Experimental,analytical,and finite element study of stress concentration factors for composite materials

Abstract

This article focuses on the evaluation of stress concentration factors for composite materials with varying hole diameters. An innovative method is presented for measuring the stress concentration around a hole using the digital image correlation method. This method can provide a quicker and cheaper testing technique. The digital image correlation method is utilized to evaluate the strain field and the displacement at the edge of the hole and obtain information about fracture mechanisms in the studied materials. The effect of the ratio of the hole diameter to its width (a/w ratio) on both gross and net stress concentration factors around the hole is investigated. In addition, there is a comparison between the net and the gross stress concentration factors uses of the digital image correlation method, the analytical solutions, and the finite-element analysis for both composite and steel materials.

Keywords

Stress concentration factor digital image correlation method composite materials finite element method plate with hole

Introduction

Composite materials are widely used in many industrial applications, such as automotive and safety parts. It received increasing attention in these applications because of their stiffness, toughness, light weight, and their resistance to tensile loading.

In fact, drilling holes in composite material cause damage around the hole edge, large stress concentration, and delamination at the entry and the exit of the hole.¹ The presence of holes in composite structures is considered as the potential reason for crack initiation and growth location.² This discontinuity in structures (hole, notch, etc.) leads to a high stress concentration around their edges. Also, the stress concentration is the site of initiation damage and it’s causing a wide range of effects, such as stress or strain gradients.

In the last decades, these discontinuities have been extensively studied using all the available research methods (numerical, analytical, and experimental). In this context, many analytical equations have been proposed. Pilkey³ computed the stresses in the neighborhood of a circular notch under tension loading. They put forward an empirical formula for evaluating the stress concentration factors (SCFs) around a circular hole in the centre of an isotropic plate. Lekhnitskii⁴ analyzed the SCF in orthotropic and isotropic materials for finite and infinite plates. Tan⁵ developed analytical equations for infinite-length and finite-width composite plates with both elliptical and circular shapes. Heywood⁶ proposed two equations for a finite isotropic and orthotropic specimen with different openings. It was shown that the presence of notch leads to a high stress concentration. Jones⁷ listed approximate values of stress concentrations around a circular hole in a variety of composite materials. They deduced that the effect of holes on the laminate behavior was much more complex than on a lamina or a plate behavior.

A numerical approach was carried out by Bin and Hwai-Chung.⁸ They suggested a general method for evaluating the stress concentrations of a composite material and isotropic plate with holes using the Finite Element method (FEM). They investigated that a constant value of SCF would be present when the ratio a/w was less than 0.5 and the SCF would tend to reach its real value when a/w was greater than 0.5. They defined two types of SCF: net and gross factors. Later, Mittal and Jain⁹ proved the effect of hole’s diameter on the gross SCF in laminated composite (orthotropic behavior) and isotropic specimens under different transverse loading conditions using the FEM. They deduced that for different loading conditions, an increasing value of a hole diameter would lead to increase the gross SCF in isotropic and orthotropic plates. In addition, Darwich¹⁰ put forward an empirical solution for stress concentration around holes in a symmetric laminated composite under uniaxial loading, in which it is updating the formulation given by Tan⁵ and compared with the FEM.

On the other hand, several works^11–15 have dealt with the problem of stress concentration using experimental techniques. Interferometry or non-interferometry techniques have been implemented in those investigations. In case of interferometry techniques, an electronic speckle pattern interferometry was used by Toubal¹² to determine the tensile strain field of a composite specimen containing a central opening. Galiotis¹³ proved with another experimental technique, called laser Raman spectroscopy that it could provide information for the fibers, stress, and strain at the microscopic level. However, these interferometric techniques are very sensitive to vibration and optics is quite involved. Accordingly, non-interferometry techniques, like the grid method by Mathias et al.¹⁴ and the digital image correlation (DIC) developed by Sutton¹⁵ have been widely used in the recent years.

The DIC method is easy to use, less sensitive to vibration, involves simple optics, reliable, and it can be applied to any class of material. Moreover, it is truly a whole field non-contact measurement method. For the above reasons, it is now becoming more and more popular and widely applied in a wide range of applications related to composite materials, including the observation of crack growth¹⁶ or the determination of residual stress measurement.¹⁷ It has also been successfully used to capture the stacking sequence effect in composites made with an epoxy matrix reinforced with natural fibers.¹⁸ Further, it is worth mentioning that the authors have successfully employed DIC techniques for measuring the strain arising and the field surface displacement at the hole level.^19,20 However, any previous studies have been applied to determine the stress concentration by the extrapolation of DIC strains to the hole boundary.

