Stress concentration around a central hole as affected by material nonlinearity in fibrous composite laminated plates subject to in-plane loading

1 Introduction

In engineering science, composite materials can be generally defined as any material that has been physically assembled from two or more materials to form one single bulk without chemical blending on atomic scale to form a material with new properties. The resulting material would still have components identifiable as the constituent of the different materials. One of the advantages of the composites is that two or more materials could be combined to take advantage of the good characteristics of each material. Composites are generally nonhomogeneous, and the resulting properties will be the combination of the properties of the constituent materials. The different types of loading may call on different components of the composite to take the load. This implies that the composite material properties may be different in tension and in compression as well as in bending. In addition, the direction of loading has a great effect on how material would respond to deformation shape and value.

A general review of laminated composite plates is presented by Zhang and Yang [1], where they listed many recent developments in utilizing the finite element method in various types of analysis such as buckling, post-buckling, free vibration, dynamics, failure, and damage of composite plates. The effect of geometric and material nonlinearities and large deformations is included in the survey. In conclusion, the authors suggested performing further research on the effect of material nonlinearity on the behavior of composite laminated plates, especially the effect of in-plane shear.

Zhen and Wanji [2] presented a single-layer higher-order model for predicting stresses at curved free boundaries of laminated composite plates subjected to in-plane loading and having circular holes. A three-node triangular finite element having 26 degrees of freedom per node was presented based on the methodology of the discrete Kirchhoff plate bending element. Numerical results showed that the proposed model is capable of predicting in-plane and interlaminar stresses around the circular hole.

Baltaci et al. [3] investigated the effect of hole size on buckling of laminated composite circular plates having circular holes and subjected to uniform radial load using the finite element method through static stability analysis. Karakuzu et al. [4] carried out failure analysis of laminated woven glass-vinylester composite plates with two parallel circular holes and subjected to in-plane traction forces. A parametric study considering hole size and hole location was performed numerically using the finite element method and verified experimentally. Iarve et al. [5] predicted stress redistribution due to damage near open holes in composite laminates using finite element analysis method combined with material property degradation model.

The distribution of stresses and deflections in rectangular isotropic, orthotropic, and laminated composite plates were studied by Jain and Mittal [6]. The plates have a central circular hole and were analyzed using finite element method under transverse static loading. The aim of the study was to investigate the effect of the ratio of the hole diameter to its width (D/b ratio) on stress concentration factor (SCF) and deflection for various plate boundary conditions. Han et al. [7] studied experimentally and analytically the effect of stitching reinforcement on the strain concentration of composite laminates containing a circular hole. There existed a notch-strengthening effect, which implied that the maximum strain at the hole edge became higher.

Jones [8] showed that the effect of holes on laminate behavior was much more complex than on a lamina or a plate behavior. In his study, it was concluded that the key factor in failure of isotropic plates with holes was the magnitude of the SCF. Gibson [9] showed that interlaminar stresses and delamination may occur at free edges and other discontinuities such as holes and joints.

The main objectives of the present study are to investigate the effect of a central hole on stress redistribution and SCFs in composite laminated plates subjected to in-plane axial forces at plate ends. The finite element computer program ANSYS^® [10] was utilized, accounting for material nonlinearity behavior using a new technique, where each layer of the laminate is defined as a new material type and then properly incorporated into the ANSYS as will be explained later in detail. Different plate dimensions and hole sizes are considered.

2 Description of the problem

Several examples of four-layered symmetric angle-ply laminated plates are presented, with various stacking sequences [+θ°/-θ°/-θ°/+θ°] or in short form as [±θ°]s. All layers have equal thicknesses of 2.5 mm and a total thickness of 10 mm for the whole laminate. Different fiber-orientation angles (θ) are considered: θ=0°, 30°, 45°, 60°, and 90°. At θ=0°, fibers are aligned unidirectionally with the geometric x-axis of the plate, which is also referred to as the axial load direction. The plate boundary conditions are set to be (U_x=0) at one end and applied uniform axial tension load at the other end. The circular hole has a diameter D, composite material fiber orientation θ°, and a plate width b=100 mm. The lamination details and geometric dimensions of the problem are as indicated in Figure 1.

Figure 1

Schematic representation showing laminated geometric dimensions.

