Dynamics of a generally layered composite beam with single delamination based on the shear deformation theory

2.1 Geometrical modeling

Figure 1 shows a laminated beam having a single through-width delamination. The composite beam has a length of L with a rectangular cross-section of b×h. The delamination dimension is L₂×b, and it is located at L₁ with respect to the left end of the LCB as shown in Figure 1. It should be noted that in this study, only a single delamination is considered. However, this study can easily be extended to include the multiple delaminations.

Figure 1

A schematic of generally LCB with single delamination.

In order to model geometrically this single delamination, consider Figure 2 in which the LCB can be viewed as a combination of four intact beams (i.e., four sub-beams of 1–4) connected at the delamination boundaries x=L₁ and x=L₁+L₂ and a common surface between sub-beams 2 and 3.

Figure 2

The delaminated beam is modeled by four interconnected sub-beams.

In this way, we will have four sub-beams of 1 to 4 with lengths and thicknesses of L_i×h_i(i=1 to 4), where L₂=L₃, L₄=L-L₁-L₂, h₁=h₄=h, and h₂ and h₃ are the thicknesses of sub-beams 2 and 3, respectively.

2.2 Kinetic and potential energies for each sub-beam

In this study, we consider an intact LCB comprising of an orthotropic material. The laminate is made of many unidirectional plies stacked up in different orientations with respect to a reference axis. The length, width, thickness, and number of layers of the intact LCB are represented by L, b, h, and n, respectively.

The laminated plate constitutive equations based on the first-order shear deformation theory can be expressed as [28]:

and

where

Aij=∑k=1nQ¯ijk(zk-zk-1), Bij=12∑k=1nQ¯ijk(zk2-zk-12),Dij=13∑k=1nQ¯ijk(zk3-zk-13) (i,j=1, 2, 6)Aij=∑k=1nksQ¯ijk(zk-zk-1) (i,j=4, 5)

In the above equations, N_x, N_y and N_xy are the in-plane forces, M_x and M_y are the bending, and M_xy is the twisting moments, Q_yz and Q_xz are the resultant shear forces, (ε_x, ε_y, ε_xy) are the midplane strains, κ_x and κ_y are the bending, and κ_xy is the twisting curvatures, γ_yz and γ_xz are the shear strains, A_ij, B_ij and D_ij (i, j=1, 2, 6) are the extensional, bending-extension coupling, and bending stiffnesses, respectively. Also, Q¯ij is the transformed material constant, k_s is the shear correction factor, and A_ij (i, j=4, 5) are the transverse shear stiffnesses.

Because of the free traction at y=±b/2 for a laminated beam, the membrane forces N_y and N_xy and the bending moment M_y are zero [29–31], and equation (1) can be rewritten as

in which

Using equation (1), the (ε_y, ε_xy, κ_y) components can be replaced by (ε_x, κ_x, κ_xy), and by substituting the results in equation (3), one obtains

where (A¯11, B¯11, etc.) are the coefficients of the matrix ([a]-[b][c]^-1[b]^T), and the matrix [c] is given by

[c]=[A22A26B22A26A66B26B22B26D22]

Using equation (2), the transverse shear force-strain relation for the LCB can be also expressed as [29–31]

The strain-displacement relationship can be written as [28]

Now, the potential energy U for the beam can be calculated using the following relationship [29, 31]

In the next step, we can express the potential energy in terms of displacement components using equations (5–7) as follows:

The above equation is used to obtain the stiffness matrices of the sub-beams 1 and 4. Moreover, referring to Figure 2, the interaction between the delaminated sub-beams 2 and 3 can be modeled as a distributed soft spring with a stiffness of k [19] to obtain the vibration characteristics based on the constrained mode. In this way, the potential energy for the sub-beams 2 and 3 can be expressed by inclusion of this soft spring in terms of displacements.

Next, we turn to the kinetic energy of the LCB using the following relation [29, 31]:

in which

(I1, I3)=∫-h/2h/2ρ(1, z2)dz

It should be mentioned that in all above relations, the symbol “,” used as a subscript stands for the differentiation with respect to any variable followed after it.

