Edge effects of uniformly loaded cross-ply composite laminates

Interlaminar stresses resulting from bending of rectangular cross-ply composite laminates are determined using a layer wise laminate theory. Two types of laminates are considered. First a fully simply supported laminate subjected to bi-directional bending is analyzed. The results obtained from this theory are compared with those of the published three-dimensional elasticity solutions to verify the validity and accuracy of the present theory. Then laminates with two edges simply supported and the other two edges free are examined. The results indicate the presence of significant interlaminar stresses near the free edges.     

Model for interlaminar normal stress in singly curved laminates

The paper presents a new model for interlaminar normal stress distribution in moderately thick and singly curved laminates. It requires in-plane strain information which must also map the kinematical effects caused by the curved laminate shape. The number of free parameters of the exact solutions for each layer is reduced and determined by a finite-element procedure which is also explained. The results of the mechanical model are compared with FEM results for unidirectional and cross-ply laminates. The new model can be developed for element-level post processing in a FEM program with simple finite structural elements.     

A general solution for a bolt loaded-hole in piezoelectric composite laminates

In this paper the two-dimensional problem of a bolt loaded-hole in an infinite piezoelectric laminate is considered in terms of the complex variable method. Firstly, an explicit form Green function for a generalized point load acted at an arbitrary point outside the hole is derived. Secondly, the Green function for a generalized point load acted at the rim of the hole, as a special case, is obtained, and then a general solution for the case of arbitrarily distributed mechanical and electric loading on the hole surface is presented based on the superposition principle. In general, the solution is in the form of series, but its novel feature is that the coefficients involved in the solution can be easily written out once the distributed loading is specified. Finally, several examples useful for engineering are given.     

Interlaminar stresses in thick rectangular laminated plates with arbitrary laminations and boundary conditions under transverse loads

In the present study, interlaminar stresses resulting from bending of thick rectangular laminated plates with arbitrary laminations and boundary conditions are analyzed analytically based on a three-dimensional multi-term extended Kantorovich method (3DMTEKM). Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations with non-homogeneous boundary conditions are obtained. Then an iterative procedure is established to achieve analytical solution. The results obtained from this theory are compared with those of analytical solutions existing in the literature. It is found that the present results have excellent agreements with those obtained by layerwise theory. The results show that the multi-term EKM converges within only three terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlaminar stresses in thick rectangular laminated plates with general types of boundary conditions and lay-ups is examined.     

Higher order shear deformation effects on analysis of laminates with piezoelectric fibre reinforced composite actuators

A complete analytical solution for cross-ply composite laminates integrated with piezoelectric fiber-reinforced composite (PFRC) actuators under bi-directional bending is presented in this paper. A higher order shear and normal deformation theory (HOSNT12) is used to analyze such hybrid or smart laminates subjected to electromechanical loading. The displacement function of the present model is approximated by employing Taylor’s series in the thickness coordinate, while the electro-static potential is assumed to be layer wise (LW) linear through the thickness of PFRC. The equations of equilibrium are obtained using principle of minimum potential energy and solution is by Navier’s technique. Transverse shear stresses are presented at the interface of PFRC actuator and laminate under the action of electrostatic potentials. Results are compared with first order shear deformation theory (FOST) and exact solution.     

A study of interlaminar stresses in variable stiffness plates

Variable stiffness composite panels have been in continuous development for the last two decades. Several studies have been carried out to evaluate their structural response under different hypotheses. It is known that correct prediction of the onset of delamination in multilayer composite laminates requires an accurate evaluation of interlaminar stresses. Although finite element codes provide results for interlaminar stresses, they are not continuous both across and along layer interfaces due to the use of C⁰ interpolated elements. In this work, a methodology for obtaining interlaminar stresses is extended and applied to the case of variable stiffness composite panels. Pagano’s three layer reference case is investigated for both constant stiffness and variable stiffness cases. A set of analyses are carried out to demonstrate the performance of the method and the variation in the stress response of the panel due to changes in composite layup. Results show that the distribution of shear stresses in the panel presents an significant variation depending on the stacking sequence. A debonding failure criteria is used to evaluate the performance of different variable stiffness configurations, obtaining improvements compared with constant stiffness panels.     