3 Finite element analysis

An eight-node structural three-dimensional solid element (specified as Solid46 in the ANSYS package) with three degrees of freedom per node is used, making a total of 24 degrees of freedom per element. In order to construct the graphical image of various geometries of plate models for different a/b plate ratios and D/b hole ratios, plates were examined using the ANSYS finite element computer package. Mapped meshing is used for all models so that more elements are employed near the hole boundary. Mapped meshing technique divides any area of complex boundaries to subareas where each one of these subareas is bounded only by three or four lines (straight or curved), and meshes them after that. This allows a better control over meshing and reduces elements with poor aspect ratios. A number of checks and convergence tests were made to select the appropriate element type and mesh size. One convergence check is shown in Figure 2, where maximum Von Mises stress is plotted versus maximum element edge size for a [±θ°]_s 10-mm-thick square laminate with D/b=1/5. Convergence indicates that good results are expected when the element edge size is 2.5 mm or lower, which is equal to one lamina thickness. Hence, this element size will be used for the rest of this study.

Figure 2

Convergence of maximum Von Mises stress versus maximum element edge size.

4 Numerical results and discussion

Numerical results are presented for laminated composite plates with a central circular hole. Two important and widely known types of composite materials are used in this study as representative types of composite materials: graphite/epoxy (AS/3501) with the following material properties [9]: E₁=138 GPa, E₂=9 GPa, G₁₂=6.9 GPa, and ν₁₂=0.3 for linear material behavior study and boron/epoxy (Boron-Narmco 5505) for nonlinear material behavior. Material property data for boron/epoxy are as indicated in Table 1 [11, 12].

4.1 Linear material behavior

Different plate geometry ratios a/b=1.0, 1.5, 2.0, 2.5, and 3.0; various hole size ratios D/b=1/5, 2/5, and 3/5; and different stacking sequences [±θ°]_s were used. Seventy-five plate cases are considered in the present study. In general, the peak stress of each case becomes higher as the hole-size parameter is increased. The plate geometric dimensions influence the peak stresses around the hole, but the circumferential location of the peak stress is nearly the same for the different geometry ratios.

Defining the SCF as:

(1)SCF=σmaxσo (1)

where σ_max is the maximum in-plane tensile stress for the with hole case, while σ_o is the uniform in-plane tensile stress without hole case.

As shown in Figure 3, the SCFs, as defined in Eq. (1), reached the peak values of 10.27, 9.4, and 7.6 for fiber orientation angles θ=0°, 30°, and 45°, respectively, at a/b plate aspect ratio of 1.5. However, at θ=60° and 90°, the corresponding SCFs were about 10.25 and 7.5, respectively, at a/b ratio of 2. For higher plate aspect ratios, lower SCFs than those indicated in Figure 3 were obtained. For all the results indicated in Figure 3, the hole size ratio D/b of 3/5 was the dominant ratio to give the maximum SCFs for all a/b ratios.

Figure 3

Effect of SCF versus plate aspect ratio (a/b) at D/b=3/5 for graphite/epoxy material.

The effect of D/b ratio on the SCF in angle-ply laminated composite plates with a circular hole and subjected to a uniformly distributed loading is illustrated in Figure 4. At all fiber orientation angles, the effect on the SCF increased with increasing D/b ratio for graphite/epoxy material. The indicated SCFs correspond to the maximum SCF values for various a/b plate ratios of 1, 1.5, 2, 2.5, and 3.

Figure 4

Effect of maximum SCF versus D/b hole size ratio for graphite/epoxy material.

In Figures 5–9, the effect of fiber orientation angle (θ) on the SCF is illustrated for different D/b ratios of 1/5, 2/5, and 3/5. The angle-ply laminated plates were made of graphite/epoxy composite material with a central circular hole and subjected to in-plane uniformly distributed axial tension end loads. In general, the SCFs showed an increasing trend as θ is increased from θ=0° up to 90° with increasing D/b ratio for a/b plate ratios of 1, 1.5, and 2. For larger plate ratios (a/b=2.5 and 3), the behavior has changed, where D/b ratio of 2/5 gave higher values than those corresponding to 3/5, especially in the range of 25°<θ°<65°.

Figure 5

Effect of fiber orientation angle (θ) on SCF for a/b=1.0 using graphite/epoxy material.

Figure 6

Effect of fiber orientation angle (θ) on SCF for a/b=1.5 using graphite/epoxy material.

Figure 7

Effect of fiber orientation angle (θ) on SCF for a/b=2 using graphite/epoxy material.

Figure 8

Effect of fiber orientation angle (θ) on SCF for a/b=2.5 using graphite/epoxy material.

Figure 9

Effect of fiber orientation angle (θ) on SCF for a/b=3 using graphite/epoxy material.