This article investigates the stress concentration in composite and steel materials with a circular hole using the DIC method and shows that this method is an original experimental work about non-contacting test to identify the stress concentration.

The second part of the article evaluates the net and gross SCF of orthotropic and isotropic materials around a circular hole. Different tests are realized with different hole diameters using the DIC and FEM, comparing them with the analytical method.

Experimental study

Specimen

CFRP plate

The plate has been made from woven ply laminates (CFRP) with 2 mm of total thickness and $[\pm 45 \deg] 2 s$ of sequences. The specimen has eight woven ply with a thickness of 0.22 mm for each ply under a fiber volume ratio $V f$ equals to 65%. The specimen dimension for each test is shown in Figure 1. Tensile test is performed as per ASTM D-3039 standard with 2 mm/min tensile speed at room temperature of 26℃. The length (L) and width (W) of plate are 250 and 25 mm, respectively. A circular hole is drilled at the center of the panel using a diamond drill. The specimens have been made by bonding the end of glass fiber/epoxy laminate legs. Figure 2 shows the plate specimens with different hole diameters.

To eliminate the error in the experimental study, the elastic coefficients for the CFRP are obtained with the DIC method (the relative stresses are derived by the experimental strains through the elastic coefficients). Material properties of the carbon/epoxy composite laminates are determined by tensile tests and performed as per ASTM standard D-3039 at room temperature. Five images per second are grabbed at displacement control rate of 2 mm/min. In-plane Young's modulus in longitudinal (

E xx

) and transverse direction (

E yy

) are obtained from initial slope of stress–strain curve. Unidirectional (0°) specimens are tested to evaluate longitudinal properties. Unidirectional (90°) specimens are tested to evaluate transverse properties. In-plane Poisson's ratio (

ν xy

) is obtained from lateral to linear strain plot witch determined from DIC for (0°) test specimen. The tensile properties of this laminate are given in Table 1.

Table 1.

Mechanical properties of CFRP ( $[\pm 45 \deg] 2 s$ ) obtained using the DIC technique.

		Woven ply laminates
Young’s modulus	E_xx	60 GPa
	E_yy	50 GPa
Poisson’s ratio	ν_xy	0.06
Shear modulus	G_xy	3.24 GPa
Total laminate thickness	e	2 mm

Steel plate

The steel plate has a constant rectangular cross-sectional dimension: 250 mm long × 25 mm wide ×2 mm thick test (The most current specimens to ASTM D3039). A typical speed for standard test samples is about 2 mm/min for each family of test. The mechanical properties of steel material are presented in Table 2.

Table 2.

Mechanical properties of 1045 steel.

		1045 steel
Young’s modulus	E	210 GPa
Poisson’s ratio	ν	0.3
Mass density	ρ	7800 kg/m³
Tensile strength	R_m	594 MPa

Experimental setup and procedure

The experimental setup comprising a DIC camera and EZ20 servo-hydraulic machine of 50 kN capacity is shown in Figure 3. The plate is properly aligned and fixed in hydraulic wedge grips. Uniform illumination of specimen surface is ensured by keeping a light source (30 watt capacity) on the side of the camera and positioned facing the test. The camera is then calibrated for its orientation and position using an appropriate calibration grid plate. It is connected to workstation laptop and five images per second are grabbed using Icasoft software.²² The features of the camera are illustrated in Table 3.

Table 3.

The features of the DIC camera.

Technique used: Single camera	Pixel to mm conversion: 12 pixels per mm
Resolution/sensor & digitization: 2448_2048 pixels²/8 bit.	Subset and step: subset size of 36–36 pixels² along with a step size of 7 pixels
Lens and imaging distance: 35 mm focal length/730 mm from the specimen.	Interpolation, shape functions, correlation criterion: gauge length of extensometer is selected as ROI for correlation
Camera noise (% of range): 0.02%	Method for extrapolation to edges: Bilinear extrapolation method (Max point).
Total number of images and capture speed: 5 image/s and 5 Hz	Image pre-smoothing: With image grabbing card.
ROI (pixels or mm): 240–480 pixels² which correspond to 20–40 mm	Strain information: 2-mm long virtual strain gauges.
Displacement resolution: 0.083 mm/pixel	Size of strain window: 2.8 pixels

Before the testing, the specimen is covered with a white paint, then a random speckle pattern over a specimen surface is obtained by spraying paint with a black spray having a nozzle of 0.5 mm diameter. All the tests are performed in displacement controlled mode at room temperature of 26℃ and a testing rate of 2 mm/min in accordance with ASTM standards.

