.     

Boundary element method for long-time water wave propagation over rapidly varying bottom topography

We study numerically the linear water wave equations for shallow channels with rapidly varying bottom topography. We do not use the shallow water approximation because it is not valid when the bottom is rapidly varying. We use the boundary element method because it allows accurate tracking of the surface waves for long times. We present the results of a range of numerical validation experiments and a comparison between propagation over a periodic and a random rough bottom topography.     

Predicting the onset of DNA supercoiling using a non-linear hemitropic elastic rod

Elastic rod models provide a means to interpret single molecule DNA experiments as well as predict DNA behavior under physiological conditions. Here we use an elastic rod model to predict the stability boundary (critical torque vs. applied tension) for single molecule DNA experiments in which the molecule is subjected to applied tension and twist. We discuss the shortcomings of the usual isotropic rod model. We then derive a consistent non-linear material law from the general representation for a hemitropic (chiral) rod. Finally, we present results of a standard bifurcation analysis predicting the stability boundary. We find results from the non-linear hemitropic rod to match the data closely.     

Synchronization in the Lorenz system: Stability and robustness

We present necessary and sufficient conditions for the existence of synchronization in a class of continuous-time nonlinear systems: the so-called -affine systems. We apply the results to the Lorenz attractor. The robustness of the synchronization against parameter value variations is discussed using the Lyapunov stability theory for perturbed systems. We obtain sufficient conditions that guarantee a bounded steady-state error. This technique gives conservative results; however, in some systems like that of Lorenz, it provides definitive results about the existence of the synchronization. Furthermore, we give estimates of the maximal error as a function of the difference between the parameter values of the systems to be synchronized.     

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.     

Observations of modal interactions in resonantly forced beam-mass structures

We present a collection of experimental results on the influence of modal interactions (i.e., internal or autoparametric resonances) on the nonlinear response of flexible metallic and composite structures subjected to a range of resonant excitations. The experimental results are provided in the form of frequency spectra, Poincaré sections, pseudo-phase planes, dimension calculations, and response curves. Experimental observations of transitions from periodic to chaotically modulated motions are also presented. We also discuss relevant analytical results. The current study is also relevant to other internally resonant structural systems.     

Postbuckling behaviors of open section composite struts with edge delamination using a layerwise theory

We investigate the mechanical stability of L-section and T-section composite struts with single edge delamination. We propose a solution procedure based on a layerwise theory and the first order shear deformation theory by taking into consideration of the von Karman geometrical nonlinearity. We derive the nonlinear equilibrium equations according to the minimum total potential energy principle, and solve them using Rayleigh–Ritz method and Newton–Raphson method. In modeling the delaminated L-section and T-section struts, we divide the structures into regions, and exert continuity conditions between different regions. The proposed model is capable of analyzing both local buckling of the base laminate and sublaminate as well as the global buckling of the whole structure. We present numerical results to provide an insight into effects of size of delamination on buckling mode and post-buckling behaviors of the struts. We perform the three-dimensional finite element analysis using the ABAQUS commercial software. The results show a very good agreement with those obtained by the analytical method. The results indicate that the presence of delaminations not only reduces the load-carrying capacity of open section struts remarkably, but also plays a pivotal role in the critical buckling load and buckling mode shape of the struts.     

FLASH magnetohydrodynamic simulations of shock-generated magnetic field experiments

We report the results of benchmark FLASH magnetohydrodynamic (MHD) simulations of experiments conducted by the University of Oxford High Energy Density Laboratory Astrophysics group and its collaborators at the Laboratoire pour l'Utilisation des Lasers Intenses (LULI). In these experiments, a long-pulse laser illuminates a target in a chamber filled with Argon gas, producing shock waves that generate magnetic fields via the Biermann battery mechanism. We first outline the implementation of 2D cylindrical geometry in the unsplit MHD solver in FLASH and present results of verification tests. We then describe the results of benchmark 2D cylindrical MHD simulations of the LULI experiments using FLASH that explore the impact of external fields along with the possibility of magnetic field amplification by turbulence that is associated with the shock waves and that is induced by a grid placed in the gas-filled chamber.     

Layers in the Presence of Conservation Laws

We study standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. Our results give spectral stability and instability results depending only on relative monotonicity of the two components of the system. We also prove the robustness of layers and their stability properties. Our results classify stability properties of layers in most such systems. Our method is based on tracking the point spectrum during a homotopy to a simple, decoupled system. Main difficulty is the possibility of eigenvalues disappearing in a branch point of the essential spectrum. This phenomenon is investigated using a Lyapunov?CSchmidt reduction method on exponentially weighted spaces combined with a matching procedure for the far-field.