Energy release rate for interlaminar cracks in graded laminates

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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 1480–1488 www.elsevier.com/locate/compscitech

Energy release rate for interlaminar cracks in graded laminates Ulaganathan Jagan, Preeti S. Chauhan, Venkitanarayanan Parameswaran * Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India Received 27 April 2007; received in revised form 26 July 2007; accepted 24 October 2007 Available online 30 October 2007

Abstract In this study, the energy release rate for interlaminar cracks in a laminate beam, graded along the depth, has been derived using beam theory. The correction for accounting crack-tip flexibility, used for conventional composites, is incorporated into the derived expressions and the variation of the energy release rate as a function of crack location and elastic gradient is investigated for the load configurations of double cantilever beam (DCB), end notched flexure (ENF) and mixed mode bending (MMB). The results of the study indicate that the energy release rate calculated using the derived expressions is in very good agreement with that obtained through finite element analysis (FEA) whereas the mode partitioning calculated from the beam theory analysis was not in agreement with that obtained from FEA. As the position of the crack varies along the depth of the beam, there is a considerable change in the energy release rate and mode partitioning. Further, it is shown that by proper load configuration, pure mode (mode-I or mode-II) conditions can be achieved for crack located at any depth along the beam. The effect of the gradation type and its strength on the energy release rate and mode partitioning is also investigated. It was observed that the energy release rate is strongly sensitive to both the strength and type of gradation, whereas, the mode partitioning is influenced more by the gradation strength than by the gradation type. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Graded materials; A. Layered structures; B. Fracture; A. Delamination; C. Failure criterion

1. Introduction Since their introduction in 1987 [1], functionally graded materials (FGM) have attracted considerable attention from researchers working in the area of materials and mechanics [2]. Due to the gradual and progressive change in the material composition and properties in a preferred direction, FGMs have an enormous potential for satisfying multiple functionalities. One of the methods of making bulk graded materials is building them layer by layer with each layer having an incrementally different composition from the adjacent one. Individual layers of different composition can be processed first by tape casting and then laminated or the layers can be deposited one by one as in rapid prototyping to produce composites graded over a

*

Corresponding author. Tel.: +91 512 2597528; fax: +91 512 2597408. E-mail address: [email protected] (V. Parameswaran).

0266-3538/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.10.027

thickness of few millimeters or more [3–5]. In such functionally graded laminates, delamination can be a dominant failure mode. Delamination has been analyzed using fracture mechanics concepts in the case of conventional composites. There exist several studies addressing the effect of material inhomogeneity on fracture parameters and the crack-tip stress field [6–8] for cracks in continuously graded materials. These studies unanimously conclude that the concept of stress intensity factor and energy release rate can be used to characterize the behavior of cracks in graded materials. While these studies provide the basis for evaluating the crack-tip severity in graded materials, the fracture toughness or the critical energy release rate, which itself could vary with position of the crack-tip in a graded composite, has to be determined experimentally. In conventional laminate composites, the resistance to delamination is characterized by the interlaminar fracture toughness (GC), determined experimentally [9,10]. The mode-I toughness (GIC) is determined through the double

U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

cantilever beam (DCB) loading, the mode-II toughness (GIIC) by the end notched flexure (ENF) test and the mixed mode toughness (GC) by the mixed mode bending (MMB) test. In conventional composites, usually the starter delamination (crack) is at mid-depth of the beam. The situation in a graded laminate is different from that of a conventional laminate for the following reasons. The behavior of delamination between different sets of layers could be different due to the change in material composition along the depth of the beam. There is a toughness profile instead of a single toughness value and obviously a single test with crack at mid-depth of the beam is not sufficient to characterize this toughness profile. Therefore tests have to be conducted with the initial crack located at different depths in the beam. Due to the variation of the elastic properties perpendicular to the crack faces (along beam depth), mixed mode conditions should be expected at the cracktip even for a symmetric geometry and loading [6]. While there exist well established test and analysis procedures for determining GIC, GIIC or GC for conventional laminates, no such method is available for graded laminates. The objective of this study is to theoretically explore the applicability of the DCB/ENF/MMB loading schemes for determining the interlaminar toughness profile of graded laminates. 2. Analysis of a functionally graded beam In the following analysis, the origin of the axis of reference coincides with the bottommost layer of the beam with the positive direction of z-axis oriented along the depth of the beam and the x-axis along the span (see Fig. 1). The elastic modulus of the beam E(z) is assumed to be a continuous function of z and each layer is assumed to be isotropic. The curvature j of a beam of depth h and width b is related to the bending moment M as j ¼ MD     1 A B A B ¼ B D B D Z h EðzÞf1; z; z2 gdz fA; B; Dg ¼ b

