Asymptotic analysis of layered elastic beams

We consider an elastic beam formed by three layers, fixed at one end and loaded at the free end. We call adherents the upper and lower layers ${\Omega }_{+}^{?}$ and ${\Omega }_{?}^{?}$ and an adhesive layer ${\Omega }_m^{?}$. We denote by $?h_{\pm ,m}$ the thickness of each layer and we suppose that the stiffness of the adhesive layer is ${?}^2$, with respect to that of the adherents. By an asymptotic analysis we obtain the zeroth order limit problem and the form of the second order displacements. To cite this article: M. Serpilli, C. R. Mecanique 333 (2005).     

Coupled analysis of elastic plate-infinite nonhomogeneous layered medium

The static interaction problem between an elastic plate and infinite medium is studied by the semi-numerical (in one direction) and semi-analytical (in two directions) method-semianalytical element method, a simple practical and effective method for coupled analysis between above-ground structure and infinite soil medium with complex material character in engineering.     

Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies.     

On compatible finite element models for elastic plastic analysis

Summary Some implicit or explicit assumptions on which the formulation of compatible finite element models for elastic plastic analysis is based are considered. It is shown that it is possible to formulate global elastic plastic element laws by independently assuming interpolation functions governing the displacement and the plastic strain fields. Some consistency requirements are discussed, which must be enforced in order to avoid artificial stiffening or fictitious redundancy. When these requirements are met, a «consistent» element is produced, whose properties are analyzed. It is also emphasized that, in spite of the use of compatible models, upper bounds to the collapse load are not always obtained for the limit analysis problem.For simplicity, beam elements are considered. However, the formulation holds for continua as well, as discussed in the last section.

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Sommario Vengono analizzate alcune ipotesi esplicitmente o implicitamente assunte per la formulazione di elementi finiti usati nell'analisi elasto-plastica. Si mostra come l'assunzione indipendente di funzioni di interpolazione che governano i campi di spostamento e di deformazioni plastiche permetta di formulare legami costitutivi elasto-plastici globali per l'elemento. Vengono anche discusse alcune condizioni che devono essere rispettate per non irrigidire artificialmente l'elemento o non introdurvi una iperstaticità fittizia. Queste condizioni portano a definire elementi «consistenti», le cui proprietà vengono analizzate. Si fa anche notare come, nonostante l'uso di modelli congruenti, non vi sia sempre la garanzia di ottenere delimitazioni superiori al carico di collasso per problemi perfettamente plastici.Per semplicità, la trattazione viene svolta con riferimento ad elementi di trave. La possibilità di estensione al continuo è messa in luce nell'ultimo paragrafo.

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Research supported by the C.N.R. (National (Italian) Research Council).     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Theoretical analysis,numerical calculation and experimental measurement of transient elastic waves in strips subjected to dynamic loadings

In this study, the transient response of an elastic strip subjected to dynamic in-plane loadings on the surface is investigated in detail. One of the objectives of this study is to develop an effective analytical method for determining transient solutions in a strip. By applying Laplace transform, the analytical solution in the transformed domain is derived and expressed in matrix form. The solution is then decomposed into infinite wave groups in which the multiple reflected waves with the same reflection are involved. Each multi-reflected wave can be identified by a coding method and be verified by the theory of generalized ray. The inverse transform is performed by using the well-known Cagniard method. The transient solutions in time domain for stresses and displacements are expressed in a closed form and are discussed in detail by an example. The experimental results show that the early time transient responses of displacements on the surface agree very well with the numerical calculations based on the theoretical solutions.     

Stress analysis of a layered elastic solid in contact with a rough surface exhibiting fractal behavior

A contact stress analysis is presented for a layered elastic half-space in contact with a rough surface exhibiting self-affine (fractal) behavior. Relationships for the mean contact pressure versus representative strain and the real half-contact width versus elastic properties of the layer and the substrate, asperity radius, layer thickness, and truncated half-contact width were derived from finite element simulations of a layered medium compressed elastically by a rigid cylindrical asperity. These relationships were incorporated in a numerical algorithm that was used to obtain the contact pressure distributions and stresses generated by the asperity contacts formed at the interface of the layered medium and the fractal surface. Analytical solutions illustrate the significance of the elastic material properties, layer thickness, and surface topography (roughness) on global parameters such as normal load and real contact area. Results for the contact pressure distribution and the surface and subsurface stresses provide insight into the initiation of yielding and the tendency for cracking in the layered medium. It is shown that cracking at the surface and the layer/substrate interface is more likely to occur in the case of a stiff layer, whereas surface cracking is more prominent for a relatively compliant layer.     

