PROPAGATION OF CURVED CRACKS IN HOMOGENEOUS AND GRADED MATERIALS

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1 PROPAGATION OF CURVED CRACKS IN HOMOGENEOUS AND GRADED MATERIALS Abstract Matthew T. Tilbrook, Robert J. Moon and Mark Hoffman School of Materials Science and Engineering University of New South Wales, Sydney NSW Australia Deflection and deviation of cracks commonly occurs due to asymmetry in loading conditions or material properties, however the analysis of curved cracks has been limited to simplistic crack shapes, or has involved extensive levels of computation. This is pertinent to fracture in functionally graded materials (FGMs), in which spatial property variation can cause crack deflection and curvature. In this work, the approximation of curved cracks using simplified shapes is investigated. A simple model for crack-curvature in homogeneous materials is presented, which relates the volume of relaxed material around the crack to the effective crack length. Predictions of energy release rate (G) and propagation direction (θ k ) compared well with results from finite element simulations, for a range of crack shapes. The approximation of curved cracks in FGMs was investigated with finite-element simulations of crack propagation using exact and simplified crack shapes. The use of an approximated crack shape was observed to significantly influence subsequent predictions. It was concluded that solutions for straight cracks are not adequate for approximating curved cracks in FGMs, however piecewise-linear shapes provide a much closer approximation to exact curved-crack shapes. Introduction Deflection and curvature of cracks frequently occurs in situations in which non-uniformity in structure or mechanical property distribution leads to mixed-mode loading. An example of this is in functionally graded materials (FGMs), in which asymmetrical property distribution can lead to high mode-mixity []. The calculation of stress intensity factors or mechanical energy release rate for a curved crack is not straightforward, generally requiring a complex potential formulation [,] or a numerically intensive approach, such as weight functions [] or finite element method []. It has been suggested [-] that a circular arc crack may be approximated to an equivalent straight crack. Approximation to an equivalent straight crack simplifies calculation appreciably and is often sufficiently accurate []. In fact, this approximation is made when treating any macroscopically straight crack as being perfectly straight, despite the fairly ubiquitous presence of crack deflection at a microscopic level. The validity of the approximation in this case is generally not questioned. It has been demonstrated [] that cracks of arbitrary shape, even those exhibiting significant deflection, may be reasonably approximated to equivalent straight cracks. An argument was presented, based on the concept of relaxation volume, and a range of cracks were simulated with finite element (FE) analysis. The concept may conceivably be extended to curved cracks in functionally graded materials (FGMs). This is particularly relevant, as the spatial variation in composition and properties, which characterises FGMs, often leads to crack deflection and curvature [].

2 Analytical and computational models of fracture in FGMs have focussed almost exclusively to date on straight cracks []. This paper addresses the prediction of energy release rate (G) and propagation direction (θ k ) values for curved cracks by approximation to simplified crack shapes. Curved cracks in homogeneous materials were considered initially [], and these findings are outlined. The approach was subsequently applied to graded materials, and investigated via FE simulations of crack propagation. Curved Cracks in Homogeneous Materials The energy release rate for a straight crack under uniaxial loading may be described in terms of the volume of material that undergoes stress relaxation []. This is usually assumed to be an elliptical shape as in Figure (a). Assuming unit thickness, the energy relaxation around a crack of length a in a linear elastic material, in which strain energy density is ½ σ ε, is: U rel λπσ a = () E' where E is the effective Young s modulus for plane stress (E = E) or plane strain (E = E/(-ν )), and k describes the aspect ratio of the relaxation volume, as in Figure (a). An incremental extension of the crack leads to an incremental increase in relaxation volume, from which energy release rate may be calculated: du rel G = da du rel = da da da = kπσ a E () This expression is correct for k =. Though this is a very crude approximation, it works because the real stress distribution, though far more complex than assumed, scales with the square of crack size, so the rate of energy release scales with crack length. That the expression is correct for a simple value of k suggests that this physical interpretation has some validity. It has been suggested that an inclined or curved crack may be reasonably approximated to an equivalent straight crack [-]. This may be understood in terms of relaxation volume, as crack curvature is unlikely to affect the relaxation of stresses around the crack, except near the crack-tip. An expression for relaxation volume around a curved crack would be similar to that in Eq. (), though the volume will be dependent on the effective crack length, a eff, as in Figure (b). This is the projected length normal to the loading direction, η, and is related to the length, a, and transverse deviation, d, by a eff = a cos η + d a sin η. The volume for an edge crack as in Fig. (b) is thus: V = π () rel ka eff The value of k is assumed to be similar to the value for an internal crack, adjusted for the edge crack configuration, ie. k = (.).. An extension of c along the crack-tip direction, θ, leads to an increase in a eff given by a eff = c cos(θ- η). Hence the new relaxation volume is:

