Shock and Vibration

Volume 2016, Article ID 3707658, 14 pages

http://dx.doi.org/10.1155/2016/3707658

## On the Finite Element Free Vibration Analysis of Delaminated Layered Beams: A New Assembly Technique

Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario, Canada M5B 2K3

Received 20 July 2015; Revised 16 October 2015; Accepted 21 October 2015

Academic Editor: Longjun Dong

Copyright © 2016 Nicholas H. Erdelyi and Seyed M. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamic analysis of flexible delaminated layered beams is revisited. Exploiting Boolean vectors, a novel assembly scheme is developed which can be used to enforce the continuity requirements at the edges of delamination region, leading to a delamination stiffness term. The proposed assembly technique can be used to form various beam configurations with through-width delaminations, irrespective of the formulation used to model each beam segment. The proposed assembly system and the Galerkin Finite Element Method (FEM) formulation are subsequently used to investigate the natural frequencies and modes of 2- and 3-layer beam configurations. Using the Euler-Bernoulli bending beam theory and free mode delamination, the governing differential equations are exploited and two beam finite elements are developed. The free bending vibration of three illustrative example problems, characterized by delamination zones of variable length, is investigated. The intact and defective beam natural frequencies and modes obtained from the proposed assembly/FEM beam formulations are presented along with the analytical results and those available in the literature.

#### 1. Introduction

Layered structures have seen greatly increased use in civil, shipbuilding, mechanical, and aerospace structural applications in recent decades. Delamination, a common failure mode in layered structures, may arise from loss of adhesion between two layers of the structure, from interlaminar stresses arising from geometric or material discontinuities, or from mechanical loadings. The presence of delamination may significantly reduce the stiffness and strength of the structures. A reduction in the stiffness, in turn, will affect the vibration characteristics of the structures. Changes in the natural frequency, as a direct result of the reduction of stiffness, may lead to resonance if the reduced frequency is close to an excitation frequency.

Several experimental methods exist to predict the onset, size, and growth of delamination as a failure mode in composite materials. Using acoustic emission (AE) sensors, different levels of amplitude signals emitted by the materials can be monitored [1]. The different amplitudes correspond to loading types and are assigned damage mechanisms. Using this technique, continuous monitoring of damage is possible experimentally. Another research [2] has shown that acoustic emission is a viable and effective tool for identifying damage and distinguishing damage types in self-reinforced polyethylene composites. More recently, further research has been done using neural networks and unsupervised learning techniques applied to the data set of acoustic emission signals [3]. The signals were successfully used to classify the AE patterns caused by different damage mechanisms in carbon-reinforced composites (delamination and matrix cracking).

The time-domain stability of vibrating delaminated systems has also been an area of study. Particularly, the mechanisms that cause the different laminates to separate from each other are typically not defined in most theoretical applications. They are described as physically inadmissible mode shapes whose existence in the frequency domain is a product of delamination tip boundary conditions. The study of the real phenomenon, however, has shown that time-dependent normal forces in the delaminated segments do not influence the global free vibration frequencies but may contribute to localized buckling [4, 5]. Instability and critical dynamic forces can be predicted, allowing for study of the onset of delamination opening.

The vibration modelling and analysis of delaminated multilayer beams has been a topic of interest for many researchers. The earliest delamination models formulated in the 1980s [6] addressed the vibration of two-layer beams, where each layer was modelled using Euler-Bernoulli bending beam theory. The upper and lower portions of the delaminated segment were assumed to vibrate independent of each other, that is, “free mode” delamination. The free mode, however, underpredicts natural frequencies for off-midplane delaminations due to unrestricted penetration of the beams into each other. In 1988 [7], Mujumdar and Suryanarayan proposed the “constrained mode” delamination model, which assumes equal transverse displacements for the top and bottom beams and the rigid connector assumption. The rigid connector assumption states that, for the beam models presented, the delamination faces, which are planar and normal to the neutral axis of the undeformed beam, remain planar and normal to the neutral axis of the deformed beam. This assumption produces a set of kinematic and force continuity conditions at the delamination tips. In a recent work by Szekrényes [8], an extensive literature survey of the research works related to the vibrations of delaminated elements was presented and, based on coupled flexural-longitudinal vibration model, the equality of axial forces in the top and bottom beams was derived and shown in an exact way. Also, the continuity of the effective bending moments was related to the equilibrium equations and it was also concluded that delamination buckling can take place if the normal force is compressive in one of the half periods of the vibration and reaches a critical value [8].

