3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

Formulation of Shell Elements Based on the Motion Formalism

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

EXTRACTION OF ELASTIC, VISCOUS AND INERTIAL COMPONENTS FROM THE TOTAL REACTION OF AN ADDITIONAL SUPPORT ATTACHED TO A RECTANGULAR PLATE

The paper deals with a mechanical system consisting of a hinged rectangular plate and an additional viscoelastic support with considering its mass-inertia. The impact of the characteristics of additional support on the plate strained state is studied by an original approach of extracting elastic, viscous and inertial components from the total reaction. The plate is assumed to be medium thickness, elastic and isotropic. The Timoshenko hypothesis is used for deformation equations. The external non-stationary force initiates plate vibrations. The impact of the additional support is replaced by the action of three unknown independent non-stationary concentrated forces. The basic formulas for deriving system of three Volterra integral equations are proposed. The system is then solved by numerical and analytical method. By discretizing in time the system of Volterra integral equations is reduced to a system of matrix equations. The system of matrix equations is solved with using generalized Kramer’s algorithm for block matrices and Tikhonov’s regularization method. Note that the approach proposed is applicable for other objects with additional supports, such as beams, plates and shells having various boundary contour and boundary supporting. The results of computing elastic, viscous and inertial components of total reactions on the plate are given. The approach proposed is verified by matching the results of computations by two different methods, namely numerical and analytical for one total reaction and numerical for the total reaction obtained by adding elastic, viscous and inertial components.

Multi-resonant piezoelectric metamaterials for broadband isolation of elastic wave transmission

This paper proposes a general method to design multi-resonant piezoelectric metamaterials. Such metamaterials contain periodically distributed piezoelectric patches bonded on the surfaces of a host structure. The patches are assumed to be shunted with digital circuits. A transfer function is designed to realize multi-resonance. The transfer function is derived only using the parameters of the patches. Consequently, it can be used to realize any type of multi-resonant metamaterial structures, like beams, plates and shells. The mechanism of generating multi-bandgaps by the transfer function is explained by analytically studying the effective bending stiffness of a multi-resonant piezo-metamaterial plate. It is shown that the transfer function induces multiple frequency ranges in which the effective bending stiffness becomes negative, consequently results in multiple bandgaps. The characteristics of these bandgaps are investigated, coupling and merging phenomena between them are observed and analyzed. Isolation effects of vibration transmission (elastic wave) in the metamaterials at multiple line frequencies or within a broad frequency band are numerically verified. The proposed multi-resonant piezoelectric metamaterials may open new opportunities in vibration mitigation of transport vehicles and underwater equipment.

Two-field formulations for isogeometric Reissner–Mindlin plates and shells with global and local condensation

In this paper, mixed formulations are presented in the framework of isogeometric Reissner–Mindlin plates and shells with the aim of alleviating membrane and shear locking. The formulations are based on the Hellinger-Reissner functional and use the stress resultants as additional unknowns, which have to be interpolated in appropriate approximation spaces. The additional unknowns can be eliminated by static condensation. In the framework of isogeometric analysis static condensation is performed globally on the patch level, which leads to a high computational cost. Thus, two additional local approaches to the existing continuous method are presented, an approach with discontinuous stress resultant fields at the element boundaries and a reconstructed approach which is blending the local control variables by using weights in order to compute the global ones. Both approaches allow for a static condensation on the element level instead of the patch level. Various numerical examples are investigated in order to verify the accuracy and effectiveness of the different approaches and a comparison to existing elements that include mechanisms against locking is carried out.

The family of multilayered finite elements for the analysis of plates and shells of variable thickness

Urban development requires careful attitude to environment on the one hand and protection of the population from the natural phenomena on the other. To solve these problems, various building structures are used, in which slabs and shells of variable thickness find the wide application. In this work, the family of multilayered finite elements for the analysis of plates and shells of variable thickness is described. The family is based on the simplest flat triangular element of the Kirchhoff type. The lateral displacements in this element are approximated by an incomplete cubic polynomial. Such an element is unsuitable for practical use, but on its basis, improved elements of triangular and quadrilateral shape are built. Particular attention is paid to taking into account the variability of the cross-section. The results of the developed elements testing are presented, and the advantages of their use in the practice of designing and calculating the structures are shown. El desarrollo urbano requiere una actitud cuidadosa con el medio ambiente, por un lado, y la protección de la población frente a los fenómenos naturales, por otro. Para resolver estos problemas, se utilizan diversas estructuras de edificios, en las que las placas y cáscaras de espesor variable encuentran una amplia aplicación. En este trabajo se describe la familia de elementos finitos multicapa para el análisis de placas y cáscaras de espesor variable. La familia se basa en el elemento triangular plano más simple del tipo Kirchhoff. Los desplazamientos laterales en este elemento se aproximan mediante un polinomio cúbico incompleto. Este elemento es inadecuado para su uso práctico, pero sobre su base se construyen elementos mejorados de forma triangular y cuadrilátera. Se presta especial atención a tener en cuenta la variabilidad de la sección transversal. Se presentan los resultados de las pruebas de los elementos desarrollados y se muestran las ventajas de su uso en la práctica del diseño y el cálculo de las estructuras.