Determination of the effective elasticity tensor of a folded sandwich core

This document describes the homogenization of a folded sandwich core. By using a numerical homogenization concept the components of the elasticity tensor of the foldcore continuum are determined. The foldcore exhibits an effective orthotropic behaviour with the particularity that it can be auxetic. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape memory and phase transitions for auxetic materials

We present a mathematical model describing the auxetic‐austenitic phase transition phenomenon by a second order shape memory phase transition. The typical properties of auxetic materials, as the negative Poisson ratio ν, are described by a function of the phase ?, called order parameter, which relates the phase transition with a change of the internal order structure of the material. In our model, the auxetic phase is represented by an order parameter ? = 1, which provides a negative Poisson's ratio, while the austenitic phase will be denoted by ? = 0. Copyright © 2013 John Wiley & Sons, Ltd.     

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.      

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.     

Effective Dynamic Material Properties of Foam-like Microstructures

The effective material parameters of a microstructured material can be found using homogenization procedures based on calculations of a Representative Volume Element (RVE) of the material. In our approach the RVE is calculated in frequency domain and inertia is taken into account, leading to a frequency dependent behavior of the RVE.With the frequency response of the RVE, effective dynamic properties of the material are calculated using an optimization procedure. Due to the frequency dependent material behavior on the microscale a viscoelastic constitutive equation is applied on the macroscale. An example calculation is presented for an auxetic 2-D foam-like microstructure which is modelled as a plane frame structure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the piezotronics and tactile sensing virtual simulation

A flexible finger with muscles made of Nitinol wires and the skin made of auxetic material is analyzed from the tactile sensing point of view. The recognizing of the shape and texture of 3D objects is performed by simulation the action of an array of nanopiezotronic transistors integrated into the skin. The array of nanopiezotronic transistors makes possible the detection of the pressure-induced changes in the auxetic skin. The shape and texture of the objects is best estimated by determining the surface and texture as an n-ellipsoid defined by 12 parameters. An inverse problem is solved in order to find these parameters from the condition that the n-ellipsoid best fits the set of data points probed by touch with the finger. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the sonic composites subjected to severe acoustic loads

Transformation acoustics opens a new avenue towards the fabrication of a new class of sonic composites with scatterers made of auxetic materials embedded in the epoxy matrix. The design of the acoustic scatterers is based on the property of Helmoltz equations to be invariant under coordinate transformations, i.e. a specific spatial compression is equivalent to a variation of the material parameters in the original space. In this paper, the noise suppression for a wide full band-gap of frequencies is discussed for spherical shell scatterers made of auxetic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nano-scale frictional contact problem incorporating the size dependency and the surface effects

In this paper, the nano-scale sliding contact problem of an isotropic semi-infinite medium is considered. The small scale effects are taken into account by means of the couple stresses and the surface energy theories. Employing the Fourier integral method, the plane elasticity problem is formulated in terms of the integral equations. The numerical results are provided for the plane strain state. The parametric study reveals that the surface material constants remarkably alter the surface stresses and the contact length. The results indicate that the stress singularity substantially decreases due to size effect compared with the classical theory predictions. For example, the in-plane stress tensile peak decreases by a %80 with respect to the Hertzian contact.     

What is the complexity of a distributed computing system?

Distributed computing systems are becoming bigger and more complex. Although the complexity of large‐scale distributed systems has been acknowledged to be an important challenge, there has not been much work in defining or measuring system complexity. Thus, today, it is difficult to compare the complexities of different systems, or to state that one system is easier to program, to manage, or to use than another. In this article, we try to understand the factors that cause computing systems to appear very complex to people. We define different aspects of system complexity and propose metrics for measuring these aspects. We also show how these aspects affect different kinds of people—viz. developers, administrators, and end‐users. On the basis of the aspects and metrics of complexity that we identify, we propose general guidelines that can help reduce the complexity of systems. © 2007 Wiley Periodicals, Inc. Complexity 12: 37–45, 2007     

An analysis of contact deformation of auxetic composites

The adaptive mode of frictional interaction has been studied as a self-locking effect upon contact deformation of isotropic and anisotropic auxetic materials with a negative Poisson ratio. This effect manifests itself in the fact that the bearing capacity of the joint rises with increasing shear load. In particular, the parameters of stress state (contact load, tangential stresses, slippage, etc.) were determined for a double-lap joint under conditions of compression with or with out shear. The contact interaction was analyzed by the finite-element method for three profiles of symmetrically located contact elements (plane, cylindrical, and wedge-shaped). The Poisson ratio was varied within the range theoretically admissible for isotropic elastic media. Analogous calculations were also performed for a joint with a deformed element made of an anisotropic auxetic composite, whose reinforcement angle was varied. The maximum loads, tangential stresses, and slippage are obtained as nonlinear functions of Poisson ratio (in the isotropic case) and reinforcement angle of the composite material. The stress concentration and the increased ultimate shear forces are also estimated. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 681–692, September–October, 2006.     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.     