The constrained mode delamination model, predicting vibration behaviour much more accurately for off-midplane delamination, is in fact simply a limiting case of the free mode delamination model. However, opening delamination modes, that is, where the layers separate from each other, commonly seen in experimental analysis [9–11], cannot be captured using the constrained model. Therefore, in the present study, the free mode delamination model will be investigated and the constrained mode delamination model can be derived in a similar manner.

The accuracy of dynamic/forced response analysis of a flexible structure depends greatly on the reliability of the modal analysis method used and the resulting natural frequencies and modes. There are various numerical, semianalytical, and analytical methods to predict the natural frequencies and mode shapes of such a system. Several exact solution methods exist for well-defined systems, such as delaminated isotropic beams with constant geometric and material properties. Single [9, 12, 13], multiple [14], and various overlapping and enveloped delamination conditions in space and on various elastic media, such as Pasternak soil [15], have been studied using analytical solution methods. Some work has also been done on delaminated sandwich structures [16], albeit with some mathematical simplification. These solution methods generally use the same procedure as Mujumdar and Suryanarayan [7] to formulate the kinematic continuity conditions across the delamination tips. The power of this type of formulation lies in the ability to be applied to any number of different system configurations. However, a potential drawback to this procedure is that the system equation must be reformulated after any configuration change, potentially limiting its applicability.

The conventional Finite Element Method (FEM) has a long, well-established history and with the advent of digital computers it is commonly used for structural analysis. The FEM is a general and systematic approach to formulate the element matrices for a given system and is easily adaptable to complex systems, such as nonuniform geometry, often modeled as a stepped, piecewise-uniform configuration. Exploiting polynomial interpolation (shape) functions, the FEM leads to constant element mass and stiffness matrices and ultimately a linear eigenvalue problem from which the natural frequencies and modes of the system can be readily extracted. The FEM method for a single beam can be modified to accurately model delaminated multilayer beams. Among others, Lee [17–19] used the layerwise FEM theory to investigate the free vibration of delaminated beams. In the recent years, layered, sandwich, and composite elements have been integrated in certain commercial software and are used to analyze the vibration of composite structures. However, modelling a delaminated configuration in commercial software packages such as ANSYS is not straightforward and can involve cumbersome, complex, time-consuming and error-prone processes. It requires manual model creation, involving the use of, for example, multipoint constraint rigid link (ANSYS element MPC184) [20] to enforce the displacement and slope continuity at the edges of delamination region [21].

Semianalytical formulations, such as the so-called Dynamic Finite Element (DFE) method [22], have also been developed to carry out structural modal analysis. The hybrid DFE formulation results in a more accurate prediction method than traditional and FEM modeling techniques, allowing for a reduced mesh size. The DFE technique follows the same typical procedure as the FEM by formulating the element equations discretized to a local domain, where element stiffness matrices are constructed and then assembled into a single global matrix. The application of the DFE to the preliminary free vibration analysis of a delaminated 2-layer beam has been reported in an earlier work by the authors [23].

Analytical methods, namely, the Dynamic Stiffness Matrix (DSM), have also been used for the vibrational analysis of isotropic, sandwich, and composite structural elements and beam-structures. The DSM approach exploits the general, closed-form solution to the governing differential equations of motion of the system to formulate a frequency-dependent stiffness matrix. The DSM produces exact results for simple structural elements, such as uniform beams, and Banerjee and his coworkers [24, 25] have developed a number of DSM formulations for various beam configurations. The DSM method for a single beam can be modified to accurately model delaminated multilayer beams. A DSM-based analysis of a two-layer split beam has also been presented in earlier works by the authors [26, 27].

