Multilayered Plates Research Articles

Nonlinear thermal analysis of multilayered composite and FGM plates with temperature-dependent properties based on an asymptotic numerical method

Nonlinear thermal analysis of multilayered composite and FGM plates with temperature-dependent properties based on an asymptotic numerical method

Mixed Problem for the Differential Equation of Parabolic Type with Measures

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.

Axisymmetric thermoelastic deformation of a multilayer plate with imperfect thermal contact of its layers

An axisymmetric stationary problem of thermoelasticity for a multilayer plate with imperfect thermal contact between its layers is solved by using the method of compliance functions along with the Hankel transform. It is assumed that the conditions of perfect mechanical contact are satisfied on their common boundaries. The Hankel transforms of displacements, stresses, and temperature at the points of the layer can be represented in the form of the linear combinations of the six auxiliary functions. The auxiliary functions are connected with the Hankel transforms of displacements, stresses, temperature and flow at the points of the upper boundary of the corresponding layer. For the problem considered six auxiliary functions can be found from the boundary conditions. Using the conditions on the common boundaries of the layers and entering a dummy layer the recurrent formulas for finding other auxiliary functions are constructed. The auxiliary functions of each layer are dependence. This dependence is represented in the matrix form using so-called compliance functions. The recurrence relations for the compliance functions of the thermoelastic multilayer plate are constructed. The algorithm for solving the сonsidered problem is formulated. The numerical calculations are made for a two-layer plate subjected to the action of the thermal loads. The influence of the coefficient of thermal resistance on the distribution of temperature at the points of the lower boundary of the upper layer and at the points of the upper boundary of the lower layer is analyzed. Also the influence of this coefficient on the distribution of normal stresses at the common boundary of layers is analyzed.

Guided wave propagation in multilayered periodic piezoelectric plate with a mirror plane

The multi-mode and dispersion natures of guided waves in plate bring big challenges of functional phononic crystal (PC) plate design for wave control. For designing a periodic multilayer piezoelectric plate to control specific guided wave propagation, it is vital to study the dispersion relation of each guided wave mode. In this paper, a plane wave expansion method is developed to calculate the dispersion relation of each guided wave mode in the periodic multilayer piezoelectric plate with a mirror plane. To quantitatively demonstrate the effectivity of the developed method, the band structures of PC plates with different kinds of piezoelectric scatterers are discussed to reveal the influences of geometry and polarization effect of piezoelectric scatterer on the guided wave propagation. Results show that the mode bandgaps of symmetric mode Lamb waves, anti-symmetric mode Lamb waves, and shear horizontal waves can be obtained separately by the proposed method, and they exhibit different responses with the changes of piezoelectric scatterer. Therefore, the proposed method can provide an effective analysis way of band structures for designing smart guided wave control devices based on multilayered piezoelectric PC plate.

Control of hygrothermal vibration of viscoelastic magnetostrictive laminates resting on Kerr’s foundation

Many composite structural applications undergo several environmental conditions, so, for the optimal design of the composite structural systems and efficient use of the engineering applications, the accurate vibrational behavior prediction of the composite structures subjected to thermal/hygrothermal environments effect is required. In the current study, vibration response and damping behavior of a multilayered composite plate with viscoelastic faces and homogenous core, resting on three-parameter Kerr’s foundation are analyzed based on a higher-order shear deformation plate theory with an exponential shape function. Simple velocity feedback control and two magnetostrictive (Terfenol-D) actuators are utilized for controlling the vibration of the sandwich plate. The governing system of the vibration problem is formulated using Kelvin-Voigt viscoelastic relation and Hamilton’s principle as well as the system is solved analytically according to Navier’s approach. To assess influences of the significant parameters such as the magnitude of the feedback control gain, the magnetostrictive layer location, the geometric coefficients, the magnetostrictive layer thickness-to-core thickness ratio, the viscoelastic layer thickness-to-core thickness ratio, orientations of the viscoelastic layer’s fiber, the half wave numbers, the lower and upper spring stiffness, the shear layer stiffness, the temperature and moisture concentrations on frequencies and deflection of the structure, comprehensive parametric examples are discussed. Findings of the study indicate that control systems of the structural applications can be improved excellently by combining the passive and active strategies for vibration damping of the structures.

