DESIGNING DYNAMIC TESTS TO ASSESS RATE DEPENDENCE IN LARGE-SCALE CRACK BRIDGING C. Lundsgaard-Larsen, 1 R. Massabò, 2 and B. N. Cox 3 1 Technical University of Denmark, Lyngby, Denmark 2 DICAT, University of Genova, Genova, Italy 3 Teledyne Scientific Co LLC, Thousand Oaks, CA, U.S.A bcox@teledyne.com SUMMARY A numerical study is used to design test geometries and loading histories that can probe the mode II bridging effect of through-thickness reinforcement in composite laminates loaded at high strain rates. Rate-dependence in the assumed cohesive law of a cohesive fracture model causes large enough changes to calibrate the law, if tests are properly selected to vary the crack sliding displacement rate. Keywords: dynamic fracture, delamination, rate effects, cohesive model INTRODUCTION Through-thickness reinforcement such as stitches or pins can increase the delamination resistance of laminated polymer or ceramic matrix composites by an order of magnitude or more [1-9]. Provided higher loads are not pre-empted by adverse effects of the reinforcement on other failure mechanisms [5], the regime of linear structural response can be extended and the damage tolerance of the structure can be enhanced, offering the materials designer significant opportunities for matching the performance of a material to specific applications. Many test data have now been published on the crack bridging effects of stitches and pins under static loading, both for single stitches or pins in model specimens and for arrays of stitches or pins bridging cracks in delamination specimens [5, 9-21]. The observed mechanisms of deformation of the reinforcement under mixed mode loading have been modeled with reasonable success, so that a good understanding exists of the main material factors influencing the efficacy of pins or stitches as bridging entities [16, 22, 23]. The bridging effect can be represented accurately by a cohesive traction law in a nonlinear fracture model. Accurate accounts of static delamination test data ensue. Some indications already exist that rate effects should be expected in bridging due to through-thickness reinforcement. Experiments in which fibres were pulled out of a substrate at rates of 1 mm/min and 100 mm/min show that the friction between the pulled out fibres and the laminate increases with speed [24]. A possible physical explanation given by the authors is visco-elastic behaviour of the fibre/matrix interface, coupled with a temperature increase due to friction. Theoretical studies of the micromechanics of fiber pullout also show that the inertia of the fibers will create rate effects in the fibre pull-out problem [25, 26].
However, no work has been reported in which the bridging tractions have been deduced from fracture tests in the dynamic regime, where the inertia or dynamic material properties might change the bridging effect. This lack can frustrate optimal design, because optimality does not necessarily result from inserting the maximum throughthickness reinforcement density. While some properties rise monotonically as the density increases, cost, in-plane properties, and energy absorption can all be affected unfavorably. Through-thickness reinforcement is expensive and causes harm to inplane fibers, compromising strength and fatigue life. If the through-thickness reinforcement is too effective in suppressing delamination, it can cause the structure to exhibit brittle failure characteristics, with minimal energy absorption, a disadvantage for ballistic performance. The question therefore arises of what experiments will be most useful for acquiring quantitative knowledge of the effects of the through-thickness reinforcement, specifically data that determine the cohesive tractions supplied by pins, etc., during dynamic loading and crack propagation. Numerical studies are used in this paper to find information-rich experiments. CHOICE OF SPECIMEN GEOMETRY AND MATERIAL PROPERTIES The subject specimen is an ENF specimen loaded in three-point bending (Fig. 1). Choosing an ENF configuration avoids the complexity of mixed mode crack conditions, which are unnecessary here, and allows relation to the well-studied static case of large scale bridging under mode II conditions [10, 27, 28]. The material is assumed to be linear elastic with material properties typical of a carbon fibre reinforced epoxy with a quadraxial lay-up (90,±45,0) (in-plane quasi-isotropic). Specimen dimensions must satisfy the following considerations, which reflect the length scales expected in large scale bridging [28]. The specimen length should be longer than the process zone length to accommodate steady state propagation of a mature zone. But if the specimen becomes very long this will lead to awkwardly large deflections. Simulations show that if the loading displacement is applied at a high acceleration at the midpoint of a very long slender specimen, the material close to the load point will move much faster than the material distant from the load point, which is constrained by inertia. This leads to a deflection shape remarkably different from the quasi-static case, and this non-equilibrium state will eventually lead to oscillatory behaviour of the specimen. This effect is increased with specimen length, leading to preference for a relatively short specimen of length (between supports) of 2L = 100 mm when the thickness 2h = 10 mm. As for specimen thickness, a thicker arm reduces the maximum stresses, but increases the size of the crack process zone. For the chosen thickness of 10 mm, the maximum stress predicted in the longitudinal direction of the beams is approximately 410 MPa in tension and 360 MPa in compression; for anticipated composite strengths, unwanted failure mechanisms should therefore be absent. The precrack length should be long enough to avoid unstable crack growth, yet short enough for the process zone to develop and reach interesting speeds before the crack tip reaches the end of the specimen. Based on numerical analysis for representative large scale bridging, the notch length is chosen to be 20 mm, i.e. 0.4L. FINITE ELEMENT FORMULATION The 2-D finite element model consists of two beams connected through zero thickness