The aim of this paper is to present an FEM formulation for the linear, free vibration analysis of a delaminated two-layer beam, using the free mode delamination model. The delamination is represented by two intact beam segments; one for each of the top and bottom sections of the delamination. The delaminated region is bounded on either side by intact, full-height beams. The beams transverse displacements are governed by the Euler-Bernoulli slender beam bending theory, and shear deformation and rotary inertia are neglected. Continuity conditions for forces, moments, displacements, and slopes at the delamination tips are enforced through a novel Boolean vector assembly scheme, leading to the integral FEM model of the system. In fact, through the presented method, one obtains specific matrices for an intact, fully delaminated, and delaminated elements attached to an intact segment from left/right. Therefore, a direct assembly method can be directly used to form various multiple-delaminated beam configurations, without the need to manually create the model and to use a constraint element (e.g., ANSYS element MPC184 [20]), to enforce the displacement and slope continuity at the edges of delamination region. Thence, the direct assembly of element matrices and the application of system’s global boundary conditions results in the linear eigenvalue problem of the defective system. In addition, two MATLAB-based computer codes, based on the Dynamic Stiffness Matrix (DSM) method [26, 27] and the analytical solutions reported in the literature [28–31], are developed and used as a benchmark for comparison. Two 2-noded and 3-noded beam FEM elements are presented, where cubic Hermite and quartic interpolation functions of approximation, respectively, are used to express the flexural displacement functions, that is, field variables and weighting functions [32]. The FEM models are used to compute the natural frequencies of an illustrative defective beam example, characterized by a single delamination zone of variable length. The frequency values are then compared with DSM data and those from the literature. Certain modal characteristics of the system are also discussed. It is worth noting that while the model used in this study assumes isotropic materials, further research is underway to extend it to sandwich [33] and fibre-reinforced laminated composite beams, characterized by an extensional response coupled with flexural/torsional and coupled bending-torsion vibration [34, 35].

#### 2. Mathematical Model

Figure 1 shows the general coordinate system and notation for a single-delaminated beam, with total length , intact beam segment lengths and , delamination length , and total height . This model incorporates a general delamination, which can include laminated composites or bilayered isotropic materials, with different material and geometric properties above and below the delamination plane. Thus, the top layer has thickness , Young’s modulus , density , cross-sectional area , and second moment of area . The bottom layer has corresponding properties, with subscript 3. The delamination tips occur at stations and , and torsion, shear deformation, axial (warping effects and axial deformation), and out-of-plane delamination are ignored. Following this notation, the general equation of motion for the th Euler-Bernoulli beam in free vibration is written as [26–28, 31]For harmonic oscillations, the transverse displacements can be written aswhere is the amplitude of the displacement , subscript “” represents the beam segment number, and is the circular frequency of excitation of the system. Back-substituting (2) into (1), the equations of motion reduce toThe general solution to the 4th-order, homogeneous differential equation (3) can be written in the following form:which represents the bending displacement of beam segment “”, is the beam segment length, and stands for nondimensional frequency of oscillation, defined asCoefficients , , , and are evaluated to satisfy the displacement continuity requirements of the beam segments and the system boundary conditions. As also reported by several researchers [8, 26–28, 31], the inclusion of delamination into the beam model leads to coupled axial-transverse motion of the delaminated beam portions, primarily associated with the continuity requirements imposed at the delamination endpoints. In order for the delamination tip cross sections to remain planar after deformation, the ends of the top and bottom beams must have the same relative axial location after deformation, preventing interlaminar slip. The midplanes, that is, the neutral axes of the beam segments, in the delaminated region are located at a distance from the midplanes of the intact segments. Hence, they will not have the same axial deformation unless some internal axial force is imposed. As mentioned earlier in this paper, based on coupled flexural-longitudinal vibration model [8], the equality of axial forces in the top and bottom beams has been recently derived and shown in an exact way and the continuity of the effective bending moments was related to the equilibrium equations. However, in what follows, this imposed axial force is briefly presented for completeness, following the method derived and discussed in [7].