4.2 Nonlinear material behavior

The nonlinear mechanical properties of the composite material are expressed as [11, 12]:

(2)Ei=Eo(1-Bi(U¯p))Ci+Di(U¯p)) (2)

where E_i and E_o are the secant and tangential mechanical property, respectively, and B_i, C_i, and D_i are material property constants for mechanical property i.

Three in-plane mechanical properties are expressed using Eq. (2): E₁, E₂, and G₁₂. The initial values of mechanical properties and the corresponding material constants are indicated in Table 1. The plastic strain energy term U¯p is normalized as:

(3)(U¯p)=UpUo=Us-UeUo. (3)

For in-plane stress case, the total secant strain energy density (U_s) may be expressed in terms of principal material directions (1) and (2) as:

(4)Us=12(σ12E1+σ22E2+τ122G12), (4)

where E₁, E₂, and G₁₂ are the secant moduli at each stress level.

Similarly, the initial elastic moduli E_o1, E_o2, and G_o12 can be used to obtain the elastic strain energy density, such that:

(5)Ue=12(σ12Eo1+σ22Eo2+τ122Go12). (5)

The plastic strain energy density (U_p) is obtained (as shown in Figure 10) as:

Figure 10

Typical nonlinear stress-strain curve for the ith mechanical property showing the strain energy density terms.

(6)Up=Us−Ue. (6)

The term (U_o) is used to normalize the plastic strain energy density, and it is usually taken equal to unity (i.e., 1 MPa).

The stress-strain relation (σ_x-ε_x) was predicted using the nonlinear material model that was originally developed by the first author [11]. A computer program MCOMP was developed for this purpose. In order to account for the nonlinear material effect, each layer with a specific orientation angle (θ) was defined as a new material type and then properly incorporated in the ANSYS computer program in the material property section. The stress-strain relation was dealt with as a multilinear fitting technique for the supplied stress-strain curve. The program utilizes the Newton-Raphson technique to achieve step convergence at each increment.

An alternative composite material is considered herein in order to compare linear with nonlinear behaviors for boron/epoxy Narmco 5505 material. Various a/b plate ratios of 1.0, 1.5, and 2.0 and D/b hole ratios of 1/5 and 3/5 are used in the analysis. A parametric study was carried out using different stacking sequences [±θ°]_s where θ=0°, 30°, 45°, 60°, and 90°. The effect of fiber orientation angle (θ) on the SCF is shown in Figures 11–13. For a square plate (a/b=1), the SCFs using linear analysis gave higher values than the corresponding nonlinear ones, except at θ=0°, the nonlinear value was 6.8 but greater than the linear one which was 5.8 for D/b ratio of 3/5, as illustrated in Figure 11, whereas the linear and nonlinear material behaviors gave almost the same highest SCF of 3.5 for all a/b plate ratios and at hole ratio D/b=1/5. For higher plate aspect ratios (a/b=1.5 and 2), the previous behavior was reversed, such that the SCFs resulting from using linear material behavior were higher than the corresponding values obtained from nonlinear material analysis for D/b ratio of 1/5, as shown in Figures 12 and 13, except at θ=90°. At this orientation angle and plate aspect ratio of 1.5, the SCFs were 5.91 and 5.38 for linear and nonlinear behaviors, respectively, for D/b=3/5, as shown in Figure 12. Moreover, the linear value was 7.05, whereas the corresponding nonlinear value was 6.23 for D/b=3/5 and for a/b plate ratio of 2.0, as shown in Figure 13.

Figure 11

Effect of fiber orientation angle (θ) on SCF for a/b=1 using boron/epoxy material.

Figure 12

Effect of fiber orientation angle (θ) on SCF for a/b=1.5 using boron/epoxy material.

Figure 13

Effect of fiber orientation angle (θ) on SCF for a/b=2 using boron/epoxy material.

5 Conclusions

1. The SCF in fiber-reinforced composite laminated plates with a central hole and subjected to in-plane axial uniform tensile loading depends greatly on fiber-orientation angle, where it could increase or decrease depending on the hole size and laminate aspect ratio.

2. The size of hole (D/b ratio) has a considerable effect on SCFs at the hole boundary. For all a/b plate aspect ratios, as D/b ratio increases the SCF increases, approaching very high values of 10 and 12 in certain cases of linear and nonlinear material analyses, respectively.

3. The problem of material nonlinearity has a great effect on the SCFs as affected by fiber orientation angles ranging between 0° and 90°. As the plate aspect ratio (a/b) and the hole size ratio (D/b) increase, the influence of nonlinear material behavior becomes more pronounced and has a significant effect on increasing the SCFs.