2.3 Element description

As shown in Figure 3, the beam element has three nodes, and each node has four degrees of freedom, namely, u_i, w_i, ψ_xi and ψ_yi for the ith node in which u and w are the LCB midplane displacements in the x and y directions, and ψ_x and ψ_y are the midplane bending slopes. The displacements u and w and the rotations ψ_x and ψ_y can thus be interpolated in terms of the intrinsic coordinate as

Figure 3

A three-noded beam element and its intrinsic coordinates.

where N_i(ξ), with i=1-3, which are the Lagrangian interpolation or shape functions associated with node i given by

where ξ denotes

ξ=x/L_e

and L_e is the element length of the beam. Thus, the vector of element degrees of freedom {δ} is given by

where superscript T denotes the transpose of a vector or a matrix.

The displacements and rotations of the beam can be related to the nodal degrees of freedom throughout the use of the shape functions to give

One of the efficient ways of deriving dynamic characteristics of a system using FEM is to employ the energy principle [32]. In implementing this method, one has to derive the kinetic and potential energies of the system that the detailed procedures are coming in the following sections.

2.4 Stiffness and mass matrices of the element

To obtain the stiffness matrix of the beam element, we start by substituting equations (14) in equation (9) to get the following expression for the stiffness matrix [32]:

in which the element stiffness matrix is given by

On the other hand, substituting from equations (14) into equation (10) and integrating over the element length gives the element mass matrix as

where the element mass matrix are:

2.5 Displacement continuity conditions

At the junction of the undelaminated segments (i.e., sub-beams 1 and 4) with delaminated segments (i.e., sub-beams 2 and 3), the displacement continuity conditions has to be satisfied. Consider the whole beam’s elements at the connecting nodes i, j, k, and l for sub-beams 1, 2, and 4 as shown in Figure 4. The overall element nodal displacement vectors and the stiffness and mass matrices of sub-beams 1–4 are {Δ}_i, [K]_i and [M]_i (i=1,2,3,4), respectively.

Figure 4

Nodes at the delamination boundaries.

At the connection nodes i-j and k-l, the displacement continuity conditions are as follows:

where e₂ is the distance between the midplanes of the sub-beams 1 and 2. The following transformation relations for the element stiffness and mass matrices can be established [32]:

Assuming the dimension c₁×c₁ for the stiffness and mass matrices of sub-beam 2 and based on equation (19), the transformation matrix T₁ has the dimension of c₁×c₁ and is given as

All T₁(i,j)=0 except

A similar treatment can be carried out for the connection nodes i-m and n-l, and the following transformation relations for the element stiffness and mass matrices can be established:

in which the transformation matrix T₂ has the dimension of the stiffness and mass matrices of sub-beam 3 (i.e., c₂×c₂) and is given by

All T₂(i,j)=0 except

in which e₃ is the distance between the midplanes of sub-beams 1 and 3.

The matrices [K¯]i and [M¯]i (i=2,3) are used to assemble the global stiffness and mass matrices.

4.1 Natural frequencies

Example 1: In order to show the accuracy of the presented method for a delaminated LCB, checking on the validity of the results is carried out. The beam is 266.7 mm long, 25.4 mm wide, and 1.778 mm thick in which the delamination is located at the midplane and has a length of 101.6 mm with L₁=117.5 mm. The eight-ply laminated beam with lay-up of [0/90/90/0]_s glass/epoxy is considered having material properties taken from [17].

The obtained results for the natural frequencies of the above beam under immovable clamped-free (I: C-F) boundary conditions (clamped at x=0) and free mode are given in Table 1. It should be mentioned that the axial displacement in the boundary of the beam for the immovable and movable boundaries is fixed and released, respectively. As one can see in this table, the obtained results for the first four natural frequencies are calculated and compared with some experimental and analytical results reported in the literature. The comparison of our results indicates very good agreement with other references.

To check further on the accuracy of the presented method, other cases are considered, which are analyzed in the following example.

Example 2: Consider a cantilever eight-ply LCBs with lay-up of [0/90]_2s and dimensions of 127×12.7×1.016 mm³. The beam is made of T300/934 graphite/epoxy with material properties adopted from [19]. In our following analysis, four different delamination lengths, namely, 25.4 mm, 50.8 mm, 76.2 mm, and 101.6 mm are considered one at the time in Figure 5.