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.     

An analytical study of the effect of slamming pressures on the interlaminar behaviour of composite panels

This paper discusses the effect of material properties on the interlaminar behaviour of ship panels made of composite materials under stress waves caused by slamming loads. First, a brief reference to the analytical model developed to simulate the propagation of stress waves caused by slamming impact on composite materials which has been recently published in Composite Structures is made. Then, some parametric studies are done, discussing the influence of material Young’s modulus on the interlaminar peak stresses. The influence of strain rates is also discussed.     

Geometrically non-linear analysis of composite structures with integrated piezoelectric sensors and actuators

This paper deals with the geometrically non-linear analysis of thin plate/shell laminated structures with embedded integrated piezoelectric actuators or sensors layers and/or patches. The motivation for the present developments is the lack of studies in the behavior of adaptive structures using geometrically non-linear models, where only very few published works were found in the open literature.

The model is based on the Kirchhoff classical laminated theory and can be applied to plate and shell adaptive structures with arbitrary shape, general mechanical and electrical loadings.

The finite element model is a non-conforming single layer triangular plate/shell element with 18 degrees of freedom for the generalized displacements and one electrical potential degree of freedom for each piezoelectric layer or patch.

An updated Lagrangian formulation associated to Newton–Raphson technique is used to solve incrementally and iteratively the equilibrium equations.

The model is applied in the solution of four illustrative cases, and the results are compared and discussed with alternative solutions when available.     

Three-dimensional piezoelasticity solution for dynamics of cross-ply cylindrical shells integrated with piezoelectric fiber reinforced composite actuators and sensors

A benchmark three-dimensional (3D) exact piezoelasticity solution is presented for free vibration and steady state forced response of simply supported smart cross-ply circular cylindrical shells of revolutions and panels integrated with surface-bonded or embedded monolithic piezoelectric or piezoelectric fiber reinforced composite (PFRC) layers. The effective properties of PFRC laminas for the 3D case are obtained based on a fully coupled iso-field model. The governing partial differential equations are reduced to ordinary differential equations in the thickness coordinate by expanding all entities for each layer in double Fourier series in span coordinates, which identically satisfy the boundary conditions at the simply-supported ends. These equations with variable coefficients are solved using the modified Frobenius method, wherein the solution is constructed as a product of an exponential function and a power series. The unknown constants of the general solution are finally obtained by employing the transfer matrix method across the layers. Results for natural frequencies and the forced response are presented for single layer piezoelectric and multilayered hybrid composite and sandwich shells of revolution and shell panels integrated with monolithic piezoelectric and PFRC actuator/sensor layers. The present benchmark solution would help assess 2D shell theories for dynamic response of hybrid cylindrical shells.     

Interlaminar stress analysis in general thick composite cylinder subjected to nonuniform distributed radial pressure

Interlaminar stresses in thick a composite cylinder with general layer stacking subjected to uniform and nonuniform distributed radial pressure are studied. The layerwise theory of Reddy is employed for formulation of the problem. An analytical method is presented for solving the governing equations. To increase accuracy, interlaminar stresses are obtained by integrating the equilibrium equation of elasticity. After a convergence study, the accuracy of the layerwise laminate theory is investigated using the predictions of finite element method. Predictions of Hooke's law and integration method for interlaminar stresses are compared. Uniform and nonuniform internal and external loads are considered and a parametric study is done for various cylinders.     

