Evaluation of stress intensity factor for cracks in graded materials using digital image correlation

Evaluation of stress intensity factor for cracks in graded materials using digital image correlation Amit Kumar and Venkitanarayanan Parameswaran * Department of Mechanical Engineering Indian Institute of Technology Kanpur, Kanpur ABSTRACT This study focuses on the determination of stress intensity factor (SIF) for edge cracks in a functionally graded plate using the technique of Digital Image Correlation (DIC). Graded composites having nearly exponential variation of elastic modulus were prepared by gravity casting technique. The elastic modulus profile of the prepared material was characterized. The material had an elastic gradient of two times across a width of 50 mm. Edge cracked plates having different crack lengths were loaded in bending and the displacements were measured using DIC. The SIF was extracted by fitting the theoretical crack-tip displacement field available for homogeneous materials to the experimental data. The experimentally determined SIF for edge cracks subjected to four-point bending agreed well with published theoretical results. For cracks subjected to three-point bending, the results indicated that the SIF is higher than that for an identical crack in a homogeneous material when the crack is located in the stiff side of the graded plate whereas it is lower than that for homogeneous materials when the crack is located in the compliant side of the plate. Keywords: Digital image correlation, Functionally graded material, Stress intensity factor 1. INTRODUCTION Functionally graded materials (FGMs) are a special class of composite materials in which the composition and microstructure is not uniform in space, but varies in a gradual progressive manner. These materials were first introduced as high performance thermal barrier coatings, but they have much wider scope for application in many fields; for example aerospace, microelectronics and biomedical engineering. FGMs owe their success to two main features: the gradual change in composition allows better exploitation of the constituent phase properties and unlike traditional bimaterials junctions, FGMs avoid the abrupt coupling of heterogeneous phases. A gradual transition in material properties has been observed to alleviate the problem of thermal cracking, particularly in thermal protection systems. Moreover the composition gradient can be tailored to suite the application requirements, even when the loading conditions differ from point to point. FGMs are now being regarded as one of the most promising candidates for future multi-functional composites in many engineering sectors. Since the introduction of the FGM concept, the design and analysis of this material has received considerable attention. An FGM used for thermal protection is most likely to have a gradual transition from pure metal to pure ceramic and will combine the high temperature properties of the ceramic with the ductile behavior of the metal. However ceramic being brittle in nature, the system is fracture prone. In a layered metal-ceramic system, a crack present in the brittle layer upon reaching the interface will most likely deflect along the weak interface leaving the second layer intact. In a continuous transition without interfaces as in an FGM, the crack could propagate through the entire thickness causing complete failure of the system. Hence understanding the behavior of cracks in a graded material is important to evaluate the integrity of structures made of graded materials. Recent years have witnessed many investigations on the fracture mechanics of graded materials. These studies have unanimously established the inverse square root nature of stresses very close to the crack tip in graded materials; however, the stress intensity factor is influenced by the elastic gradients [1]. The theoretical values of stress intensity factor for embedded and edge cracks in infinite FGM plates are available [2]. While the theoretical study of fracture of * Further author information: (Send correspondence to Venkitanarayanan Parameswaran) Venkitanarayanan Parameswaran.: E-mail: venkit@iitk.ac.in, Telephone: 1 512 259 7528

FGMs is well established, experimental investigations are very few. FGM plates have been manufactured using epoxy and glass beads and their fracture behavior has been investigated by Butcher et al. [3]. Lambros et al. have experimentally investigated crack growth in polymeric FGMs having a non-linear constitutive response using the technique of digital image correlation [4]. The present study focuses on the application of the technique of Digital Image Correlation (DIC) to evaluate the stress intensity factor (SIF) for edge cracks in FGMs. DIC is a full field technique for measuring the surface displacements of deforming objects [5, 6]. DIC needs minimal surface preparation, is non-invasive and can be applied to any type of material. In the present study, FGMs with near exponential variation of elastic modulus were prepared and the displacement field around edge cracks loaded in bending was measured using DIC. The SIF was determined by fitting the theoretical displacement fields to the experimental data. The results for edge cracks subjected to pure bending are compared with published theoretical results. Further the SIF for edge cracks subjected to three point bending is also estimated. The details of the FGM preparation, characterization of the modulus profile, experimental setup and analysis of the experimental data to determine the SIF are presented in the following sections. 