Direct Calculation of the Group Velocity for Two-Dimensional Complex, Composite and Periodic Structures Using a Wave and Finite Element Scheme

Guided waves have immense potential for structural health monitoring applications in numerous industries including aerospace. It is necessary to evaluate guided wave dispersion characteristics, i.e., group velocity and phase velocity profiles, for using them effectively. For complex structures, the profiles can have highly irregular shapes. In this work, a direct method for calculating the group velocity profiles for complex, composite, and periodic structures using a wave and finite element scheme is presented. The group velocity calculation technique is easy to implement, highly computationally efficient, and works with the standard finite element formulation. The major contribution is summarised in the form of a comprehensive algorithm for calculating the group velocity profiles. The method is compared with the existing analytical and numerical methods for calculation of dispersion curves. Finally, an experimental study in a multilayered composite plate is conducted and the results are found to be in good agreement. The technique is suitable to be used in all guided wave application areas such as material characterisation, non-destructive testing, and structural health monitoring.

Quasi-3D theory for the vibration and deflection of a magnetostrictive composite plate resting on a viscoelastic medium

In this study, the vibration and deflection of a multilayered composite plate are investigated based on simple and refined sinusoidal shear deformation plate theories. The plate’s core is made of a homogeneous material. Hamilton’s principle is used to derive the dynamic system of equations. The interaction between the smart composite sandwich plate and the surrounding medium is simulated by the visco-elastic foundations model. The analytical solution for the vibration problem of the simply supported plate is obtained by Navier’s approach. Influence of the magnitude of feedback control gain, magnetostrictive layer location, half-wave numbers, lamination schemes, thickness ratios, aspect ratios, core-thickness ratios, magnetostrictive-thickness ratio, and viscoelastic foundations on eigenfrequency values and deflections of the sandwich plate are discussed. Numerical results illustrate that the vibration behavior of the smart sandwich plate is dependent on the stiffness of the viscoelastic foundations, the feedback control gain value, and the magnetostrictive-thickness ratio. Also, the vibration suppression process can be improved by changing the location of the smart layers in the structure.

An Enhancement of the Warping Shear Functions of Refined Zigzag Theory

Abstract The paper presents an enhancement in Refined Zigzag Theory (RZT) for the analysis of multilayered composite plates. In standard RZT, the zigzag functions cannot predict the coupling effect of in-plane displacements for anisotropic multilayered plates, such as angle-ply laminates. From a computational point of view, this undesirable effect leads to a singular stiffness matrix. In this work, the local kinematic field of RZT is enhanced with the other two zigzag functions that allow the coupling effect. In order to assess the accuracy of these new zigzag functions for RZT, results obtained from bending of angle-ply laminated plates are compared with the three-dimensional exact elasticity solutions and other plate models used in the open literature. The numerical results highlight that the enhanced zigzag functions extend the range of applicability of RZT to the study of general angle-ply multilayered structures, maintaining the same seven kinematic unknowns of standard RZT.

MATHEMATICAL MODELLING OF NON-STATIONARY HEAT TRANSFER IN A MULTILAYER COMPOSITE MATERIAL

The article considers the issue of designing a composite textile material based on the use of a 3D textile matrix for firefighter combat clothing with improved performance characteristics. To reduce labour and material costs for design and create an alternative to the experimental selection of the structure and composition of the material, a mathematical model of non-stationary heat transfer in the “environment – composite material – human” system is proposed. The problem of temperature distribution at any time for the outer and inner layers is presented in the form of heat transfer in a multilayer plate. The problem of temperature distribution in the heat-insulating layer of the material is presented in the form of heat transfer through a limited rod in the air. The developed mathematical model allows calculating the distribution of temperature fields in the layers of the material at different values of the effective heat flow and determine the limit parameters of its thermal protection effect.

