Determination of Stress Intensity Factor for Cracks in 1

Orthotropic Composite Materials using Digital Image Correlation 2

Gajanan P. Mogadpalli and Venkitanarayanan Parameswaran§ 3

Department of Mechanical Engineering 4

Indian Institute of Technology Kanpur, India 208 016 5

6

ABSTRACT: 7

This paper focuses on the application of the Digital Image Correlation (DIC) 8 technique to determine the stress intensity factor (SIF) for cracks in orthotropic composites. 9 DIC is a full field technique for measuring the surface displacements of a deforming object 10 and can be applied to any type of material. To determine the SIF from full field displacement 11 data, the asymptotic expansion of the crack-tip displacement field is required. In this paper 12 the expansion of the crack tip displacement field, is derived from an existing solution for 13 strain fields. Unidirectional fiber composite panels with an edge crack aligned along the fiber 14 were tested under remote tensile loading and the displacements were recorded using DIC. 15 The SIF was calculated from the experimental data by fitting the theoretical displacement 16 field using the least square method. The SIF thus determined was in good agreement with 17 theoretical results and therefore demonstrates the applicability of the derived displacement 18 field and DIC technique for studying fracture in composites. 19

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KEY WORDS: Digital Image Correlation, Stress Intensity Factor, Orthotropic materials, 21 Composites, Fracture 22

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31 § Corresponding author- venkit@iitk.ac.in, Phone: +91-512-2597528, Fax: +91-512-2597408 32

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Introduction 33

Due to their high strength to weight and stiffness to weight ratio, composite materials 34 are being used extensively in the making of automobile parts, aerospace structures and even 35 in construction and repair of buildings and bridges. Additional attributes like low thermal and 36 electrical conductivity makes them attractive replacement for many conventional metals. One 37 area in which these materials lag behind metals is the resistance to fracture. Most often the 38 thermosetting resins used as the matrix in fiber composites are inherently brittle and hence 39 susceptible to fracture during service. The orthotropic nature of composites adds complexity 40 to the analysis of these materials; however, there exist enough literature addressing the 41 analysis procedure for structures made of composites [1]. 42

According to linear elastic fracture mechanics (LEFM), the behavior of a crack in any 43 brittle material can be characterized completely by a single parameter known as the stress 44 intensity factor (SIF). The crack becomes unstable once a critical level of the SIF is reached 45 and hence the determination of the SIF for a given loading and geometry is very important in 46 the analysis of cracked structures. In unidirectional fiber composites, the behavior of cracks 47 aligned along the fiber can be very well characterized by the SIF [2]. Different experimental 48 techniques have been used in the past to determine the SIF for cracks in composites by 49 various researchers. 50

Shukla et al. [3] used strain gages installed near the crack tip to determine the SIF for 51 an edge cracked composite panel. They have shown that by judiciously placing the gage at an 52 appropriate angle to the crack, the SIF can be determined with good accuracy. In another 53 study, Liu et al. [4] have used the technique of Coherent Gradient Sensing (CGS) to 54 experimentally determine SIF for cracked composite panels under bending. CGS is a full 55 field technique, however, requires extensive surface preparation and elaborate optical setup. 56 Recently Khanna, et al. [5] used the technique of transmission photoelasticity to determine 57 the SIF for cracks in composites. They have prepared transparent composites by matching the 58 refractive index of the fibers and the matrix. However, the requirement of transparency of the 59 specimen may not allow high fiber volume fractions. 60

In the last couple of decades, the technique of Digital Image Correlation (DIC) has 61 become very popular for measuring the surface displacements of deforming objects [6-8]. 62 DIC is a full field technique, needs minimal surface preparation, is non-invasive and can be 63 applied to any type of material. McNeill et al. [9] was the first to use DIC to determine the 64 SIF for cracks in isotropic materials. Recently, Yonemaya et al. [10] have reported a detailed 65 study on the application of DIC to determine mixed mode SIF in homogeneous materials. Dai 66 et al. [11] have used DIC to study the displacement fields ahead of a stably growing crack in 67 fiber composites; however, their study was not aimed at determining the SIF. To the best of 68 our knowledge, the application of DIC to determine SIF for cracks in orthotropic composites 69 is not yet reported. 70