2. PREPARATION AND CHARACTERIZATION OF GRADED MATERIAL The graded material required for the study was prepared in the form of a particulate composite with continuously varying particle volume fraction. The matrix material used was an epoxy system consisting of the Araldite, LY556 and Hardener HY951, both supplied by Vantico Performance Polymers Pvt. Ltd., India. Glass beads of size 60-100 µm supplied by Hindustan Glass Beads were used as the reinforcement. Adding rigid particles to the resin will improve the elastic modulus of the composite. In order to characterize this effect, homogeneous composites with glass bead volume fraction varying from 0% to 50% in steps of 10% were first prepared and their elastic properties were measured as per ASTM D 638. The elastic modulus of the composite with 50% glass beads was three times that of pure epoxy. FGMs sheets, of size 200 mm in length, 90 mm in width and 6.4 mm in thickness were prepared by a gravity casting technique. A homogeneous mixture of epoxy resin and glass beads in the required proportion was first prepared. After degassing the mixture in vacuum, the hardener in the required amount was added and mixed gently without trapping air bubbles. Simultaneously another mixture of epoxy and hardener was prepared. The mixture consisting epoxy, glass beads and hardener was first poured into an acrylic mould kept in vertical position. Then the mixture of epoxy and hardener was poured into the acrylic mould till the mold is completely filled and then the mold was sealed. After 20 minutes the mold was flipped upside down and kept in vertical position. Glass beads have a higher specific gravity of 2.2 as compared to 1.17 for the epoxy resin. When the mold is flipped upside down, the top layer of the mixture that is rich in glass beads starts moving down due to its higher specific gravity. Simultaneously the resin gels, its viscosity increases and this will arrest the settling of the glass beads. This results in a casting with a glass bead rich region at the bottom, a resin rich region at the top, and an intermediate region with continuously varying glass bead content. The sheet was allowed to cure in the mould at room temperature for 36 hours. After this the sheet was post cured in an air circulating oven for 4 hours at 100 o C. In order to determine the composition profile of the prepared FGM sheet, small samples of size 25 x 6.4 x 5 mm were sliced along the width of the sheet. The density of these samples was measured by hydrostatic weighing. The glass bead volume fraction (V f ) was calculated from the density using the rule of mixtures. The composition profile of a typical graded sheet is shown in figure 1. It can be noticed that there is a region of continuously varying glass bead content in the middle portion of the sheet. This portion was used for the fracture experiments. From the composition profile and the variation of the elastic modulus as function of the glass bead content, the elastic modulus profile of the graded sheet was determined. The elastic modulus profile for a typical sheet, shown in figure 2, indicates that the elastic modulus varies by a factor of two over a length of 50 mm. The continuous line in figure 2 is an exponential fit to the elastic modulus data. The composition profile and elastic modulus profile was characterized for each casting and this indicated that the gradation achieved by the gravity casting technique is fairly repeatable. Plates of length 170 mm, width 45 mm and thickness 6.4 mm were cut out of the prepared sheets. A notch of width 200 µm was first made on one of the edges using a water cooled diamond cutter. A natural crack of the required length was extended from this notch tip by forcing a sharp blade into the notch and tapping it in a controlled manner (see figure 3). The length of the crack was measured precisely using a traveling microscope equipped with a micrometer. During this measurement a slight crack face opening load was applied by a thin wedge to facilitate locating of the crack tip. By this way the crack length could be measured to an accuracy of ± 0.1 mm.

40 8 Vf (%) 30 20 10 0 0 10 20 30 40 50 60 70 80 90 Distance (mm) Elastic modulus (GPa) 7 6 5 4 3 E = 3.6e 0.016x 0 5 10 15 20 25 30 35 40 45 50 Distance (mm) Figure 1. Composition profile of the graded sheet. Figure 2. Elastic modulus profile of the graded sheet. E 2 x x y r θ W y E 1 a Natural crack Notch Figure 3. Natural crack and axis convention used in the analysis Figure 4. Random dot pattern for DIC technique 3. EXPERIMENTAL DETAILS The DIC set up used in the study comprised of a 2 Mega pixel CCD camera, a zoom lens and the image acquisition and correlation software (Vic2D) supplied by Correlated Solutions Inc., USA. The technique of DIC involves applying a random black and white pattern on the surface of the specimen and then capturing digital images of this pattern before and after loading. The in plane displacement components are then calculated through intensity correlation of the images taken before and after deformation. The details of the correlation procedure can be found in [5]. A random dot pattern was applied to the specimen surface using black and white spray paint. A photograph of the random dot pattern is given in figure 4. The specimen was loaded in a 25 kn testing machine and the load was measured using a 1 kn load cell. A subset size of 34 x 34 pixels was used for the correlation. Proper care was taken to align the horizontal axis of the camera with the crack line. The u and v displacement maps from a typical test, shown in figure 5, indicate a smooth displacement field. The displacements shown in figure 5 include rigid body components (translation and rotation) that arise during the loading. Tests were conducted for different crack lengths for the case two cases 1) E 2 /E 1 =2 (crack is in the compliant side) and 2) E 2 /E 1 =0.5 (crack is in the stiff side). The displacement data from the correlation was exported and the stress intensity factor was calculated from the displacement field as explained in the next section. 4. EXTRACTION OF STRESS INTENSITY FACTOR The SIF was determined by fitting the asymptotic crack tip displacement field to the experimentally measured displacement field. Crack tip displacement field specific to graded materials are not yet available. Bueno and Lambros [4] have reported the use of the displacement field available for homogeneous materials for estimating SIF in graded materials. The expansion of the displacement field for homogeneous materials is the following.

