TOUGHNESS OF PLASTICALLY-DEFORMING ASYMMETRIC JOINTS M. D. Thouless, M. S. Kafkalidis, S. M. Ward and Y. Bankowski Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 4809 U.S.A. Ford Research Laboratory, Ford Motor Company, Dearborn, MI 48, U.S.A.. Introduction Owing to current trends towards light-weight energy-efficient vehicles, the use of adhesives as a replacement for traditional joining techniques has many advantages for the automotive industry. For example, spot-welding is difficult in light-weight materials such as aluminum and impossible in composites and polymers. These materials can be effectively joined with adhesives. Furthermore, adhesively-bonded joints can exhibit substantially greater stiffnesses than spot-welded joints (). The use of Linear Elastic Fracture Mechanics (LEFM) to describe the failure of adhesive joints is now wellestablished (-4). These techniques are, of course, only appropriate when plasticity is limited to a zone that is small in comparison with the specimen geometry. However, an important design consideration for the use of adhesive joints in the automotive industry is how they fail during gross plastic deformation of the structural members. Energy absorption during a crash is provided by this plastic deformation. Design for crash-worthiness with adhesive joints requires that the properties of an adhesive be optimized so that the energy absorption can be maximized by controlled failure of the interfaces. Techniques to assess adhesive failure in the presence of plasticity is therefore required for design purposes. In a previous paper, a testing geometry for adhesive joints in which steady-state crack propagation occurred in the presence of gross plastic deformation was described (5). An analysis based on a steady-state energy balance was presented that described the conditions required for crack propagation. This analysis was used to compute the toughness of symmetrical adhesive joints; and it was shown that the resulting value of toughness was independent of the thickness of the plastically-deforming substrate material. In the present paper, it is demonstrated that the previous result can also be derived from Hutchinson s I-integral [6,7]. The results of additional experiments are presented to show that the geometry-independence of the technique applies also for asymmetrical joints.. Analysis The general problem considered here is illustrated in Figure. An adhesively-bonded double-cantilever beam has a width b and arms of thickness h and h, respectively. A moment M is applied to each arm. For this geometry, crack propagation occurs under steady-state conditions, and the path-independent I-integral introduced by Hutchinson can then be used. The I-integral is defined by $ I = & u Wn σ ij n i % j ) dl () C x ( where W is the history-dependent work density of the material at any point relative to its initial state of zero strain, n is the unit outward normal to the contour C, σ is the stress, and u is the displacement [7]. Hutchinson has shown that this is a path-independent integral that can be used as a crack-driving force even in the presence of non-proportional loading, providing steady-state conditions apply [7]. In the particular geometry considered here, the only contribution to the I-integral is from the segments parallel to the x -axis and for which x is negative. The I-integral then simplifies to
h [ ] I = W +σ ε h dx () If the material on both sides of the joint are assumed to be identical and to follow a power-law constitutive equation so that σ = Aε n, (3) W is given by ( ) = σ dε W x ε ( x ) = 0 A ( n + ) ε n + ( x ). (4) For pure bending in the arms, ε (x ) is given by # ε ( x ) = % x + h & ( $ R R x 0 + (5a) " ε ( x ) = $ x + h % # R R & x 0 (5b) where R is the radius of curvature along the neutral axis for the arm above the x -axis, and R is the radius of curvature along the neutral axis for the arm below the x -axis. In addition, from simple-beam theory, it can be shown that the radius of curvature, R, of a plastically-deforming beam is related to the thickness of the beam, h, and the applied moment, M, by /n! bah n+ $ R =. (6) "# n+ ( n + )M %& Therefore, substituting Eqns. (3-6) into Eqn. (), and integrating, results in an expression for the I-integral, I = Mn /n! ( n + )M $! b( n + ) " # Ab % & ( +n)/ n h + $ (+ n)/n (7) " # h % & which is identical to the expression derived for the crack-driving force using a steady-state energy-balance argument in Ref. (5). In this reference, it was further demonstrated that, by using Eqn. (6), it is possible to deduce the critical moment required to fracture the joint by measuring the radius of curvature of the two arms, R and R, after failure. The toughness of the joint is given by n+ An h Γ = n + ( n + ) ( n + ) R + h n+ " % n + n + (8) # $ R &. Experiments The details of the sample preparation and of the experiments are provided in Ref. (5). Rectangular, flat coupons were cut from sheets of different thicknesses of a 5754 aluminum alloy. All the coupons were 0.0 mm wide and 90.0 mm long. These coupons were bonded together to form test samples with different ratios of arm thicknesses, h and h. The samples were bonded over a distance of 30.0 mm from one end of the coupons by a 0.5 mm layer of a commercial structural adhesive. Three different adhesives were used, and six different ratios of h/h were examined. After curing, the sample was placed over a wedge, and a drop weight was released onto the bonded end of