The present work explores the application of DIC to determine SIF for cracks in 71 orthotropic composites. The crack-tip displacement fields for an orthotropic material, 72 required for analyzing the experimental data, are derived from the existing strain fields. The 73 details of the preparation and characterization of the unidirectional fiber composite, required 74 for the study, are presented. The crack tip displacements for an edge crack oriented along the 75 fiber and loaded in tension are measured using DIC. The SIF is then calculated from the 76 experimental data using the derived displacement field. The experimentally determined SIF is 77 compared with the theoretical predictions by Bao et al. [12]. The details of the entire work 78 are presented sequentially in the following sections. 79

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Expansion of crack-tip displacement field for orthotropic composites 80

For determining the SIF from full field experimental data, which in this case would be 81 the displacements, the theoretical expansion of the crack-tip displacement field is required. 82 The complete expansion of the crack-tip displacement field for an orthotropic material is 83 developed in this section. Liu et al. [4] have given a thorough analysis of the deformation 84 field for cracks in orthotropic solids. They have also provided the general form of the 85 displacements with explicit expression for the leading term of the expansion. For accurate 86 estimation of the SIF from the experimental data, more than one term in the expansion has to 87 be used [10] and hence explicit expression for the higher order terms in the expansion are 88 required. In the present study these are developed from the strain field equations given by 89 Shukla et al. [3]. For the case of a crack aligned along one of the principal material axes in an 90 orthotropic material, and subjected to opening mode loading, the strain components can be 91 written as follows [3]. 92

93

( ) ( ){ } ( ) ( ){ }( ){ } ( ){ }

2 212 11 1 12 11 2

2 211 12 1 12 11 2

Re Re2 2

Re Re

xx a a Z a a Z

a a Y a a Y

α β α βε α β β α

α αβ βα β β αα α

− += − + + − −

+ + − + − −2 2

(1) 94

( ) ( ){ } ( ) ( ){ }( ){ } ( ){ }

2 222 12 1 22 12 2

2 212 22 1 22 12 2

Re Re2 2

Re Re

yy a a Z a a Z

a a Y a a Y

α β α βε α β β α

α αβ βα β β αα α

− += − + + − −

+ + − + − −2 2

(2) 95

( ){ } ( ) ( ){ }266 661 2 1 2Im Im Im Im

2xya aZ Z Y Yβγ α β β α β αα α

2= − − − + − −2

(3) 96

where 66 12 66 1222 22

11 11 11 11

2 22 ; 22 2

a a a aa aa a a a

β α2 2+ += + = − and 97

11 12 22 661 1 1, , and LT TL

L L T T LT

a a a aE E E E G

ν ν= = − = − = = 98

In equations (1-3), Z1, Z2, Y1 and Y2 are complex functions. The subscripts L and T 99 refer to the primary material axes and E, ν and G refer to the elastic modulus, Poisson’s ratio 100 and the shear modulus respectively. The reader is referred to [3] for the detailed derivation of 101 equations (1-3). For a finite plate with an edge crack, the complex functions Z1, Z2, Y1 and Y2 102 are represented by a series as follows [3]. 103

( ) ( )1 1

2 21 1 1 2 2 2

0 0

and N N

n nn n

n n

Z z A z Z z A z− −

= =

= =∑ ∑ (4) 104

( ) ( )1 1 1 2 2 20 0

and M M

m mm m

m m

Y z B z Y z B z= =

= =∑ ∑ (5) 105

where ( ) ( ) 21 1 2 2; ; 1i iz x i y re z x i y r e iθ θβ α β α1= + + = = + − = = − . In equation (4) 106

and (5), N and M are the number of terms required for accurate representation of the strain 107 field. In general, for points farther from the crack tip, the number of terms required in the 108 series for accurate representation of the strain field will be larger. 109

