Composite materials consist of various fibrous reinforcements coupled with a compatible matrix to achieve superior structural performance. The selection of composite materials for specific applications is generally determined by the physical and mechanical properties of the materials, evaluated for both function and fabrication [1]. Knowledge of stress distributions in anisotropic materials is very important for proper use of these new high-performance materials in structural applications [2].

Kuwamura [3] analyzed the stresses around the circular cutout by means of Ikeda’s formula in consideration of orthotropic elasticity of the woods, which revealed that the stresses distribution around the circular hole is nearly equal to that of an isotropic plate. Selivanov [4] studied the time variation in the stresses around an elliptic hole in a composite plate. Kumar et al. [5] presented an approximate solution in the form of a polynomial for the normal stress distribution adjacent to a class of optimum holes in symmetrically laminated infinite composite plates under uniaxial loading. Giare and Shabahang [6] used a finite element analysis to calculate the stress distribution around a hole in a finite isotropic plate reinforced by composite material. Tsai et al. [7] developed a novel procedure for predicting the notched strengths of composite plates each with a center hole. And the stress distribution of a composite plate with a center hole is obtained by a finite element analysis.

In this paper, the effect of lamina material properties on the stress distribution of the orthotropic composite plate containing a circular cutout under hydrostatic tension is investigated. Based on the generalized Hooke’s law, the generalized plane stress in an orthotropic composite plate and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E₁, E₂, G₁₂ and ν₁₂ on the stress distribution along the boundary of the circular cutout in the composite plate under hydrostatic tension.

Tangential Stresses on the Boundary of the Circular Opening

The stress-strain distribution of an infinite anisotropic plate containing a through-the-thickness cutout has been derived using a complex variable method [8-10]. For an orthotropic plate subjected to equal tensile stress p in two principal directions which are applied at a considerable distance from the circular opening as shown in Figure 1 (and this is equivalent to hydrostatic tension in plane xy). The normal stress component σ_θ for an element tangential to the opening is [9]

Figure 1: An orthotropic plate containing a circular hole under hydrostatictension with principal material axes 1-2 and x-y axes.

(1)

where

(2)

(3)

The θ is the polar angle measured from the x-axis; E_θ is the Young’s modulus in tension (compression) along the direction tangent to the opening contour, which is related to elastic constants in the principal directions by the formula

(4)

For an isotropic plate [9]

(5)

Consider the orthotropic plate containing a circular hole under hydrostatic tension. For a dimensionless analysis, the three lamina material constants E₁, E₂ and G₁₂ are represented by two dimensionless ratios E₂/E₁ and G₁₂/E₁. Totally, the lamina material has three parameters E₂/E₁, G₁₂/E₁ and ν₁₂. For an unidirectional lamina with different values of E₂/E₁, G₁₂/E₁ and ν₁₂, the normal stress component σ_θ of the orthotropic plate under the hydrostatic tension will be varied and provided different values.

Various cases of different combinations of the three material parameters are considered in this study. In general case, the ranges of E₂/E₁ and G₁₂/E₁ are between zero and one, the ranges of ν₁₂ are between 0 and 0.6. For a lamina with given material properties E₂/E₁, G₁₂/E₁ and ν₁₂, the normal stress component σ_θ of the orthotropic plate under the hydrostatic tension can be evaluated.

In the (Figures 2-4), the bold solid line represents the circular hole and the bold dotted line shows the stress distribution σ_θ in an isotropic plate subjected to identical load as in an orthotropic plate. The stress distribution for an orthotropic plate is symmetrical with respect to the opening center. Thus, the variation of the stress distribution θ is considered only in the range of 0° ≤ θ ≤ 180° for an orthotropic plate. In an isotropic plate given p=100 MPa, then calculated from Equation (5), the constant stress σ_θ is 200 MPa. Therefore, the stress concentration factor in an isotropic plate is S_c=2.0. It is well known that the stress concentration factor for a circular cutout in the isotropic thin plate under uni-axial tensile load is “three”. However, in this study, tensile stresses in two perpendicular directions equal to p are applied in the thin plate with a circular cutout, therefore, through the linear superposition, it is found the stress concentration factor for this isotropic plate is S_c=2.0.

