On the geometric phase in the spatial equilibria of nonlinear rods

Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arising from the deformed patterns exhibited by a rod subjected to end moments. This mechanical problem is classical and has a long tradition dating back to Kirchhoff. However, the perspective from geometric phases seems to go more deeply into relations between local strain states and global geometry of shapes, and infuses genuinely new insights and better understand-ing, which enable one to describe this kind of deformation in a neat and elegant way. On the other hand, visual represen-tations of these deformations provide beautiful illustrations of geometric phases and render the meaning of the abstract concept of holonomy more direct and transparent.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.     

On the Gibbs conditions of stable equilibrium,convexity and the second-order variations of thermodynamic potentials in nonlinear thermoelasticity

The Gibbs conditions of stable thermodynamic equilibrium are formulated for nonlinear thermoelastic materials, based on the constrained minimization of four fundamental thermodynamic potentials. Sufficient conditions for incremental stability are stated in each case. The previously unexplored connections between the second-order variations of thermodynamic potentials are used to establish the convexity or concavity properties of all thermodynamic potentials in relation to each other, and to derive the relationships between the specific heats at constant stress and deformation, and between the isentropic and isothermal elastic moduli and compliances. The comparison with the derivation based on the classical thermodynamic approach is also given.     

On the complex variable displacement method in plane isotropic elasticity

A modified formulation of the complex variable displacement method in plane isotropic elasticity is presented. It makes use of two equations deduced from the planar Navier equations in terms of the complex variable, which differs from England’s original formulation based on only one equation. This formulation is more direct and complements the one by England.     

冲击拉伸实验试件几何尺寸的研究

应用分离式Hopkinson拉伸实验技术研究了圆柱形冲击拉伸试件的长径比(L/D)对实验结果的影响,其中长径比从1到5。研究结果表明：对于LY12材料,L/D2.67的试件满足材料实验的要求,L/D2不能得到准确的材料参数。     

On initial,boundary conditions and viscosity coefficient control for Burgers' equation

In order to use the optimal control techniques in models of geophysical flow circulation, an application to a 1D advection–diffusion equation, the so-called Burgers' equation, is described. The aim of optimal control is to find the best parameters of the model which ensure the closest simulation to the observed values. In a more general case, the continuous problem and the corresponding discrete form are formulated. Three kinds of simulation are realized to validate the method. Optimal control processes by initial and boundary conditions require an implicit discretization scheme on the first time step and a decentered one for the non-linear advection term on boundaries. The robustness of the method is tested with a noised dataset and random values of the initial controls. The optimization process of the viscosity coefficient as a time- and space-dependent variable is more difficult. A numerical study of the model sensitivity is carried out. Finally, the numerical application of the simultaneous control by the initial conditions, the boundary conditions and the viscosity coefficient allows a possible influence between controls to be taken into account. These numerical experiments give methodological rules for applications to more complex situations. © 1998 John Wiley & Sons, Ltd.     

Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λ$\lambda$-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.     

On the geometric conservation law in transient flow calculations on deforming domains

This note revisits the derivation of the ALE form of the incompressible Navier–Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context. Copyright © 2005 John Wiley & Sons, Ltd.     

On pressure boundary conditions for the incompressible Navier-Stokes equations

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.     

On the boundary conditions for transversely isotropic piezoelectric plates

By invoking the theorem of work reciprocity for piezoelectric media, necessary conditions, which the prescribed edge data of the plate must fulfill in order that it should generate a decaying state within the plate, are established through generalizing the method proposed by Gregory and Wan. These decaying state conditions for the case of axisymmetric deformation of a transversely isotropic piezoelectric circular plate when stress and electric displacement conditions are imposed on the plate edge are derived explicitly, which are then used for the formulation of boundary conditions for the plate theory solution (or the interior solution). Also an analytical solution of the axisymmetric decaying state of transversely isotropic piezoelectric circular plates is derived. Furthermore, the corresponding necessary conditions for the axisymmetric deformation of elastic circular plates are indeed reproduced directly.     