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The x-y coordinate reference has its origin at the crack tip with the x-axis aligned 110 along the crack. An and Bm are real constants and the constant A0 is related to the opening 111 mode SIF, (KI) as 0 2IK A π= . The strains are related to the displacement components, u 112 and v through the relations 113

; ;xx yy xyu v u vx y y x

ε ε γ∂ ∂ ∂ ∂= = = +

∂ ∂ ∂ ∂ (6) 114

By integrating equation (1) with respect to x and equation (2) with respect to y, we 115 will obtain the displacement components u and v as follows. 116

117

( ) ( ){ } ( ) ( ){ }( ){ } ( ){ } ( )

2 212 11 1 12 11 2

2 211 12 1 12 11 2

Re Re2 2

Re Re

u a a Z a a Z

a a Y a a Y g y

α β α βα β β α

α αβ βα β β αα α

− += − + + − −

+ + − + − − +2 2

(7) 118

( ) ( ){ } ( )( ) ( ){ } ( )

( ){ } ( )( ){ } ( ) ( )

2 21 222 12 22 12

2 21 212 22 22 12

Im Im2 2

Im Im

Z Zv a a a a

Y Ya a a a f x

α β α βα β β α

α β α α β α

β βα β β αα β α α β α

− += − + + − −

+ −

+ + − + − − +2 + 2 −

(8) 119

1 12 21 21 1 2 2

0 0

1 11 21 1 2 2

0 0

2 2( ) ; ( )2 1 2 1

1 1( ) ; ( )1 1

N Nn n

n nn nM M

m mm m

m m

Z z A z Z z A zn n

Y z B z Y z B zm m

+ +

= =

+ +

= =

= =+ +

= =+ +

∑ ∑

∑ ∑ (9) 120

121

The unknown functions of integration g(y) and f(x) can be determined by substituting 122 equations (7) and (8) in equation (6) and then equating the resulting expression for shear 123 strain to equation (3). After some simplifications (see Appendix) we can obtain g(y) and f(x) 124 as follows. 125

( )( )

1

2

f x Cx D

g y Cy D

= +

= − + (10) 126

In equation (10), D1 and D2 are the rigid body translations in the x and y directions 127 and C corresponds to the rigid body rotation in the x-y plane. If these are set to zero, equation 128 (7-9) give the complete expansion of the displacement field due to deformation alone. This 129 expansion will be used subsequently for extraction of SIF and the other higher order 130 constants An and Bm from the experimental data. 131

Experimental procedure 132

Preparation of the composite material 133 The unidirectional composite panels required for the study were prepared as follows. 134 Unidirectional prepregs consisting of E-glass fiber in epoxy (LY556) resin were prepared 135 first using an in-house prepreg making machine. Thirty prepregs were subsequently 136 laminated using a hydraulic press to make a unidirectional composite panel of size 200 x 200 137

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x 3.6 mm3. The fiber volume fraction of the laminate, determined as per ASTM D3171-06 138 was 0.62. Three tensile test specimens, conforming to ASTM standard D 3039M-00, were cut 139 from the panel such that the reinforcement directions are 0o, 90o and 45o with the loading 140 direction. Tensile tests were conducted using a 25 kN tensile testing machine to determine the 141 elastic constants. Strain gages were used to measure the strain. The longitudinal modulus, EL, 142 Poisson’s ratio, νLT, and the transverse modulus ET were determined by testing the 0o and 90o 143 specimens. By measuring E45 and using the transformation relations, the shear modulus GLT 144 was determined. The elastic properties of the composite are given in table 1. 145

After elastic characterization of the prepared composite, single edge notched tension 146 (SENT) specimens of the size shown in figure 1, were cut using a water cooled diamond saw. 147 After drying the specimen, an edge notch of width 250 µm and length 5 mm was cut into the 148 specimen using a diamond cutter. A natural crack was extended from this notch by lightly 149 tapping a sharp razor blade into the initially cut notch. By carefully tapping the blade the 150 crack was made to grow to the required size (see figure 1b) without causing any significant 151 secondary damage such as fiber de-bonding or delimitation. The length of the crack was 152 measured precisely using a traveling microscope equipped with a micrometer. During this 153 measurement a slight crack face opening load was applied by a thin wedge to facilitate the 154 locating of the crack tip. By this way the crack length could be measured to an accuracy of ± 155 0.1 mm. 156

