Modal Characteristics of Composite Beams with Single Delamination- A Simple Analytical Technique

Modal Characteristics of Composite Beams with Single Delamination- A Simple Analytical Technique

K. Torabi^a, M. Shariati-Nia^a*, M. Heidari-Rarani^b

^a Department of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

^bDepartment of Mechanical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, 81746-73441, Iran

1. Introduction

Advanced composite materials are increasingly used in structural designs of aircraft, helicopters, spacecraft, automobiles, marine and submarine vehicles because of the desirable properties such as high strength and stiffness, lightweight, fatigue resistance, and damage tolerance, etc. However, composites are very sensitive to the anomalies induced during their fabrication or service life. One of the commonly encountered types of defects or damage in laminated composite structures is delamination. Delaminations may originate during fabrication or may be service-induced, such as by impact or fatigue loading. Delaminations not only affect the strength and integrity of the structure but also cause the reduction in the stiffness, thus affecting its vibration and stability characteristics. Reflections of these effects in dynamic response are the alteration of natural frequencies and damping ratios. As a result, considerable analytical, numerical and experimental efforts have been expended to capture these phenomena.

One of the earliest models for vibration analysis of composite beams including delaminations was proposed by Ramkumar et al. [1]. They modeled a beam with one through-thickness delamination simply using four Timoshenko beams connected at delamination edges. Natural frequencies and mode shapes were solved by a boundary eigenvalue problem. Using this model, the predicted natural frequencies were consistently lower than the results reported in experimental measurements. Authors attributed this discrepancy to the effect of contact with the delaminated "free" surfaces during vibrations. They suggest that the inclusion of the contact effect may improve the analytical prediction.

To study the effect of a through-thickness delamination on the free vibration of an isotropic beam, Wang et al. [2] present an analytical model using four Euler-Bernoulli beams that are joined together. They assum that the delaminated layers deform "freely" without touching each other ("free mode" model) and will have different transverse deformations.

Later, Mujumdar and Suryanarayan [3] presented two models namely the free mode model and the constrained mode model for the flexural vibrations of isotropic beams with delamination at the mid-plane as well as at the off-mid-plane locations. Experimental results have also been presented for various cases of delaminations in the beams. This paper concludes that the constrained mode model in which the transverse displacement and the normal stress of the upper and lower layer at the delaminated interface have been constrained to be the same, gives results in good agreement with the experimental results. The free mode model (similar to the one developed by Wang [2]) underestimates the frequencies, particularly at the higher modes.

The "constrained model" fails to explain the delamination opening modes found in experiments [4]. In these experiments conducted by Shen and Grady, the opening modes were even found in the first bending mode of the beam for some delamination cases. However, their finite element formulations (Model A and B) were essentially followed by the "constrained model" by Mujumdar and Suryanarayan [3] and the "free model" by Wang et al. [2]. In this paper the "Model A" is corresponding to the "constrained model" and the "Model B" is corresponding to the "free model". The discrepancy between the results predicted by the two models is significant even in cases where mode shapes do not show any opening in the delamination region. Furthermore, in some cases, opening delamination modes are shown clearly in their experiment, while the "constrained model" frequency prediction has a better match with the corresponding experimental results for these modes, even though the delamination cannot open using the "constrained model".

An analytical model based on the Timoshenko beam theory is presented by Hu and Hwu [5] for the free vibrations of delaminated sandwich beams. The natural frequencies and the mode shapes of the delaminated composite sandwich beams are presented in this paper. Lee [6] presents a displacement-based layer-wise finite element model for the analysis of free vibration of delaminated beam. In which the effects of the fiber angle, location, size and number of delamination are investigated numerically. A review paper on the vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures is presented by Zou et al. [7]. This paper deals with various models proposed for the free vibrations of delaminated beams.

