Proc. (CD), 12th Intern. Con. on Fracture (ICF-12) (held in Ottawa, Canada, July 2009), Robert Bell, ed., Carleton University, pp. 1 10. Size Eect on Strength o Bimaterial Joints: Computational Approach versus Analysis and Experiment Jia-Liang Le 1, Ferhun C. Caner 2, Qiang Yu 1, and Zdeněk P. Bažant 1 1 Northwestern University, IL, USA; 2 Technical University o Catalonia, Spain Abstract Metal-composite joints connecting conventional steel to an advanced iber composite are an attractive structural system or hybrid ship hulls as well as aircrat rames. The objective o this study is to analyze the size eect on the strength o hybrid joints numerically, theoretically and experimentally. In the numerical analysis, linear elastic continuum elements are used or the composite and steel while cohesive racturing zero-thickness elements are used or the steelcomposite interace. The numerically simulated size eect on the strength o the joints is veriied by analytical and experimental studies. Analytical ormulation o the asymptote o size eect law is anchored at the large size limit by LEFM (linear elastic racture mechanics) solutions with complex singularity. A general approximate size eect law, spanning all sizes, is urther derived via asymptotic matching. The experimental studies involve testing o geometrically similar specimens with size ratio o 1:4:12. The analytical, numerical and experimental studies all indicate that the strength o metal-composite joints exhibits a strong size eect. 1. Introduction Bimaterial joints o metal and iber composite, with a very thin adhesive between them, are the crucial component o an eective structural system or large ships [1]. They are also needed or large uel-eicient aircrat. Due to the stress concentration caused by the corner geometry and material mismatch, the bimaterial corners are the critical locations or the design o hybrid joints. Fracture mechanics o various bimaterial joints has been extensively investigated [2, 3]. However, the scaling o racture and associated size eect or such joints have not yet been explored. Previous studies have shown that iber composite structures exhibit a strong size eect [4-7], and so one may expect the same to occur in the hybrid joints as well. Correct understanding o the size eect on the strength o joints is important or correct extrapolation o small-scale laboratory test results to ull-size joints. In this study, we consider a double-lap hybrid joint specimen shown in Fig. 1a. We deine the nominal strength σ N = P max /bd, where P max = maximum load, b = width o the specimen (in the third dimension), and D = in-plane characteristic dimension. Here we chose D = interace length (Fig. 1a). To avoid small 1
secondary eects o the length o crack ront edge in the third dimension (stemming rom a transition rom plane strain to plane stress along the edge), it is better to consider two-dimensional similarity, i.e., set the width b = constant. This paper aims to study numerically, analytically and experimentally the eect o structure size D on the nominal strength σ N o the hybrid joint. 2. Computational Simulation o Metal-Composite Joints Computational simulation o the metal-composite joints plays an important role in this study, not only to obtain the size eect law or small size structures, but also to design the specimen dimensions or experimental studies. For accurate determination o the size eect on the strength o joints, the experimental investigations should span a large size range. Thus large variations o the skin thickness are required. In this study, we are interested in interacial ailure; hence all the specimens must be designed to ail in this manner. Thereore, it is important to determine the maximum load below which the skin and steel block thicknesses would suice to avoid axial tensile ailure o the skins as well as plastic yielding o the steel block. This maximum load depends not only on the shear strength o the interace, but also on the racture energy. It is well known that, or the purposes o size eect calculations (i.e. calculation o peak loads), only the initial tangent o the cohesive law is relevant. Thus, the cohesive law assumed in the calculations is a linear sotening law governing the racture o the steel-composite interace, as shown in Fig. 2. In the inite element model, the steel block o the hybrid joint is discretized by hexahedral inite elements with an isotropic linear elastic material law. For the continuum element that discretizes the composite skin, orthotropic linear elasticity is assumed. At the steel-composite interace, cohesive racturing zero-thickness interace elements made up o two our-node quadrilaterals, which connect the aces o two adjacent hexahedral continuum elements, are inserted. In these elements, the suraces o the two quadrilaterals are initially coincident. During the loading, these suraces are allowed to separate or slip, to simulate crack opening or shearing racture, and their relative normal and shear displacements produce normal and shear stresses, captured at Gauss points o these inite elements according to the prescribed constitutive law. The detailed FEM ormulation and its implementation can be ound in [8]. The simulations or mesh size sensitivity o the ailure load o specimens o all sizes were perormed irst. It was suspected that when the LEFM limit is approached or large sizes, the inite element model had to employ a ine mesh at the racturing interace; otherwise the cohesive racturing interace elements would not have been able to capture the nearly singular stress ield at the crack ront. It has been ound that, indeed, using a dierent number o elements in the 2