A random dot pattern was applied to the specimen surface using black and white 157 spray paint. A photograph of the random dot pattern is given in figure 1 (c). Photographs of 158 this pattern were taken before loading and during loading for 50 N load increments up to a 159 load of 350 N. The load was measured using a 1 kN load cell. A region of 30 x 40 mm2 160 around the crack tip was photographed using a 2 Mega Pixel CCD camera through a zoom 161 lens. The digital photographs corresponding to each load was correlated with the photograph 162 of the unloaded specimen using the Vic2D DIC software (Correlation Solutions Inc., USA). 163 The principle of DIC is detailed in [6-8] and hence the same is not provided here. A subset 164 size of 34 x 34 pixels was used for the correlation. Proper care was taken to align the 165 horizontal axis of the camera with the crack line. The u and v displacement maps for a load 166 level of 300 N, shown in figure 2, indicate a smooth displacement field. The displacements 167 shown in figure 2 include rigid body components (translation and rotation) that arise during 168 the loading. 169

Extraction of stress intensity factor from displacement data 170

The experimental data comprising of the coordinates (x and y) of each point in the 171 imaged area along with the corresponding displacement components (ue and ve) has to be 172 analyzed using the displacement field derived in the earlier section (equations 7-10). In the 173 present study, the radial displacement ur was used to extract the SIF. From the experimentally 174 measured displacements, the radial component of the displacement ure was calculated for 175 each data point as 176

cos sinre e eu u vθ θ= + (11) 177

where θ, (−π < θ < π) is the angle made by the position vector of the data point with 178 the x-axis shown in figure 1. The experimentally measured displacements include rigid body 179 contributions also. Using equations (7-9) and considering the rigid body translations xT and 180

yT in x and y directions respectively, the theoretical displacement field can be written as 181

cos sin cos sinr x yu u v T Tθ θ θ θ= + + + (12) 182

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Note that in the polar coordinates, rigid body rotation in the x-y plane will not induce 183 any radial displacement [10] and hence does not appear in equation (12). Equation (12) has 184 (N+M+2) unknowns where N and M are the number of terms used in the series expansion of 185 the displacement (equation 7-9). The number of data points will be much more than the 186 number of unknowns. Therefore the unknowns (A0…AN, B0….BM, Tx, Ty) are determined by 187 fitting equation (12) to the experimental data (equation 11) data in a least square sense. 188 Though any of the displacement components u or v could be used for extraction of the SIF as 189 indicated by Yonemaya [10], ur was used in this study as it is not sensitive to rigid body 190 rotation of the specimen. 191

In the present study, the displacements were measured over a region of 30 x 40 mm2 192 around the crack-tip. One could use this entire data set in the analysis, however, this will add 193 to computation time. Instead, the data points sampled from an annular region of inner radius 194 R1 and outer radius R2 were used to extract the SIF. The inner radius R1 has to be larger than 195 half the plate thickness [3] to avoid three-dimensional effects and hence R1 was set to 2 mm. 196 The outer radius R2 was varied from 5 to 15 mm in steps of 2.5 mm and the effect on SIF was 197 investigated. For each value of R2, the SIF was determined by increasing the number of terms 198 in the series (equation 7 & 8) starting from 2 terms (N+M=2) as suggested by Yonemaya 199 [10]. This procedure was continued until the addition of further terms did not produce any 200 change in the SIF. As an additional confirmation, the displacement map was regenerated 201 using the constants (A0…AN, B0….BM, Tx, Ty) and was visually compared with the 202 experimental displacement maps to see the match between the two. 203

Results and Discussion 204

The variation of SIF as a function of number of terms in the series (N+M) for the 205 different values of R2 is shown in figure 3 for a load of 300 N. The continuous line in figure 3 206 is the SIF for an orthotropic SENT specimen calculated using equation (13) [12]. 207

3

2 3

( / ) ( )