To perform DIC experiment, we use a load–unload cycle to verify that our material remained in the linear elastic range. Then, we replicate three tests for one hole size and we choice the average, because it is very difficult to generate a suitable pattern with manual paint. This procedure was repeated for all the hole size to obtain correct displacement results and to eliminate the error due to the speckle pattern.

Tensile-test results

At room temperature, there have been two sets of tests (composite and steel plates) with the same plate size, the same loading rate and having different hole diameters (Figure 4). In the case of steel, we can see that the crack growth is perpendicular to the loading direction and is still at the edge of the hole, whereas the crack growth in composite material is around the hole but the orientation of the crack follows the direction of the fiber.

In our example, we have an orientation tissue of $[\pm 45 \deg] 2 s$ . Consequently, the crack growth is almost under an angle of 45° to the load direction.

In Figure 5, the monitoring of tensile test for the composite plate (with traction machine) gives three different zones. This evolution is carried out according to three phases:

A linear elastic phase characterized by an elastic module E.

Curve’s linearity loss and start of composite damage.

A brutal rupture of fibers and thus a total damage of the specimen.

In an experimental study, we can investigate the stress in function of the strain for the composite and steel plate with different hole diameters (Figures 6 and 7).

It can be concluded that the hole size has a very significant effect on the tensile strength of the plate. That, if the hole size increases the tensile strength decreases. Figure 8 shows this evolution of the tensile strength as a function of hole diameter for CFRP and steel material. It’s clear that for a holed plate specimen, the CFRP is more resisted than the steel material.

This observation is in good agreement with the previous results given by Salleh et al.,²³ in which they have found that the tensile strength declines when the size of hole increases, with losing more than 60% of the tensile strength when the hole was bigger than 8 mm. Figures 9 and 10 show the fracture of both composite and metal plates.

The SCF

Definition of the SCF

Theoretical background of the gross SCF

The theoretical coefficient of the gross SCF is defined according to Tan⁵ (the ratio of the maximum stress in the zone of discontinuity to the nominal stress in the section):

Kt = σ max / σ nom

(1)

where

σ max

is measured by the numerical methods like the analytical methods or by finite element method. For our study, we will calculate by another method like the DIC camera.

σ nom

is nominal or background stress, it is calculated without taking into account the hole by the following equation:

σ nom = \frac{F}{wh}

(2)

where F is the traction force, uniform on edges of the plate, h is the plate thickness, and w is the width of the plate.

Theoretical background of net SCF

According to Hwai-Chung,⁸ the theoretical net SCF is the application of the reference nominal stress across the net section in place of the whole width of the plate (gross area). $σ nom$ based on this definition will be defined as:

σ nom = \frac{F}{(w - a) h}

(3)

According to equations (1) and (2), the net SCF can be related by the $K tg$ with the following equation:

K tn = K tg (1 - a / w)

(4)

Analytical methods

Orthotropic material

For an infinite orthotropic plate (w ≫ a) with central hole, the coefficient $K_{t}^{\infty}$ is obtained by the following equation:⁴

K_{t}^{\infty} = 1 + \sqrt{2 (\sqrt{E 11 / E 22} - v 12) + E 22 / G 12)}

(5)

where

E 11

is the Young's modulus in longitudinal direction (along the x-axis) and

E 22

is the Young’s modulus in transverse direction (along the y-axis).

This study focuses on a plate made of woven ply laminates, supposed to have an orthotropic behavior. The mechanical properties of the composite material used in this study are illustrated in Table 1. $K_{t}^{\infty}$ is equal to 5.03 for our infinite composite plate.

