Piezoelectricity with polarization gradient: homogenization

The aim of this paper is to perform homogenization of the equation of linear piezoelectricity with the polarization gradient. We assume that the material coefficients are microperiodic. This assumption can be weakened. One can also consider nonuniform homogenization. Then the homogenized (macroscopic) moduli depend on macroscopic variable.     

Random Vibrations with Impacts: A Review

Analytical methods and specific results are presented for random vibrations of systems with lumped parameters and “classical” impacts whereby finite relations between impact/rebound velocities are imposed at the impact instants that are not known in advance but rather governed by the equations of motion. Emphasis is placed on the procedures using special piecewise-linear transformation of state variables that exclude the velocity jumps at impacts or makes them small if impact losses are present. In the former case, exact analyses for stationary probability densities of the response to white-noise excitation are possible, whereas the stochastic averaging method is applied in the latter case. Furthermore, the special case of an isochronous system permits a more profound response analysis, such as predicting the spectral density of the response or subharmonic response to narrow-band excitation. The method of direct energy balance is also illustrated, based on direct application of the stochastic differential equation calculus between impacts. Certain two-degree-of-freedom impacting systems are considered, with application to moored systems, as used in ocean engineering.     

Elastic metamaterials with local resonances: an overview

Metamaterials are artificial composite materials engineered to have properties that may not be found in nature. By exploring locally resonant effect of the building units, elastic metamaterials are able to possess negative values of effective mass, effective bulk or shear modulus. Mass-spring and continuum material versions of these elastic metamaterials are reported and the physical mechanisms of negative effective parameters are demonstrated. Applications of metamaterials to acoustic cloaking and superlensing are also discussed.     

Local intrinsic modes: Layer with nonplanar interface

Laterally inhomogeneous layered media serve as models for realistic propagation environments in electromagnetics, optics, underwater acoustics, seismology and other disciplines. When the inhomogeneities extend over long distances compared to the local wavelength, general methods (like those based on integral equations) for these nonseparable boundary value problems become intractable. For weak lateral variations, the propagation process can be localized around intrinsic modal fields, which are synthesized by plane wave spectra capable of describing propagation properties uniformly in guiding, cutoff and radiating regions. These local intrinsic modes, specified by modal invariants, traverse the medium along lateral trajectories, which may refract and form caustics similar to those for ordinary ray fields in an inhomogeneous environment. In the present investigation, these concepts are applied to a single homogeneous layer separated from a homogeneous half space by an interface with arbitrary but slowly changing surface profile, as exemplified by a shallow ocean waveguide. The previously derived intrinsic modes for a two-dimensional wedge geometry (J.M. Arnold and L.B. Felsen, “Intrinsic modes in a nonseparable ocean waveguide”, Journal of Acoustical Society of America 76, 850–860 (1984)), serve as the prototype for the construction of the wave spectra, first, for the two-dimensional configuration with nonplanar interface, and then for the general three-dimensional case.     

Continua with self-rotation nuclei: evolution of asymmetric fields

We present a consistent theory of continua with defect distribution including the density of rotation nuclei. The elastic and self-fields of stresses and strains become asymmetric; the tensor of incompatibility also becomes asymmetric. We derive the dislocation–stress relations and the equations of motion related to the momentum and moment of momentum. Some applications important for earthquake engineering are presented.     

Overall moduli of solids with microcracks: Load-induced anisotropy

For a linearly elastic brittle solid containing microcracks that may be closed or may undergo frictional sliding, a general method is developed for estimating the overall instantaneous moduli which depend on the loading conditions. When the cracks are all open and when they are randomly distributed, then the overall response is isotropic. The moduli for this case have been obtained by Budiansky and O'C onnell (1976). On the other hand, when some cracks close, and when some closed cracks undergo frictional sliding, then the overall response becomes anisotropic and dependent on the loading conditions, as well as on the loading path. The self-consistent method is used to estimate the overall moduli. The effects of crack closure and loadinduced anisotropy are included. Several illustrative examples are worked out, showing the important influence of the load path on the overall response when crack closure and frictional sliding are involved.     

Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.     

Gradient elasticity with surface energy: mode-III crack problem

In this paper an anisotropic strain-gradient dependent theory of elasticity is exploited, which contains both volumetric and surface energy gradient dependent terms. The theory is applied to the solution of the mode-III crack problem and is extending previous results by Aifantis and co-workers. The two boundary value problems corresponding to the “unclamped” and “clamped” crack tips, respectively, are solved analytically. It turns out that the first problem is physically questionable for some values of the surface energy parameter, whereas the second boundary value problem is leading to a cusping crack, which is consistent with Barenblatt's theory without the incorporation of artificial assumptions.     

Energy Pumping with Various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small nonlinear attachment. The essential nonlinearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment. Different nonlinear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to civil engineering. As a specific application, attenuation of vibrations of a building is studied with two building models. In particular, the case of stochastic excitations is analyzed to examine if it is possible to process energy pumping when a seism occurs and an indicator of efficiency has been introduced.     

Water hammer with column separation: A historical review

Column separation refers to the breaking of liquid columns in fully filled pipelines. This may occur in a water-hammer event when the pressure in a pipeline drops to the vapor pressure at specific locations such as closed ends, high points or knees (changes in pipe slope). The liquid columns are separated by a vapor cavity that grows and diminishes according to the dynamics of the system. The collision of two liquid columns, or of one liquid column with a closed end, may cause a large and nearly instantaneous rise in pressure. This pressure rise travels through the entire pipeline and forms a severe load for hydraulic machinery, individual pipes and supporting structures. The situation is even worse: in one water-hammer event many repetitions of cavity formation and collapse may occur.This paper reviews water hammer with column separation from the discovery of the phenomenon in the late 19th century, the recognition of its danger in the 1930s, the development of numerical methods in the 1960s and 1970s, to the standard models used in commercial software packages in the late 20th century. A comprehensive survey of laboratory tests and field measurements is given. The review focuses on transient vaporous cavitation. Gaseous cavitation and steam condensation are beyond the scope of the paper.     

Microstructures with finite surface energy: the two-well problem

We study solutions of the two-well problem, i.e., maps which satisfy uSO(n)ASO(n)B a.c. in ⁿ , where A and B are n×n matrices with positive determinants. This problem arises in the study of microstructure in solid-solid phase transitions. Under the additional hypothesis that the set E where the gradient lies in SO(n) A has finite perimeter, we show that u is locally only a function of one variable and that the boundary of E consists of (subsets of) hyperplanes which extend to and which do not intersect in . This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.     

Correlated scale-free network with community: modeling and transportation dynamics

Many transport processes on network depend crucially on the underlying network topology. In this paper, we propose a model to generate correlated scale free transportation networks with community structure by considering the mechanisms of dynamical network evolution and rewiring links. With the introduction of congestion effects, we investigate the performance and carrying capacity of this network. The results show that congestion in the uncorrelated network is more serious than the assortative or disassortative ones. Therefore, the correlated network with communities can bear much more traffic flow. In addition, the networks with lager modularity can enhance the transportation efficiently.     

Cluster statistics in a bidimensional suspension: Comparison with percolation

We study the cluster statistics and the viscosity of a two-dimensional suspension of passive macroscopic spheres undergoing shear. The second moment of the finite cluster statistics exhibits a maximum for a 2-D concentrationΦ _S near 0.67 without measurable anomaly in the viscosity. The results of the cluster statistics are compared to those obtained in percolation.     

Flux in Porous Media with Memory: Models and Experiments

The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient–flux and the pressure–density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux.