Homogeneous Anisotropic Plates Research Articles

Generalized thermoelastic waves in a homogeneous anisotropic plate with voids

AbstractIn this contribution, the thermoelastic Lamb waves in the homogeneous, transversely isotropic, plate with voids have been investigated in the context of classical and non‐classical theories of thermoelasticity. To investigate the problem, the fundamental governing equations have been developed in a unified way. On the solutions of these governing equations, the thermal and mechanical stress‐free boundary conditions, as well as constraints due to voids, have been imposed. The solutions of governing equations indicate that there exists a coupled system of wave motion called thermal waves (T‐mode), void wave motion (V‐mode), and elastic waves (E‐mode). However, the shear horizontal component of elastic wave decouples from the system is unaffected by elasto‐poro‐thermal coupling. For the symmetric and anti‐symmetric families of vibrations, the secular equation for the wave motion along with its special cases has been derived. To discover the wave characteristics, the secular equation has been solved using the numerical‐functional iteration method in MATLAB software, and the findings have been manifested by plotting the graphs. It has been concluded that the presence of voids affects the magnitudes of phase velocity, attenuation coefficient, and specific loss. This work may be helpful for geophysicists and engineers in the field of earthquake engineering. The developers of modern facilitating things such as acoustic touch screens, liquid viscosity sensors, etc. which are based on Lamb‐type wave propagation may also get benefitted.

Poro-Thermoelastic Waves in a Homogeneous Anisotropic Plate Plunged in the Inviscid Fluid

This paper inspects the behavior of thermoelastic waves in the homogeneous, transversely isotropic plate containing voids immersed in the inviscid fluid in reference to the one-temperature generalized model of thermoelasticity. The basic governing equations for the solid plate have been developed in the context of the linear theory of poro-thermoelasticity. Helmholtz decomposition principle has been employed to solve the equations of motion for liquid. For the stress-free solid-liquid interfaces, the isothermal and thermally insulated boundary conditions have been applied simultaneously on the obtained solutions. The solutions of governing equations reveal that there exists a coupled system of waves namely thermal waves, void wave motion, and elastic waves, and a decoupled purely transverse wave. Apart from that, one mechanical wave in each liquid layer also exists. The secular equation for anti-symmetric and symmetric modes of vibration has been derived which better explains wave motion. To unveil the wave characteristics, the numerical–functional iteration technique has been employed for generating numerical data and results have been validated by tracing out the various graphs. The effects of temperature change, as well as voids in the solid plate and inviscid liquid in the neighborhood of the plate, have been noticed on phase velocity, attenuation coefficient, etc

A general isogeometric polar approach for the optimisation of variable stiffness composites: Application to eigenvalue buckling problems

This study presents a general approach for the multi-scale design of variable stiffness composites (VSCs). The first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) is solved to determine the optimal distribution of the VSC stiffness properties at the macroscopic scale satisfying the requirements of the problem at hand. In this phase, the VSC laminate is modelled as an equivalent homogeneous anisotropic plate whose behaviour is described in terms of polar parameters (PPs), which vary locally over the structure. The First-order Shear Deformation Theory is used to take into account the influence of the transverse shear stiffness on the mechanical response of the VSC and Basis Spline (B-Spline) surfaces are employed to represent the PPs fields. In this background, the expression of the gradient of the buckling factor is determined analytically by exploiting the properties of the polar formalism and of the B-Spline surfaces. Moreover, the effect of the discrete variables, involved in the definition of the B-Spline surfaces, on the performances of the optimised solution is investigated. The effectiveness of the approach is proven on two benchmark problems dealing with the maximisation of the first buckling load of a VSC laminate, subject to feasibility and geometric requirements, taken from the literature. The results obtained by means of the MS2LOS based on the polar formalism outperform those reported in the literature, which are obtained through an optimisation strategy based on lamination parameters.

A general global-local modelling framework for the deterministic optimisation of composite structures

This work deals with the multi-scale optimisation of composite structures by adopting a general global-local (GL) modelling strategy to assess the structure responses at different scales. The GL modelling approach is integrated into the multi-scale two-level optimisation strategy (MS2LOS) for composite structures. The resulting design strategy is, thus, called GL-MS2LOS and aims at proposing a very general formulation of the design problem, without introducing simplifying hypotheses on the laminate stack and by considering, as design variables, the full set of geometric and mechanical parameters defining the behaviour of the composite structure at each pertinent scale. By employing a GL modelling approach, most of the limitations of well-established design strategies, based on analytical or semi-empirical models, are overcome. The GL-MS2LOS makes use of the polar formalism to describe the anisotropy of the composite at the macroscopic scale (where it is modelled as an equivalent homogeneous anisotropic plate). In this work, deterministic algorithms are exploited during the solution search phase. The challenge, when dealing with such a design problem, is to develop a suitable formulation and dedicated operators, to link global and local models physical responses and their gradients. Closed-form expressions of structural responses gradients are rigorously derived by taking into account for the coupling effects when passing from global to local models. The effectiveness of the GL-MS2LOS is proven on a meaningful benchmark: the least-weight design of a cantilever wing subject to different design requirements. Constraints include maximum allowable displacements, maximum allowable strains, blending, manufacturability requirements and buckling factor.