3 V πk + V = [ a(cosη + d sinη) + c cos( θ η)] rel rel () The energy release rate is calculated from energy release associated with this crack extension: U rel πkσ = E' [ a(cosη + d sinη) c cos( θ η) + ( c) cos ( θ η) ], () G = lim c U A rel = πkσ E' [ a(cosη + d sinη) cos( θ η) ] () The optimum propagation direction, θ k, relative to the crack-tip is that which maximises energy release rate []. The analysis above may be modified slightly to include the possibility of crack kinking: πkσ G( θ k ) = [ a(cosη + d sinη) cos( θ η + θk )] () E' This is maximised for: dg( θk ) dθ k = ie. θ k = η θ () The relaxation volume approximation may therefore be used to predict mechanical energy release rate and propagation direction. (a) Relaxed material a σ k a (b) η σ Loading angle k a eff a eff d a Transverse Deviation a σ σ Relaxed material FIGURE. (a) and (b) curved cracks showing approximate relaxation volumes. To confirm this, cracks with a range of shapes were simulated using ANSYS, a commercial finite element analysis package. The mesh was created under free-meshing conditions, with significant refinement around the crack-tip, resulting in approximately elements. Quarter-point elements were used at the crack-tip. Specimens were modeled in two dimensions under plane stress conditions. Mode I and II stress intensity factors, K I and K II,

4 were calculated from correlation of crack-opening displacement near the crack-tip and mechanical energy release rate was calculated from: G = ( K I + KII ) () E The propagation direction, θ k, was calculated using the maximum strain energy release rate criterion, with a small test kink from the crack-tip []. Various crack shapes were examined, and FE results were compared with predictions from the theoretical model. The effect of transverse deviation is illustrated in Fig. (a), whilst the influence of loading direction is shown in Fig. (b). Generally, good agreement was obtained between results of the simple analytical model and FE simulation, particularly for mechanical energy release rate values. Some deviation was noted, the variation in propagation direction with transverse deviation in Fig. (a) for example, which was attributed to curvature effects near the crack-tip. Further validation was obtained by comparison with the more rigorously calculated predictions of Noda et al. [] for several curved edge-crack configurations and good agreement was observed []. Energy Release Rate, G [J/m ] G - FEA θ k - FEA G - Analytical θ k - Analytical Propagation Direction, θ k [deg] Energy Release Rate, G [J/m ] θ k - Analytical G - Analytical θ k - FEA G - FEA - Propagation Direction, θ k [deg]..... Relative Transverse Deviation - Loading Direction, η [deg] FIGURE. Comparison of analytical approximation and FE results for G and θ k forcurved cracks, showing effect of variation in (a) transverse deviation, d, and (b) load angle, η. Default parameter values are a/b =. where b is specimen height, d =., θ =., η =. Curved Cracks in Graded Materials While this simple model could not be applied to FGMs due to the spatial property variation, it was assumed that curved cracks in FGMs, by analogy with those in homogeneous materials, could also be approximated by simple-shaped cracks. This was investigated with FE simulations, with a similar approach as for homogeneous materials, with automatic remeshing for crack propagation []. Propagation directions were predicted using the maximum energy release rate (G max ) and maximum tangential stress (MTS) criteria [].

5 The effect of using approximated crack shapes on crack-path predictions was observed. Cracks were approximated with straight, diagonal and piecewise-linear shapes, as shown in Fig. (a). For each crack configuration, crack propagation was simulated such that, at each increment, an approximated crack shape was used for calculation of the propagation direction for the subsequent increment. The general specimen configuration is shown schematically in Fig. (b) while an example of the finite element meshing used is given in Fig. (c). (a) Actual Crack (b) Material Material Material -Linear (c) Interface Regions Notch Tip FIGURE. Crack-path predictions. (a) Illustration of crack-shapes approximations used (b) Schematic of specimen and loading geometry. (c) Representative FE mesh. m] y [m y [mm] MTS MTS MTS MTS MTS MTS Diag. MTS P/wise MTS x [mm ] Bimaterial Interface x [mm] FIGURE. Crack propagation trajectories predicted using exact and approximated crack-shapes for (a) an asymmetrically notched homogeneous beam under three-point loading and (b) a bimaterial specimen under four-point loading. Specimens containing the following material interface configurations were examined: () no interface (asymmetrically-notched homogeneous specimen); () sudden interface (bimaterial