An Analysis of the Decay of Saint-Venant End Effects in a Transversely Isotropic Matrix with an Isotropic Covering

A calculation model is proposed for a numerical analysis of the decay of Saint-Venant end effects in a laminate material of irregular structure to which corresponds a transversely isotropic matrix with an isotropic covering. The elastic properties of the matrix correspond to those of a unidirectional glass-fiber-reinforced plastic. The problem is investigated within the framework of the concept of a representative element of the material. The decay of the end effect in the direction perpendicular to the isotropy plane of a transversely isotropic matrix for the case of symmetric deformation of the material is considered. The source of the end effect is simulated by a piecewise-constant periodic surface load. This load is local for the calculated region and changes within a part of the boundary comparable with the typical size of the structural heterogeneity of the material. The equations of the linear elasticity theory of orthotropic bodies, a model of piecewise-homogeneous media, and quantitative criteria of decay of the end effect are used. Starting from the base system of equations, a discrete problem is constructed and solved. The results of a computing experiment for the choice of a representative element and the results of determining the maximum extent of the end-effect zone for this element are presented.     

On the sonic composites with scatterers made from auxetic material

A finite size periodic array of resonators made from auxetic material embedded into an epoxy matrix is analyzed in this paper. According to the Bragg's theory, the sound attenuation band is due to the superposition of multiple reflected waves inverse proportional to the central distance between resonators. The sound attenuation in such composites is studied using a method that combines the features of the cnoidal method and the genetic algorithm [1-3]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

For linear elastostatics, the Lagrange multiplier to couple the displacement (i.e., Dirichlet) condition is well known in mathematics community, but the Lagrange multiplier to couple the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the Hybrid Trefftz method (HTM). However, there has not been any analysis for these Lagrange multipliers to couple the traction condition so far. New error analysis of the HTM for elasticity problems is explored in this paper, to derive error bounds with the optimal convergence rates. Numerical experiments are reported to support this analysis. The error analysis of the HTM for linear elastostatics is the main aim of this paper. In this paper, the collocation Trefftz method (CTM) without a multiplier is also introduced, accompanied with error analysis. Numerical comparisons are made for HTM and CTM using fundamental solutions (FS) and particular solutions (PS). The error analysis and numerical computations show that the accuracy of the HTM is equivalent to that of the CTM, but the stability of the CTM is good. For elasticity and other complicated problems, the simplicity of algorithms and programming grants the CTM a remarkable advantage. More numerical comparisons show that using PS is more efficient than using FS in both HTM and CTM. However, since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM. The comparisons of HTM and CTM using FS and PS are the next aim of this article. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

重建极性连续统理论的基本定律和原理(Ⅲ)--Noether定理

对现有的各种偶应力理论进行了认真的再研究，目的是要提出一个耦合型的Noether定理并由此重新建立偶应力弹性动力学的较为完整的守恒定律和相应的均衡方程．这里给出了新的各种守恒定律和均衡方程的具体形式，并建立起从给定的不变性要求所得到的这类守恒定律的确切属性．最后，由这里的结果自然地推导出各种特殊情形；并可自然地过渡到微极连续统的结果．     

Positive definite balancing Neumann–Neumann preconditioners for nearly incompressible elasticity

In this paper, a positive definite Balancing Neumann–Neumann (BNN) solver for the linear elasticity system is constructed and analyzed. The solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting Schur complement, involving only interface displacements, using a BNN preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions. While the Schur complement becomes increasingly ill-conditioned as the materials becomes almost incompressible, the BNN preconditioned operator remains well conditioned. The main theoretical result of the paper shows that the proposed BNN method is scalable and quasi-optimal in the constant coefficient case. This bound holds for material parameters arbitrarily close to the incompressible limit. While this result is due to an underlying mixed formulation of the problem, both the interface problem and the preconditioner are positive definite. Numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material.     

Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

The existence of riemann invariants in the one-dimensional equations of the non-linear theory of elasticity

The conditions for the existence of Riemann invariants of a one-dimensional system of equations of the non-linear theory of elasticity are investigated. Haantjes' diagonalization criterion is used to determine the form of the elastic potential for which the system has six Riemann invariants or three Riemann invariants (for waves which propagate in one direction). In particular, it is shown that the Haantjes criterion is satisfied and there are three Riemann invariants in the case of the elastic potential for slightly-non-linear weakly-anisotropic elastic media [1–3]. A procedure for computing Riemann invariants is described. The Riemann invariants are computed approximately for a form of elastic potential which satisfies the Haantjes criterion.     

Dynamic Pricing via Dynamic Programming<Superscript>1</Superscript>

This article specifies an efficient numerical scheme for computing optimal dynamic prices in a setting where the demand in a given period depends on the price in that period, cumulative sales up to the current period, and remaining market potential. The problem is studied in a deterministic and monopolistic context with a general form of the demand function. While traditional approaches produce closed-form equations that are difficult to solve due to the boundary conditions, we specify a computationally tractable numerical procedure by converting the problem to an initial-value problem based on a dynamic programming formulation. We find also that the optimal price dynamics preserves certain properties over the planning horizon: the unit revenue is linearly proportional to the demand elasticity of price; the unit revenue is constant over time when the demand elasticity is constant; and the sales rate is constant over time when the demand elasticity is linear in the price. ¹We acknowledge professor robert e. kalaba for initiating this work and suggesting solution methods.