Analysis Of Multilayered Plates Research Articles

Mesh-adapted stress analysis of multilayered plates using a layerwise model

This paper proposes a new finite-element modelling of a recent layerwise model for multilayered plates. This layerwise model is built from a specific 3D stress-field expansion along the thickness direction and involves, in particular, interlaminar transverse shear and out-of-plane stresses as generalized stresses. Its main feature is that 3D equilibrium equations and free-edge boundary conditions are directly taken into account into the stress-based construction of the model. A dual displacement-based finite-element discretization is implemented using the FEniCS software package and a remeshing strategy is proposed based on a novel error indicator. The error indicator is built based on the 3D stress field directly deduced from the layerwise generalized stresses and compared to a reconstructed stress field based on the model generalized displacements. The proposed error indicator is shown to identify the most critical parts of a laminate structure associated with complex 3D stress fields such as boundaries or stress concentration/singularity regions (near free-edges or delamination fronts). Through the combination of thickness discretization and in-plane mesh refinement in regions of interest, the proposed framework therefore offers an attractive alternative to 3D solid finite elements for an accurate prediction of stress states in composite laminates.

Hygro-thermo-mechanical modelling and analysis of multilayered plates with embedded functionally graded material layers

This work addresses the modelling and analysis of multilayered plates with embedded functionally graded material (FGM) layer(s) under hygro-thermo-mechanical loadings. The hygroscopic, thermal and mechanical problems are all solved simultaneously using a new layerwise mixed model based on least-squares formulation with multi-field independent variables, namely, displacements, temperature, moisture, transverse stresses, transverse heat flux, transverse moisture flux, in-plane strains and in-plane components of both thermal and moisture gradients. This mixed formulation ensures that interlaminar C0 continuity requirements, where the material properties may actually change, are fully fulfilled a priori. An added feature is included to fully describe the FGM layer z-continuous effective properties through-thickness, using any homogenization method, by applying a high-order z-expansion to its effective properties, similarly to finite element approximations. The numerical results demonstrate the effects of hygrothermal environments in the analysis of distinct multilayered plates with embedded FGM layers, considering different side-to-thickness ratios, under a series of hygro-thermo-mechanical loadings. The rule of mixtures is used to estimate the FGM layer effective properties, including different material gradation profiles. Three-dimensional (3D) approximate solutions corroborate this model’s capability to predict accurately a quasi-3D hygro-thermo-mechanical description of the through-thickness distributions of displacements and stresses, temperature and heat flux, moisture and moisture flux.

Equivalent single-layer models in deformation analysis of laminated multilayered plates

The performance of selected equivalent single-layer (ESL) models is evaluated within several classical benchmark tests for linear static analysis of multilayered plates. The authors developed their own finite element software based on the first-order shear deformation theory (FOSD) with some modifications incorporated including a correction of the transverse shear stiffness and an application of zigzag-type functions. Seven different ESL models were considered in the study; beside the classical FOSD model, there were three FOSD models with various transverse shear corrections and three ESL models enhanced by the application of zigzag functions. In addition, particular attention was paid to investigation of differences related to the “soft” and “hard” variants of the simple support.

Thermomechanical analysis of multilayered plates in terms of Reddy-type higher-order theory

ABSTRACTA Reddy-type higher-order plate theory considering transverse normal thermal strain (HPTC) has been presented for thermomechanical analysis of multilayered plates. Displacement variables in model HPTC have not increased although transverse normal deformation is considered. Based on the proposed model, analytical solutions of simply-supported plates under thermal and mechanical combined loads are presented. Analyzing thermomechanical behaviors of composite plates, effects of number of layers, transverse normal strain and thermal expansion coefficients on displacements, and stresses of composite plates have been studied. Numerical results showed that transverse normal strain and thermal expansion coefficients have a significant impact on behaviors of composite plates.

Axiomatic/Asymptotic Evaluation of Refined Plate Models for Thermomechanical Analysis

This work deals with refined models for the thermal stress analysis of multilayered plates. The Carrera Unified Formulation has been used in order to generate refined models of any order, and both equivalent single layer (ESL) and layer-wise (LW) schemes have been adopted. A Navier-type solution has been employed and, as a result, only simply supported orthotropic plates have been considered. A linear temperature distribution along the thickness direction is considered. The purpose of this work is to establish the relevance of the displacement variables and to discard the irrelevant terms in order to obtain refined models with the same accuracy as full theories, but with a lower computational cost. The axiomatic/asymptotic technique has been employed. The effectiveness of each displacement variable has been measured considering the influence of several parameters, such as the length-to-thickness ratio (a/h), the stacking sequence and the kind of material (isotropic and orthotropic). The “best” reduced models have been proposed and their relative stress and displacement components distributions along the thickness direction have been discussed. It has been found that the relevance of each displacement variable is affected to a great extent by the problem considered. In addition, it has been demonstrated that the nature of the load (mechanical or thermal) leads to the necessity of retaining different displacement variables for a given problem.