Figure 5

Typical interface locations of the delamination for a [0/90]_2s graphite/epoxy composite beam.

The fundamental frequencies of centrally located delamination in a LCB with delamination positioned at the interfaces 1 to 4 one at the time are compared in Tables 2–5, respectively, both for the free mode and constrained mode. Good agreement is seen between the calculated by the present method, the experimental and analytical results of [7, 18, 19], and finite element results [34].

Table 6 shows the second bending frequencies (Mode 2) of the delaminated beam, with various delamination lengths located centrally. Again, good agreement is observed between the present results, the results obtained using finite element models based on the higher-order theory (HOT), 3D (NEi Software Inc., Westminster, USA) [20], and analytical results by [7]. These comparisons further validate the capability of the present method for reliable frequency prediction of the LCB.

Example 3: In this example and examples 4–6, the LCB material is taken as AS4/3501 graphite-epoxy, having the following mechanical properties [29]:

E₁₁=144.8 GPa, E₂₂=9.65 GPa, G₁₂=4.14 GPa, G₁₃=4.14 GPa, G₂₃=3.45 GPa, υ₁₂=0.33, ρ=1389.23 kg/m³

For all the problems, the width of the beam is taken as unity, and the thickness of each layer in the LCB is equal. Also, the calculated natural frequencies in these examples are presented in a dimensionless form (Ω=ω/E11H2ρL4) with the shear correction factor of k=5/6 [29], and the following are other non-dimensional parameters used in our analysis:

L¯1=L1L, L¯2=L2L,  h¯2=h2h

The first five normalized frequencies of an unsymmetric delaminated LCB [90/0] with slenderness ratio (L/h=15) under various boundary conditions referred to the constrained mode are presented in Table 7. In this calculations, a central delamination with L¯2=0.2 is considered. A close inspection of the results in this table reveals that the longitudinal vibration does not play a significant role on the vibration analysis in this case until the fourth and fifth modes (see Table 7).

Example 4: Consider an immovable cantilever angle-ply LCB with a central delamination located at the midplane, i.e., interface 1 and symmetric stacking sequence of [θ/-θ/θ/-θ]_s (see Figure 6). Figure 7 illustrates the variation of nondimensional frequency (ω_w0-ω_w)/ω₀ vs. L¯2 on the free mode (Figure 7A) and the constrained mode (Figure 7B) as the layout angle changes. In this figure, ω₀, ω_w0, and ω_w are the fundamental frequency of an intact beam, of the LCB without Poisson’s effect, and of the LCB with Poisson’s effect included, respectively.

Figure 6

Interface location of the delamination for a [θ/-θ/θ/-θ]_s graphite/epoxy composite laminate.

Figure 7

The influence of Poisson’s effect on the fundamental natural frequencies of the delaminated LCB (L/h=15). (A) Free mode. (B) Constrained mode.

For the free and constrained modes, the inclusion of the Poisson’s effect causes more pronounced increase on the fundamental frequency in the LCB as the delamination length L¯2 decreases. Moreover, from these figures as it is expected, the Poisson’s effect produces no significant changes on the fundamental frequency for the unidirectional (θ=0°) or cross-ply (θ=90°) LCB. However, the fundamental frequency for an angle-ply beam where Poisson’s effect is not considered deviates significantly from the exact value (i.e., considering Poisson’s effect), especially for the layout angle between 30° and 60°. Note that the maximum difference occurs at (θ=45°), and for both cases of free and constrained modes, this difference becomes 61.5%.

Example 5: Consider a delaminated beam as shown in Figure 6 with θ=10°. The first five fundamental frequencies of such beam based on the free mode under various boundary conditions with movable boundary are calculated and presented in Table 8. The beam has the slenderness ratio of L/h=15, and the central delamination is located at interface 1 with L¯2=0.2. A close inspection of the numbers given in this table reveals that retaining the longitudinal and torsional deformations in some cases will play a major role on the beam natural frequencies (the modes with predominance of longitudinal and torsional vibrations). Another important aspect is that the order of natural frequency modes changes when the boundary conditions change. For example, the first longitudinal mode shifts from the third natural frequency for clamped-clamped, clamped-hinged, and hinged-hinged boundary conditions to the fifth natural frequency for clamped-free boundary conditions. This is due to the change in the system stiffness caused by the change in the boundary conditions.

