Cosserat interphase models for elasticity with application to the interphase bonding a spherical inclusion to an infinite matrix

Interphases are often modeled as interfaces with zero thickness using jump conditions that can be developed based on approximate shell or membrane models which are valid for specific limited ranges of the elastic material parameters. For a two-dimensional problem it has been shown (Rubin and Benveniste, 2004) that the Cosserat model of a finite thickness interphase is a unified model that is accurate over the full range of elastic parameters. In contrast, many other interphase models are valid for only limited ranges of the elastic parameters. In this paper, the accuracy of different Cosserat models of a finite thickness interphase that connects a spherical inclusion to an infinite matrix is examined. Specifically, four Cosserat interphase models are considered: a general shell (GS)$(\mathit{GS})$, a membrane-like shell (MS)$(\mathit{MS})$, a simple shell (SS)$(\mathit{SS})$ and a generalized membrane (GM)$(\mathit{GM})$. The models (GS)$(\mathit{GS})$ and (MS)$(\mathit{MS})$ both satisfy restrictions on the strain energy function of the interphase that ensure exact solutions for all homogeneous three-dimensional deformations, while the other models (SS)$(\mathit{SS})$ and (GM)$(\mathit{GM})$ do not satisfy these restrictions. The importance of these restrictions is examined for the three-dimensional inhomogeneous inclusion problem being considered. This is the first test of the accuracy of an elastic interphase model for a spherical interphase.     

Bifurcation indicator for geometrically nonlinear elasticity using the Method of Fundamental Solutions

In the present work, we propose a numerical analysis of instability and bifurcations for geometrically nonlinear elasticity problems. These latter are solved by using the Asymptotic Numerical Method (ANM) associated with the Method of Fundamental Solutions (MFS). To compute bifurcation points and to determine the critical loads, we propose three techniques. The first one is based on a geometrical indicator obtained by analyzing the Taylor series. The second one exploits the properties of the Padé approximants, and the last technique uses an analytical bifurcation indicator. Numerical examples are studied to show the efficiency and the reliability of the proposed algorithms.     

Stress-difference elasticity and its application to photomechanics

The two-dimensional elasticity equations have been reformulated in terms of the obtainable photoelastic data and the first stress invariant. The result is a new group of partial-differential equations which can be used to either separate the stresses along a line segment or to generate photoelastic data within a region. The equations can also be used to check the quality of the data used in the shear-difference method. Examples which illustrate the use of these equations are presented and discussed.     

Bifurcation analysis for T system with delayed feedback and its application to control of chaos

In this paper, a three-dimensional autonomous nonlinear system called the T system which has potential application in secure communications is considered. Regarding the delay as parameter, we investigate the effect of delay on the dynamics of T system with delayed feedback. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associated characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.     

Bifurcation and stability of spatially periodic solutions of nonlinear evolution equations with integral operators

A more general kind of nonlinear evolution equations with integral operators is discussed in order to study the spatially periodic static bifurcating solutions and their stability. At first, the necessary condition and the sufficient condition for the existence of bifurcation are studied respectively. The stability of the equilibrium solutions is analyzed by the method of semigroups of linear operators. We also obtain the principle of exchange of stability in this case. As an example of application, a concrete result for a special case with integral operators of exponential type is presented. This work was supported by the Chinese National Foundation of Natural Science     

Some properties of the set of fourth-order tensors,with application to elasticity

In this paper the algebraic structure of the set of fourth-order tensors is examined. Special emphasis is given to the two different tensor products which can be defined over this set, when it is regarded as a vector space and as the set of all linear transformations over a vector space, respectively.The new formalism introduced here makes it possible to write in a simple form the restrictions imposed on the elasticity tensors, in finite elasticity, by balance of angular momentum and by the principle of material frame-indifference.A reassessment of the various elasticity tensors used in the literature is presented, and some classical results are re-stated in a simple and natural way.