ð1Þ ð2Þ ð3Þ

0

The shear stress distribution in the beam will be Z z fB þ D~zgEð~zÞd~z ¼ VQðzÞ sxz ðzÞ ¼ V

1481

ð4Þ

0

where V is the shear force resultant for the cross-section concerned. These equations are taken directly from Sankar [11], however, the elastic modulus E(z) is used instead of the plane strain elastic modulus EðzÞ. 3. Energy release rate for graded laminate with a crack In this section, the energy release rate for a cracked laminate with elastic modulus graded along the depth is derived. The analysis follows the procedure outlined in the paper by Williams [12]. Fig. 1 shows the schematic of the cracked laminate and the bending moments applied on the crack arms and the un-cracked section of the beam. The change in slope of the crack arms and the un-cracked beam during a crack extension of da is also indicated in Fig. 1. Subscripts 1 and 2 will henceforth refer to the lower crack arm and upper crack arm, respectively. For a cracked laminate of width b, the energy release rate is given by the expression   1 dU e dU s G¼  ð5Þ b da da where Ue is the work done by the external load and Us is the strain energy. Using Eq. (1) we can easily write down the quantities in Eq. (5) as (for details see Appendix) dU e 2 ¼ M 21 D1 þ M 22 D2  ðM 1 þ M 2 Þ D da dU s 1 2 ¼ fM 21 D1 þ M 22 D2  ðM 1 þ M 2 Þ D g 2 da

ð6Þ

Use of Eq. (6) in Eq. (5) will give the energy release rate, Gb from flexural deformation as Gb ¼

1 2 fM 21 D1 þ M 22 D2  ðM 1 þ M 2 Þ D g 2b

ð7Þ

The effects of shear can be considered following the same approach used in [12] and the energy release from shear deformation Gs can be written as Z Z Z 1 h1 s2xz 1 h2 s2xz 1 h s2xz bdz þ bdz  bdz Gs ¼ b 0 2lðzÞ b 0 2lðzÞ b 0 2lðzÞ ð8Þ

dφ2 δa da

φ0

z

M

δa x

dφ φ0 + 0 δ a da

M2

h2

M1

h1

d φ1 δa da

Fig. 1. Schematic of a cracked laminate showing crack-tip moments and rotations.

In Eq. (8) the shear stress sxz is determined for the crack arms and the un-cracked section from Eq. (4). Noting that, the shear force V ¼ dM , Eq. (8) can be re-written as follows da  2 Z h1  2 Z h2 dM 1 Q21 dM 2 Q22 dz þ dz Gs ¼ da 2lðzÞ da 2lðzÞ 0 0  2 Z h dM 1 dM 2 Q2 þ dz ð9Þ  da da 0 2lðzÞ where l(z) is the shear modulus. The total energy release rate will be the sum of Gb and Gs. In the following sections, the energy release rate for specific cases of DCB, ENF and MMB will be derived.

