Planar isotropic structures with negative Poisson’s ratio

A new design principle is suggested for constructing auxetic structures – the structures that exhibit negative Poisson’s ratio (NPR) at macroscopic level. We propose 2D assemblies of identical units made of a flexible frame with a sufficiently rigid reinforcing core at the centre. The core increases the frame resistance to the tangential movement thus ensuring high shear stiffness, whereas the normal stiffness is low being controlled by the local bending response of the frame. The structures considered have hexagonal symmetry, which delivers macroscopically isotropic elastic properties in the plane perpendicular to the axis of the symmetry. We determine the macroscopic Poisson’s ratio as a ratio of corresponding relative displacements computed using the direct microstructural approach. It is demonstrated that the proposed design can produce a macroscopically isotropic system with NPR close to the lower bound of ?1. We also developed a 2D elastic Cosserat continuum model, which represents the microstructure as a regular assembly of rigid particles connected by elastic springs. The comparison of values of NPRs computed using both structural models and the continuum approach shows that the continuum model gives a healthy balance between the simplicity and accuracy and can be used as a simple tool for design of auxetics.     

Determining the Mie potential parameters using Poisson’s ratio

The known value of Poisson’s ratio specifying the relation between the strains along the principal directions in the case of uniaxial strain is used to propose an approach to derive an equation relating this ratio to the exponents of the Mie pair potential. An example of determining one of these exponents is discussed when the other exponent is given.     

FE analysis of the in-plane mechanical properties of a novel Voronoi-type lattice with positive and negative Poisson’s ratio configurations

This work presents a novel formulation for a Voronoi-type cellular material with in-plane anisotropic behaviour, showing global positive and negative Poisson’s ratio effects under uniaxial tensile loading. The effects of the cell geometry and relative density over the global stiffness, equivalent in-plane Poisson’s ratios and shear modulus of the Voronoi-type structure are evaluated with a parametric analysis. Empirical formulas are identified to reproduce the mechanical trends of the equivalent homogeneous orthotropic material representing the Voronoi-type structure and its geometry parameters.     

Rayleigh and Love surface waves in isotropic media with negative Poisson’s ratio

The behavior of Rayleigh surface waves and the first mode of the Love waves in isotropic media with positive and negative Poisson’s ratio is compared. It is shown that the Rayleigh wave velocity increases with decreasing Poisson’s ratio, and it increases especially rapidly for negative Poisson’s ratios less than ?0.75. It is demonstrated that, for positive Poisson’s ratios, the vertical component of the Rayleigh wave displacements decays with depth after some initial increase, while for negative Poisson’s ratios, there is a monotone decrease. The Rayleigh waves are characterized by elliptic trajectories of the particle motion with the change of the rotation direction at critical depths and by the linear vertical polarization at these depths. It is found that the elliptic orbits are less elongated and the critical depths are greater for negative Poisson’s ratios. It is shown that the stress distribution in the Rayleighwaves varies nonmonotonically with the dimensionless depth as (positive or negative) Poisson’s ratio varies. The stresses increase strongly only as Poisson’s ratio tends to?1. It is shown that, in the case of an incompressible thin covering layer, the velocity of the first mode of the Love waves strongly increases for negative Poisson’s ratios of the half-space material. If the thickness of the incompressible layer is large, then the wave very weakly penetrates into the halfspace for any value of its Poisson’s ratio. For negative Poisson’s ratios, the Love wave in a layer and a half-space is mainly localized in the covering layer for any values of its thickness and weakly penetrates into the half-space. For the first mode of the Love waves, it was discovered that there is a strong increase in the maximum of one of the shear stresses on the interface between the covering layer and the half-space as Poisson’s ratios of both materials decrease. For the other shear stress, there is a stress jump on the interface and a more complicated dependence of the stress on Poisson’s ratio on both sides of the interface.     

A study of shallow water waves with Gardner’s equation

In this paper the dynamics of solitary waves governed by Gardner’s equation for shallow water waves is studied. The mapping method is employed to carry out the integration of the equation. Subsequently, the perturbed Gardner equation is studied, and the fixed point of the soliton width is obtained. This fixed point is then classified. The integration of the perturbed Gardner equation is also carried out with the aid of He’s semi-inverse variational principle. Finally, Gardner’s equation with full nonlinearity is solved with the aid of the solitary wave ansatz method.     

Influence of the spatial variation of Poisson’s ratio upon the elastic field in nonhomogeneous axisymmetric bodies

The effect of a nonconstant Poisson’s ratio upon the elastic field in functionally graded axisymmetric solids is analyzed. Both of the elastic coefficients, i.e. Young’s modulus and Poisson’s ratio, are permitted to vary in the radial direction. These elastic coefficients are considered to be functions of composition and are related on this basis. This allows a closed form solution for the stress function to be obtained. Two cases are discussed in this investigation: first, both Young’s modulus and Poisson’s ratio are allowed to vary across the radius and the effect of spatial variation of Poisson’s ratio upon the maximum radial displacement is investigated; secondly, Young’s modulus is taken as constant and the change in the maximum hoop stress resulting from a variable Poisson’s ratio is calculated.     