the sample (Fig. ). The impact of the weight forced the wedge through the sample, causing the arms to bend and the adhesive to fracture at a constant rate of m/s. It was observed that, after failure, the aluminum substrates were deformed into arcs of circles in the region of the adhesive (Fig. 3). The radii of curvature of these arcs were measured using a ruler on the magnified shadows of the specimens projected on a screen. The values for the power-law hardening exponents, A and n, were obtained by fitting power laws to tensile tests performed on the aluminum substrates. The best-fit values for A and n depended slightly on the particular thickness of the aluminum used (see Ref. (5)). The exponent n varied between 0.3 and 0.7, while A varied between 350 MPa and 40 MPa. As noted in Ref. (5), the possibility that the power-law fits may have introduced errors into the calculation of Γ has been eliminated by comparing the results to those obtained by numerically integrating the stress-strain data directly. Furthermore, it was observed that Γ is not very sensitive to particular choices of A and n provided the area under the moment-curvature curve is well-described by the choice in the appropriate range. The
appropriate values of A and n for each arm were used with the measured values of the radii of curvature to compute the toughness of the bonded joints. 3. Discussion The mean values of the measured curvatures (κ), the corresponding values of the moments deduced to have acted on each arm, and the toughness of the joints are shown in Table. Typically, for large aspect ratios, one arm did not deform plastically, and no radius of curvature could be determined. In these cases, the moment was measured from the deformed arm, and it was assumed that this same moment acted on the undeformed arm. As expected, it was found that in these cases, the deduced moment was of a magnitude that would not cause plastic deformation of the thicker arm. When the toughness was calculated, it was seen that the contribution from the strain energy in the elastic arm to the crack-driving force was very small. No reliable results could be obtained for the joints with the two most extreme aspect ratios formed with the toughest adhesive. This was because the deformation was so extreme that the arms formed a tight enough arc of a circle to interfere with the hammer as it descended (Fig. 3). This illustrates a limitation of this technique if the bonded region is too long compared with the radius of curvature. The results of Table show that, for a particular adhesive, Γ is independent of the aspect ratio of the two arms. Furthermore, the present results are in excellent agreement with those obtained in Ref. (5) for the same adhesives. It should be noted that, although the measured curvatures could be quite different for the two arms, the corresponding moments that were calculated from these radii were, to a very close approximation, the same. Any difference in the moments is attributed to the fact that the samples may not have been placed perfectly symmetrically on the wedge, so that rotation of the samples occurred during the test. The difference in moments acting on the two arms implies that an additional counter-balancing moment must have been exerted on the bonded portion of the sample. However, a close examination of the results shows that, even for the largest imbalance, the additional moment makes a negligible contribution to the calculation of the overall value of toughness. From Eqn. (6), it can be seen that, if both arms of the wedge-impact specimen are made of the same material and an equal moment is applied to each, the ratio of the resulting radii of the arms is given by R = h ( +n )/n! # $ &. (9) R " h % In other words, for the aluminum sample, the ratio of the radii increases approximately as the ninth power of the thickness ratio. This demonstrates why the deformation occurs predominately in the thinner arm, even for relatively symmetrical samples. Furthermore, from Eqn. (6) it can be seen that the relative contributions to the crack driving force has the same dependence on the thickness ratio. It is also of interest to compare the results obtained in these experiments with interfacial fracture studies made under linear-elastic conditions. A dependence of the toughness on the symmetry of loading ( mixedmode effects) is often found (8-). However, the loading geometry that most closely mimics the situation described here is where an asymmetrical double cantilever beam is opened by means of a wedge (3, 4). The extent of the mode-mixedness is limited in this case, and loading effects on toughness do not appear to be substantial. However, it was observed in this geometry that, owing to the asymmetry of the loading, the crack path always ran near the interface between the adhesive and the thinner arm (3, 4). Similar effects of the mode-mixedness on crack path have been observed and explained in other adhesive systems (8,, 5, 6). In the present experiments it was also observed that the crack inevitably ran close to, or at, the interface of the thinnest arm. 4. Conclusions A wedge-impact technique has been used to determine the toughness of asymmetric joints made of aluminum bonded with three different commercial adhesives. It was found that the toughness did not vary The adhesives tested here were from a different batch and of a different age from those tested in Ref. (5). They therefore exhibited a small difference in the toughness.