Based on the results given by Tan⁵ on a finite composite plate containing central hole subjected to uniaxial load, the ratio $\frac{K_{tg}^{\infty}}{K tg}$ is introduced by:

\frac{K_{tg}^{\infty}}{K tg} = \frac{3 (1 - \frac{a}{w})}{2 + (1 - \frac{a}{w}) 3} + \frac{1}{2} (\frac{a}{w} M) 6 (K_{t}^{\infty} - 3) (1 - (\frac{a}{w} M) 2)

(6)

with

M 2 = \frac{\sqrt{1 - 8 [\frac{3 (1 - \frac{a}{w})}{2 + (1 - \frac{a}{w}) 3} - 1]} - 1}{2 (\frac{a}{w}) 2}

(7)

Isotropic material

In an isotopic material, E_xx = E_yy and G_xy = $\frac{E yy}{2 (1 + ν xy)}$ . The $K_{t}^{\infty}$ for infinite plate is equal to 3, and according to equations (6) and (7), the equation for isotropic material is simplified in the following form:

\frac{K_{tg}^{\infty}}{K tg} = \frac{3 (1 - \frac{a}{w})}{2 + (1 - \frac{a}{w}) 3}

(8)

DIC method

DIC is a full-field optical strain measurement technique that works by measuring the optical flow between the loaded and unloaded states of a specimen. It works by tracking the same points between two consecutive images (reference and deformed image) of the material specimen at different stage of its deformation. The basic principle of DIC is to search for maximum correlation between small square zones within ROI over the specimen surface in deformed and un-deformed state, as illustrated in Figure 11. The small square zone containing a set of pixels is called as subset and the distance between two subsets is defined as a step. The subset size determines the area of the image being traced between the successive images for displacement measurement and step size determines the number of pixels over which the subset should be shifted for estimating displacement. This displacement field can be differentiated to find the strain field. However, there are several parameters like step size, subset size, and strain window size, which can influence the accuracy of measurements.²⁴

In the directions of loading, the DIC technique uses random speckle patterns on the surface of a specimen to track the deformation via comparison of the original and distorted images by a pixel-by-pixel. It is very critical to generate suitable patterns for DIC to obtain correct displacement results. The speckle patterns for DIC are normally random gray scale patterns generated by spray painting. There are some general guidelines for generating ideal patterns.

First, the specimen surface is cleaned using isopropyl alcohol. Acrylic paint of titanium white color is applied over the specimen surface using an air brush. The white paint is allowed to dry for 1 h. Then, acrylic paint of carbon black color is applied over the specimen surface in a random fashion using the airbrush to get a random speckle pattern. An air pressure of 0.15 MPa is chosen at which adequate size and density of the black dots are obtained (An area of 100 mm² contains 130–150 black dots). The specimen containing the speckle pattern is shown in Figure 12 by using the standard software ICASOFT®.²² Strains are obtained from DIC system, interpolating all the strain values of the grid points located in the 40 mm × 20 mm specimen central area.

The DIC method is used to visualize the deformation processes with their numerical estimation in real time. Image correlation with a DIC camera is then used to determine the local stresses of the hole, for each point around it and at the level geometric discontinuity (Figures 13 and 14). Then, we can measure the evolution of the strain field (ɛ11) along the y-axis passing through the hole by a DIC camera and finite element analysis (FEA) (Figure 15).

For a steel material, the maximum strain is around the hole and exponentially decreases until reaching a minimum value in the edge of the plate. Whereas, for a composite material, the strain also increase at the hole edge and then fluctuates down, when getting near the edge, until having a minimum value. This phenomenon could be explained by fibers cracks mechanism around the hole. Although small differences in magnitude, ɛ_xx distribution from both FEA and DIC has a similar trend and relatively shows a close quantitative agreement.

In order to calculate the maximum stress in the zone of discontinuity, we can use the following stress tensor for planar structures:

[σ xx σ yy σ xy] = [Q 11 Q 12 0 Q 12 Q 22 000 Q 66] [ɛ xx ɛ yy 2 ɛ xy]

(9)

where

Q 11 = \frac{E xx}{1 - ν xy ν yx}

Q 22 = \frac{E yy}{1 - ν xy ν yx}

Q 12 = \frac{ν xy E yy}{1 - ν xy ν yx} = \frac{ν yx E xx}{1 - ν xy ν yx}

, and

Q 66 = G xy

Subsequently, when the deformation field ( $ɛ xx$ and $ɛ xx$ ) are obtained from the DIC method, we can calculate the maximum stress around the hole $(with ν yx = ν xy (\frac{E yy}{E xx}))$ :

σ xx = \frac{E xx ɛ xx}{[1 - ν_{xy}^{2} (E yy / E xx)]} + \frac{ν xy E yy ɛ yy}{[1 - ν_{xy}^{2} (E yy / E xx)]}

(10)

Accordingly, the stress concentration (Kt = $σ max$ / $σ nom with σ max = σ xx$ ) can be plotted with various ratios of the hole diameter to its width (a/w) for woven ply laminates and steel materials.