2 tan( / ) 0.752 2.02 0.37 1 sin

cos

( ) 1 0.1( 1) 0.016( 1) 0.002( 1) ;2

L TLT TL

LT

K aF a W Y

W aa W a aF a W

a W WW

E EY

G

σ π ρ

ππ π

π

ρ ρ ρ ρ ρ ν ν

=

2 = + + − 2 2

= + − − − + − = −

(13) 208

where σ is the applied far field stress, a is the crack length and W is the specimen 209 width (see figure 1). It can be observed from figure 3 that, for each value of R2, the SIF 210 decreases with increasing number of terms and then stabilize after a certain number of terms. 211 However, even the converged value of SIF for R2=5 mm has an error of -12%. With 212 increasing R2 the error in the converged value of SIF decreases and the SIF approaches the 213 theoretical value. Higher values of R2 also require more number of terms for the SIF to 214 converge. Figure 4 shows the regenerated displacement field and the experimental 215 displacement field for the four different R2 values. It can be observed that the regenerated and 216 experimental displacement contours match very well in the case of R2=15 mm and not so well 217 for R2=5 mm. 218

The converged SIF for as a function of the applied load is shown in figure 5. There is 219 excellent agreement between the experimentally determined SIF and that calculated using 220

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equation 13 except for the case of 50 N and 350 N. In the case of 50 N the overall 221 displacement level is very small and hence the inherent resolution limitation of the DIC can 222 be the reason for the error of 9%. At 350 N, the SIF is close to 1.2 MPa-√m which is twice 223 the KIC

of epoxy. At this load level, damage in the material in the form of fiber de-bonding 224 might alleviate the crack tip severity. Thus the LEFM assumptions may not be justified for 225 this load case. For all other loads, the error is within 2% which is acceptable for all practical 226 purposes. The specimen failed in a brittle manner at a load of 380N. 227

Conclusions 228

The stress intensity factor for an edge crack along the fiber in unidirectional 229 composites was determined for far field tensile loading using the digital image correlation 230 technique. Complete expansion of the displacement field for cracks oriented along one of the 231 principal material axes in orthotropic composites was derived. Tests were conducted by 232 loading edge cracked composite panel in remote tension and the SIF was determined from the 233 displacement data using the derived displacement field. The SIF determined using DIC 234 measurements of the displacement field is in good agreement with available theoretical 235 results. The experimentally determined SIF was strongly sensitive to the number of terms in 236 the series expansion of the displacement field as well as to the extent of area from which the 237 date is sampled. The study indicated that up to 12 terms in the series expansion of the 238 displacement are required to evaluate the SIF with good accuracy. 239

Acknowledgements 240

The authors acknowledge the financial support of Aeronautical Research and 241 Development Board through the grant number DARO/08/1051282/M/I. 242

References 243

[1] Agarwal, B. D. and Broutman, L. J. (1990) Analysis and Performance of Fiber 244 Composites, John Wiley and Sons, New Jersey. 245

[2] Anderson, T. L. (1995) Fracture Mechanics-Fundamentals and Applications, CRC Press, 246 Boca Ralton. 247

[3] Shukla, A., Agarwal, B. D., and Bhushan, B. (1989) Determination of Stress Intensity 248 Factor in Orthotropic Composite Materials Using Strain Gages, Engineering Fracture 249 Mechanics, 32(3), 467-477. 250

[4] Liu, C. Rosakis, A. J., Ellis, R. W. and Stout, A. (1998) Study of the Fracture Behavior 251 of Unidirectional Fiber Reinforced Composites Using Coherent Gradient Sensing, 252 International Journal of Fracture, 90, 355-382. 253

[5] Khanna, S. K., Ellingsen, M. D. and Winter, R. M. (2004) Investigation of Fracture in a 254 Transparent Glass Fiber Reinforced Polymer Composites using Photoelasticity, Journal 255 of Engineering Materials and Technology, 126, 1–7. 256

[6] Peters, W. H. and Ranson, W. F. (1982) Digital Imaging Techniques in Experimental 257 Stress Analysis, Optical Engineering, 21(3), 427-431. 258