Cauchy formalism for Lamb waves in functionally graded plates

Propagation of harmonic Lamb waves in plates made of functionally graded materials with transverse inhomogeneity is analyzed by applying Cauchy six-dimensional formalism previously developed for the study of Lamb wave propagation in homogeneous or stratified anisotropic plates. For anisotropic plates with arbitrary transverse inhomogeneity a closed form implicit solution for the dispersion equation is derived and analyzed.

An improved algorithm for detecting point of impact in anisotropic inhomogeneous plates

Conventional triangulation techniques fail to correctly predict the acoustic source location in anisotropic plates due to the direction dependent nature of the elastic wave speeds. To overcome this problem, Kundu et al. [1] proposed an alternative method for acoustic source prediction based on optimizing an objective function. They defined an objective function that uses the time of flight information of the acoustic waves to the passive transducers attached to the plate and the wave propagation direction ( θ) from the source point to the receiving sensors. Some weaknesses of the original algorithm proposed in Ref. [1] were later overcome by developing a modified objective function [2]. A new objective function is introduced here to further simplify the optimization procedure and improve the computational efficiency. A new algorithm for source location is also introduced here to increase the source location accuracy. The performance of the objective function and source location algorithm were experimentally verified on a homogeneous anisotropic plate and a non-homogeneous anisotropic plate with a doubler patch. Results from these experiments indicate that the new objective function and source location algorithm have improved performance when compared with those discussed in Refs. [1,2].

Characterization of composite plates using the virtual fields method with optimized loading conditions

This paper deals with the direct identification of bending stiffnesses of thin homogeneous anisotropic plates . The idea is to determine the unknown parameters from a single test giving a unique heterogeneous strain field. For the purpose of identification, the virtual fields method (VFM) is used. Here, the VFM is used taking into account two very recent developments: the piecewise construction of virtual fields and the noise minimization effect. Using these new aspects, improvements in terms of result accuracy are shown in comparison to previous studies. These new aspects are used as a tool to grade and to optimize different testing configurations with respect to the location of the applied loading. Some preliminary experimental results are then presented. They are in good agreement with theoretical expectations found in the first part of the study.

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].

A quasi-three-dimensional approach for homogeneous anisotropic plates undergoing extensional deformations

A quasi-three-dimensional approach for homogeneous anisotropic plates undergoing extensional deformations

A quasi-three-dimensional approach for homogeneous anisotropic plates undergoing extensional deformations

In present paper, the Kane-Mindlin assumptions are applied to homogeneous anisotropic plates undergoing extensional deformations. A complex potential approach is formulated;to that end, a third complex potential is added to the two Lekhniskii complex potentials, [2]. The components of displacement and stress as well as boundary conditions are represented in terms of these three complex potentials. The advantage of the proposed approach is that problems of anisotropic plates under extensional deformations are treated as a quasi-three-dimensional problem by considering the effect of the transverse normal strain.

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

Fully Isotropic Laminates and Quasi-Homogeneous Anisotropic Laminates*

This work pertains to the development of the fully isotropic laminates (FIL) and the quasi-homogeneous anisotropic laminates (QHAL). An FIL exhibits con comitant isotropy in the extensional and bending stiffnesses; its macromechanical be havior matches that of a homogeneous isotropic plate. A QHAL is a multidirectional lami nate with identical anisotropy in the extensional and bending stiffnesses, which matches that of a homogeneous anisotropic plate. Because of these characteristics, classical theo ries of homogeneous (isotropic and anisotropic) thin plates are applicable to the macro mechanical analysis of FIL and QHAL, which significantly reduces the laminate design effort.

Acoustic emission from a nucleating point fracture in a composite plate. Part I: Theory

A symmetric cross-ply laminated sheet in a state of plane stress is considered. The mathematical equations that describe this can be interpreted as the elastodynamic equations for a homogeneous anisotropic thin plate. The source of acoustic emission is taken in the first instance to be a point force and in the second instance to be a point explosion with a given pressure-time relationship. In the third example an infinitesimal shear dislocation with a prescribed slip direction and time dependence is considered. Expressions are developed for the in-plane displacement field in the frequency domain. These expressions for the frequency spectra appear as proper finite integrals. Starting from these asymptotic estimates for both the near and far fields are also developed. The normal displacement is seen as a Poisson effect resulting from the in-plane deformations. Numerical results are presented to show the variation of the P and S wave numbers with respect to direction. Furthermore the location of stationary points for the purposes of asymptotic evaluation is graphically displayed. An interesting result is the occurrence of second-order stationary points in the propagation of S waves. In Part II of this two part paper we will present numerical results for the displacement field due to the various source types discussed here. It is hoped that these results will assist in the ultrasonic nondestructive evaluation of composite plates to monitor the occurrence of quasistatic crack growth.