A new fundamental diagram theory with the individual difference of the driver’s perception ability

Based on the driver’s individual difference of the driver’s perception ability, we in this paper develop a new fundamental diagram with the driver’s perceived error and speed deviation difference. The analytical and numerical results show that the speed-density and flow-density data are divided into three prominent regions. In the first region, the speed-density and flow-density data are scattered around the equilibrium speed-density and flow-density curves of the classical fundamental diagram theory, where the widths of these scattered data are very narrow and slightly increase with the real density (i.e., the scattered data appear as two thicker lines); the running speed is approximately equal to the free flow speed and the real flow approximately linearly increases with the real density. In the second region, the speed-density and flow-density data are scattered widely in a two-dimensional region, but the shapes of these widely scattered data are related to the properties of the driver’s perceived error and speed deviation difference. In the third region, the scattered speed-density and flow-density data appear but these scattered data will quickly degenerate into the equilibrium speed-density and flow-density curves with the increase of the real density. Finally, the numerical results illustrate that the new fundamental diagram theory also produces the F-line, U-line, and L-line. The shapes of the scattered data, F-line, U-line, and L-line are relevant to the properties of the driver’s perceived error and speed deviation difference. These results are qualitatively accordant with the real traffic, which shows that the new fundamental diagram theory can better describe some complex traffic phenomena in the real traffic system. In addition, the above results can help us to further explain why the widely scattered speed-density and flow-density data appear in the real traffic system and better understand the effects of the driver’s individual difference on traffic flow.     

Resonant-Based Identification of the Poisson’s Ratio of Orthotropic Materials

The resonant-based identification of the in-plane elastic properties of orthotropic materials implies the estimation of four principal elastic parameters: E ₁ , E ₂ , G ₁₂ , and ν ₁₂ . The two elastic moduli and the shear modulus can easily be derived from the resonant frequencies of the flexural and torsional vibration modes, respectively. The identification of the Poisson’s ratio, however, is much more challenging, since most frequencies are not sufficiently sensitive to it. The present work addresses this problem by determining the test specimen specifications that create the optimal conditions for the identification of the Poisson’s ratio. Two methods are suggested for the determination of the Poisson’s ratio of orthotropic materials: the first employs the resonant frequencies of a plate-shaped specimen, while the second uses the resonant frequencies of a set of beam-shaped specimens. Both methods are experimentally validated using a stainless steel sheet.     

A new type of multiple-lump and interaction solution of the Kadomtsev–Petviashvili I equation

Nonlinear Dynamics - In this paper, the solution in the form of Grammian of the Kadomtsev–Petviashvili I equation is employed to investigate a new type of multiple-lump solution. The bound...     

A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg–Landau equation

Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg–Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted.     

Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects

The integro-partial differential equations governing the dynamic behavior of viscoelastic plates taking account of higher-order shear effects and finite deformations are presented. From the matrix formulas of differential quadrature, the special matrix product and the domain decoupled technique presented in this work, the nonlinear governing equations are converted into an explicit matrix form in the spatial domain. The dynamic behaviors of viscoelastic plates are numerically analyzed by introducing new variables in the time domain. The methods in nonlinear dynamics are synthetically applied to reveal plenty and complex dynamical phenomena of viscoelastic plates. The numerical convergence and comparison studies are carried out to validate the present solutions. At the same time, the influences of load and material parameters on dynamic behaviors are investigated. One can see that the system will enter into the chaotic state with a paroxysm form or quasi-periodic bifurcation with changing of parameters.     

A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.     

Group preserving scheme and reproducing kernel method for the Poisson–Boltzmann equation for semiconductor devices

This paper introduces that the nonlinear Poisson–Boltzmann equation for semiconductor devices describing potential distribution in a double-gate metal oxide semiconductor field effect transistor (DG-MOSFET) is exactly solvable. The DG-MOSFET shows one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry. Lie symmetry properties of this model is investigated in order to extract some exact solutions. The reproducing kernel Hilbert space method and group preserving scheme also have been applied to the nonlinear equation. Numerical results show that the present methods are very effective.

     

Decoupling coefficients of dilatational wave for Biot’s dynamic equation and its Green’s functions in frequency domain

Green’s functions for Biot’s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green’s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term “decoupling coefficient” for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green’s functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng’s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green’s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method (BEM) and other applications.     

Moment Lyapunov exponents of the stochastic parametrical Hill’s equation

The Lyapunov exponent and moment Lyapunov exponents of Hill’s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time-varying damping which are small intensity stochastic excitations.     

Do general viscoelastic stresses for the flow of an upper convected Maxwell fluid satisfy the momentum equation?

Given a general velocity field consistent with the stagnation point flow, can the viscoelastic stresses arising in the flow of an upper convected Maxwell fluid found by solving the constitutive equation also satisfy the momentum equation? Consideration is given to the study of the stress tensor arising in the steady flow of an upper convected Maxwell (UCM) fluid with a velocity field consistent with the stagnation point flow. By the method of characteristics, exact solutions to the partial differential equations arising in the approximating model of the viscoelastic stresses in the flow of an upper convected Maxwell (UCM) fluid are obtained for the three components of the stress tensor, for reasonably general velocity fields. We are able to account for the effects of variable boundary data at the inflow by considering the viscoelastic stresses over two spatial variables. Furthermore, we assume a relatively general velocity field. As a special case, some results present in the recent literature are obtained; it is known that these special case solutions do not satisfy the momentum equation. In the general case we consider, we find that the general solution will not satisfy the momentum equation except in a limited restricted case. We discuss how this shortcoming might be rectified by use of a more general velocity field.