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3.1. Energy release rate for graded DCB The elastic modulus for the bottom most layer (z = 0) is denoted as E0, that for the top most layer is denoted as E2 and E2/E0 represents the strength of the gradation. For the DCB loading (see Fig. 2a), M1 = M2 = Pa and the total energy release rate G can be written as Z h 1  Z h2 P2  2 Q21 Q22  2 2 dz þ dz G ¼ ½D1 a1 þ D2 a2  þ P 2b 2lðzÞ 2lðzÞ 0 0 ð10Þ The stiffness coefficients (A, B, D)1,2 are evaluated for each arm independently by setting appropriate integration limits in Eq. (3). The constants (A*, B*, D*)1,2 used in Eq. (10) are then calculated using Eq. (2). In Eq. (10) a1 = a2 = a, is the crack length. 3.2. Energy release rate for graded ENF loading For the ENF loading (see Fig. 2b), the crack-tip moments to be used in Eq. (7) are calculated as follows. As a first approximation, the two crack arms are assumed to deform with the same curvature and this will give us the following condition M 1 D1 ¼ M 2 D2 ð11Þ Noting that M = M1 + M2 = Pa/2 and using Eq. (11) the total energy release rate can be obtained from Eqs. (7) and (9) as" # P 2 a2 D1 þ R2 D2  D G¼ 2 8b ð1 þ RÞ Z h1  Z h2 P2 Q21 Q22 2 dz þ R dz þ 2lðzÞ 2lðzÞ 4ð1 þ RÞ2 0 0 Z P 2 h Q2  dz ð12Þ 4 0 2lðzÞ z (w)

where R ¼ D1 =D2 . The assumption of identical curvature for both arms precludes any opening of the crack faces and hence corresponds to pure mode-II conditions at the crack-tip. 3.3. Energy release rate for MMB test The loading configuration for the mixed mode bending test is shown schematically in Fig. 2(c). The load applied at mid span is P and the opening load applied at point A in Fig. 2(c) is assumed to be kP. By proper choice of the loading lever lengths, different values of k in the range (0 < k < 1) can be achieved [10]. For this arrangement, the moments are as follows M 1 ¼ Pað0:5  kÞ;

h

a

x (u)

B

h2

The effect of shear can be easily incorporated using Eq. (9). In the case of the MMB loading, mode partitioning can be calculated by apportioning the mode-I and mode-II parts of the applied moments based on the assumption that in the case of pure mode-II the two crack arms will deform with the same curvature. The moments MI1,2 and MII1,2, corresponding to mode-I and mode-II part of the energy release rate are calculated as M I1 ¼ M I2 ¼ M II1 ¼

P D

B

C

L

L

E

A a

h

h2

h1 F

M1 þ M2 ; 1þR

M II2 ¼ R

ð15Þ M1 þ M2 1þR

ð16Þ

P 2 f0:5R  ð1 þ RÞkg2

fD1 a21 þ D2 a22 g 2 2bð1 þ RÞ ( ) P 2 a2 D1 þ R2 D2  ¼ D 2 8b ð1 þ RÞ

GIb ¼ GIIb

P

(a) DCB loading

RM 1  M 2 1þR

Using Eqs. (13)–(16) in Eq. (7) the mode-I and mode-II components of the total energy release rate GI and GII are obtained as

h1

2L

ð13Þ

Using Eq. (13) in Eq. (7) the energy release rate can be calculated as   P 2 a2 D 2 Gb ¼ ð0:5  kÞ D1 þ k 2 D2  ð14Þ 2b 4

P A

M 2 ¼ kPa

ð17a; bÞ

At this point, it should be made clear that GIb and GIIb obtained from Eq. (17a and b) based on a global analysis should be used only as an indicator of the mode partitioning and the actual mode contributions will be different even for a homogeneous material [13].

(b) ENF loading

D

C

B

A a

h E

4. Correction for crack-tip flexibility

kP

P

L

L

h1 F

(c) MMB loading Fig. 2. Graded laminate with unequal crack arms.

h2

In the case of conventional composites, a correction is employed while calculating the energy release rate to account for the crack-tip flexibility. For DCB with equal arms, this correction has been derived by considering the un-cracked portion of each crack arm to be supported on an elastic foundation [14]. The stiffness (rotational and extensional) of the