Figure 5. In plane displacement contours (u & v) from DIC technique for four point bending, a/w=0.55 Load=40 N, E 2 /E 1 =2 1+ ν κ 1 κ + 1 u = Re Zn + ReYn ( yim Zn + yimyn) Ec n= 0 2 2, (1) 1+ ν κ + 1 κ 1 v= Im Zn + ImYn ( yre Zn + yreyn) Ec n= 0 2 2 where κ = (3 ν) /(1 + ν) for plane stress, ν being the Poisson s ratio and E c is the elastic modulus at the crack tip. The functions Z n and Y n are defined as follows. n 1/ 2 n Z = Az ; Y = Bz ; Z = Zdz; Y = Ydz n n n n n n n n, (2) θ where z = x + iy = re i. The SIF K I is related to the coefficient A 0 as K1 = A0 2π. From the experimentally measured displacements (u e & v e ), the radial component of the displacement u re was calculated for each data point as ure = ue cosθ + ve sinθ, (3) where θ, ( π < θ < π) is the angle made by the position vector of the data point with the x-axis shown in figure 3. The experimentally measured displacements include rigid body contributions also. Using equation (1) and considering the rigid body translations T x and T y in x and y directions respectively, the theoretical displacement field can be written as ur = ucosθ + vsinθ + Txcosθ + Tysinθ (4) Note that in the polar coordinates, rigid body rotation in the x-y plane will not induce any radial displacement and hence does not appear in equation (3). Equation (3) has (N+2) unknowns where N is the number of terms used in the series expansion of the displacement (equation 1). The number of data points will be much more than the number of unknowns. Therefore the unknowns (A 0 A N, B 0.B M, T x, T y ) are determined by fitting equation (4) to the experimental data (equation 3) in a least square sense. Though any of the displacement components u or v could be used for extraction of the SIF, u r was used in this study as it is not sensitive to rigid body rotation of the specimen. In the present study, the displacements were measured over a region of 80 x 40 mm 2 around the crack-tip. One could use this entire data set in the analysis, however, this will add to computation time. Instead, the data points sampled from an annular region of inner radius R 1 and outer radius R 2 were used to extract the SIF. The inner radius R 1 has to be larger than half the plate thickness to avoid three-dimensional effects and hence R 1 was set to 3 mm. The outer radius R 2 was 10 mm. The SIF was determined by increasing the number of terms in the series (equation 1) starting from 2 terms (N=2). This procedure was continued until the addition of further terms did not produce any change in the SIF. As an additional confirmation, the displacement field, u r, was regenerated using the constants (A 0 A N, B 0.B M, T x, T y ) and was visually compared with the experimental displacement field, u re, to see the match between the two. Further discussion on these aspects is provided in the next section.