Overall dynamic constitutive relations of layered elastic composites

A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity.The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.     

An exact theory of interfacial debonding in layered elastic composites

An exact theory of interfacial debonding is developed for a layered composite system consisting of distinct linear elastic slabs separated by nonlinear, nonuniform decohesive interfaces. Loading of the top and bottom external surfaces is defined pointwise while loading of the side surfaces is prescribed in the form of resultants. The work is motivated by the desire to develop a general tool to analyze the detailed features of debonding along uniform and nonuniform straight interfaces in slab systems subject to general loading. The methodology allows for the investigation of both solitary defect as well as multiple defect interaction problems. Interfacial integral equations, governing the normal and tangential displacement jump components at an interface of a slab system are developed from the Fourier series solution for the single slab subject to arbitrary loading on its surfaces. Interfaces are characterized by distinct interface force–displacement jump relations with crack-like defects modeled by an interface strength which varies with interface coordinate. Infinitesimal strain equilibrium solutions, which account for rigid body translation and rotation, are sought by eigenfunction expansion of the solution of the governing interfacial integral equations. Applications of the theory to the bilayer problem with a solitary defect or a defect pair, in both peeling and mixed load configurations are presented.     

Fractional order models for the homogenization and wave propagation analysis in periodic elastic beams

Meccanica - The advancement of manufacturing techniques has led to a rapid increase in the design and fabrication of periodic and architectured materials for a variety of dynamics and wave...     

Free oscillations of layered elastic composite shells of revolution

Kemerovo State University, Kemerovo 650043. Translated from Prikladnaya Mekhanika Tekhnicheskaya Fizika, Vol. 36, No. 5, pp. 145–154, September–October, 1995.     

An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators

Wave propagation in a piezoelectric layered structure of a film bulk acoustic resonator(FBAR) is studied. The accurate results of dispersion relation are calculated using the proposed elastic electrode model for both electroded and unelectroded layered plates. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode model(often used in the quartz resonator analysis) are illustrated in detail, which shows that an elastic electrode model is indeed needed for the accurate analysis of FBAR. These results can be used as an accurate criterion to calibrate the 2-D theoretical model for a real finite-size structure of FBAR.     

Boundary element analysis for bending elastic plates with three generalized displacements

In this paper, the two fundamental differential equations for bending elastic plates with three generalized displacements are transformed into a set of boundary integral equations by Green formula. Three kinds of boundary conditions on edges have been strictly derived. So this paper gives a satisfactory method of boundary element analysis for solving the problem of bending elastic plates.     

The plane solution for bending of joined dissimilar elastic semi-infinite strips

The problem is reduced to a system of two singular integral equations for determining the interface slope and shear stress. The dominant part of the system is analyzed to determine the order of the stress singularity and its dependence on the elastic constants. After removing logarithmic singularities from the right hand sides we solve these equations numerically for several chosen composites and the interface slope and traction are exhibited graphically. The solution should be relevant in studying adhesive joints by means of a bending test.     

Conservation properties for plane deformations of anisotropic linearly elastic curved strips

Plane deformations of a curved strip, composed of an homogeneous cylindrically anisotropic linearly elastic material, are considered. The strip is in equilibrium under the action of end loads, with the lateral sides traction-free. Two conservation properties for certain cross-sectional stress measures are established, generalizing previously known results for the case of a rectangular strip. Such conservation properties are useful in assessing the influence of material anisotropy on Saint-Venant's principle, as well as in establishing convexity properties for cross-sectional stress measures. In particular, it is anticipated that the results should be useful in determining the extent of edge effects in the testing of anisotropic and composite curved strips.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.