Finite elements method

A comparison is carried out using FEA and those given in the literature.^5,8 To validate this work, a simulation under “ABAQUS 6.12” software has been realized with a mesh-type Shell hexagonal, continuum, refined, and progressive around the hole (Figures 16 and 17). To reduce the computation time with a good accuracy, we meshed only a quarter of the plate with a refined mesh around the hole and larger elsewhere. The fineness meshing is optimized to have a constant value of maximum stress.

Results and comparisons

Variations of the SCF with different hole size (steel plate)

The results given by FE method (ABAQUS simulation) and a DIC camera are shown in Table 4.

Table 4.

Experimental data for isotropic material (10 KN).

a/w	Ktg (FEM)	Ktg (DIC)	Ktn (FEM)	Ktn (DIC)	Kt infinite
0.01	3.004	3.009	2.973	2.979	3
0.1	3.040	3.068	2.736	2.761	3
0.2	3.150	3.190	2.520	2.552	3
0.3	3.400	3.509	2.380	2.456	3
0.4	3.800	4.003	2.280	2.402	3
0.5	4.500	4.693	2.250	2.346	3

Using the equation (3), $K_{t}^{\infty}$ is equal to 3 for our infinite steel plate. Then, for finite steel plate, the Ktg increases when the hole size ratio (a/w) rises, whereas the Ktn decreases when the hole diameter grows up (Figures 18 and 19).

For the net SCF Ktn, it is found that the results obtained by the DIC camera are very coherent with the finite element results and the literature values. However, for the gross SCF Ktg, when a/w is higher than 0.4, a significant difference is noted comparing with the results given by DIC and FEA. Whereas, the accuracy of Tan’s equation⁵ is suitable for a/w < 0.4 but, with increasing a/w ratio, the accuracy decreases. Based on the experimental results, for a large hole diameters (a/w is higher than 0.4) in steel specimen, a plastic deformation can be occurred around the hole edge. However, the present FE analysis was carried out in the framework of elastic analysis. For this reason, a variation between the DIC and the numerical results is observed when a/w > 0.4.

Variation of SCF with different hole size (CFRP plate)

A comparison was made between the stress concentration calculated by the image correlation method and the FEM (Table 5).

Table 5.

Experimental data for composite material (10 KN).

a/w	Ktg (FEM)	Ktg (DIC)	Ktn (FEM)	Ktn (DIC)	Kt infinite
0.01	4.464	4.750	4.420	4.702	5.03
0.1	4.688	4.900	4.220	4.410	5.03
0.2	5.200	5.420	4.160	4.336	5.03
0.3	5.614	6.060	3.930	4.242	5.03
0.4	6.283	7.030	3.770	4.218	5.03
0.5	7.240	8.520	3.620	4.160	5.03

Using the equation (4), $K_{t}^{\infty}$ is equal to 5.03 for our infinite composite plate. It is shown that the gross SCF Ktg increases when the hole is near to the edges of the plate. Accordingly, the Ktg factor also follows the same trend compared with isotropic plate (Figures 20 and 21).

It’s clear that the Ktn factor declines and the Ktg factor increases when the hole size ratio (a/w) rises. The results of the FEM and the DIC camera are not very identical, because for the FEA, the composite is supposed to have an orthotropic behavior while actually it needs to study the behavior of the fiber and the matrix.

Indeed, we can note that both Ktn and Ktg factors have the same tends for steel and composite materials, but the transition length in composite materials is higher than that for a steel material.

Conclusion

In this work, the DIC method is used to determine the SCF for composite and steel materials. A comparison between the Net SCF and the Gross SCF uses the DIC method, the analytical solutions and the finite-element analysis is carried out. The following conclusions can be drawn:

The obtained results indicate that the stress concentration due to the presence of a geometrical discontinuity has a significant effect of composite and steel material. Moreover, we have seen that the evolution of the net and gross SCFs have the same tends for steel and composite materials, but the transition length in composite materials is higher than that for a steel material.

The result given by the DIC camera is practically identical with the FE results for isotropic materials. Contrariwise, for orthotropic materials, the results of the DIC method are not very coherent with the FEM and the literature values, which reflect on the anisotropic character of the composite material.

The present work has shown the potential of DIC for on-line monitoring of SCF for both composite and steel materials. This technique is revealed as an efficient and simple methodology to study possible damage around discontinuities, which implies that DIC method is able to make an approximate guess on the SCF to the hole boundary. It is still a challenge though to accurately identify internal damage.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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