Figure 2: The normal stress component σ_θvs. θ and ν₁₂ for p=100MPa,E₁=204GPa, E₂/E₁=0.6, and G₁₂/E₁=0.4.

Figure 3: The normal stress component σ_θvs. θ and G₁₂/E₁ for p=100 MPa,E₁=204 GPa, E₂/E₁=0.6, and ν₁₂=0.4.

Figure 4: The normal stress component σ_θvs. θ and E₂/E₁ for p=100 MPa,E₁=204 GPa, G₁₂/E₁=0.6, and ν₁₂=0.4.

In Table 1 given p=100 MPa, E₁=204 GPa, E₂/E₁=0.6 and G₁₂/ E₁=0.4, the variation of the stress σ_θ with respect to the polar angle θ and ν₁₂ is the following:

Table 1: The normal stress component σ_θvs. θ and ν₁₂ for p=100 MPa, E₁=204 GPa, E₂/E₁=0.6, and G₁₂/E₁=0.4.

With ν₁₂=0.2, 0.4, and 0.6, the minimum corresponding stresses σ_θ are 187.28 MPa, 177.83 MPa, and 167.93 MPa at θ=90° respectively, and the maximum stresses σ_θ are 211.62 MPa, 221.17 MPa, and 232.14 MPa at θ=50° as well as θ=130° respectively. Also, the stress concentration factors are S_c=2.12, 2.21, and 2.32 respectively, for orthotropic case.

Table 1 shows that within the ranges of the polar angle 0° ≤ θ ≤ 20°, 70° θ ≤ 110°, and 160° ≤ θ ≤ 180°, the stress σ_θ decreases along with the increased values of ν₁₂. But, within the ranges of the polar angle 30° ≤ θ ≤ 60° and 120° ≤ θ ≤ 150°, the stress θ increases along with the increased values of ν₁₂. A summary of Table 1 is shown in Figure 2.

In Table 2 given p=100 MPa, E₁=204 GPa, E₂/E₁=0.6 and ν₁₂=0.4, the variation of the stress σ_θ with respect to the polar angle θ and G₁₂/ E₁ is following:

Table 2: The normal stress component σ_θvs. θ and G₁₂/E₁ for p=100 MPa,E₁=204 GPa, E₂/E₁=0.6, and ν₁₂=0.4.

With G₁₂/E₁=0.2, 0.4, 0.6, 0.8 and 1.0, the minimum corresponding stresses σ_θ are 176.16 MPa at θ=50° as well as =130°, and other minimum corresponding stresses σ_θ are 177.83 MPa, 156.61 MPa, 145.03 MPa and 137.69 MPa at θ=90° respectively, and the maximum stresses σ_θ are 231.32 MPa at θ=90°, 221.17 MPa, 246.09 MPa, 262.21 MPa and 273.54 MPa at θ=50° as well as θ=130° respectively. Also, the stress concentration factors are Sc=2.31, 2.21, 2.46, 2.62, and 2.73 respectively, for orthotropic case.

Table 2 shows that within the ranges of the polar angle 0° ≤ θ ≤ 20°, 70° ≤ θ ≤ 110° and 160° ≤ θ180° the stress σ decreases along with the increased values of G₁₂/E₁. But, within the ranges of the polar angle 30° ≤ θ ≤ 60° and 120° ≤ θ 150°, the stress σ_θ increases along with the increased values of G₁₂/E₁. A summary of Table 2 is shown in Figure 3.

In Table 3 given p=100 MPa, E₁=204 GPa, G₁₂/E₁=0.6 and ν₁₂=0.4, the variation of the stress σ_θ with respect to the polar angle θ and E₂/E₁ is following:

Table 3: The normal stress component σ_θvs. θ and E₂/E₁ for p=100 MPa,E₁=204 GPa, G₁₂/E₁=0.6, and ν₁₂=0.4.