cohesive elements that partly cover the beam length. The pre-crack constitutes the part of the beam without cohesive elements. A frictionless contact interaction is formulated in a penalty master-slave configuration between the two beams to prevent interpenetration. The mesh is mainly uniform with a refinement near the interface. Nodes between bulk and cohesive elements are coincident and the side length of the cohesive elements is 0.125 mm. Mesh convergence is found by varying the element size. Three idealised cohesive laws are considered (Fig. 2), one representing a laminate without z-pins (triangular) and two representing z-pinned laminates (triangular and trapezoidal). The cohesive laws with an area below the traction-separation curve of 5000 J/m 2 and critical displacement δ c = 0.5 mm correspond to z-pinned laminates, whereas area and critical displacement for laminates with through-thickness reinforcement are reduced from this model by a factor of 10, which is an estimate based on previous experience. The three cohesive laws are used as examples to test the influence of the cohesive law on the fracture behaviour of the specimen. The triangular and trapezoidal shapes are chosen since they are simple and adequate for initial analysis. The properties of the cohesive laws are illustrated in Fig. 2 and listed in Table 1. Only selected results from all cases are presented in this paper. LOADING HISTORY In simulations, the specimen is loaded in displacement control. The loading history is an idealisation of experimental loading conditions similar to those of a split Hopkinson pressure bar with a special pulse shaper. The velocity of the load point is zero at time t = 0 and increases linearly to a constant speed v 0 over the time t 0 (Fig. 3). If v 0 is small and t 0 large, the loading situation is quasi-static. If v 0 is large and t 0 approaches zero, the loading corresponds to a hard impact and the acceleration in the interval (0, t 0 ) is large. The ramp time t 0 will have a large influence on the dynamic fracture behaviour of the specimen. If the ramp time is very short natural vibrations of the specimen may be excited, leading to an oscillating stress state near the crack tip. The challenge is to choose a ramp time sufficiently small to obtain rapid crack growth, yet sufficiently large to avoid a highly fluctuating fracture behaviour. Andrews et al. analysed the effect of ramp time on the dynamic amplification of the energy release rate for specimens with a stationary crack [29]. They found that if the ramp time was equal to the first natural vibration period T 1, there was no dynamic amplification, and no subsequent oscillations. Conversely, smaller rise times led to oscillations in the specimen. The cases studied by Andrews et al. are different from the present, since loading is force controlled (the force is increased linearly instead of displacement), and the crack is stationary. However, by considering the fracture behaviour of the tested specimens at various ramp times, T 1 was found to be a suitable compromise to obtain rapid crack growth while limiting stress wave effects. An eigenvalue analysis of the specimen was performed. The bond-line outside the precrack was assumed intact and the nodes between the two beams were fixed except along the pre-crack where cohesive elements were inserted. The cohesive elements were given a very large stiffness in the normal direction and a negligible stiffness in the tangential direction, to allow sliding but restrain normal opening of the pre-crack. This ensures that only the relevant natural vibration modes will appear in the analysis, i.e., normal opening of the pre-crack is prohibited. For a specimen of length 2L = 100 mm the
oscillation period is T 1 = 188 μs, i.e., a ramp time t 0 = 188 μs was chosen. For a longer specimen with 2L = 210 mm, by the same considerations T 1 = t 0 = 558 μs. RESULTS FOR HIGH-SPEED LOADING Results are presented for specimens with z-pins loaded at high speed. An approximation is suggested for relating the process zone sliding displacement rate to the crack tip speed and displacement profile, simplifying assessment of the information content of tests. Laminates with Z-Pins Loaded at High Speed Consider representative results at a load point velocity v 0 = 20 m/s (Fig. 4). Comparison with results for quasi-static loading reveal dynamic effects (effects only found for dynamic loading). The reaction force (Fig. 4a) shows strong oscillations, even though the crack length history (Fig. 4b) remains relatively smooth. In the loading displacement interval 4 5 mm, the leading and trailing edges move at approximately the same constant velocity (140 m/s) (Fig.4b), and furthermore, the sliding displacement profile is also constant within this load point displacement interval, a useful characteristic for experimental analysis (see below). A similar interval appears for other loading rates between 10 and 50 m/s; the crack propagation speeds and crack sliding displacement rates within the constant interval are given in Table 2. The maximum crack propagation velocity in Table 2 is 273 m/s, which is reached at a loading velocity of 40 m/s. An increase in loading velocity to 50 m/s does not lead to increased propagation speed; instead, the higher loading speed generates multiple process zones and increased oscillations, which make data difficult to interpret. The sliding displacement rate in the process zone (as reported in the third column of Table 2) is that predicted at the trailing edge of the zone. The velocity varies within the process zone from a minimum at the leading edge to a maximum at the trailing edge. The maximum sliding rate in the process zone within the considered loading interval is approximately 4.7 m/s for a loading speed of 20 m/s. The sliding rate at the trailing edge of the process zone for loading speeds 10 50 m/s are listed in Table 2. A Simplified Approach to Data Analysis The process zone sliding rate is useful as a rate measure, since it describes the rate of the fracture processes (e.g., those by which z-pins are pulled out). The following presents a simple method for relating the sliding rate in the process zone to the crack displacement profile and crack tip speed, which are both are experimentally measurable using a digital image correlation system. Figure 4b shows that the crack speed becomes approximately constant within a small range of load point displacements between 4 and 5 mm. Under conditions where the crack displacement profile and the crack velocity are time-invariant, the crack displacement rate of a fixed point in the process zone is given for either the opening or sliding displacement, u i, i = 1 or 2, by where v tip is the crack velocity (Fig. 5). u ui η = i i = v tip (1) t x