U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

elastic foundation is related to the elastic constants and cross-sectional properties of the beam. This analysis indicates that the effect of crack-tip flexibility for a DCB can be captured by increasing the crack length, a, by an amount (vh/2). The same approach is extended for ENF loading and the formula for v for DCB and ENF is provided by Hashemi et al. [15]. For DCB and ENF with unequal arms, such correction schemes are not available at present even for conventional composites. Hence the following approach has been used in the present study. As a reasonable approximation the value of elastic modulus and shear modulus at the crack plane, Ec and lc are used in the formula provided in [15]. The expression for v then becomes sffiffiffiffiffi"  2 #1=2 Ec C 1:18Ec v ¼ k1 ; C¼ ð18Þ 32 Cþ1 lc lc pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi The factors k 1 ¼ 1=11 for DCB and k 1 ¼ 1=63 for ENF of conventional laminates [15] are retained as such. For the case of DCB with unequal arms, it is proposed to increase the crack length differently as a1 = a + vh1; a2 = a + vh2 for the lower and upper arms, respectively in Eq. (10). For ENF specimen, the crack length is modified as (a + vheq) in Eq. (12), where heq is defined as   h1 h1 heq ¼ h1 þ 1  ð19Þ h2 h h In the case of the MMB loading, the crack-tip correction similar to the DCB loading is incorporated for calculating GIb in Eq. (17) and for calculating GIIb the correction is incorporated similar to that for the ENF loading. For graded laminates with unequal arms, the proposed approach is approximate; however, it accounts for the effect of crack-tip flexibility reasonably well as will be shown later in Section 6. 5. Mode partitioning through finite element analysis A beam of depth h, and length of 2L = 15h was analyzed for a crack length a/2L = 1/3. The elastic modulus and shear modulus were assumed to vary exponentially as EðzÞ ¼ E0 ekz ; lðzÞ ¼ l0 ekz

ð20Þ

1483

Gradations corresponding to (E2/E0) of 2 and 5 were considered such that in both cases the mid plane modulus Em was the same. Similar analysis was also performed for a homogeneous beam having an elastic modulus of Em. The Poisson’s ratio was taken as 0.3 for the entire beam. For all the cases, the position of crack plane was varied from h1/h = 0.2 to h1/h = 0.8, in increments of 0.1. The commercial finite element code ANSYS was used to perform the finite element analysis. The entire beam was modeled using four node quadrilateral plane stress elements of uniform size equal to h/100. This size was arrived at after performing a convergence study starting with a size of h/ 40. The elastic modulus gradation was modeled by assigning different elastic modulus to each layer of elements, calculated at the centroid of each layer. The boundary conditions used for the analysis of DCB loading were (i) u = 0 and w = 0, at x = 0 and z = 0 (point B in Fig. 2a), (ii) u = 0, at x = 0 and z = h (point A in Fig. 2a). For the ENF/MMB loading, the boundary conditions were (i) u = 0 and w = 0, at x = 0 and z = 0 (point E in Fig. 2b and c), (ii) w = 0, at x = 2L and z = 0 (point F in Fig. 2b and c). Additionally in the case of the ENF/MMB model, surface to surface contact elements were used all along the crack faces to prevent penetration of the crack faces. The individual mode-I and mode-II energy release rate components (GI and GII) were calculated using the modified virtual crack closure integral [16]. 6. Results for exponentially graded laminates 6.1. Exponentially graded DCB loading The energy release rate was calculated using beam theory (BT), beam theory including shear (BTS) and beam theory including shear and crack tip correction (BTSC) for exponentially graded DCB of length 2L = 15h, a/2L = 1/3, E2/E0 = 2 and 5, for crack location (h1/h) varying from 0.2 to 0.8. The difference in G (%) between these estimates and that obtained from FEA is given in Table 1. It can be observed that the G values, calculated using BT and BTS are not in agreement with that obtained from FEA. The inclusion of the shear effects does not improve the G values either. This is because

Table 1 Comparison of energy release rate obtained using theory with that obtained from FEA for DCB loading h1/h 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Difference (%) in Ga E2/E0 = 1

