# Energy release rate analysis for adhesive and laminate double cantilever beam specimens emphasizing the effect of residual stresses

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6 64 J. A. Nairn / International Journal of Adhesion & Adhesives 20 (1999) Coefficient Error (%) C m Error C r Error Arm Axial Ratio (a/h) Fig. 3. The average or the absolute values of the errors between the theoretical coefficients C m and C r and the mechanical and residual stress results calculated by FEA as a function of the arm axial ratio. These results are for R = 28. For each arm axial ratio, the error was averaged for λ from 1 to 128. Because typical adhesive DCB specimens strive to use long, slender arms (large a/h), the residual stress correction evaluated here should be sufficiently accurate for all commonly-used adhesive fracture specimens. The errors in C m normally exceeded 1% and got very large for small a/h. This observed error in simple beam theory for calculating mechanical energy release rate has been discussed before [7 10]. The error is caused by an over-simplification in the simple beam theory analysis that prevents rotation of the specimen arms at the crack tip; for accurate analysis, this rotation can not be ignored. Williams analyzed an adhesive DCB specimen ignoring residual stresses [10]. He wrote a corrected mechanical beam theory result as G I ( T = 0) = (C m (a + W ) P ) 2 where C m = lim λ C m = 12 EB 2 h 3 1 is the limiting mechanical energy release rate coefficient for a very thin adhesive. In other words, the corrected analysis replaces the true crack length, a, by an effective crack length, a + W. Using the beamon-elastic-foundation model of Kanninen [16], Williams derived the effective crack length correction in terms of specimen properties to be ( ) 1/4 1 + R λ W = h 1 (18) 6 If G F I EA is the energy release rate calculated by finite element analysis with T = 0, a numericallydetermined effective crack length correction for equation (17) can be found from F EA = (17) G F EA I C m P a (19) When including specimens with possibly thick adhesive layers, however, it is preferable to find the effective crack length correction relative to the full composite beam analysis for C m rather then for the thin-adhesive beam result of C m. In other words, here the FEA effective crack length correction was found from F EA = G F EA I C m P a (20)

13 J. A. Nairn / International Journal of Adhesion & Adhesives 20 (1999) Table 1. Errors caused by ignoring residual stresses in delamination experiments on a series of graphite/epoxy laminates. The experiments are assumed to be on 16-ply laminates with delamination in the middle of the laminate. The laminates listed here are the layups for the 8-ply laminates in each DCB specimens. 8-Ply Arm G IC Error 8-Ply Arm Layup G IC Error (%) Layup (%) [0 8 ] 0.00 [0 2 /90 2 /0 2 /90 2 ] [90/0 7 ] [0/90/0/90/0/90/0/90] [90 2 /0 6 ] [90/0/90/0/90/0/90/0] [90 4 /0 4 ] [90 2 /0 2 /90 2 /0 2 ] [90 6 /0 2 ] [0 2 /90 2 /45 2 / 45 2 ] [90 8 ] 0.00 [90 2 /0 2 /45 2 / 45 2 ] [0 2 /90 6 ] [0 2 /45 2 / 45 2 /90 2 ] [0 4 /90 4 ] [90 2 /45 2 / 45 2 /0 2 ] [0 6 /90 2 ] [45 2 / 45 2 /0 2 /90 2 ] [0 7 /90] [ 45 2 /45 2 /90 2 /0 2 ] Consequence of ignoring residual stresses From equation (24), the error caused by ignoring residual stresses is a function only of C r, T, and G IC. We considered a series of graphite/epoxy laminates with G IC = 200 J/m 2 and T = 125 C, which are typical values for graphite epoxy laminates [20 22]. The value of C r depends on the layup of the specimen arms and can be calculated by the laminated beam analysis in the Appendix followed by substitution into equation (8). Table 1 lists the error caused by ignoring residual stresses for delamination down the middle of a series of 16-ply laminates which leads to 8-ply laminate DCB arms. In all examples, the undelaminated 16- ply laminate is symmetric; the 8-ply arms, however, may be unsymmetric. The ply properties used in these calculations were E 1 = MPa, E 2 = 8970 MPa, ν 12 = 0.30, G 12 = 6900 MPa, α 1 = K 1, α 2 = K 1, and h i = mm. The first column of Table 1 gives a series of DCB arms with 0 and 90 plies. The two special cases of [0 8 ] and [90 8 ] are not affected by residual stresses and thus introduce no error. The arms in these specimens are themselves symmetric laminates and therefore do not curve due to residual stresses. Because there is no thermal curvature, there is no external work due to residual stresses and no residual stress effect on G I. These layups will have residual stresses caused by differential shrinkage between the fibers and the matrix. These residual stresses, however, do not release any energy as the delamination crack propagates. Many previous delamination experiments have been done on unidirectional laminates. It is correct to ignore residual stresses when analyzing such experiments. All other arrangements of 0 and 90 plies, besides [0 8 ] and [90 8 ], have an effect of residual stresses that cannot be ignored. The smallest error, caused by a change of a single ply on the surface, is 6.21% ([90/0 7 ]); the largest error is 76.36% ([0 2 /90 6 ]). An interesting example is the [0 4 /90 4 ] laminate. If the delamination growth is parallel to the fibers in the surface plies, the apparent G IC will be 46.54% too high (293 J/m 2 in this example). But, if the laminate is rotated 90 and the delamination is propagated parallel to the fibers in the central plies, the apparent G IC will be 37.68% too low (125 J/m 2 in this example). The two apparent toughnesses will differ by more than a factor of two. The second column of Table 1 lists some alternate layups with four 0 and four 90 plies and lists some quasi-isotropic layups. Among the 0/90 laminates, the sign of the error depends on the stacking sequence and the magnitude depends on the ply groupings. Positive errors result when the 90 plies are in the middle while negative errors result when 0 plies are in the middle. The errors get smaller as the ply groupings get smaller; the smallest error is for alternating 0 and 90 plies. Among the quasi-isotropic layups, the error ranges from % to %. It is not possible to construct laminate arms made from 0 2, 90 2, 45 2, and 45 2 ply pairs having a residual stress effect that causes less than a 14% error. If the ply pairs are split apart, the errors can be made smaller. Symmetric arm layups such as [0/90/ + 45/ 45/ 45/ + 45/90/0]

