Elastic properties of Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–NiAl: a modified version of Hashin–Shtrikman bounds

The modified nonlinear relations for the estimation of elastic constants of Al₂O₃–NiAl composite material are developed. The concept of microstructure and interconnectivity of phases at the interface is used. Hashin–Shtrikman relations are described in their actual form and modified version of Hashin–Shtrikman relations for bulk and shear moduli are discussed. These relations for elastic and mechanical properties are applied mainly for Al₂O₃–NiAl composite material. Theoretical predictions using modified relations are compared with Hashin–Shtrikman bounds and experimental results of elastic properties for Al₂O₃–NiAl matrix-inclusion-based composite. It is found that the predicted values of elastic and mechanical properties using modified relations are quite close to the experimental results.     

Estimation of very narrow bounds to the behavior of nonlinear incompressible elastic composites

Variational bounds for the effective behavior of nonlinear composites are improved by incorporating more-detailed morphological information. Such bounds, which are obtained from the generalized Hashin–Shtrikman variational principles, make use of a reference material with the same microstructure as the nonlinear composite. The geometrical information is contained in the effective properties of the reference material, which are explicitly present in the analytical formulae of the nonlinear bounds. In this paper, the variational approach is combined with estimates for the effective properties of the reference composite via the asymptotic homogenization method (AHM), and applied to a hexagonally periodic fiber-reinforced incompressible nonlinear elastic composite, significantly improving some recent results.     

Continuum Dyson's equation and defect Green's function in a heterogeneous anisotropic solid

A continuum Dyson's equation and a defect Green's function (GF) in a heterogeneous, anisotropic and linearly elastic solid under homogeneous boundary conditions have been introduced. The continuum Dyson's equation relates the point-force Green's responses of two systems of identical geometry and boundary conditions but of different media. Given the GF of either system (i.e., a reference), the GF of the other (i.e., a defect system with “defect” change of materials property relative to the reference) can be obtained by solving the Dyson's equation. The defect GF is applied to solve the eigenstrain problem of a heterogeneous solid. In particular, the problem of slightly inhomogeneous inclusions is examined in detail. Based on the Dyson's equation, approximate schemes are proposed to efficiently evaluate the elastic field. Numerical results are reported for inhomogeneous inclusions in a semi-infinite substrate with a traction-free surface to demonstrate the validity of the present formulation.     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Finite deformation of incompressible fiber-reinforced elastomers: A computational micromechanics approach

The in-plane finite deformation of incompressible fiber-reinforced elastomers was studied using computational micromechanics. Composite microstructure was made up of a random and homogeneous dispersion of aligned rigid fibers within a hyperelastic matrix. Different matrices (Neo-Hookean and Gent), fibers (monodisperse or polydisperse, circular or elliptical section) and reinforcement volume fractions (10–40%) were analyzed through the finite element simulation of a representative volume element of the microstructure. A successive remeshing strategy was employed when necessary to reach the large deformation regime in which the evolution of the microstructure influences the effective properties. The simulations provided for the first time “quasi-exact” results of the in-plane finite deformation for this class of composites, which were used to assess the accuracy of the available homogenization estimates for incompressible hyperelastic composites.     

Geomechatronics – Interaction between ground and construction machinery and its application to construction robotics

“Geomechatronics” is a technical field in which “Geotechniques” is fused with “Mechatronics” that is the technical field to promote the automatic control of machines by using the electronics. In the field of “Geomechatronics”, a construction machine, which treats geotechnical materials such as soil and rock, automatically evaluates the properties and conditions of the ground and determines the optimum controlling method of itself for the ground with the base of the machine–ground interaction. Some researches for practical use in the field of geomechatronics are introduced, and then the progressing view of this research and technical filed is explained in this paper.     

Studying the Vibrations of In-plane Loaded Plates of Variable Thickness

A technique for analyzing the natural vibrations of variable-thickness plates under in-plane loading has been developed. The technique is based on variational and R-function methods. It is used to study the dependence of the natural frequencies of the plates on their shape and boundary and loading conditions__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 85–93, January 2005.     