Difference (%) in G E2/E0 = 2

Difference (%) in G E2/E0 = 5

BT

BTS

BTSC

BT

BTS

BTSC

BT

BTS

BTSC

5.07 8.03 10.85 12.13 10.85 8.03 5.07

5.02 7.92 10.66 11.90 10.66 7.92 5.02

0.34 0.31 0.21 0.15 0.21 0.31 0.34

4.89 7.65 10.44 12.10 11.29 8.50 5.29

4.85 7.54 10.26 11.88 11.09 8.38 5.24

0.44 0.51 0.38 0.17 0.06 0.09 0.20

4.69 7.21 9.81 11.78 11.68 9.08 5.62

4.65 7.11 9.64 11.56 11.46 8.94 5.56

0.59 0.79 0.79 0.54 0.16 0.01 0.06

BT-beam theory, BTS-beam theory with shear, BTSC-beam theory with shear and crack-tip correction. a % difference = (GBT  GFEA)/GFEA*100.

U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

the material is isotropic with in each layer and the span to depth ratio is large. However, after incorporating the correction for crack-tip flexibility (Eq. (18)), the energy release rate (BTSC) calculated using Eq. (10) is in excellent agreement with that obtained from FEA. The variation of energy release rate (BTSC) as a function of the crack location is shown in Fig. 3 along with that obtained from FEA. It can be observed from Fig. 3 that for the same applied load, G decreases as the crack location approaches the mid plane. For homogeneous material G is lowest when the crack is located at the mid plane. In the case of graded DCB, the trend is similar; however, the crack location corresponding to Gmin does not coincide with the mid plane of the beam. The effect of gradation is the least for cracks near the mid plane of the beam and highest for cracks located towards the top or bottom surfaces. It can be observed that G is also strongly influenced by the strength of the gradient (E2/E0). Due to the asymmetry of the specimen and the gradation of the modulus, the crack will be in mixed mode condition. The variation of GII component in G as a function of crack location and gradation strength is shown in Fig. 4. The open symbols are from FEA and the closed symbols are the values obtained from the work of Gu and Asaro [6]. It can be seen that the contribution of mode-II decreases as the crack location moves towards the middle region of the beam. Pure opening mode conditions are achieved for a particular crack location which depends on the strength of the gradient (E2/E0) as indicated in Fig. 4. The crack face sliding displacements (mode-II) change their directions, as shown in Fig. 4, beyond this particular crack location. Fig. 4 also indicates that the GII/G ratio obtained through FEA is in agreement with that obtained by Gu and Asaro [6] for a semi-infinite crack by integral transform method.

40

E2 /E0

35

1 2

30

G II /G (%)

1484

5

25 20 15 10 5 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.9

Fig. 4. Percentage GII for different crack location and E2/E0 ratio for DCB. Open symbols are from FEA and closed symbols are from Ref. [6].

6.2. Exponentially graded ENF test A comparison of the energy release rate calculated using Eq. (12) after including the crack-tip flexibility correction (Eqs. (18) and (19)) with that obtained from FEA is shown in Fig. 5 for gradations of E2/E0 = 2, 1/2, 5 and 1/5 along with that for a homogeneous material. The case of E2/E0 = 1/5 is the beam having E2/E0 = 5 inverted with respect to the loading. It can be noticed that, for the same applied load the energy release rate is highest when the crack is located close to the mid-depth of the beam. The GI component in G obtained from FEA is shown in Fig. 6, as a function of crack location, for homogeneous material and for gradations of E2/E0 = 2, 1/2, 5 and 1/5. It can be noticed that for the case of homogeneous

18

1.2

E2 /E0 15

E2 /E 0

Normalized energy release rate

Normalized energy release rate

0.8

h1 /h

1 2 5

12

9

6

3

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1 /h Fig. 3. Normalized energy release rate for DCB as a function of normalized crack location and gradation strength (E2/E0) for a/2L = 1/3. Energy release rate is normalized by the energy release rate (BT) for a homogeneous DCB with equal arms and modulus equal to Em.

