Alberto Milazzo

Multidomain boundary integral formulation for piezoelectric materials fracture mechanics

Abstract A boundary element method and its numerical implementation for the analysis of piezoelectric materials are presented with the aim to exploit their features in linear electroelastic fracture mechanics. The problem is formulated employing generalized displacements, that is displacements and electric potential, and generalized tractions, that is tractions and electric displacement. The generalized displacements boundary integral equation is obtained by using the closed form of the piezoelasticity fundamental solutions. These are derived through a displacement based modified Lekhnitskii’s functions approach. The multidomain boundary element technique is implemented to achieve the numerical solution. Results are presented for typical fracture mechanics problems. The generalized stress intensity factors and the generalized relative crack displacements are computed to show the soundness of the approach. The results obtained are also analyzed to highlight some interesting features of the electroelastic coupling effects.

An analytical solution for the magneto-electro-elastic bimorph beam forced vibrations problem

Based on the Timoshenko beam theory and on the assumption that the electric and magnetic fields can be treated as steady, since elastic waves propagate very slowly with respect to electromagnetic ones, a general analytical solution for the transient analysis of a magneto-electro-elastic bimorph beam is obtained. General magneto-electric boundary conditions can be applied on the top and bottom surfaces of the beam, allowing us to study the response of the bilayer structure to electromagnetic stimuli. The model reveals that the magneto-electric loads enter the solution as an equivalent external bending moment per unit length and as time-dependent mechanical boundary conditions through the definition of the bending moment. Moreover, the influences of the electro-mechanic, magneto-mechanic and electromagnetic coupling on the stiffness of the bimorph stem from the computation of the beam equivalent stiffness constants. Free and forced vibration analyses of both multiphase and laminated magneto-electro-elastic composite beams are carried out to check the effectiveness and reliability of the proposed analytic solution.

An equivalent single-layer approach for free vibration analysis of smart laminated thick composite plates

An equivalent single-layer model for the free vibration analysis of smart laminated plates is presented. The electric and magnetic fields are assumed to be quasi-static, and third order in-plane kinematics is employed to adequately take the shear influence into account when the plate thickness increases. The model governing equations are the plate equations of motion written in terms of mechanical primary variables and effective stiffness coefficients, which take the multi-field coupling effects into account. The model shows that the surface magneto-electric boundary conditions enter the definitions of the laminate forces and moment resultants. Moreover, it reveals that new stiffness terms, which are related to the derivatives of the transverse displacement component and are exclusively associated with the piezoelectric and piezomagnetic couplings, are involved. Free vibration solutions for simply supported plates are presented to validate the model by comparing the present results with benchmark 3D solutions. Comparison of the results obtained by lower order models, namely zero and first order shear deformation theories, is presented and discussed, focusing on the adequateness of the obtained models with respect to the plate thickness. Some characteristic features of smart laminate behavior have also been addressed.

A symmetric and positive definite variational BEM for 2-D free vibration analysis

Abstract A general BEM model for structural dynamics is derived by using a symmetric and positive definite variational formulation. The functional employed involves domain displacements and boundary tractions and displacements. These variables are taken to be independent of one another. The boundary variables are expressed in terms of their nodal values while the domain displacement field is approximated by a linear combination of static fundamental solutions. The source point of the latter is located outside the domain. The resolving system is a linear system and for free vibration a classic linear algebraic eigenvalue problem is inferred. The stiffness and mass matrices are symmetric and positive definite and the domain integral, when associated with the inertial term, can be transformed into a boundary integral. Numerical results are presented to prove the efficiency of the method.

Bending stress fields in composite laminate beams by a boundary integral formulation

Abstract The elasticity of a composite laminate under bending loads is approached through a boundary integral formulation and solved by the boundary element method. The integral equations governing the behaviour of each layer within the laminate, are deduced using the reciprocity theorem. Exact analytical singular solutions of the generalized orthotropic elasticity, i.e. the fundamental solutions of the problem, are employed as the kernels of the integral equation. The formulation does not make any assumption as to the nature of the elastic response and it allows consideration of general section geometries and stacking sequences. The solution is obtained through the enforcement of the interface continuity conditions and the prescribed boundary conditions in such a way that all of the elasticity relations involved in the problem description are exactly satisfied. Numerical solutions for different laminate configurations are presented which show that the present approach gives accurate stress distributions with meaningful advantages and without requiring too much computational effort.

Boundary Element Solution for Free Edge Stresses in Composite Laminates

The edge-stress problem in multilayered composite laminates under uniform axial extension is analyzed through an alternative method based on a boundary integral formulation. The basic equations of the formulation are discussed and solved by the multiregion boundary element method. Generalized orthotropic elasticity analytic fundamental solutions are employed to establish the integral equations governing the problem. The formulation is absolutely general with regard to the laminate stacking sequence and the section geometry and it does not require any aprioristic assumption on the elastic response nature. This makes the formulation suitable for an investigation of the singular behavior of the stress field at the free edge in composite laminates. The interlaminar normal and shear stress distributions are examined in detail with the aim of calculating the stress singularity at the interlaminar free edge. The singularity parameters, i.e., power and strength, are determined for two family of laminates in order to ascertain the effectiveness of the method for the free edge-stress problem.