[7] Sutton, M.A., Wolaters, W.J., Peters, W.H., Ranson, W.F. and Mcneill, S.R. (1983) 259 Determination of Displacements using an Improved Digital Correlation Method, Image 260 and Vision Computing, 1 (3), 133-139. 261

[8] Bruck, H. A., McNeill, S. R., Sutton , M.A. and Peter, W. H. (1989) Digital Image 262 Correlation Using Newton–Raphson Method of Partial Differential Correlation, 263 Experimental Mechanics, 28 (3), 261-267 264

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[9] McNeill, S. R., Peters, W. H. and Sutton, M. A. (1987) Estimation of Stress Intensity 265 Factor by Digital Image Correlation, Engineering Fracture Mechanics, 28 (1), 101-112. 266

[10] Yoneyama, S. Y., Morimota, and Takashi, M. (2006) Automatic Evaluation of Mixed–267 mode Stress Intensity Factors Utilizing Digital Image Correlation., Strain, 42, 21-29. 268

[11] Jian Jun Dai, Xi Dong Liang, Xue Feng Yao, Yeh, H. Y. (2005), Study of Cracked 269 Unidirectional Glass Fiber-reinforced Composites by Digital Speckle Correlation 270 Method, Journal of Reinforced Plastics And Composites, 24 (16), 1737-46. 271

[12] Bao, G., Ho, S., Suo, Z. and Fan, B. (1992) The role of material orthotropy in fracture 272 specimens for composites, International Journal of Solids and Structures, 29, 1105–273 1116. 274

APPENDIX

Using equations 7 & 8 to determine the shear strain will give after algebraic simplification the following

2 266 22

11 12 12

2 266 22

11 12 22

66 2211 12 12

66 2211 12 22

( ) 2 Im2 2 ( )

( ) 2 Im2 2 ( )

( ) 2 Im2 2 ( )

( ) 2 Im2 2 ( )

a au v a a Zy x

a aa a Z

a aa a Y

a a df dga a Ydx dy

α β αβα β α

α β αβα β α

β β α αβα β α

β β α αβα β α

∂ ∂ −+ = + + − ∂ ∂ +

−− − + − −

− + + + − +

+ − − + − + + −

(A1)

6622 2211 12 11 122 22 2

( ) ( ) 2aa aa a a aαβ αβ

β α β α− + − = + − =

− + (A2)

Use of A2 in A1 and then equating to equation (3) gives 0df dgdx dy

+ = from which

follows equation (10)

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Table1. Elastic properties of the composite

Longitudinal modulus, EL (GPa) 39.8

Transverse modulus, ET (GPa) 17.7

Shear modulus, GLT (GPa) 5.0

Poisson’s ratio, νLT 0.29

Figure 1. Specimen geometry (a), natural crack (b) and random dot pattern (c) (S=130 mm, W= 30 mm, a/W=0.37)

Natural crack Notch

a =11.2 mm

(b)

x

y

W =30 mm

(c)

(a)

x

y

Fiber direction

θ r

S

W

a

11

x

y

x

y

Figure 2. Displacement maps obtained from DIC for an applied load of 300N

12

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

2 4 6 8 10 12 14 16 18 20

Numer of terms (N+M)

SIF

(MPa

-m1/

2 )

5.07.510.012.515.0

Figure 3. Variation of error and SIF with total number of terms for different data sets. (the broken line is the theoretical SIF from equation 13).

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(a) R2 = 5 mm, (M+N)=12 (b) R2 = 7.5 mm, (M+N)=14

(c) R2 = 12.5 mm, (M+N)=14 (d) R2 = 15 mm, (M+N)=18

Figure 4. Experimental contours of ur (solid lines) and contours regenerated using the minimum number of terms for SIF convergence (broken line) for different data sets. (Contour numbers are in µm).

14

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 50 100 150 200 250 300 350 400

Load (N)

K (M

Pa-m

1/2 )

Figure 5. Comparison of SIF obtained from DIC (circles) with that calculated using equation 13 (broken line).