In the analytical research done by Luo and Hanagud [8], a piecewise-linear spring model is used to simulate the behavior between delaminated surfaces. Shear and rotary inertia effects, as well as bending-extension coupling, are included in the governing equations. Frequencies and mode shapes are solved through a boundary eigenvalue problem. The proposed model includes the "free model" and the "constrained model" as special cases. The nonlinear response simulated by this model shows good agreement with the experiment results.

Karmakar et al. [9] studied the effect of delamination on free vibration characteristics of graphite-epoxy composite pre-twisted shallow shells of various stacking sequences considering length of delamination as a parameter. An exhaustive review on the vibration of delaminated composites has been presented by Della et al. [10]. The paper deals with various analytical models and numerical analysis for the free vibration of composite laminates. Della et al. [11] develop analytical solutions to study the free vibrations of multiple delaminated beams under axial compressive loadings. The Euler–Bernoulli beam theory and free mode and constrained mode assumptions in delamination buckling and vibration are used in the analysis. Ramtekkar [12] presented free vibration analysis of laminated beams with delamination using mixed finite element model. Analytical solutions for beams with multiple delaminations were presented by some researchers. Shu and Della [13, 14] and Della and Shu [15] used the "free mode" and "constrained mode" assumptions to study a composite beam with various multiple delamination configurations. Their study emphasized the influence of a second delamination on the first and second natural frequencies and the corresponding mode shapes of a delaminated beam. Shariati Nia et al. [16] presented an analytical method for calculating natural frequencies of a delaminated composite beam from both free and constrained mode frequencies. An improved combined natural frequency is proposed in this new formulation based on the breathing of delamination.

In this paper, at first, the delaminated beam is modeled as four interconnected Euler-Bernoulli beams. The free mode and constrained mode models are considered. Furthermore, the expressions for displacement functions are presented in a more convenient form based on concerning basic standard trigonometric and hyperbolic functions. The proposed technique considerably simplifies the calculation, and regardless of the number of delaminations, the problem can be solved by a 2×2 system of equations. Also, delaminated composite beams with different sizes and locations of delaminations are modeled in commercial finite element software, ABAQUS, and natural frequencies are extracted from the modal analysis.

2. Analytical Modelling of a Delaminated Composite Beam

Fig 1.(a) shows a cantilever beam with a single delamination at an arbitrary location and various lengths. L, h and a are beam length, beam thickness and delamination length, respectively.

The beam is separated along the interface by a delamination with length a located at the centre of the beam. The beam can be subdivided into three span-wise regions, a delamination region and two integral regions. The delamination region is comprised of two segments (delaminated layers), beam 2 and beam 3, which are joined at their ends to the integral segments, beam 1 and beam 4. Each of the four beams is treated as beams with different boundary conditions.

2.1. Free Mode Model

In "free mode" model, it is assumed that the delaminated layers deform "freely" without touching each other and have different transverse deformations. The governing equations for the free vibration of a delaminated beam using the Euler–Bernoulli beam theory are as the following:   

where D_i is the equivalent bending stiffness of the ith beam, m_i is the mass per unit length and is equal to ,  ρ_i is the mass density and A_i is the cross-sectional area of the beam. The bending stiffness for homogeneous and isotropic beams is given by D_i=EI_i, where E denotes the Young’s modulus and I is the moment of inertia. The mechanical properties of the composite beams are determined using the classical lamination theory (CLT) as follows:

where  is the extensional stiffness,  is the bending-extension coupling stiffness, and  is the bending stiffness of the ith beam [13].

For free vibration, the solution of Eq. (1) is given as follows:

where ω is the natural frequency and W_i is the mode shape. Substituting Eq. (3) for Eq. (1) and eliminating the trivial solution sin(ωt)=0, one can obtain the general solutions of the differential equation in Eq. (1) as follows:

where,

and λ_i are the non-dimensional frequencies. The lowest eigenvalue λ is the nondimensional primary frequency of the beam. The 16 unknown coefficients C_i, S_i, CH_i and SH_i are determined by four boundary conditions and twelve continuity conditions.