Nonlinear vibration analysis of elastically supported multi-layer composite plates using efficient quadrature techniques

Different numerical schemes are introduced to formulate and solve nonlinear vibration analysis of elastically supported multilayer composite plate problems. The type of elastic foundation is Winkler - Pasternak foundation model. The governing equations are formulated according to a first order transverse shear theory. Examined schemes are based on polynomial, sinc, discrete singular convolution differential quadrature (DSCDQ) methods. These schemes are appointed to reduce the problem to nonlinear Eigen-value problem. The reduced system is solved iteratively. Numerical analysis is performed to explore influence of different computational characteristics on convergence and efficiency of the obtained results. Comprehensive numerical results validate the solutions by comparison with those obtained by the exact and numerical ones. Moreover, a parametric study is introduced to investigate the influence of supporting conditions, foundation parameters, elastic and geometric characteristics of the vibrated plate, on natural frequencies and mode shapes. The obtained results exhibit that the (DSCDQM) is an accurate efficient method in the dynamic analysis of discontinuity plates resting on nonlinear elastic foundation.

3D analytical solution for multilayer functionally graded plates with viscoelastic interlayers

In this study, an analytical model is developed to research on the time-varying performance of multilayer functionally graded plates glued by viscoelastic interlayers. In the plate, the elastic modulus in each graded layer is exponentially distributed along the thickness, and its mechanical property is modeled on the basis of the three-dimensional (3D) elasticity theory. The viscoelastic property in the interlayer is described by the standard linear solid model considering hereditary nature of viscoelasticity. By using the series expansion and the Laplace transformation, the solutions of stress and displacement for the multilayer plate are analytically derived out. The comparison analysis indicates that the finite element solution and the present one have a good agreement. The Kirchhoff-Love solution becomes imprecise as the thickness or the gradient index increases, because the transverse shear deformation is ignored. In the parameter analysis, the effects of the time variable, the gradient index and the viscoelastic constants on the time-varying performance of the multilayer functionally graded plate are analyzed in detail. The results show that the stress and displacement distributions of the multilayer functionally graded plate can be optimized by changing the gradient index.

DYNAMIC BIAXIAL STRESS ANALYSIS OF FLAT LAYERED CERAMIC COMPOSITES

We study theoretically the biaxial bending of symmetric, flat layered ceramic composites (laminates) due to external loading. We focus on three-layered alumina/zirconia laminates. We compare the principal stresses in the samples in the case of static and harmonic dynamic loading. The dynamic equation within the Kirchhoff theory for thin homogeneous plates is first generalized to the case of multilayered plates. It is solved numerically with the relaxation method, which we have developed for this purpose.

Size-dependent postbuckling analysis of graphene reinforced composite microtubes with geometrical imperfection

Graphene reinforced composites have attracted a great deal of attention in recent years due to their outstanding mechanical properties. The present work focuses on the size-dependent postbuckling characteristics of functionally graded (FG) graphene platelets reinforced composite (GPLRC) multilayer microtubes containing initial geometrical imperfection. It is assumed that the distribution of GPL reinforcements in microtubes is uniform or layer-wise change across the radial direction, and five typical distribution patterns (i.e., UD, FG-X, FG-O, FG-V and FG-A) are considered. Based on the modified couple stress theory and a refined higher-order beam theory, the non-classical governing equations with the account of von Kármán geometrical non‐linearity for postbuckling analysis are derived by employing the principle of minimum potential energy and solved by using an analytical method. After validating the analytical solutions by comparing with results reported in the literature, the influences of various important parameters on the postbuckling of FG-GPLRC multilayer microtubes are investigated. It is found that the microstructure effect on the postbuckling behavior is significant when the outer radius of the microtube is comparable to the material length scale parameter. Moreover, the initial geometrical imperfection decreases the postbuckling load-carrying capacity at small deflection, but increases it at large deflection. Additionally, our results indicate that the FG-V distribution pattern produces the highest postbuckling load-carrying capacity for FG-GPLRC multilayer tubes, which differs from the cases for FG-GPLRC multilayer beams, plates and thin shells where the FG-X distribution pattern always has the best reinforcing effect. The present work could provide theoretical guidelines for the optimal design and safety assessment of graphene reinforced composite tubular structures, and may find potential application in microscale engineering devices and systems.