0.2

1.0

0.5 1.0 2.0

0.8

5.0

0.6

0.4

0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1 /h Fig. 5. Normalized energy release rate for ENF as a function of normalized crack location and gradation strength (E2/E0) for a/2L = 1/3. Energy release rate is normalized by the energy release rate (BT) for a homogeneous ENF with equal arms and modulus equal to Em.

U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

1485

40

h1/h 0.2 0.3 0.4 0.5 0.6 0.7 0.8

100

E2 /E 0

35

0.2 0.5

30

1.0

80

25

5.0

G II /G (%)

G I /G (%)

2.0

20 15

60

40 10

h1 /h=0.2 5

20

h1 /h=0.5

h1 /h=0.8

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1 /h Fig. 6. Percentage GI for different crack location and E2/E0 ratios in graded ENF.

material, there is 32% mode-I contribution in G for h1/ h = 0.2 and the mode-I contribution becomes zero for crack locations h1/h P 0.5. For graded ENF, a similar trend is seen except that the crack location at and beyond which the mode-I contribution is zero is dependent on the E2/E0 ratio. For E2/E0 = 5, pure mode-II conditions are obtained for crack locations h1/h P 0.6 where as for E2/ E0 = 1/5, pure mode-II conditions are obtained for crack locations h1/h P 0.4 Note that h1/h = 0.3 in the case of E2/E0 = 5 is the same crack location as h1/h = 0.7 in the case of E2/E0 = 1/5. This would indicate that pure modeII conditions can be generated for more or less any crack location by appropriately orienting the direction of increasing modulus towards the load. It was observed from FEA results that for crack locations giving pure mode-II conditions, the two crack arms deform with the same curvature as assumed in Eq. (12). For locations of the crack for which mixed mode conditions are obtained, opening of the crack faces close to the crack-tip was observed in the FEA. This indicates that the crack arms do not have the same curvature as assumed in Eq. (12). Despite this the total energy release rate calculated from Eq. (12) with crack-tip correction (Eqs. (18) and (19)) is in good agreement with that obtained from FEA. The maximum difference between the G estimated from Eq. (12) without the crack-tip correction and that obtained from FEA was 7%. Inclusion of the crack-tip correction reduced the maximum error to 2%. 6.3. Exponentially graded MMB loading In MMB loading different mixed mode conditions can be created by changing the value of k. The percentage of GII in the total energy release rate G, estimated from FEA, for values of k in the range of 0.1–0.9 is shown in Fig. 7 for a gradation of E2/E0 = 5. Fig. 7 shows some interesting observations regarding creation of pure modeI conditions. For the crack locations, h1/h = 0.2, 0.3, 0.4

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

k Fig. 7. Percentage GII for different load ratio (k) and crack location (h1/h) obtained from FEA for a graded MMB of E2/E0 = 5, (Broken lines are estimates from (17)).

and 0.5, GII becomes zero or nearly zero at particular values of k (k > 0.5), indicating pure mode-I conditions. For the case of h1/h = 0.6, recall that the DCB loading gives pure mode-I conditions (see Fig. 4). For h1/h = 0.7 and above pure mode-I conditions cannot be achieved for values of k in the range of 0.1–0.9 as indicated in Fig. 7. The reason for the pure mode-I conditions in MMB loading is the following. The MMB loading is a combination of DCB and ENF loading. The overall crack surface displacement therefore has two parts, one from the DCB part of the loading and the other from the ENF part. For the crack locations, 0 < h1/h < 0.6, the sliding displacement from DCB part and that from ENF part are of opposite nature and will exactly cancel each other at a particular value of k, leading to pure mode-I conditions. For E2/E0 = 5, the sliding displacements changes direction after h1/h = 0.6 in the case of DCB loading (see Fig. 4) where as the sliding displacements have the same direction for all crack locations in ENF loading. Therefore, the two sliding displacements add up for 0.6 < h1/h < 1.0 and mode-I conditions cannot be achieved for any value of k. Pure mode-I conditions can however be created for 0.6 < h1/h < 1.0 by changing the orientation of the gradation with respect to the loading and supports. Fig. 8 shows the percentage of GII in G for the graded beam of E2/E0 = 1/5 (elastic modulus decreasing by five times from bottom to top). In this figure the crack location (h1/h) is still measured from the bottom of the beam. One can see that for 0 < h1/h 6 0.3, pure mode-I condition is achieved. These crack locations would correspond to 0.7 6 h1/h < 1.0 for the beam orientation of Fig. 7 (E2/E0 = 5). Therefore, pure mode-I conditions could be generated for more or less any crack location by appropriate choice of the load ratio k and the orientation of the gradation with respect to the loading and supports.