6 specimen); () wide continuously-graded interface; () narrow continuously-graded interface () wide discontinuously-graded interface ( steps). Four-point bend loading was used for all configurations except the homogeneous specimen. This was subjected to symmetric threepoint loading which resulted in mixed-mode loading. Crack paths obtained using each of the approximated shapes are compared with those obtained using the exact shape in Figures to. Fig. shows the paths in the asymmetrically-notched homogeneous specimen, and the bimaterial specimen. Fig. shows the paths for the two continuously-graded specimens with differing gradient steepness, whilst Fig. shows the path for discontinuously graded specimen, which contained three steps. y [mm] MTS MTS Diag. MTS P/wise MTS y [mm] MTS MTS MTS P/wise MTS Gradient/Material Interface x [mm] x [mm] FIGURE. Crack propagation trajectories predicted using exact and approximated crackshapes for graded specimens with (a) linear and (b) concave variation in Young s modulus, under four-point loading. Deviation was observed between predictions from the G max and MTS criteria even when the exact crack shape was used. The G max criterion is understood to provide more reliable predictions as it takes T-stresses into account []. Accordingly, the trajectory predicted with the exact crack shape and the G max criterion was regarded as the reference, or exact path, for each configuration. The MTS criterion however is generally easier to implement however, so paths predicted with this criterion are also included in the plots. Generally, the cracks predicted using straight and diagonal approximations deviated significantly from the exact path. These approximations tended to result in slightly less deflection. This could be attributed to a slight skewing of the crack-tip stress field, caused by the difference in crack-tip orientation of the approximated and exact crack shapes. The piecewise-linear approximation gave good predictions for all the configurations examined. It should be noted however that the crack paths in the configurations considered did not deviate significantly from a piece-wise linear shape. In other configurations, this may not be the case. Gradient/Material Interface

7 The inclusion of the MTS criterion predictions, which generally differed slightly from the G max criterion predictions, serves to demonstrate that the choice of deflection criterion can influence crack path predictions as significantly, if not more so, than the use of an approximated crack path. y [mm] MTS MTS MTS P/wise MTS Gradient/Material Interface x [mm] FIGURE. Crack propagation trajectories predicted using exact and approximated crackshapes for a graded specimen, with a stepped linear variation in Young s modulus, under four-point bend loading. A key motivation for this study was to determine whether the existing solutions for straight cracks in graded materials could be relevantly applied to deflected or curved cracks in graded materials. It appears that reasonable estimates may be obtained using solutions for straight or diagonal (inclined straight) crack shapes, though the deflection angle will tend to be underestimated slightly. The superiority of predictions from piecewise linear crack shape approximations indicates that the development of analytical solutions for branched or piecewise linear cracks in graded materials is a useful area for further work. It may be inferred that crack shape does have a significant effect on the predicted optimum propagation direction for a particular crack. While solutions for straight cracks may be adequate for crudely approximating curved cracks in FGMs, piecewise-linear shapes provide a much better approximation. Conclusions A range of crack shapes in homogeneous and graded materials were examined using finite element analysis, and the following conclusions were reached:

8 . Curved cracks in homogeneous materials may be reasonably approximated by equivalent straight cracks. This may be understood in terms of the volume of material around the crack from which strain energy will be released during crack propagation.. Cracks in graded materials may be reasonably approximated by equivalent straight cracks also, although systematic divergence occurs between those simulated exactly and those simulated using approximated crack shapes.. linear crack shapes provide a significantly better approximation than diagonal or straight crack shapes. Accordingly, analytical solutions for piecewise linear cracks in graded materials would be very useful, and should be a focus of further theoretical work in this area. References. Gu P., Asaro R.J., Int. J. Sol. Struct., vol., -,.. Cotterell, B., Rice, J.R., Int. J. Fract., vol., -,.. Chen, Y.Z., Theor. Appl. Frac. Mech., vol., -,.. Guagliano, M., Vergani, L., Fatigue Fract. Engng. Mater. Struct. ; vol., -.. Bittencourt T.N., Wawrzynek, P.A., Ingraffea, A.R., Sousa, J.L.A., Engng. Fract. Mech. vol., -,.. Noda, N., Oda, K., Ishi, K., JSME Int. J. A, vol. (), -,.. Kitagawa, H., Yuuki, R., Ohira, T., Eng. Fract. Mech., vol., -,.. Noda, N., Oda, K., Int. J. Fract., vol., -,.. Tilbrook, M.T., Hoffman, M., Int. J. Fract. (submitted).. Tilbrook, M.T., Moon, R.J., Hoffman, M., Comp. Sci. and Technol., (accepted).. Ashby, M.F., Jones, D.R.H., Engineering Materials, ed., Butterworth Heinemann,.. Qian, J., Fatemi, A., Eng. Fract. Mech., vol., -,.. Tilbrook M.T., Moon R.J., Hoffman M., Eng. Fract. Mech. (submitted).

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