On the theory of constitutive equations for composites with different resistance in compression and tension

The paper proposes a new approach to the construction of nonlinear constitutive equations for the description of elastic deformation of fibrous composites with different resistance in compression and tension. The framework of the approach is the generalized rheological method of constructing constitutive equations of the uniaxial stress–strain state. Transition to the tensor equations uses the hypothesis of existence of a special cone of all possible compression states in the space of strains. The cone structure is governed by the reinforcement pattern of a composite. When imposed with strains in the cone, the composite behaves as a linearly elastic medium, while beyond the cone, the composite is nonlinear, with higher stiffness conditioned by tension of high-modulus reinforcement fibers. It is highly critical that the constitutive equations obtained using this approach belong to the theory of hyperelasticity. The elastic potentials of stresses and strains are obtained for these equations, which guarantees implementation of fundamental principles of equilibrium thermodynamics. Application of the developed general theory is exemplified by construction of the constitutive equations for the plane stress state of a thin unidirectional fiber reinforced composite plate and for a multilayer plate with different-direction reinforcement of layers. Using the finite element method in combination with the initial stress method, the stress and strain patterns are analyzed in rectangular one-, two- and three-layer carbon fiber reinforcement polymer plates with a circular cut under the action of external stresses at the boundary.

The calculation of the thermal stressed state of multilayer platesofa non-canonical shape

A method for solving the problem of stationary thermoelasticity of multilayer plates of non-canonical shape in plan is proposed, which is based on the method of immersing a complex domain into the domain of a canonical form. A non-canonical plate with arbitrary boundary conditions "immerses" into a canonical plate. To ensure that the specified boundary conditions are met, additional compensating loads are added to the auxiliary structure, distributed along the contour of the initial structure. The intensities of compensating loads are determined from a system of integral equations. Deformation of the layers of the plates are described within the framework of the first-order theory, taking into account the transverse shear strains in each layer. The field of temperature loads is obtained by solving the non-stationary problem of thermal conductivity of a multilayer plate. The temperature stresses in a five-layer plate when heated by a film heat source are investigated.

Applicability ranges for four approaches to determination of bending stiffness of multilayer plates

A linear static problem of bending of a multilayer plate with homogeneous isotropic layers is considered. The deflection is assumed to have harmonic shape in the tangential directions. For calculation of the bending stiffness we introduce two dimensionless parameters: a small thickness parameter equal to the ratio of the thickness to the deformation wavelength in the tangential directions, and a large inhomogeneity parameter equal to the ratio of the maximum and minimum Young’s moduli of the layers. The plane of these parameters is split into four regions, in which different models are available for calculating the bending stiffness. The first of these is the Kirchhoff - Love model based on the hypothesis of straight non-deformable normal. In the second region, the Timoshenko - Reissner model with a shear parameter is used, calculated by an asymptotic formula of the second order of accuracy. The third region is based on assumptions on inextensible normal fiber and large heterogeneity. Finally, in the fourth region, compression of the normal is taken into account, and an approximate formula for bending stiffness is proposed for a three-layer plate. Error estimation of the models is carried out on test examples by comparison with the exact numerical solution of the three-dimensional problem of the elasticity theory. The possibility of suitability of the bending stiffness for calculating the plate eigenfrequencies is discussed.