On the dynamic behavior of piezoelectric active repair by the boundary element method

The dynamic behavior of piezoelectric active repair bonded on cracked structures is analyzed in this article. The boundary element code used to perform the simulations is implemented in the framework of piezoelectricity in order to model the coupling between the elastic and the electric fields, which represents the most important feature of piezoelectric media. The fracture mechanics problem, i.e. the crack, as well as the bonding layer between the host structure and the active patch is modeled by means of the multidomain technique provided with an interface spring model. More particularly, the spring interface model allows considering the bonding layer as a zero-thickness elastic ply characterized by normal and tangential stiffness constants. The crack is also modeled as an elastic interface characterized by vanishing stiffness. The dual reciprocity method (DRM) has been used in the present time-dependent application for the approximation of the domain inertia terms. Numerical analyses have been carried out in order to characterize the dynamic repairing mechanism of the assembled structure by means of the computation of the dynamic stress intensity factors and discussions are presented to highlight the effect of the inertial forces on the fracture mechanics behavior of the overall assembled structure.

Magneto-Electro-Elastic Bimorph Analysis by the Boundary Element Method.

The influence of the magnetic configuration on the behavior of magneto-electro-elastic bimorph beams is analyzed by using a boundary element approach. The problem is formulated by using the generalized displacements and generalized tractions. The boundary integral equation formulation is obtained by extending the reciprocity theorem to magneto-electro-elastic problems; it is numerically implemented by using the boundary element method multidomain technique to address problems involving nonhomogeneous configurations. Results under different magnetic configurations are compared highlighting the characteristic features of magnetopiezoelectric behavior particularly focusing on the link between interlaminar stress and magnetic induction.

A meshfree method for transverse vibrations of anisotropic plates

A meshfree approach, called displacement boundary method, for anisotropic Kirchhoff plate dynamic analysis is presented. This method is deduced from a variational principle, which uses a modified hybrid functional involving the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain as independent variables. The discretization process is based on the employment of the fundamental solutions of the static problem operator for the expression of the variables involved in the functional. The stiffness and mass matrices obtained for the dynamic model are frequency-independent, symmetric and positive definite and their computation involves boundary integrals of regular kernels only. Due to its features, the final resolving system can be solved with the classical approaches by using standard numerical procedures. To assess the formulation, the free vibrations of some anisotropic plates were calculated and the results compared with those obtained using other solution techniques. The present results are in good agreement with those found in the literature showing the accuracy and effectiveness of the proposed approach.

Electroelastic Analysis of Piezoelectric Composite Laminates by Boundary Integral Equations

A boundary integral representation for the electroelastic state in piezoelectric composite laminates subjected to axial extension, bending, torsion, shear/bending, and electric loadings is proposed. The governing equations are presented in terms of electromechanical generalized variables by the use of a suitable matrix notation. Thus, the three-dimensional electroelasticity solution for piezoelectric composite laminates is generated from a set of two partially coupled differential equations defined on the cross section of each individual ply within the laminate. These ply equations are linked through the interface conditions, which allow restoration of the model of the laminate as a whole. For this model, the corresponding boundary integral representation and the relative boundary integral equations are deduced, and their features are discussed. The formulation presented lays out the analytical foundation for the development of the multidomain boundary element method to determine numerically the electromechanical response of piezoelectric composite laminates. Numerical results showing the characteristics of the method are given, and the fundamental behavior of piezoelectric composite laminates is pointed out for both mechanical and electrical loads. I. Introduction P IEZOELECTRIC materials generate an electric field when subjected to strain fields and undergo deformation when an electric field is applied. This inherent electromechanical coupling, known as direct and converse piezoelectric effects, is widely exploited in the design of many devices working as transducers, sensors, and actuators. In addition, piezoelectric materials are of primary concern in the field of advanced lightweight structures, where the smart structure technology is now emerging. 1−4 When piezoelectric members are bonded or merged within a structure, it is possible to combine the mechanical properties of the host structure with the additional capabilities to sense deformation and to adapt the structural response accordingly. The first attempt at the application of smart structures with sensing and control capabilities was concerned with the application of piezoelectric patches on the surfaces of beams and plates to induce strain actions on the passive structure or detect its deformation. The development of this approach, together with the improvement of composite material technology, led to the concept of distributed sensing and control, which can be accomplished by the introduction of piezoelectric layers within composite laminates. More recently, the idea of distributed structural control properties has been fully developed by the use of active fiber composites. 5 In these fiber-reinforced composites, the fiber and the matrix have additional functions besides their typical roles. The fiber, which generally exhibits a piezoelectric behavior, not only accomplishes the task of structural reinforcement, but it also has the function of both sensor and actuator. In this kind of smart structure, an active control of the mechanical response is performed on the basis of the intrinsic properties of the structure material. Some of the fundamental problems involved in the mechanical testing, analysis, and design, namely, failure modes and strength and stiffness degradation under static and cyclic fatigue loads, are not tractable without a thorough knowledge of the