Shape control of smart composite plate with non-rectangular piezoelectric actuators

Quasi-static shape control of a smart structure may be achieved through optimizing the applied electric fields, loci, shapes and sizes of piezoelectric actuators attached to the structure. In this paper, a finite element analysis (FEA) software has been developed for analyzing static deformation of smart composite plate structures with non-rectangular shaped PZT patches as actuators. The mechanical deformation of the smart composite plate is modeled using a 3rd order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation is verified by comparing with experimentally measured deformation. Numerical results are obtained for the optimum values of the electric field in the PZT actuators to achieve the desired shape using the linear least square (LLS) method. The numerical results demonstrate the influence of the shapes of actuators.     

Micromechanics of piezoelectric composites with improved effective piezoelectric constant

This paper is concerned with the derivation of a micromechanics model of a new type of piezoelectric fiber reinforced composite (PFRC) materials. A continuum mechanics approach is employed to determine the effective properties of these composites. The piezoelectric fibers of these composites are considered to be electroded at the fiber–matrix interface such that the electric fields in the fiber and matrix become equal in the direction transverse to the fiber direction. The model has been verified with the existing models. The present model also predicts that the effective piezoelectric coefficient of these PFRC which accounts for the actuating capability in the fiber direction due to the applied field in the direction transverse to the fiber direction improves over the corresponding coefficient of the material of the piezoelectric fibers if the fiber volume fraction exceeds a critical fiber volume fraction.     

Three-dimensional analysis for rectangular 1–3 piezoelectric fiber-reinforced composite laminates with the interdigitated: electrodes under electromechanical loadings

In this paper, an exact analysis for the rectangular composite laminated plate consisting of 1–3 piezoelectric fiber-reinforced composite layer and orthotropic elastic composite layer subjected to the electric field with the interdigitated electrode (IDE) and arbitrary loads is presented without any simplification. The solution of the derived governing differential equations is obtained through the power series expansion and Fourier expansion methods. An illustrative analysis is carried out to investigate the influence of the stiffness anisotropy, free-strain anisotropy and electric field on the shear curvature of the laminated plate with 1–3 active fiber-reinforced composite layer. The numerical results show that, the magnitude of shear curvature gradually increases as the stiffness anisotropy and the free-strain anisotropy increase, and increasing the electrical field also leads to an increase in shear curvature. Results presented here can be used to assess various approximate theories and enhance the understanding of the static and dynamic response behavior of the 1–3 piezoelectric composite structures.     

Effect of stacking sequence and thickness on the interlaminar stresses for quasi-isotropic laminated plates with a hole

This paper discusses the evaluation of interlaminar stresses, in the presence of in-plane stress gradients, for composite laminates by using three-dimensional equilibrium equations. The stress gradients are calculated by means of an improved finite difference scheme. A quasi-isotropic laminated plate with a circular hole subjected to a uniform tension stress is considered. The effects of stacking sequence and the thickness of the laminate on the interlaminar stresses have been studied.     

A new efficient numerical method and the dynamic analysis of composite laminates with piezoelectric layers

Firstly, a numerical method for the inversion of Laplace transform is developed and its accuracy is shown through examples. Then, a state-vector equation for the dynamic problems of piezoelectric plates is deduced directly from a modified mixed variational principle for piezoelectric bodies and its exact solution for the dynamic problems of simply supported rectangle piezoelectric plate is simply given. For multilayered hybrid plates, we derive the solution in terms of the propagator matrices. The techniques accounts for the compatibility of generalized displacements and generalized stresses on the interface both the elastic layers and piezoelectric layers, and the transverse shear deformation and the rotary inertia of laminate are also considered in the global algebraic equation of structure. Meanwhile, there is no restriction on the thickness and the number of layers. As an application of the numerical inversion of Laplace transform presented in this paper, typical numerical examples of the harmonic vibration and transient response are proposed and discussed. Since the highly accurate numerical results, they can serve as benchmarks to test various thick plate theories and various numerical methods, such as the finite and boundary element methods for transient response problems.     

Cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation

Based on the three-dimensional (3D) piezoelectricity, two asymptotic formulations for the cylindrical bending vibration of simply supported, functionally graded (FG) piezoelectric cylindrical shells with open-circuit and closed-circuit surface conditions are presented. The normal electric displacement and electric potential are prescribed to be zero on the lateral surfaces. In the present asymptotic formulations the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constant in single-layer shells, to be layerwise constant in multilayered shells and to obey an identical exponent-law distribution in FG shells. The method of multiple time scales is used to eliminate the secular terms arising from the regular asymptotic expansion. The orthonormality and solvability conditions for various orders are derived. The recursive property among the motion equations of various order problems is shown. The present asymptotic formulations are applied to several illustrative examples. The accuracy and the rate of convergence of the present asymptotic solutions are evaluated. The coupled electro–elastic effect and the influence of the material-property gradient index on the free-vibration behavior of FG piezoelectric shells are studied.     

Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials

Interface and interfacial cracks interacting with screw dislocations in piezoelectric bimaterials subjected to antiplane mechanical and in-plane electrical loadings are studied within the framework of linear piezoelectricity theory. Straight dislocations with the Burgers vector normal to the isotropic basal plane near the interface or interfacial crack are considered. The dislocations are characterized by a discontinuous electric potential across the slip plane and are subjected to a line-force and a line-charge at the core. An explicit solution for the screw dislocation in piezoelectric bimaterial with straight interface is found based on the solution of a similar problem for infinite homogenous medium. The obtained relation is independent of the nature of singularity. This fundamental result is used to analyze dislocation interacting with a set of collinear interfacial cracks in piezoelectric bimaterials. Three solutions for the screw dislocation interacting with a semi-infinite crack, finite crack, and edge crack between two bonded dissimilar piezoelectric materials are obtained in closed-form. These solutions can be used as Green’s functions for the analyses of interfacial cracks in piezoelectric bimaterials.     

Interlaminar stresses in antisymmetric angle-ply cylindrical shell panels

Layerwise theory of Reddy is utilized for investigating free-edge effects in antisymmetric angle-ply laminated shell panels under uniform axial extension. Following some physical arguments, governing displacement field is divided into local and global parts. The former is discretized through the shell thickness by a zig-zag interpolation function while the latter is calculated by a first-order shear deformation theory. Local equilibrium equations are then solved through a state space approach. Accuracy of the proposed technical solution is subsequently verified by a novel analytical elasticity solution. For this end, the problem is analytically solved for specific boundary conditions along the edges. The numerical results show excellent agreement between two theories for various composite shell panels.     

Multiscale analysis of laminated plates with integrated piezoelectric fiber composite actuators

This study is concerned with the detailed analysis of fiber-reinforced composite plates with integrated piezoceramic fiber composite actuators. A multiscale framework based on the asymptotic expansion homogenization method is used to couple the microscale and macroscale field variables. The microscale fluctuations in the mechanical displacement and electric potential are related to the macroscale deformation and electric fields through 36 distinct characteristic functions. The local mechanical and charge equilibrium equations yield a system of partial differential equations for the characteristic functions that are solved using the finite element method. The homogenized electroelastic properties of a representative material element are computed using the characteristic functions and the material properties of the fiber and matrix. The three-dimensional macroscopic equilibrium equations for a laminated piezoelectric plate are solved analytically using the Eshelby-Stroh formalism. The formulation admits different boundary conditions at the edges and is applicable to thick and thin laminated plates. The microscale stresses and electric displacement in the fibers and matrix are computed from the macroscale fields through interscale transfer operators. The multiscale analysis procedure is illustrated using two model problems. In the first model problem, a simply-supported sandwich plate consisting of a piezoceramic fiber composite shear actuator embedded between two graphite/polymer layers is studied. The second model problem concerns a cantilever graphite/polymer substrate with segmented piezoceramic fiber composite extension actuators attached to its top and bottom surfaces. Results are presented for the homogenized material properties, macroscale deformation, macroscale average stresses and microscale stress distributions.