Mode jumping in the buckling of struts and plates: a comparative study

A detailed investigation of the phenomenon of mode jumping in compressed struts on stiffening foundations and elastic plates of varying lengths is performed, with emphasis on the effects of altering boundary conditions. The variety of possible modal interactions is presented in a concise form using the parameter space of Arnol'd tongues, borrowed from non-linear dynamical systems theory. For the strut system, a full range of end conditions from simply supported to clamped is examined. For the plate, simply supported and clamped flexural conditions along both long (unloaded) and short (loaded) edges are considered, together with in-plane conditions ranging from free to pull in, to fully restrained. For each system, simply supported end conditions are found to provide protection against early mode jumping in a so-called “safety envelope”, but this is eroded as the end conditions are systematically altered from simply supported to clamped. For the plate system, mode jumping is induced at an earlier stage in the loading process by restricting the long (unloaded) edges against in-plane movement, but is delayed by clamping the same edges against rotation.     

On bending of strain gradient elastic micro-plates

Bending of strain gradient elastic thin plates is studied, adopting Kirchhoff’s theory of plates. Simple linear strain gradient elastic theory with surface energy is employed. The governing plate equation with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient plate theories; however, they strongly increase the stiffness of the thin plate.     

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

剪切变形对FGM圆板轴对称屈曲的影响

基于一阶剪切变形板理论，推导了功能梯度材料圆形板在边界面内均布压力作用下的轴对称屈曲方程。在推导过程中，忽略了前屈曲耦合变形。利用一阶板理论与经典板理论屈曲方程之间在数学形式上的相似性，得到了一阶板理论下功能梯度材料圆板与经典板理论下均匀圆板临界屈曲载荷之间的解析关系。利用这个解析关系，可以直接从已有的较为简单的经典理论的结果，获得一阶板理论下功能梯度材料板的临界屈曲载荷。     

On the necessity of including the turbulence experienced by an inertial particle in Lagrangian random-walk models

Many Lagrangian models have been developed in the literature in order to simulate the dispersion of particles in turbulent gas and liquid flows. The purpose of the present study is to critically analyze the impact of different fluid autocorrelation functions on the behavior of the dispersed phase in homogeneous isotropic turbulence. The “purely Lagrangian” autocorrelation, well-appropriate for turbulent diffusion problems, needs to be modified by other more sophisticated autocorrelation coefficients, including either space–time characteristics or better particle parameters to obtain appropriate numerical dispersion results in concordance with a recent theory.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

Crack kinking from an interface crack with initial contact between the crack lips

The authors recently theoretically studied crack kinking and opening from an initially closed crack (without friction) in some homogeneous medium. The same problem, but for an interface crack, is considered here. Comninou has shown that the asymptotic stress field prior to kinking is governed by a single, mode II stress intensity factor (SIF). Using this result, plus a homogeneity property of the problems of elastic fracture mechanics with unilateral contact envisaged, a change of scale, and two reasonable hypotheses, we establish the expression of the SIF at the tip of the small, open crack extension. It is shown that whatever the geometry of the external boundary and the crack and whatever the loading, these SIF depend solely upon the initial (mode II) SIF (in a linear way), the kink angle and Dundurs' parameters α and β. Using this result and Goldstein and Salganik's “principle of local symmetry” to predict the kink angle, one finds that it is independent of the loading but does depend upon Dundurs' parameters α and β. This contrasts with the case of an ordinary (initially closed) crack in some homogeneous medium, for which the kink angle was not only independent of the loading but an absolute constant. However, it is numerically found that the influence of the mismatch of elastic properties upon the kink angle is rather weak.     

Nonlinear dynamic response of nanotube-reinforced composite plates resting on elastic foundations in thermal environments

This paper presents an investigation on the nonlinear dynamic response of carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations in thermal environments. Two configurations, i.e., single-layer CNTRC plate and three-layer plate that is composed of a homogeneous core layer and two CNTRC surface sheets, are considered. The single-walled carbon nanotube (SWCNT) reinforcement is either uniformly distributed (UD) or functionally graded (FG) in the thickness direction. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion that includes plate-foundation interaction are solved by a two-step perturbation technique. Two cases of the in-plane boundary conditions are considered. Initial stresses caused by thermal loads or in-plane edge loads are introduced. The effects of material property gradient, the volume fraction distribution, the foundation stiffness, the temperature change, the initial stress, and the core-to-face sheet thickness ratio on the dynamic response of CNTRC plates are discussed in detail through a parametric study.     

任意外型薄板非线性后屈曲问题的边界元解

本文就薄板后屈曲问题建立一组新型的边界元计算公式，用这组公式求解能方便处理各种边界问题，另外文中将面内应力分解成基本部份和附加部份，并利用微分算子分解理论导得了挠度的一个不同形式的基本解，由于计算公式中，实现了面内位移和挠度的解耦，从而使迭代过程得到简化，文末还对圆板后屈曲路径进行了计算，得到了满意的结果。