To test the usefulness of the assumption of uniform profile and velocity conditions, a comparison was made of the sliding velocity computed from simulations by: (1) the right-hand side of Eq. (1), where the computed displacement profile u(x) of the process zone at time t n is differentiated with respected to the x-direction and multiplied by the crack tip speed, and (2) from the difference in the computed displacements at two time steps divided by the time step, i.e., (u(x,t n )-u(x,t n-1 ))/(t n -t n-1 ), where (t n -t n-1 ) is the time step between outputs from the simulation, equal to 1/200 of the total time domain. The sliding velocity profiles computed by the two methods are plotted in Figure 6. The actual sliding rate profile and that estimated assuming time-invariance correlate well. The approach is only valid when the crack tip speed and the displacement profile are constant, which is approximately true only for a limited interval in the simulation. However, the presence of this interval opens a simple route to testing for rate dependence. SENSITIVITY TO RATE DEPENDENCE IN THE COHESIVE LAW The question of whether a particular dynamic test will yield data that contain information about possible rate dependence in the cohesive law can be addressed by analyzing solutions such as those presented in the previous sections as though they were the outcome of actual tests. Even this theoretical exercise is complex if undertaken most generally. Therefore a simplified approach is taken here by restricting the domains of the solutions to those where profiles and the crack speed are approximately invariant and assuming idealized cohesive laws. Considering a limited solution domain is conservative with respect to assessing information content, because the full domain must contain more information. Assuming idealized cohesive laws is also conservative, because even if a more general form of the law proves partly indeterminate when actual data are analyzed, the degrees of freedom that can be determined in it cannot be less than those inferred for the simpler law. As a practical note, with the simplified approach one can also work around the limitation of the ABAQUS code that it cannot accept a general rate-dependent cohesive law. Consider a separable linear cohesive law of the form p = q + s u i i i i (i = 1,2; u i w i ) (2a) p i = 0 (δ i > w i ) (2b) which experiments have shown to be a useful approximation for stitches in tension [9] or shear [10] and pins in tension or shear [15] in cases of pure mode I or pure mode II loading. Assume that rate dependence consists of linear changes in the parameters of Eq. (2): q i s i (0) q i = q i + α i η i (i = 1 or 2) (3a) (0) s i = s i + β i η i (i = 1 or 2) (3b) (0) w i = w i + γ i η i (i = 1 or 2) (3c) w i (0) (0) (0) where,, and are the parameter values for the static case and η i = du i /dt. The discussion above reveals that the crack velocity is approximately constant with some value v tip for certain intervals during a delamination test, e.g., for load point
displacements ranging from approximately 4 to 5 mm in the cases of Fig. 4; and that the opening and sliding displacement profiles are approximately linear functions of location along the crack for locations that are 10 50 mm behind the crack tip. Indications of the sensitivity of an experiment to rate dependence can be based on analyzing the domains where the crack speed is uniform and the profile linear. In this regime, the capacity of an experiment for testing rate dependence is indicated by the range of values of the product - u i / x v that can be attained, e.g., by varying the load point velocity. Table 2 in conjunction with the profile data of Fig. 6 show that, by this measure, significant rate dependence does in fact arise in the dynamic ENF tests, provided the loading rate is chosen as indicated by the range in Table 2 where variation arises in v tip. CONCLUSIONS The design of a test procedure must be based on many different and often contrasting requirements that include: - obtaining data that probe rate dependence in the cohesive law, which implies shorter ramp times and higher loading speeds. - avoiding the large crack speeds for which the process is dominated by kinetic energy and the effects of the reinforcement become negligible. - obtaining crack propagation both in the domain were the process zone is developing and in that where it is fully developed to maximize the information content of the experimental measurements on all model parameters. To satisfy this requirement the half-length of the ENF specimen should be several times longer than the characteristic bridging or cohesive zone length scale. - avoiding large oscillations of the specimen and high-order, high-frequency waves, which implies shorter specimens with shorter notches and larger ramp times. - avoiding failure by mechanisms other than delamination fracture, which implies thicker specimens. For typical carbon/epoxy properties, a test geometry and a range of loading speeds and ramp times can be selected that meet the above conditions. The analyses revealed domains with simple characteristics, where crack growth is approximately steady state and the crack profile is at least partially linear. In these domains, the crack sliding speed is constant along the linear crack profile, enabling a simple method for inferring rate dependence in cohesive laws from test data. Investigations confirmed that different crack sliding speeds could be obtained in the designed specimens on varying the loading parameters in the chosen range. Information-rich experiments can be designed for typical material parameters. ACKNOWLEDGMENTS BNC, CLL and RM supported by the U.S. Army Research Office through contract number DAAD19-99-C-0042, administered by Dr. David Stepp. RM supported by the U.S. Office of Naval Research through contract no. N00014-05-1-0098, administered by Dr. Yapa D.S. Rajapakse.