5. RESULTS AND DISCUSSION The stress intensity factor was determined for different crack lengths in the range of 0.3 < a/w < 0.75 and two different gradations corresponding to E 2 /E 1 = 0.5 and 2 for four-point bending and three-point bending. For each case, the SIF was determined as explained in the last section by increasing the number of terms in equation (1 & 2) until convergence. Figure 6, shows the pattern of convergence of SIF for four-point bending. The SIF is normalized as K / σ π a, where I 2 σ = 6 M / BW, M is the bending moment at the crack plane and B is the specimen thickness. It can be seen that at least 12 terms are required in equation (1) for convergence. Figure 7 gives the comparison of SIF for four-point bending for E 2 /E 1 = 2 with theoretical results available for exponential variation of elastic modulus [2]. The experimentally determined SIF is in good agreement with the theoretical values with a maximum deviation of 6%. The SIF for cracks subjected to three point bending is shown in figure 8. The continuous line in figure 8 is the SIF for a homogeneous material having the same geometry. The effect of material nonhomogeneity on SIF can be clearly seen in figure 8. The SIF for the case E 2 /E 1 =0.5 is higher than that for a homogeneous material where as the SIF is lower than that for a homogeneous material for E 2 /E 1 =2. When the crack is in the compliant side of the plate (E 2 /E 1 =2), the SIF is about 15% lower than that for a homogeneous material whereas when the crack is in the stiff side of the plate (E 2 /E 1 =0.5), the SIF is about 15% higher than that for a homogeneous material. This difference seems to be more prominent for crack lengths a/w < 0.5. When a/w=0.5 the crack tip is at the same location irrespective of whether the crack is on the stiff side or on the compliant side, however, the SIF is different for the two cases. A comparison of the experimentally measured displacement u re and that calculated using equation (1) is shown in figure 9. The continuous lines are the regenerated displacement field and the broken lines are the experimentally measured displacement. It should be noted here that the rigid body components are removed while plotting the displacement contours in figure 9. The contours regenerated using four terms (N=4) in equation (1), shown in figure 9 do not match well with the experimental contours. When the number of terms is increased to 12, the agreement between the measured displacement field and calculated displacement field improves considerably. It was pointed out earlier that the SIF value also converges for N=12 (see figure 6). The convergence of SIF along with the matching of the regenerated and experimental displacement fields is an indicator of the accuracy of the determined SIF. 6. CONCLUSIONS An experimental investigation into the effect of elastic gradient on the SIF for edge cracks in graded plates subjected to bending has been carried out. Graded sheets required for the study was prepared using a gravity casting technique. The technique of DIC was employed to make full field measurement of the displacement field. The SIF was calculated by fitting the theoretical displacement field to the measured data. The results of the study indicated that, the SIF is higher than that for a homogeneous material when the crack is located in the stiff side of the material. In a ceramic-metal FGMs used for thermal protection applications, when the crack is in the ceramic rich side; the SIF will be higher compared to when it is in the metallic side. Ceramics have much lower resistance to fracture and combined with the increase in the SIF due to the gradation, the chances of a crack in the ceramic side becoming critical is higher in such FGMs. KI/(σ πa) 1.6 1.4 1.2 1.0 0.8 0.6 a/w=0.42 a/w=0.51 a/w=0.57 0 4 8 12 16 20 24 Number of terms (N) Figure 6. Convergence of SIF with number of terms (E 2 /E 1 =2). KI/(σ πa) 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Theoretical Expertiment 0.3 0.4 0.5 0.6 0.7 0.8 a/w Figure 7. Stress intensity factor for four-point bending (E 2 /E 1 =2)

3.5 KI/(σ πa) 3.0 2.5 2.0 1.5 E2/E1=0.5 E2/E1=2 1.0 0.5 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 a/w Figure 8. Stress intensity factor for edge cracks in graded sheets subjected to three point bending. N=4 N=12 Figure 9. Contours of regenerated and measured displacement field (E 2 /E 1 =0.5, a/w=0.7) for N=4 and 12. Displacements are in (µm). ACKNOWLEDGMENTS The authors acknowledge the financial support for procuring the DIC system used in this study through project from Aeronautical Research and Development Board through the project DARO/08/1051282/M/I. REFERENCES 1. F. Erdogan, Fracture Mechanics of Functionally Graded Materials, Comp. Engng., 5:7, pp. 753-770, 1995. 2. F. Erdogan, and B. H. Wu, The Surface Crack Problem for a Plate with Functionally Graded Properties, Jl. of App. Mech., 64, pp. 449-456, 1997. 3. R. J. Butcher, C. E. Rousseau, and H. V. Tippur, A Functionally Graded Particulate Composite: Preparation, Measurements and Failure Analysis, Acta Mater., 47:1, pp. 259-268, 1999. 4. J. A. Bueno and J. Lambros, Investigation of Crack Growth in Functionally Graded Materials using Digital Image Correlation, Engng. Frac. Mech., 69, pp. 1695-1711, 2002. 5. W. H. Peters and W. F. Ranson, Digital Imaging Techniques in Experimental Stress Analysis. Opt Engng; 21(3), pp. 427 431, 1982. 6. S. Yoneyama, Y. Morimota, and M. Takashi, Automatic Evaluation of Mixed mode Stress Intensity Factors Utilizing Digital Image Correlation, Strain, 42, pp. 21-29, 2006.