Free vibration analysis of delaminated bimaterial beams

Abstract

Non-dimensional parameters, namely, axial stiffness ratio and bending stiffness ratio, are introduced to study the vibration of bimaterial beams with a single delamination. The use of these parameters provide a better understanding on the influence of the axial stiffness and bending stiffness of the delaminated layers on the vibration of a delaminated beam. Both the ‘free mode’ and ‘constrained mode’ models in the study of the vibration of delaminated beams are used in the present analysis. A relative slenderness ratio specific for delamination vibration is further introduced, which is shown to dominate the vibration characteristics of the beam. For lower slenderness ratios, the influence of the delamination decreases and a global vibration mode dominates. For higher slenderness ratios, the influence of the delamination increases and a local vibration mode prevails.

Introduction

Delaminations are common defects in composite laminates due to their weak interlaminar strength. They may occur as a consequence of imperfect fabrication processes, such as incomplete wetting and air entrapment, or of external effects during the operational life of the composites, such as impact by foreign objects. Delaminations may not be visible or barely visible on the surface, since they are embedded within the composite structures. However, they may significantly reduce the stiffness and strength of the structures [1]. A reduction in the stiffness will affect the vibration characteristics of the structures, such as the natural frequency and mode shape. Delaminations reduce the natural frequency, as a direct result of the reduction of stiffness, which may cause resonance if the reduced frequency is close to the working frequency. It is important to be able to predict this change of frequencies and mode shapes in a dynamic environment.

To study the influence of a through-width delamination on the free vibration of an isotropic beam, Wang et al. [2] presented an analytical model using four Euler–Bernoulli beams that are joined together. They assumed that the delaminated layers deform ‘freely’ without touching each other (‘free mode’ model) and will have different transverse deformations. In contrast, Mujumdar and Suryanarayan [3] assumed that the delaminated layers are in touch along their whole length all the time, but are allowed to slide over each other (‘constrained mode’ model). Thus, the delaminated layers are ‘constrained’ to have identical transverse deformations. This ‘constrained mode’ model was extended by Shu and Fan [4] on a bimaterial beam and Hu and Hwu [5] on a sandwich beam to include the effects of the rotary inertia and transverse shear deformation. A similar ‘constrained mode’ model was proposed by Tracy and Pardoen [6] on a composite beam. Valoor and Chandrashekhara [7] extended this model for thick composites to include the effects of the transverse shear deformation and the rotary inertia. In addition, the Poisson effect was included due to its significance in the analysis of angle-ply laminated beams. However, the ‘constrained mode’ model, failed to predict the opening in the mode shapes found in the experiments by Shen and Grady [8] and Lestari and Hanagud [9]. To simulate the ‘open’ and ‘closed’ behavior between the delaminated surfaces, Luo and Hanagud [10] presented an analytical model based on the Timoshenko beam theory, which uses piecewise-linear springs. The spring stiffness would then be equal to zero (0) for the ‘free mode’ and infinity (∞) for the ‘constrained mode’. The influence of the delamination on the modal damping, natural frequency and mode shape of a composite beam was examined by Saravanos and Hopkins [11]. They used an analytical solution based on a generalized layerwise theory which involves kinematic assumptions representing the discontinuities in the in-plane and through-the-thickness displacements across each delamination crack.

Analytical solutions for beams with multiple delaminations have been presented by many researchers. Shu [12] presented an analytical solution to study a sandwich beam with double delaminations. His study emphasized on the influence of the contact mode, ‘free’ and ‘constrained’, between the delaminated layers and the local deformations at the delamination fronts. Lestari and Hanagud [9] studied a composite beam with multiple delaminations using the Euler–Bernoulli beam theory with piecewise-linear springs to simulate the ‘open’ and ‘closed’ behavior between the delaminated surfaces. Lee et al. [13] studied a composite beam with arbitrary lateral and longitudinal multiple delaminations by using the ‘free mode’ assumption and a constant curvature assumption at the multiple-delamination tip. Shu and Della [14], [15] and Della and Shu [16] used the ‘free mode’ and ‘constrained mode’ assumptions study a composite beam with various multiple delamination configurations. Their study emphasized on the influence of a second delamination on the first and second natural frequencies and the corresponding mode shapes of a delaminated beam.

Studies using numerical methods have also been presented, mostly using finite elements. One-dimensional finite element methods were presented by Ju et al. [17], Krawczuk et al. [18], and Chakraborty et al. [19] using the first-order shear deformation theory, and Lee [20] using the layerwise theory. Whereas, two-dimensional finite element methods were presented by Ju et al. [21] and Zak et al. [22], [23] using the first-order shear deformation theory, Chattopadhyay et al. [24], Radu and Chattopadhyay [25] and Hu et al. [26] using the higher-order deformation theory, Cho and Kim [27] using the higher-order zig-zag theory, and Kim et al. [28], [29] using the layerwise theory.

In the present research, the free vibration of a delaminated bimaterial beam is studied. The study focuses on the influence of the axial stiffness and bending stiffness of the delaminated layers on the natural frequency and mode shape of the beam. Two parameters are introduced, namely, the axial stiffness and bending stiffness ratios, which provide a better understanding on the influence of the axial stiffness and bending stiffness of the delaminated layers. Both the ‘free mode’ and ‘constrained mode’ models in delamination vibration are used. A monotonic relation between the natural frequencies and the stiffness ratios is observed. A new slenderness ratio specific for vibration is further introduced, which is shown to dominate the vibration characteristics of the beam. Global, mixed and local vibration modes occur depending upon the slenderness ratios of the delaminated beams.

The research is presented as follows: first, the analytical solutions are presented. Next, the vibration of delaminated bimaterial beam is studied using the axial stiffness and bending stiffness ratios. Finally, the dominance of the relative slenderness ratio on the vibration characteristics of the delaminated beam is shown.