Figure 1: Dimensions of centre-loaded ENF specimen. Figure 2: Cohesive laws used in the simulations. Figure 3: Schematic displacement (a) and velocity (b) histories for the load point in the simulations. Figure 4. Results for specimen with 2L=100 mm, a=20 mm and v=20 m/s. Cohesive laws correspond to a z-pinned laminate, the solid line is triangular and the dashed is trapezoidal.
Figure 5: Schematic illustration of opening profile for a pure mode I crack. Figure 6: Actual computed relative sliding displacement rates in the process zone compared with the result of the simplified prediction based on Eq. (1). The simulation in subplot (a), (b), (c), or (d) corresponds to results in Figure 4 at load point displacement 4.0, 4.3, 4.6 and 4.9 mm, respectively.
Table 1: Properties defining the three cohesive laws. σ 0 [MPa] δ c [mm] J c [J/m 2 ] Not reinforced (triangular) 20 0.05 500 Reinforced (triangular) 20 0.5 5000 Reinforced (trapezoidal) 10 0.5 5000 Table 2: Crack tip velocities and process zone crack sliding displacement rates obtained during various loading rates. Loading Crack tip PZ sliding rate speed [m/s] speed [m/s] (at trailing edge) [m/s] 10 85.1 2.3 20 140 4.7 30 236 6.7 40 273 7.7 50 271 7.5 References 1. Cox, B.N., R. Massabò, and K.T. Kedward, Suppression of delaminations in curved structures by stitching. Composites Part A, 1996. 27A: p. 1133-1138. 2. Glaessgen, E.H., I.S. Raju, and C.C. Poe, Delamination and Stitch failure in stitched composite joints. AIAA Journal, 1999. paper 1247. 3. Holt, D.J., Future Composite Aircraft Structures May be Sewn Together. Automotive Engineering, 1982. 90(7). 4. Lee, C. and D. Liu, Tensile strength of stitching joint in woven glass fabrics. Journal of Engineering Materials and Technology, 1990. 112: p. 125-130. 5. Mouritz, A.P. and B.N. Cox, A mechanistic approach to the properties of stitched laminates. Composites, 2000. A31: p. 1-27. 6. Rhodes, M.D. and J.G. Williams. Concepts for Improving Damage Tolerance of Composite Compression Panels. in 5th DoD/NASA Conference on Fibrous Composites in Structural Design. 1981. 7. Sawyer, J.W., Effect of stitching on the strength of bonded composite single lap joints. AIAA Journal, 1985. 23: p. 1744-1748. 8. Tada, Y. and T. Ishikawa, Experimental evaluation of the effects of stitching on CFRP laminate specimens with various shapes and loadings. Key Engineering Materials, 1989. 37: p. 305-316. 9. Turrettini, A., in Materials Department. 1996, University of California, Santa Barbara: Santa Barbara, California. 10. Massabò, R., D.R. Mumm, and B.N. Cox, Characterizing mode II delamination cracks in stitched composites. International Journal of Fracture, 1998. 92: p. 1-38. 11. Dransfield, K., C. Baillie, and Y.-W. Mai, Improving the delamination resistance of CFRP by stitching - a review. Composites Science and Technology, 1994. 50: p. 305-317. 12. Dransfield, K., L.K. Jain, and Y.-W. Mai, On the effects of stitching in CFRPs-I. Mode I delamination toughness. Composites Science and Technology, 1998. 58: p. 815-827.
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