Multi-layered plate finite element models with node-dependent kinematics for smart structures with piezoelectric components

This article presents a type of plate Finite Element (FE) models with adaptive mathematical refinement capabilities for modeling laminated smart structures with piezoelectric layers or distributed patches. The p-version shape functions are used in combination with the higher-order Layer-Wise (LW) kinematics adopting hierarchical Legendre polynomials. Node-Dependent Kinematics (NDK) is employed to implement local LW models in the regions with piezoelectric components and simulate the global substrate structure with the Equivalent Single-Layer (ESL) approach. Through the proposed NDK FE models, the electro-mechanical behavior of smart structures can be predicted with high fidelity and numerical efficiency, and various patch configurations can be conveniently modeled through one set of mesh grids. Moreover, the effectiveness and efficiency of the NDK FE approach are assessed through numerical examples and its application is demonstrated.

Dynamic behaviour of pavement on a two-parameter viscoelastic foundation subjected to loads moving with variable speeds

ABSTRACT The dynamic response of a multilayer finite plate resting on a two-parameter viscoelastic foundation to moving variable loads is obtained analytically/numerically. The governing differential equation of a multi-layer finite plate resting on a two-parameter viscoelastic foundation is derived firstly. Then, the dynamic response is derived analytically using mode superposition and Duhamel’s formula, followed by the solving of the numerical method. Next, the proposed solution is verified by comparing the degraded solution with both the field measurement data and the known theoretical results. Three different loading models are assessed to select the loading model that would provide insight into the effects of acceleration and deceleration. The effects of foundation parameters and vehicle parameters, especially the acceleration or deceleration and initial velocity of the moving load, on the dynamic response of the multilayer plate are compared through parametric studies. The numerical results suggest that acceleration and deceleration exert considerable effects on the dynamic response of the plate, and that the stiffness of pavement exposed to frequent acceleration and deceleration should be increased by at least 15% to counteract the effects of variable speed.

Geometrically nonlinear flexural analysis of multilayered composite plate using polynomial and non-polynomial shear deformation theories

In this paper, the geometrically nonlinear bending analysis of multilayered composite plates is carried out through a rigorous comparative study employing Green-Lagrange and von Kármán nonlinearity. The responses are analyzed for various transverse loads and boundary conditions, and emphasized the impending structural condition, which requires the consideration of full geometric nonlinearity. The study is conducted using a C0 finite element plate model using third-order and non-polynomial shear deformation theories (TSDT and NPSDT). The principle of virtual work is utilized to formulate the weak-form of governing equations, and discretized using nine-noded Lagrange elements with seven degrees-of-freedom per node. The performance of the present model has been validated by comparing present results with those available in the literature and obtained ANSYS results. The results reveal that the consideration of Green-Lagrange nonlinearity is essential for plates with all sides subjected to movable simply supported boundary conditions or with one free edge. The present study also provides a clear idea about the utilization of TSDT and NPSDT for the anti-symmetric and symmetric cross-ply plate, respectively, to get more accurate solution. Further, the effect of penalty for various theories and problems is also highlighted, and its prominent impact is asserted for some cases.

Manipulation of the guided wave propagation in multilayered phononic plates by introducing interface delaminations

In this paper, the manipulation of the guided elastic waves in periodically multilayered elastic plates (PMEPs) by introducing interfacial delaminations is explored. A periodic array of interfacial delaminations (PAIDs) distributed along the direction of wave propagation is considered. The spectral element method (SEM) is adopted and applied to accurately and efficiently calculate the transmission spectra and the band structures of the symmetric and antisymmetric guided elastic waves. The central and side periodic interfacial delaminations are considered separately in this paper. Through the comparison of the band structures for the perfect PMEP and the PMEP with PAIDs, we find that not only the total frequency ranges of the band-gaps are increased significantly, but also all of the band-gaps are moved to the relatively low-frequency ranges when PAIDs exist in a PMEP. Extra band-gaps will be generated, especially some of them are in the low-frequency ranges. In addition, the presence of PAIDs in a PMEP can widen the frequency ranges where the guided waves can only propagate with a single wave mode, i.e. the symmetric wave mode or the antisymmetric wave mode. Thus, the capacity of a PMEP to block or filter the guided waves can be strengthened significantly by just intentionally introducing the interfacial delaminations or cracks, which may provide a new way for the design of novel phononic crystals.