Dynamics and thermodynamics of a model with long-range interactions

The dynamics and thermodynamics of particles/spins interacting via long-range forces display several unusual features compared with systems with short-range interactions. The Hamiltonian mean field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular, the model shows a second-order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact, the numerical simulations show the presence of quasi-stationary states (QSSs), i.e. metastable states that become stable if the thermodynamic limit is taken before the infinite time limit. The QSSs differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heats, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly decaying correlations, and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSSs have recently been interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which does not preclude to the previous one.Received: 8 July 2003, Accepted: 27 October 2003, Published online: 11 February 2004PACS: 05.70.Fh, 89.75.Fb, 64.60.Fr, 75.10.NrCorrespondence to: A. Rapisarda     

基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。     

The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity

Recently, Sherief et al. (Int. J. Solids Struct. 47:269–275, 2010) proposed a model in generalized thermoelasticity based on the fractional order time derivatives. The propagation of electro-magneto-thermoelastic disturbances in a perfectly conducting elastic half-space is investigated in the context of the above fractional order theory of generalized thermoelasticity. There acts an initial magnetic field parallel to the plane boundary of the half-space. Normal mode analysis together with the eigenvalue approach technique is used to solve the resulting non-dimensional coupled governing equations of the problem. The obtained solution is then applied to two specific problems for the half-space, whose boundary is subjected to (i) thermally isolated surfaces subjected to time-dependent compression and (ii) a time-dependent thermal shock and zero stress. The effects of fractional parameter and magnetic field on the variations of different field quantities inside the half-space are analyzed graphically.     

A free rectangular plate on elastic foundation

In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential equation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous equations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.     

A free rectangular plate on elastic foundation

This article will discuss the bending problems of the rectangular plates with free boundaries on elastic foundations. We talk over the two cases, that is, the plate acted on its center by a concentrated force and the plate subjected to by a concentrated force equally at four corner points respectively. We select a fiexural function which satisfies not only all the geometric boundary conditions on free edges wholly but also the boundary conditions of the total internal forces. We apply the variational method meanwhile and then obtain better approximate solutions.     

On one model of a transversally isotropic elastic foundation

The method of expansion of the stress and displacement components into series in terms of functions of the transverse coordinate and the method of variation with respect to the state being determined are used to construct an iterative model of a transversally isotropic elastic foundation. This model allows for all the SSS components. The order of the equations does not depend on the number of terms of the expansions, which approximate the displacements and stresses. As an example, the problem on a concentrated force applied to an elastic foundation is considered. Pridneprovskaya State Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 130–134, June, 2000.     

A non-Fickian,particle-tracking diffusion model based on fractional Brownian motion

The work is motivated by the recent discovery that ocean surface drifter trajectories contain fractal properties. This suggests that the dispersion of pollutants in coastal waters may also be described using fractal statistics. The paper describes the development of a fractional Brownian motion model for simulating pollutant dispersion using particle tracking. Numerical test cases are used to compare this new model with the results obtained from a traditional Gaussian particle-tracking model. The results seems to be significantly different, which may have implications for pollution modelling in the coastal zone. © 1997 John Wiley & Sons, Ltd.     

Optimal shape of an elastic column on elastic foundation

By using Pontryagin's maximum principle we determine the shape of an elastic compressed column on elastic, Winkler type foundation. We assume that the column has clamped ends. The optimality conditions for the case of bimodal optimization are derived. It is shown that the optimal cross-sectional area function is determined from the solution of a nonlinear boundary value problem. In the special case of a compressed column with no foundation, the optimality condition and the solution obtained earlier are recovered.     

A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity

We formulate a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. We show that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. In particular, we derive optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely, the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains and to the strain-gradient elasticity regularization. We show how the critical energy-release rate of specific materials can be determined from test data. Finally, we demonstrate the scope and fidelity of the model by means of an example of application, namely, Taylor-impact experiments of polyurea 1000 rods.     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory

A generalized refined theory including surface effects is developed for functionally graded ultra-thin films with different surface properties. The classical generalized shear deformable theory is adopted to model the film bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. As a result, the shape function also shows size-dependence on the film thickness. Since the film is non-homogeneous through the thickness, the state space method and approximate laminate model are employed to derive the variation of shape function through the thickness direction. A simply supported thin film in cylindrical bending is considered as an example to illustrate the application of the present theory. By comparing to the Kirchhoff plate theory including surface effects, the necessity of the present theory for FGM thin films is solidly validated. It is established that the present FGM thin films exhibit significant size-dependence when the thickness approaches to micro-scale. As the gradient index changes, the extent of size-dependence varies accordingly.     

A fractional calculus interpretation of the fractional volatility model

Based on criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, the model is formulated in terms of a fractional integration of stochastic processes.     

弹性地基梁损伤诊断研究

利用试验得到的振动参数评估结构的破损情况，是当前结构工程学科十分活跃的领域。由于弹性地基梁的振动模态受地基和梁两方面因素的影响，其损伤诊断问题变得十分复杂。本文通过对两靖自由弹性地基梁的灵敏性分析发现弹性地基梁的前两阶自由模态主要与地基有关，利用这一特性构造了两级识别的方法，并引入优化领域寻优能力极强的遗传算法进行识别，找到了令人满意的答案。     

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. The effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.     

Bending of beams on three-parameter elastic foundation

The bending of a Timoshenko beam resting on a Kerr-type three-parameter elastic foundation is introduced, its governing differential equations are formulated and analytically solved, and the solutions are discussed and applied to particular problems. Parametric analyses of elastically supported beams of infinite and finite length are carried out and comparisons are made between one, two or three-parameter foundation models and more accurate 2D finite element models. In order to estimate the necessary soil parameters, an analytical procedure based on the modified Vlasov model is proposed. The presented solutions and applications show the superiority of the Kerr-type foundation model compared to one or two-parameter models.     

弹性地基上四边自由Reissner矩形中厚板的有限积分变换法

将弹性地基视为Winkler模型,利用二维有限积分变换的方法推导出了弹性地基上四边自由矩形中厚板位移和内力的精确解.由于在求解过程中不需要预先人为选取位移函数,而是从弹性地基上中厚板的基本方程出发,直接利用有限积分变换的数学方法求出可以完全满足四边自由边界条件,弹性地基上矩形中厚板问题的精确解,使得问题的求解更加合理.最后通过计算实例验证了所采用方法及所推导出的公式的正确性.     

A micromechanically based couple-stress model of an elastic orthotropic two-phase composite

We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin's (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couple-stress properties from the response of a unit cell under several types of boundary conditions: displacement, displacement-periodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions' volume fraction of 18%.     

Iterative technique for circular thin plates on Gibson elastic foundation using modified Vlasov model

In this paper, to investigate the influence of soil inhomogeneity on the bending of circular thin plates on elastic foundations, the static problem of circular thin plates on Gibson elastic foundation is solved using an iterative method based on the modified Vlasov model. On the basis of the principle of minimum potential energy, the governing differential equations and boundary conditions for circular thin plates on modified Vlasov foundation considering the characteristics of Gibson soil are derived. The equations for the attenuation parameter in bending problem are also obtained, and the issue of unknown parameters being difficult to determine is solved using the iterative method. Numerical examples are analyzed and the results are in good agreement with those form other literatures. It proves that the method is practical and accurate. The inhomogeneity of modified Vlasov foundations has some influence on the deformation and internal force behavior of circular thin plates. The effects of various parameters on the bending of circular plates and characteristic parameters of the foundation are discussed. The modified model further enriches and develops the elastic foundations. Relevant conclusions that are meaningful to engineering practice are drawn.