Coupled Thermo-Electro-Mechanical Analysis of Smart Plates Embedding Composite and Piezoelectric Layers

This article considers the static analysis of multilayered piezoelectric plates subjected to mechanical pressures and to field loads such as electric potential and temperature. The Principle of Virtual Displacements has been extended to thermo-electro-mechanical cases by including the virtual internal electric and thermal works and by writing the opportune constitutive relations. The governing equations are given in terms of three main primary variables (displacements, electric potential and temperature) and they are solved in a closed-form solution. The primary variables are expanded in the thickness direction in both Equivalent Single Layer (ESL) and Layer-Wise (LW) form with higher orders of expansion. The models proposed enable to obtain a quasi-3D description of the static thermo-electro-mechanical analysis of multilayered plates and to analyze the effects of each physical field involved. The limitations of classical theories for such analyses have also been discussed.

A Comparison of Various Two-Dimensional Assumptions in Finite Element Analysis of Multilayered Plates

This work deals with a refined Finite Element (FE) analysis of multilayered plates. Various two-dimensional axiomatic assumptions in the thickness direction are illustrated and discussed by considering: 1-Taylor type expansion; 2-combinations of Legendre polynomials; 3-Lagrange polynomials. Both cases of an equivalent single layer description (the whole plate is seen as an equivalent single layer) and a layer-wise description (each layer is seen as an independent plate) have been implemented. The order N of the thickness expansions is a free parameter of the present formulation. A large variety of plate theories are therefore obtained. Related standard serendipity-type quadrilateral FEs are considered in this paper. FE matrices are written in a concise form by referring to the Carrera Unified Formulation and in terms of a few fundamental nuclei, whose form does not depend on the through-the-thickness polynomial assumption, order N, variable description or element number of nodes. The advantages and disadvantages of the various FEs are discussed by considering static and dynamic problems related to significant multilayered plate problems.

Linear analysis of laminated multilayered plates with the application of zig-zag function

An Equivalent Single Layer Model for linear static analysis of multilayered plates is considered in the paper. The displacements field of the First Order Shear Deformation Theory is supplemented with the Murakami's function to emulate the zig-zag effect. The applied Reissner's Mixed Variational Theorem allows to assume an independent shear stress field of parabolic distribution in each layer and to satisfy the equilibrium conditions at the layer interfaces. The Finite Element implementation of the model is based on the application of the 9-node Lagrangian element with a selective reduced integration. Two numerical examples of symmetrically laminated plates are presented to illustrate the accuracy of presented approach.

Mixed Finite Elements for Thermoelastic Analysis of Multilayered Anisotropic Plates

This paper presents some finite elements for the thermoelastic analysis of multilayered plates which are based on unified formulation and the Reissner mixed variational theorem (RMVT). The pure mechanical statement of the theorem is herein extended to thermoelastic analysis and the related mixed constitutive equations have been derived. Assumptions are made for the displacement fields in thickness direction and the RMVT is employed to obtain finite element (FE) matrices. The unified formulation allows to easily derive the FE matrices in terms of a few fundamental nuclei whose form is not affected by: number of the nodes of the elements; order of the expansion for the displacements-variable description (Layer-Wise, LW, and Equivalent-Single Layer, ESL, cases are both addressed). The Murakami Zig-Zag function is used to introduce Zig-Zag effects in the framework of ESL descriptions. Comparisons to exact 3D solutions as well as to classical finite elements have proved the accuracy of the proposed elements and thier capability to accurately describe the stress field of layered plates subjected to thermal loadings.

Method of initial functions: stable algorithms in the analysis of thick laminated composite structures

Exact three-dimensional elasticity solutions for some static and dynamic problems of bending multi-layered anisotropic rectangular plates are presented. They are derived by the method of initial functions without employing any hypotheses about the character of the stress-strain state of the linearly elastic anisotropic structures. Two computationally stable algorithms of the method which extend the field of its application are developed. The results presented illustrate some possibilities of the designed stable algorithms. The analysis of multi-layered plates under concentrated static and harmonic dynamic loads are presented.

A two-dimensional theory for the analysis of laminated plates

A new displacement-based two-dimensional theory for the analysis of multilayered plates is presented. The theory is based on the only kinematic constraint of transverse inextensibility, whereas no restrictions are imposed on the representation of the in-plane displacement components. A governing system of integral-differential equations is obtained which can be given a closed-form solution for a number of problems where no boundary layer are present. It is also shown that most of the 2-D plate models can be directly derived from the presented theory. The possibility of developing asymptotic solutions in the boundary layers is discussed with reference to the problem of a plate in cylindrical bending. Finally some numerical solutions are compared with those given by the plate model by Lo et al. (1977) and with F.E.M. solutions.