with the aspect ratio of the two arms for the range of thicknesses used. The values of toughness obtained were consistent with previous work in which it was shown that the toughness did not depend on the thickness of the adherends. These results indicate that, with this particular type of loading, a geometricallyindependent value for the toughness can be found, despite the presence of a significant amount of plastic deformation. Although no mixed-mode effects were observed in these experiments, the issue of whether they are important for geometries that introduce even more severe shear loading needs to be addressed in future work. Acknowledgments This work was supported by Ford Motor Company, a Rackham Faculty Grant from the University of Michigan, and NSF Grant No. CMS 96445. References. D. A. Wagner, C. M. Cunningham and M. A. Debolt, Proc. International Body Engineering Conference, 3-8 (993).. A. N. Gent and A. J. Kinloch, J. Polymer Sci. A, 9 659-668 (97). 3. A. Gledhill, A. J. Kinloch, S. Yamani and R. J. Young, Polymer, 9, 574-58 (978). 4. M. D. Chang, K. L. DeVries and M. L. Williams, J. Adhesion 4, -3 (97). 5. M. D. Thouless, J. L. Adams, M. S. Kafkalidis, S. M. Ward, R. A. Dickie and G. L. Westerbeek, submitted to J. Mater. Sci.. 6. J. W. Hutchinson, Harvard University Report DEAAP S-8 (AFSOR-TR-74-040) (974). 7. Y. Wei and J. W. Hutchinson, submitted to Acta Mater.. 8. J. W. Hutchinson and Z. Suo, Adv. Appl. Mechs., 9, 63-9 (99) 9. H. C. Cao and A. G. Evans, Mechs. Mater., 7, 95-304 (989). 0. K. M. Liechti and Y. S. Chai, J. Appl. Mechs., 4, 95-304 (99).. J. S. Wang and Z. Suo, Acta Metall. Mater., 38, 79-90 (990).. M. D. Thouless, J. W. Hutchinson and E. G. Liniger, Acta Metall. Mater., 40, 639-649 (99). 3. M. D. Thouless, Acta Metall. Mater., 38, 35-40 (990). 4. M. D. Thouless, Scripta Meter. Mater., 6, 949-95 (99). 5. N. A. Fleck, J. W. Hutchinson and Z. Suo, Int. J. Solids Structs., 7, 683-703 (99). 6 A. R. Akisanya and N. A. Fleck, Int. J. Fract., 58, 93-4 (99).
Table (Aluminum substrates) LMD-4 Adhesive h (mm) h (mm) κ (m ) - κ (m ) - M M (Nm/m) (Nm/m) Γ (kj/m ).0 3.0 N.A. < N.A. N.A. N.A..0.3 N.A. < N.A. N.A. N.A..6.3 96 ± 6.3 5 ± 04.4 ± 0.4.6.0 94 ± 6 0 ± 5 ± 09 ± 5.5 ± 0.4.0.3 53 ± 3 8± 60 ± 68 ± 5. ± 0.3.3.3 83 ± 6 66 ± 8 7 ± 67 ±.0 ± 0.3 Average.3 ± 0.3 XD-4600 Adhesive h (mm) h (mm) κ (m ) - κ (m ) M - M (Nm/m) (Nm/m) Γ (kj/m ).0 3.0 9 ± 6 < 43.5 ± 0.5 (43). ± 0..0.3 3 ± 6 < 44.7 ± 0.5 (45).3 ± 0..6.3 55 ± 3 3.6 ±.6 99 ± 6 ± 6. ± 0..6.0 5 ± 3 8.6 ±.6 98 ± 05 ± 5. ± 0..0.3 3 ± 3 ± 4 ± 55 ± 6. ± 0..3.3 6 ± 8 46 ± 7 66 ± 6 ±.3 ± 0. Average. ± 0. Essex-7330 Adhesive h (mm) h (mm) κ (m ) - κ (m ) - M M (Nm/m) (Nm/m) Γ (kj/m ).0 3.0 ± 6 < 43.7 ± 0.5 (44). ± 0..0.3 4 ±6 < 43.0 ± 0.5 (43). ± 0..6.3 54 ± 3 < 99 ± (99). ± 0..6.0 5 ± 3 3.3 ±.0 98 ± 85 ± 7. ± 0..0.3 3 ± 5 5. ±.7 4 ± 5 6 ±.0 ± 0..0.0 83 ± 5 40 ± 7 39 ± 3 ±.0 ± 0. Average. ± 0.
M x h x M C h Figure. The general problem of an asymmetric double-cantilever beam analyzed in this paper. A contour C is shown along which the I-integral is taken. Hammer Sample Wedge Figure. The test geometry in which a asymmetric double-cantilever beam is fractured by means of a wedge being forced along the interface under the impact of a hammer. Figure 3. Three aluminum samples bonded by the XD-4600 adhesive after failure.