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Riassunto Questo articolo tratta della struttura algebrica dell' insieme dei tensori del quarto ordine. Particolare rilievo è dato alla definizione di prodotto tensoriale tra tensori del secondo ordine, che è diversa a seconda che essi siano considerati transformazioni lineari su di uno spazio vettoriale, oppure essi stessi elementi di uno spazio vettoriale.Il nuovo formalismo introdotto rende semplice la determinazione delle restrizioni imposte ai tensori elastici, in Elasticità finita, dal bilancio della quantità di moto e dal principio di indifferenza materiale.Diventa anche possibile un riordinamento dei tensori elastici piò frequentemente usati nella letteratura. Ciò permette di ritrovare, in modo semplice e naturale, alcuni risultati classici.

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Dedicated to Guido Stampacchia,in mortem     

Thin interphase/imperfect interface in elasticity with application to coated fiber composites

The imperfect interface conditions which are equivalent to the effect of a thin elastic interphase are derived by a Taylor expansion method in terms of interface displacement and traction jumps. Plane and cylindrical interfaces are analyzed as special cases. The effective elastic moduli of a unidirectional coated fiber composite are obtained on the basis of the derived imperfect interface conditions. High accuracy of the method is demonstrated by comparison of solutions of several problems in terms of the imperfect interface conditions or explicit presence of interphase as a third phase. The problems considered are transverse shear of a coated infinite fiber in infinite matrix and effective transverse bulk and shear moduli and effective axial shear modulus of a coated fiber composite. Unlike previous elastic imperfect interface conditions in the literature, the present ones are valid for the entire range of interphase stiffness, from very small to very large.     

Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

It is well-known that classical continuum theory has certain deficiencies in predicting material’s behavior at the micro- and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori–Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.     

Bifurcation of non-negative solutions for an elliptic system

In the paper, we consider a nonlinear elliptic system coming from the predator-prey model with diffusion. Predator growth-rate is treated as bifurcation parameter. The range of parameter is found for which there exists nontrivial solution via the theory of bifurcation from infinity, local bifurcation and global bifurcation.     

Bifurcation analysis of an aeroelastic system with concentrated nonlinearities

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system??s instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed.     

Research on an orthogonal relationship for orthotropic elasticity

IntroductionAsymplecticsystematicmethodology[1- 3]forelasticitywasestablishedbyZhongWan_xie .Hepresentedcreativelythedualvectorsandthesymplecticorthogonalrelationshipandopenedaworkplatformparalleledtothetraditionalelasticity[4 - 9].AnewdualvectorandanewdualdifferentialmatrixLwerepresentedforasymplecticsystematicmethodologyfortwo_dimensionalelasticityandaneworthogonalrelationshipwasdiscoveredforisotropicplaneproblems[4 ]byLuoJian_hui.Theneworthogonalrelationshipisgeneralizedfororthotropicelas…     

Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics - the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

C 1 natural element method for strain gradient linear elasticity and its application to microstructures

C 1 natural element method (C 1 NEM) is applied to strain gradient linear elasticity, and size effects on microstructures are analyzed. The shape functions in C 1 NEM are built upon the natural neighbor interpolation (NNI), with interpolation realized to nodal function and nodal gradient values, so that the essential boundary conditions (EBCs) can be imposed directly in a Galerkin scheme for partial differential equations (PDEs). In the present paper, C 1 NEM for strain gradient linear elasticity is constructed, and several typical examples which have analytical solutions are presented to illustrate the effectiveness of the constructed method. In its application to microstructures, the size effects of bending stiffness and stress concentration factor (SCF) are studied for microspeciem and microgripper, respectively. It is observed that the size effects become rather strong when the width of spring for microgripper, the radius of circular perforation and the long axis of elliptical perforation for microspeciem come close to the material characteristic length scales. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch.     

Inverse three-dimensional elasticity theory problem for an isotropic medium with A deformable inclusion

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 11, pp. 44–49, November, 1990.     

Bifurcation of an elliptic vortex ring

Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied. The following crosslinking model is proposed: A thick vortical region composed of many equivalent vortex filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and propagates around the ring cross section, thus preventing further bifurcation. And it becomes a turbulent blob. This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes solution.