For cantilever beam B.C.'s at fixed end are as follows:

and at free end (x=L)

Continuity conditions at x=x₁

 (x=0)

where,

The second term on the left side of Eqs. (7-4) represents the contribution to the bending moment from the differential stretching between beams 2 and 3 and contributes to the bending stiffness of the beam [3, 13-15]. Similarly, the continuity conditions at the end tip of delamination (x=x₂) are derived. The boundary conditions and continuity conditions provide 16 homogeneous equations for 16 unknown coefficients. A non-trivial solution for the coefficients exists only when the determinant of the coefficient matrix vanishes. The frequencies and mode shapes can be obtained as eigenvalues and eigenvectors, respectively.

2.2. Constrained Mode Model

The "constrained mode" model is simplified by the assumption that the delaminated layers are in touch along their whole length all the time, but are allowed to slide over each other. Therefore, the delaminated layers have the same transverse deformations (w₃=w₂). This is reasonable since if there is no opening in the delamination region, the "free model" and "constrained model" are essentially the same. In this model the governing equations are as the following:

For beams 2 and 3

where,

The generalized solutions for the constrained model are identical in form to the free model. The unknown coefficients C_i, S_i, CH_i and SH_i, however, are reduced to twelve coefficients which can be determined by four boundary conditions and eight continuity conditions.

B.C.'s for cantilever beam at fixed end (x=0) and free end (x=L) are the same as Eqs. (6-1) and (6-2), but four continuity conditions at x=x₁ are as follows:

Similarly, four continuity conditions at the end tip of delamination (x=x₂) can be derived. These boundary and continuity conditions provide 16 homogeneous equations for 16 unknown coefficients.

According to these procedures, the characteristic equation of a beam with one delamination relies on the solution of a determinant of order 16 for free mode model and 12 for constrained mode model. Furthermore, this determinant order increases by varying the number of delamination and converting single delamination to multiple delaminations. This method is computationally expensive to analyze the presence of multiple delaminations in beams.

2.3. Basic Functions

The expressions for the displacements of beams as general solution of the differential equations in Eq. (9) can be written in a simple form based on the basic standard trigonometric and hyperbolic functions and in terms of just four factors: the displacement W₀, slop , bending moment M₀, and shear force V₀ at x=0.

where the functions of g_i(x) are expressed as follows:

2.3.1. Constrained Mode Model

According to basic functions expressed in Eq. (13) and the boundary conditions at fixed end of beam 1, transverse deformations of beam 1 can be written as follows:

Based on two continuity conditions at x=x₁ for displacement and slope, and according to Eqs. (11-1) and (11-2), deflection of beam 2 is as follows:

These unknown coefficients ( ) can be obtained using two other continuity conditions at x=x₁ for bending moment and shear force, which are expressed in Eq. (11-3) and (11-4) as follows:

Similarly, based on two continuity conditions at x=x₂ for displacement and slope, deflection of beam 4 is as the following:

Also, these unknown coefficients can be obtained from two other continuity conditions at x=x₂ for bending moment and shear force as follows:

Finally, the other two unknown coefficients (M₀, V₀), are obtained by using two equations from boundary conditions at free end of beam 4. Natural frequencies and mode shapes can be obtained by solving this simple system of equations.

The general solution proposed in this method satisfies the compatibility conditions at the delamination tips, and two of the factors are obtained directly from the type of support at x=0. The other two factors are obtained by the system of two equations from the boundary conditions at x=L. The technique used considerably simplifies the calculation, and regardless of the number of delaminations, the problem can be solved by a 2×2 system of equations.