Automated numerical simulation of the propagation of multiple cracks in a finite plane using the distributed dislocation method

In this paper, an automated numerical simulation of the propagation of multiple cracks in a finite elastic plane by the distributed dislocation method is developed. Firstly, a solution to the problem of a two-dimensional finite elastic plane containing multiple straight cracks and kinked cracks is presented. A serial of distributed dislocations in an infinite plane are used to model all the cracks and the boundary of the finite plane. The mixed-mode stress intensity factors of all the cracks can be calculated by solving a system of singular integral equations with the Gauss–Chebyshev quadrature method. Based on the solution, the propagation of multiple cracks is modeled according to the maximum circumferential stress criterion and Paris' law. Several numerical examples are presented to show the accuracy and efficiency of this method for the simulation of multiple cracks in a 2D finite plane.     

On the tangential stress anomaly in the displacement discontinuity method

It is shown that the anomaly associated with the incorrect evaluation of tangential stresses in the displacement discontinuity (DD) method, commonly used to solve crack problems, is related to the order of singularity of the fundamental solutions of linear elasticity. It is established here that a minimum of linear variation of the shear DD distribution is needed to obtain the correct tangential stress jump across a crack. Alternatively, a correction term (‘patch’) that improves tangential stresses is derived. It is also shown that need for higher functionality is a fundamental requirement rather than a convenient artifice for obtaining better accuracy.     

Recent advances in the production of controllable multiple emulsions using microfabricated devices

This review focuses on recent developments in the fabrication of multiple emulsions in micro-scale systems such as membranes, microchannel array, and microfluidic emulsification devices. Membrane and microchannel emulsification offer great potential to manufacture multiple emulsions with uniform drop sizes and high encapsulation efficiency of encapsulated active materials. Meanwhile, microfluidic devices enable an unprecedented level of control over the number, size, and type of internal droplets at each hierarchical level but suffer from low production scale. Microfluidic methods can be used to generate high-order multiple emulsions (triple, quadruple, and quintuple), non-spherical (discoidal and rod-like) drops, and asymmetric drops such as Janus and ternary drops with two or three distinct surface regions. Multiple emulsion droplets generated in microfabricated devices can be used as templates for vesicles like polymersomes, liposomes, and colloidosomes with multiple inner compartments for simultaneous encapsulation and release of incompatible active materials or reactants.     

Numerical robustness of single-layer method with Fourier basis for multiple obstacle acoustic scattering in homogeneous media

We investigate efficient methods to simulate the multiple scattering of obstacles in homogeneous media. With a large number of small obstacles on a large domain, optimized pieces of software based on spatial discretization such as Finite Element Method (FEM) or Finite Difference lose their robustness. As an alternative, we work with an integral equation method, which uses single-layer potentials and truncation of Fourier series to describe the approximate scattered field. In the theoretical part of the paper, we describe in detail the linear systems generated by the method for impenetrable obstacles, accompanied by a well-posedness study. For the numerical performance study, we limit ourselves to the case of circular obstacles. We first compare and validate our codes with the highly optimized FEM-based software Montjoie. Secondly, we investigate the efficiency of different solver types (direct and iterative of type GMRES) in solving the dense linear system generated by the method. We observe the robustness of direct solvers over iterative ones for closely-spaced obstacles, and that of GMRES with Lower–Upper Symmetric Gauss–Seidel and Symmetric Gauss–Seidel preconditioners for far-apart obstacles.     

On the shoreline boundary conditions for Boussinesq‐type models

We propose and illustrate a novel type of shoreline boundary conditions for Boussinesq‐type models. On the basis of characteristic equations of the non‐linear shallow water equations, boundary conditions are developed equations that can suitably model the motion of the instantaneous shoreline. Such boundary conditions are then implemented in a numerical solver for a specific set of Boussinesq‐type equations, which have been proved very effective for near‐shore modelling. Finally, a number of tests are performed to validate and illustrate the behaviour of the new conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

On the no-slip boundary conditions for squeeze flow of viscous fluids

A typical class of boundary conditions for squeeze flow problems in lubrication approximation is the one in which the squeezing rate and the width between the squeezing plates are constant. This hypothesis is justified by claiming that the plates moves so slowly that they can be considered static. In this short note we prove that this assumption leads to a contradiction and hence cannot be used.