Mixed-mode fracture mechanics of anisotropic plates by means of the T-criterion

A direct generalization of the T-criterion for the prediction of mixed-mode fracture characteristics of homogeneous anisotropic plates is given in this paper. This fracture criterion, introduced by one of the authors (P.S.T.), was successfully used in various LEFM applications with isotropic materials as well as for cases where small scale yielding (s.s.y) of metallic materials was encountered. Due to the principles of T-criterion, according to which the different role played by the distortional and dilatational strain-energy density in fracture is fully recognized, its eventual use with anisotropic materials was prohibited, since the decomposition of the anisotropic elastic potential was not as yet conceivable. It is in terms of such an analytical decomposition especially for transversely isotropic media, recently derived by the authors, that the present generalization of the T-criterion was under-taken. Although the example problems treated herein assume homogeneously anisotropic media, preliminary comparison of the theoretical predictions with experimental data from fibrous composites revealed satisfactory agreement.

Load and Strain Histories for CFRP Laminates under Low Velocity Impact

The purpose of this study is to confirm the validity of analytical results of CFRP laminated plates under low-velocity impact. The impulsive responses of CFRP laminated plates are analyzed by using the higher-order shear deformation theory and homogeneous anisotropic plate theory. The plate is simply supported on all four sides and is hit by a spherical steel impactor at its center. The impact load is caculated numerically by using the nonlinear integral equation drived from Hertzian contact law. The strain histories on the plate are evaluated from the higher-order theory and the numerical impact load histories obtained above. The theoretical strain histories are in good agreement with the experimental results obtained by strain gauges. Thus, it was shown that the results obtained from this study could explain not only qualitatively, but also quantitively, the impact characteristics which depend on the stacking sequence of unidirectional prepreg.

Edge and layering effects in a multilayered composite plate

Ultrasonic waves provide an efficient means of characterizing the effective mechanical properties of a nonhomogeneous composite material. Characteristics of wave propagation in an undamaged composite medium are influenced by many factors, the most important of which are: microstructure, constituent properties, interfaces, residual stress field, and ply lay-ups. The guided waves in a cross-ply laminated plate are studied. Because of the complexity of the exact dispersion equation that governs the wave propagation in a multi-layered fiber-reinforced plate, a stiffness method is presented that can be applied to any number of layers. It is shown that a sufficiently large number of layers is required for the plate to be modeled as a homogeneous anisotropic plate. The reflection of guided waves from the edge of a multi-layered plate is also studied. Results are quite different than in the case of a single homogeneous plate.

Load and strain histories for CFRP laminates under low-velocity impact.

The purpose of this study is to confirm the validity of analytical results of CFRP laminated plates under low-velocity impact. The impulsive responses of CFRP laminated plates are analyzed by using the higher-order shear deformation theory and homogeneous anisotropic plate theory. The plate is simply supported on all four sides and is hit by a spherical steel impactor at its center. The impact load is caculated numerically by using the nonlinear integral equation drived from Hertzian contact law. The strain histories on the plate are evaluated from the higher-order theory and the numerical impact load histories obtained above. The theoretical strain histories are in good agreement with the experimental results obtained by strain gauges. Thus, it was shown that the results obtained from this study could explain not only qualitatively, but also quantitively, the impact characteristics which depend on the stacking sequence of unidirectional prepreg.

A Comparative Study of Crack/Crack Interaction in Composite Materials Using Macro and Micromechanical Models

The effects of interaction between crack-like defects in fiber reinforced composite materials on the local stress concentrations in their neighbourhoods are theoretically investigated from the macro and micro and micromechanical points of view. The most simple models consisting of homogeneous and heterogeneous anisotropic elastic plates with two collinear or parallel cracks are assumed, and the mathematical formulations are made in terms of the Cauchy-type singular integral equations of the first kind by the methods of harmonic-type stress functions and Fourier transforms. Numerical calculations are carried out for the stress intensity factors, and the results thus obtained from these models are compared with each other.

A comparative study of crack/crack interaction in composite materials using macro and micromechanical models.

The effects of interaction between crack-like defects in fiber reinforced composite materials on the local stress concentrations in their neighbourhoods are theoretically investigated from the macro and micromechanical points of view. The most simple models consisting of homogeneous and heterogeneous anisotropic elastic plates with two colinear or parallel cracks are used. Plane stress and plane strain conditions are assumed, and the mathematical formulations are made in terms of the Cauchy-type singular integral equations of the first kind through the methods of harmonic-type stress functions and Fourier transforms. Numerical calculations are carried out for the stress intensity factors, and the results thus obtained from these models are campared to each other.

Theories for Elastic Plates Via Orthogonal Polynomials

A complementary energy functional is used to derive an infinite system of two-dimensional differential equations and appropriate boundary conditions for stresses and displacements in homogeneous anisotropic elastic plates. Stress boundary conditions are imposed on the faces a priori, and this introduces a weight function in the variations of the transverse normal and shear stresses. As a result the coupling between the two-dimensional differential equations is described in terms of a single difference operator. Special attention is given to a truncated system of equations for bending of transversely isotropic plates. This theory has three boundary conditions, like Reissner’s, but includes the effect of transverse normal strain, essentially through a reinterpretation of the transverse displacement function. Full agreement with general integrals to the homogeneous three-dimensional equations is established to within polynomial approximation.