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U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

Eq. (17)) and that obtained from FEA was 12%. This error reduced to 3% after including the crack-tip correction.

h1/h

100

0.2 0.3

7. Influence of gradation type on energy release rate and mode partitioning

0.4

80

0.5

G II /G (%)

0.6 0.7

60

The influence of the type of gradation for a given gradation strength (E2/E0) on the energy release rate and mode partitioning was also investigated. For this study, the property variation was assumed in the form of power law as given in Eq. (21).

0.8

40

20

EðzÞ ¼ E0 þ ðE2  E0 Þðz=hÞ

m

ð21Þ

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k Fig. 8. Percentage GII for different load ratio (k) and crack location (h1/h) obtained from FEA for a graded MMB of E2/E0 = 1/5.

h1 /h

Normalized energy release rate

2.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.8 1.6

Normalized energy release rate

3.0

1.4

Exponential

2.6

m=0.5, (Eq.21) m=1.0

2.2

m=2.0

1.8

1.4

1.0

0.6 0.1

1.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1/h

1.0

Fig. 10. Normalized energy release rate for graded DCB having different E variations. The energy release rates are normalized with that for an identical homogeneous beam having unequal arms and modulus of (E2 + E0)/2. (Continuous line is the estimate from beam theory and data points are from FEA).

0.8 0.6 0.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

k

40.0

Fig. 9. Normalized energy release rate for MMB as a function of load ratio (k) and crack location (h1/h) for (E2/E0 = 5) and a/2L = 1/3. Energy release rate (G) is normalized by the G for an identical homogeneous beam with unequal arms and modulus Em.

Exponential

35.0

m=0.5, (Eq.21) m=1.0

30.0

m=2.0

The percentage GII in G calculated using Eq. (17), (broken lines in Fig. 7) is not in agreement with that calculated from FEA. Such disagreement has been observed in the case of conventional composites also [13]. The mode partitioning from the global analysis does not account for the local crack-tip field which governs the mixed mode ratio and hence the disagreement. However, the total energy release rate (GI + GII) calculated using Eq. (17) with crack-tip correction agreed well with that obtained from FEA as shown in Fig. 9. Without crack-tip correction, the maximum difference between the calculated G (from

GII/G (%)

25.0 20.0 15.0 10.0 5.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1/h Fig. 11. Percentage GII as a function of crack location and gradation type for DCB.

U. Jagan et al. / Composites Science and Technology 68 (2008) 1480–1488

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for the same gradation strength, the energy release rate is strongly influenced by the type of gradation whereas mode partitioning is not very sensitive to the type of gradation.

50 45

GII/G (%)

40 35

Acknowledgement

30

The authors acknowledge Advanced Systems Laboratory, Defence Research and Development Organization, India, for financial support to perform this study.

25 20 15

Exponential m=0.5, (Eq.21)

10

Appendix

m=1.0

The beam curvature is related to the slope as j  d/ . dx Using Eq. (1) we get

m=2.0

5

k=0.5 k=0.8

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h1/h Fig. 12. Percentage GII as a function of crack location and gradation type for MMB.