15 J. A. Nairn / International Journal of Adhesion & Adhesives 20 (1999) where h/2 A 11 = B E(y)dy = B h/2 h/2 B 11 = B y E(y)dy = B h/2 n E(θ i )h i (41) i=1 h/2 D 11 = B y 2 E(y)dy = B h/2 n E(θ i )h i ȳ i (42) i=1 h/2 N T = B E(y)α(y)dy = B h/2 n ( ) E(θ i )h i ȳ 2 i + h2 i (43) 12 n E(θ i )α(θ i )h i (44) i=1 h/2 M T = B y E(y)α(y)dy = B h/2 i=1 n E(θ i )α(θ i )h i ȳ i (45) i=1 (46) Here E(θ i ), α(θ i ), h i, and ȳ i are the x-direction modulus, x-direction thermal expansion coefficient, thickness, and midpoint of ply i with orientation angle θ i. A 11, B 11, and D 11 are the (1, 1) elements of the usual laminated plate theory A, B, and D matrices except for the extra factor of B because F and M are total forces and moments instead of force and moment resultants. By the usual rules for rotation of thermomechanical properties, E(θ i ) and α(θ i ) are 1 E(θ i ) ( = cos4 θ i 1 + 2ν ) 12 sin 2 θ i cos 2 θ i + sin4 θ i (47) E 1 G 12 E 1 E 2 α(θ i ) = α 1 cos 2 θ i + α 2 sin 2 θ i (48) Here E 1, E 2, G 12, ν 12, α 1, and α 2 are the in-plane tensile and shear moduli, Poisson s ratio, and thermal expansion coefficients of the ply material with 1 being the fiber direction and 2 being the transverse direction. Finally, setting F = 0 and solving for κ gives where the effective beam properties are κ = C κm + α κ T (49) A 11 Cκ = A 11 D 11 B11 2 and α κ = B 11N T A 11 M T A 11 D 11 B 2 11 (50) References [1] Ripling, E. J., S. Mostovoy, and R. L. Patrick Materials Research & Standards 1964, 4, [2] Mostovoy, S. and E. J. Ripling J. Appl. Polym. Sci. 1966, 10, [3] Mostovoy, S. and E. J. Ripling J. Appl. Polymer Sci. 1971, 15, [4] ASTM D , in Annual Book of ASTM Standards 1996, 15.06, [5] ASTM D a, in Annual Book of ASTM Standards 1996, 15.03, [6] Reeder, J. R. and J. H. Crews, Jr. AIAA J. 1990, 28, [7] Hashemi, S., A. J. Kinloch, and J. G. Williams Proc. R. Soc. Lond. 1990, A347, [8] Blackman, B., A. J. Kinloch, and J. G. Williams, personal communication. [9] Williams, J. G. J. Comp. Mater. 1987, 21, [10] Williams, J. G. in Proc. Int l Mechanical Engineering Congress and Exhibition: The Winter Annual Meeting of the ASME, Symposium on Mechanics of Plastics and Plastic Composites November 1995, San Francisco, USA. [11] Williams, J. G. Comp. Sci. & Tech. 1989, 35,

16 74 J. A. Nairn / International Journal of Adhesion & Adhesives 20 (1999) [12] Nairn, J. A. J. Applied Mech. 1997, 64, [13] Crandall, S. H., N. C. Dahl, and T. J. Lardner, An Introduction to the Mechanics of Solids, McGraw- Hill Book Company, New York (1978). [14] Timoshenko, S. J. Opt. Soc. Amer. 1925, 11, [15] Krishnamurthy, T., Ramamurthy, T. S., Vijayakumar, K. and Dattaguru, B. in Proc. Int. Conf. Finite Elements in Computational Mechanics, 2-6 December 1985, Bombay, India, [16] Kanninen, M. F. Int. J. Fract. 1973, 9, [17] Nairn, J. A. and P. Zoller ASTM STP , [18] Hudson, R. C., B. D. Davidson, and J. J. Polaha Proc. ICCM , I, [19] Polaha, J. J., B. D. Davidson, R. C. Hudson, and A. Pieracci J. Reinforced Plast. and Comp. 1996, 15, [20] Hunston, D. L. Comp. Tech. Rev. 1984, 6, [21] Hunston, D. L., R. J. Moulton, N. J. Johnston, and W. D. Bascom in Toughened Composites, ed. N. J. Johnston 1987, ASTM STP 937, [22] Liu, S. and J. A. Nairn J. Reinf. Plast. & Comp. 1992, 11, [23] Davies, P., W. Cantwell, C. Moulin, and H. H. Kausch Comp. Sci. & Tech. 1989, 36,

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