Thermally induced wrinkling in orthotropic layered plates with irregular delaminations

Delaminated regions figure prominently among potential threats to the structural integrity of layered plate configurations. Under a certain thermal loading threshold, geometrically nonlinear local instabilities in the form of buckling or wrinkling across the delaminated region crop up, giving rise to markedly amplified distributions of contour peeling stresses. The present paper aims to shed light on and quantify the manifold aspects and implications of the delamination-thermal-wrinkling trio. The paper faces the challenges of handling the nature of the layered configuration, the inherent geometrical irregularity of delaminated regions, the discontinuous interfacial conditions, the 3D stress state along the delamination contour, and the nonlinear evolution of local instabilities across an orthotropic delamination. For that purpose, a specially tailored 2D multi layered plate model and a corresponding triangular finite element are derived. The original contribution of the proposed model is in its ability to capture the thermally-driven, nonlinear small scale phenomena related to geometrically nonlinear response of the layered structure, using a 2D multi-layered plate theory solved with efficient 2D multi-layered triangular finite elements, as opposed to computationally expensive 3D finite element analysis. This is accomplished via the integration and synergy of methodologies that include: multi-layered high order plate theory to account for the layered layout, geometrically nonlinear strain-displacement relations to account for geometrical nonlinearities, orthotropic and thermo-elastic constitutive laws to account for thermal loads, and interlayer interface modelling which, combined with a the shear-locking free triangular FE, allows accounting for arbitrarily shaped delaminations. The model is validated against a 1D closed form solution and a 3D continuum based finite element analysis and is then used for a numerical study. In the study, the onset and the evolution of local instabilities in an adhesively bonded orthotropic layer across an irregular delamination are looked into. Special attention is given to the significant influence of material orthotropy and the relative directionality of the delamination on the threshold thermal load, the nonlinear wrinkling patterns, and the peeling traction distribution.     

Generalized theory of multilayer plates

The layer-wise generalized theory of elastodynamic of multilayer plates is presented in this paper. This theory is based on expanding the displacement vector components of each layer into power series about the transverse coordinate. The number of terms retained in the power series is arbitrary and it is chosen depending on the problem being considered and the solution accuracy required. The system of governing equations is obtained by Hamilton's variation principle.The possibilities of the theory proposed and validity of results obtained are illustrated by examples of investigating the strain-stressed state of one- and three-layer structures. The issues of applicability of two-dimensional approximations built on the basis of the power series method are considered with respect to calculation of displacements, inplane and transverse stresses in multilayer plates under dynamic loading. Calculation results are compared with data obtained from Ambartsumyan's theory (the hypothesis of a unique non-strained normal for the pack), the layer-wise theory based on the broken line hypothesis as well as the three-dimensional elasticity theory.     

Parametric vibrations of orthotropic plates with complex shape

The paper proposes a method to study the parametric vibrations of orthotropic plates with complex shape. The method is based on the R-function theory and variational methods. Dynamic-instability domains and amplitude–frequency responses for plates with complex geometry and different types of boundary conditions are plotted     

Kirchhoff and thomson-tait transformations in the classical theory of plates

The transformation of the torque into the transverse force is considered; this transformation is traditional in the educational literature [1] and was proposed by Kirchhoff [2] and Thomson and Tait [3] to match the order of the differential equation of the classical theory of plates with the number of boundary conditions. It is shown that this transformation is not universal and its mathematical and physical justification depends on the conditions of the plate fixation and loading. It is shown that this justification is absent for the most widely used problems of bending of a rectangular plate freely supported and fixed on the contour.     

Elastic theory of unconstrained non-Euclidean plates

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.     

A perturbation solution in the nonlinear theory of circular plates

In this paper, we analyzed some problems of nonlinear circular plates by means of perturbation method. The perturbation parameters chosen here are obtained from solving the equations and are not certain mechanical quantities given precedently. This is an extension of W. Z. Chien’s perturbation method, which uses the central deflection as the perturbation parameter.     

Calculation of spatial vibrations of plates of complex shape

We consider problem of the spatial vibrations of plates with complex shape within Timoshenko's theory of plates. We write the constitutive equations in a local oblique coordinate system where all the bounding contours coincide with coordinate lines. We have demonstrated the effectiveness of the proposed method using as an example the vibration of a rectangular plate weakened by a regular hexagonal hole under the action of an impulsive surface load. We analyze the effect of nonlinear terms on the frequency and amplitude of normal displacements of the shell as a function of the load amplitude. S. P. Timoshenko Institute of Mechanics. National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 44–53, August, 1999.     

On the energy method in the non-linear theory of thin plates

Summary The equilibrium configuration of a thin plate under normal pressure is studied. A variational non-linear treatment of the problem is considered, taking into account the bending stresses and allowing large deflections.Existence, uniqueness, and regularity of solutions are obtained.

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Sommario Si studia il problema non lineare della configurazione di equilibrio di una piastra sottile caricata normalmente nel suo piano medio, tenendo conto delle tensioni di natura flessionale e in un regime di spostamenti moderatamente grandi.Con una tecnica variazionale si studiano l'esistenza, l'unicità, e la regolarità delle soluzioni del problema.

     

General theory of thin plates on the basis of nonsymmetric theory of elasticity

The paper uses the asymptotically justified hypothesis method to construct three different general refined theories of micropolar thin elastic plates, depending on the values of physical dimensionless material parameters, involving: (i) independent displacement and rotation fields, (ii) constrained rotation, and (iii) low shear stiffness. All angular shear deformations are taken into account.