For variation of E from E0 = 5 GPa to E2 = 25 GPa, three different gradations were considered by choosing three different values for m. The variation of normalized G with crack location (h1/h) for the three different gradations, (m = 0.5,1,2) and that for exponential gradation is shown in Fig. 10 for DCB. The data points in Fig. 10 are from FEA. Fig. 10 shows that, at a given crack location, G is different for the four different gradations. Similar trend was observed for ENF and MMB as well. This indicates that the type of gradation also has a significant effect on the energy release rate. Fig. 11 shows the influence of the gradation type on mode partitioning obtained from FEA for DCB. It can be noticed that the pattern of variation of GII with (h1/h) is similar for the four gradations. Additionally, for any given crack location, GII component in G does not differ much for the four gradations. Similar observations can be made for MMB from Fig. 12. This leads to the conclusion that the mode partitioning is not significantly influenced by the gradation type. 8. Conclusions A general expression for the energy release rate for graded laminates with interlaminar cracks has been derived based on beam theory. The derived expressions along with correction for crack-tip flexibility can accurately estimate the total energy release rate for graded laminates. However, the derived expressions could not predict the mode partitioning correctly. The results of the study also indicate that by using the conventional loading configurations of DCB/ ENF/MMB, along with proper orientation of the gradation, complete characterization of the interlaminar cracks located at any position along the depth of the graded beam can be performed for conditions ranging from pure mode-I to mixed mode to pure mode-II. Finally it was observed that

d/0 ¼ ðM 1 þ M 2 ÞD ; da

d/1 ¼ M 1 D1 ; da

d/2 ¼ M 2 D2 da

ðA1Þ

Referring to Fig. 1, change in slope of arm 1 and

arm 2 d/0 d/1  during crack extension da will be da and da da d/

d/0 2  da [12]. The change in external work due to this da da slope change is     d/1 d/0 d/2 d/0   dU e ¼ M 1 da þ M 2 da ðA2Þ da da da da Using (A1) in (A2) will give the first equation in (6). For linear material response the change in strain energy due to the slope change will be     M 1 d/1 d/0 M 2 d/2 d/0 dU s ¼   da þ da ðA3Þ 2 da da 2 da da Using (A1) in (A3) will give the second equation in (6). References [1] Niino M, Hirai T, Watanabe R. The functionally gradient materials. J Jpn Soc Compos Mater 1987;13(1):257. [2] Suresh S, Mortensen A. Fundamentals of functionally graded materials. London: IOM Communications Ltd; 1998. [3] Hill MR, Lin WY. Residual stress measurement in ceramic–metallic graded material. J Eng Mater Technol 2002;124:185–91. [4] Acikbas NC. Fabrication of functionally graded SiAlON ceramics by tape casting. J Am Ceramic Soc 2006;89:3255–7. [5] Zhou MY, Xi JT, Yan JQ. Modeling and processing of functionally graded materials for rapid prototyping. J Mater Process Technol 2004;146:396–402. [6] Gu P, Asaro RJ. Cracks in functionally graded materials. Int J Solids Struct 1997;34(1):1–17. [7] Erdogan F, Wu BH. The surface crack problem for a plate with functionally graded properties. J Appl Mechan 1997;64: 449–56. [8] Jain N, Rousseau C-E, Shukla A. Crack tip stress fields in functionally graded materials with linearly varying properties. Theor Appl Fract Mechan 2004;42(2):155–70. [9] ASTM D 5528-01, Standard test method for mode-I interlaminar toughness of unidirectional fiber-reinforced polymer matrix composites. [10] ASTM D 6671-01, Standard test method for mixed mode-I-mode-II interlaminar toughness of unidirectional fiber-reinforced polymer matrix composites. [11] Sankar BV. An elasticity solution for functionally graded beams. Compos Sci Technol 2001;61:689–96.

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[12] Williams JG. On the calculation of energy release rate for cracked laminates. Int J Fract 1988;36:101–19. [13] Ducept F, Gamby D, Davies P. A mixed-mode failure criterion derived from tests on symmetric and asymmetric specimens. Compos Sci Technol 1999;59:609–19. [14] Williams JG. End corrections for orthotropic DCB specimens. Compos Sci Technol 1989;35:367–76.

[15] Hashemi S, Kinloch AJ, Williams JG. The analysis of interlaminar fracture in unaxial fibre-polymer composites. Proc Royal Soc London 1990;A427:173–99. [16] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factor by a method of modified crack closure integral. Eng Fract Mechan 1977;9(4):931–8. p. 35.