With E₂/E₁=0.2, 0.4, 0.6, 0.8 and 1.0, the minimum corresponding stresses σ_θ are 107.45 MPa, 142.61 MPa, 156.61 MPa 164.34 MPa, and 169.31 MPa at θ=90° respectively, and the maximum stresses σ are 277.49 MPa, 257.60 MPa, 246.09 MPa, 238.66 MPa and 233.47 MPa at θ=50° respectively. Moreover, for E₂/E₁=1.0, the minimum stress σ occurred at θ=0° as well as θ=180° and the maximum stress σ_θ occurred at θ=40°, θ=130° and θ=140° as well. Also, the stress concentration factors are Sc=2.77, 2.58, 2.46, 2.39, and 2.33 respectively, for orthotropic case.

Table 3 shows that within the ranges of the polar angle 0° ≤ θ ≤ 30°, 80° ≤ θ 100° and 150° ≤ θ ≤ 180°, the stress σ_θ increases along with the increased values of E₂/E₁ respectively. But, in the ranges of the polar angle 40° ≤ θ ≤ 60° and 120° ≤ θ ≤ 140°, the stress σ_θ decreases along with the increased values of E₂/E₁. As for the polar angles θ=70° as well as =110°, with the values of 0.2 ≤ E₂/E₁ ≤ 0.6, the stress σ_θ decreases along with the increased values of E₂/E₁, but for the values of 0.6 ≤ E₂/ E₁ ≤ 1.0, the stress σ_θ increases along with the increased values of E₂/E₁. A summary of Table 3 is shown in Figure 4.

The effect of lamina material properties on the stress distribution of the orthotropic composite plate containing a circular cutout under hydrostatic tension is presented.

First of all, given fixed material parameters for E₁=204 GPa, E₂/ E₁=0.6, and G₁₂/E₁=0.4 Table 1 indicates that for the values of 0.2 ≤ ν₁₂ ≤ 0.6, within the ranges of the polar angle 0°≤ θ ≤ 50° and 90° ≤ θ ≤ 130°, the stress σ increases with respect to the increased values of the polar angle θ, but within the ranges of the polar angle 50° ≤ θ ≤ 90° and 130° ≤ θ ≤ 180 the stress σ_θ decreases with the increased values of the polar angle θ.

Second, given fixed material parameters for E₁=204 GPa, E₂/E₁=0.6, and ν₁₂=0.4 Table 2 indicates that for the value of G₁₂/E₁=0.2, within the ranges of the polar angle 0° ≤ θ ≤ 50° and 90° ≤ θ ≤ 130°, the stress σ_θ decreases with respect to the increased values of the polar angle θ, but within the ranges of the polar angle 50° ≤ θ ≤ 90° and 130° ≤ θ 180° the stress σ_θ increases along with the increased values of the polar angle θ.

As for the values of 0.4 ≤ G₁₂/E₁ ≤ 1.0, within the ranges of the polar angle 0° ≤ θ 50° and 90° ≤ θ ≤ 130°, the stress σ_θ increases with the increased values of polar angle θ, but within the ranges of the polar angle 50° ≤ θ ≤ 90° and 130° ≤ θ 180° the stress σ_θ decreases with respect to the increased values of the polar angle θ.

Third, given fixed material parameters for E₁=204 GPa, G₁₂/E₁=0.6, and ν₁₂=0.4 Table 3 indicates that for the values of 0.2 ≤ E₂/E₁ ≤ 1.0, within the ranges of the polar angle 0° ≤ θ ≤ 50 and 90° ≤ θ ≤ 130° the stress σ_θ increases with respect to the increased values of the polar angle θ, but within the ranges of the polar angle 50° ≤ θ ≤ 90° and 130° ≤ θ ≤ 180 the stress σ_θ decreases along with the increased values of the polar angle θ.

Fourth, it is well known that for fiber reinforced composite laminated plates, the Poisson’s ratio can be negative values. However, this case is not discussed/analyzed in the present study; such investigation will be included in the future research works.

The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high precision requirements in the engineering applications.