2.4. Numerical Investigations

Numerical investigations are undertaken for the free vibrations of cross-ply laminated beams with embedded internal delaminations of different sizes and at several different locations. For this purpose, an eight-layer symmetric cross-ply laminated [0/90]_2s beam, under the clamped-free supports is considered. Thickness direction delamination locations are defined as mentioned in work of Shen and Grady [4]. "Interface 1" implies the mid-plane delamination, while "Interface 4" implies the skin ply delamination and so on. Lengthwise delamination locations are at the middle of beams. Fig. 2 shows interfaces and dimensions for a cross-ply delaminated beam.

3. Finite Element Modelling

The delamination is introduced as a debonding of adjoining plies in the laminated composite beam. The geometry of the delaminated beam is shown in Fig 1. The finite element modelling for simulating the delaminated beams and extracting the natural frequencies is conducted using the commercial finite element software ABAQUS. The composite beam is modeled as a cantilever with length (L) 127 mm, width (b) 12.7 mm and total thickness (h) 1.016 mm with delamination of length of 25.4, 50.8, 76.2, and 101.6 mm along one of the four ply interfaces shown in Fig 2. Nominal thickness of each ply is 0.127 mm and delamination dimensionless length (a/L) is 0.2, 0.4, 0.6, and 0.8. Material properties of unidirectional composite ply are presented in Table 1.

A shell element formulation is adopted since the beam is comparatively thin (h/b is less than 10 and h/L is less than 100). The composite laminates are modeled using a double layer of shells, with reduced integration and three integration points over the thickness (ABAQUS element type S4R, Simpson’s rule for shell section integration), and an orthotropic material definition is used. Modelling a delamination generally involves nonlinearities, due to the opening and closing of the delamination during cyclic deformation of the structure and friction between the two faces of the delamination. This requires explicit solvers and time-domain simulations.

At "free mode" model, there is no interaction defined between the surfaces of the delamination. Consequently, the elements on either side of the delamination deform freely and can separate and penetrate each other, effectively, causing a lower stiffness and absence of damping induced by contact during cyclic closing of the delamination. At "constrained mode" model, standard surface-to-surface contact interactions with normal and tangential behavior are introduced between the surfaces of the two sub-laminates, which allow neither penetration nor separation between the sub-laminate structures. Therefore, the delaminated layers have the same transverse deformations. Based on the model proposed and material properties, a total of 33 cases including an intact beam and 32 delaminated beams with different delamination lengths and different delamination interfaces are calculated. Natural frequencies are compared with the experimental results provided in Ref. [4].

4. Results and Discussion

The fundamental frequencies of a delaminated composite beam with various locations and sizes of delamination are presented in Tables 2-5. The experimental and analytical results related to model A (constrained model) and model B (free model) are provided from Ref. [4]. It is observed that the fundamental frequencies obtained from analytical and finite element method, closely match the experimental results of Ref. [4] for most of the cases.

Comparing constrained mode and free mode model, for delaminations located near the beam surfaces, free mode models have better agreement with experimental results. Figures 3 and 4 show the first three mode shapes for constrained and free mode models respectively. Delamination dimensionless length (a/L) is 0.4 and is located along interface 4. Blue elements show points with no deflection and refer to nodes of mode shapes. Furthermore, the second and third frequencies obtained from finite element analysis and analytical results from basic function formulation are presented in Tables 6-9. Comparing constrained mode and free mode model, and according to first three mode shapes, when delamination locates on the node of one mode shapes, the discrepancy decreases between related free mode and constrained mode frequencies.

Table 6. 2^nd and 3^rd natural frequencies (Hz) for analytical delamination along interface 1.

^a Without bending-extension coupling.

^b With bending-extension coupling.

Table 7. 2^nd and 3^rd natural frequencies (Hz) for delamination along interface 2.

^a Without bending-extension coupling.

^b With bending-extension coupling.

Table 8. 2^nd and 3^rd natural frequencies (Hz) for delamination along interface 3.

^a Without bending-extension coupling.

^b With bending-extension coupling.

Table 9. 2^nd and 3^rd natural frequencies (Hz) for delamination along interface 4.

^a Without bending-extension coupling.

^b With bending-extension coupling.