Example 6: The influence of the length and thicknesswise location of a central delamination on the normalized fundamental frequency of the immovable clamped-clamped and clamped-free beam comprising of four laminae with unsymmetric lay-up shown in Figure 8 is presented in Tables 9 and 10. The dimensionless lengths of the delaminations are 0.2, 0.4, 0.6, and 0.8.

Figure 8

A LCB with [0/45/0/45] lay-up and a central delamination.

Lower and upper bounds of the natural frequencies are obtained out of solutions for the free and constrained modes of the delaminated layers, respectively, as pointed out in section 3.

Based on the results given in this table, if this single delamination is located at interfaces 2 or 3, the natural frequencies of the beam under free and constrained modes become the same, that is, no “opening mode” is seen. But when the delamination is located at interface 1, the so-called “opening modes” may appear more easily even for short-length delamination. It should be mentioned that though the thicknesswise distance of interface 1 and 3 is the same from the free surfaces, the opening mode is only seen when the delamination is located at interface 1. This could be due to the closeness of stiffnesses of sub-beams [0°] and [45°/0°/45°] when the delamination is placed at interface 3. On the other hand, when the delamination is placed at interface 1, the difference between the stiffnesses of the clustered sub-beams 2 with [0°/45°/0°] lay-ups and sub-beam 3 with [45°] lay-up is relatively high.

Next, we would like to see the effects of length-to-thickness ratio (L/h) and delamination length on the dimensionless fundamental frequency of a LCB with movable clamped-hinged boundary conditions. In all related calculations, the central delamination is located at interface 1. Figure 9 shows the variation of dimensionless fundamental frequency vs. length-to-thickness ratio (L/h) for various L¯2. Referring to this figure, it is clear that for large values of L/h, the Ω₁ approaches to a constant value, and moreover, as L¯2 increases, the value of Ω₁ decreases. In addition, for the L/h<15, the fundamental frequency increases with much faster rate than for L/h>15.

Figure 9

Effect of length-to-thickness ratio on dimensionless fundamental natural frequencies of the delaminated LCB with various delamination lengths.

Now, consider a similar beam as presented in Figure 8 with the movable hinged-hinged boundary condition. In this case, again, we take a central delamination with L¯2=0.4 located at interface 1 with beam slenderness ratio of L/h=15.Figure 10 shows the effect of material anisotropy on the first two natural frequencies of the delaminated LCB. It is noted that the value of E₁₁ is varied, while the other elastic constants are kept unchanged and similar to those of graphite/epoxy material coefficients. For both the free mode and constrained mode, increasing the material anisotropy (E₁₁/E₂₂) causes the decrease in natural frequencies of the beam. Moreover, when the material anisotropy is increased, the difference between the frequencies based on the free and constrained modes get more pronounced; therefore, the delamination opening becomes more likely.

Figure 10

Effect of material anisotropy on the first two frequencies of the delaminated LCB with movable hinged-hinged boundary condition.

4.2 Mode shapes

The first mode shape of a thick LCB (L/h=10) with immovable clamped-free boundary conditions where the delamination staying in different thicknesswise location under the free mode (k=0) assumption is shown in Figures 11–13 for four different delamination lengths. Note that the lay-up configuration and thicknesswise location of the delamination shown in Figure 8 is also considered in this case. From these figures, we can see that the first mode of such beam does not show any opening in the cases where the delamination is located at interfaces 2 or 3, while in cases where the delamination is located at interface 1, except for (L¯2=0.2), we can observe clearly the delamination opening mode. The reason for such behavior is the closeness or farness of sub-beam stiffnesses as was discussed in the results given in Table 8. Indeed, the orientation of clustered sub-beams will play a significant role on this issue. To obtain the vibrational characteristics of the delaminated beam by considering the dynamic adhesion of sub-beams 2 and 3 is an interesting subject that is out of the scope of this study. This subject will be considered in our future works.

Figure 11

First mode shape of a LCB with delamination located at interface 1 related to four different lengths of delamination.

Figure 12

First mode shape of a LCB with delamination located at interface 2 related to four different lengths of delamination.

Figure 13

First mode shape of a LCB with delamination located at interface 3 related to four different lengths of delamination.