racturing interace gave rise to dierent simulated scaling o ailure stress with size. For example, when all interaces were discretized using only 10 zerothickness interace elements along the interace length or all sizes, there was no size eect, i.e., the ailure happened always at the same nominal stress or all the sizes, implying that the simulated scaling o strength with size had a zero exponent. But when the number o elements was increased to more than 150, it was ound that the simulated strength scaled with a size exponent o approximately 0.5, which is very close to the LEFM scaling or the design size range. This is a demonstration o the act that, to obtain accurate inite element predictions o the peak loads o structures ailing by quasi-brittle racture, it is necessary to have knowledge about the scaling o strength with size, or else mesh sensitivity analyses need to be perormed beore the inite element results could be trusted. The simulated size eect curve obtained rom the inite element calculations is shown in Fig. 3. For the practical size range o hybrid joints, there does not seem to be a signiicant transition rom the LEFM scaling law or large sizes toward the strength based scaling with a vanishing size exponent or small sizes. Note that the computational model described here can be conveniently used or determining the cohesive racture parameters rom experiments. This can be done by calibrating the model to simultaneously it the experimental results or all the sizes. 3. Analytical Formulation o the Asymptotics o Size Eect Law The asymptotic properties o the size eect law or large sizes can be analytically obtained rom the linear elastic racture mechanics. Since both the loading and the geometry o the specimen are symmetric, the elastic ield must be symmetric beore the peak load is attained. Thereore, we analyze only one quarter o the specimen (Fig. 1b). In this quarter, there are two critical bimaterial corners where the corner geometry and material mismatch cause singularity and stress concentration. To identiy the critical corner rom which the crack propagates, the stress singularity exponents must be calculated. In this study, a complex ield method is used to calculate the displacement singularity [9]. Under plane loading conditions, the elastic ield (displacement, traction, and stress ields) in the material can be represented by two holomorphic unctions 1 (z 1 ) and 2 (z 2 ) where z i = x + μ i y; μ i is the root with positive imaginary part o the 4 th order equation λμ 4 +2ρλ 0.5 μ 2 +1 = 0 where λ = s 11 /s 22 and ρ =0.5(2s 12 +s 66 )(s 11 s 22 ) -1/2 (s ij = element o general material compliance matrix). For the near-corner elastic ield, these two holomorphic unctions can be written δ δ as: k ( zk ) = φ k r (cosθ + μk sinθ ) ( k = 1, 2), where δ = displacement singularity exponent. By imposing the traction and displacement at two exterior boundaries and the interace, one obtains a system o linear equations, which can be written in a matrix orm K(δ)ν = 0. The displacement singularity δ can be obtained by 3
setting Det(K) = 0. To obtain it numerically, we choose here to seek the value o δ or which the condition number o matrix K becomes very large. According to the experimental investigations which will be reported in the subsequent section, the orthotropic elastic constants o the iber composite used in this study are: E 1 = 21 GPa, E 2 =9.5 GPa, ν 12 = 0.2, G 12 = 3.0 GPa. For steel, which is isotropic, E = 200 GPa, ν = 0.3. Fig. 4 shows the plot o Det(K) in the complex plane o the displacement singularity. At the let corner o the joint (at which the stier material terminates), the displacement ield is ound to exhibit singularities with exponents representing a pair o complex conjugates : δ = 0.537 ± 0.07i and, at the right corner (at which the soter material terminates) a real displacement singularity δ = 0.78. So, the singularity at the let corner is much stronger than it is at the right corner. Hence, the crack is expected to start propagating rom the let corner, which agrees with experimental observations. This is the corner that governs the strength o the hybrid joint, and so the racture needs to be investigated only or that corner. Various racture criteria have been proposed to characterize the crack initiation or general bimaterial corners [10, 11]. Due to the nature o mix-mode ratcure at bimaterial corner, the use o stress intensity actors as a racture criterion necessitates an empirical equation involving modes I and II stress intensity actors. A more general and eective approach is to consider the energy release rate or the corresponding racture toughness as the ailure criterion. Consider a bimaterial corner with the strongest stress singularity λ = κ ±iη. The near-tip stress ield can be written as iη κ σ ij = Re[ Hr ij ( θ, η)] r (1) where H is the stress intensity actor, which is in general complex. Dimensional analysis shows that H must have the orm: P κ i( ω η ln D) H = D h( η, ϕ) e (2) bd where P = applied load, b = width o the joint, D = characteristic size o the joint (= interace length), h(η,ϕ) = dimensionless complex stress intensity actor, ϕ = eective loading angle (which combines the eects o loading angle and boundary conditions), and ω = phase angle o h(η,ϕ). Once the crack initiates rom the corner tip, it will propagate along the path that corresponds to the highest energy dissipation by racture. The adhesive layer, which connects the iber composite and the steel and is as thin as possible, is normally in hybrid joints designs much weaker than both materials, and so the crack is expected to propagate along the interace. The initiation o a crack, or macro-crack, requires ormation o a microcracking zone o a certain inite characteristic length l FPZ within (and possibly near) the adhesive layer. This zone, called the racture process zone (FPZ) develops stably 4
and transmits cohesive stresses. As soon as the ull FPZ develops, the maximum load is attained. Ater that, the equilibrium load will be assumed to decrease, which requires the geometry to be positive [12] (this is normally satisied but, o course, needs to be veriied). In analogy to the derivation o the size eect law or cracks in homogenous solids [13], we may assume that, not too close to the FPZ, the eect o a inite-size FPZ on the elastic ield is approximately equivalent to the eect o an interace crack o a certain inite eective length c (which is proportional to the length o the FPZ, l FPZ, and is cca l FPZ /2) [12, 13]. Thereore, what matters at the maximum load in a large enough joint is the asymptotic ield near o the interace crack, rather than o a corner. In the oregoing, a distinction is made between: 1) the near-tip asymptotic ield o the imagined eective crack substituted or the overall eect o the FPZ; 2) the near-tip ield o the corner applied not too close to the tip o the crack so that it can envelop the near-tip ield o the crack; and 3) the ar-away ield aected by the boundary conditions. These ields are here matched energetically, through the strength o the singularities. Note that the second ield corresponds to what has been conceived by Barenblatt [14] as the intermediate asymptotic. The intermediate asymptotic is relevant i D >>D >> l FPZ (D = the size o the corner tip singular ield), and the near tip ield o a crack is applicable only i the radial distance rom crack tip r << l FPZ, i.e. in the large-size limit. In the large-size asymptotic limit, the asymptotic near-tip ield o the hypothetical interace crack o length c, substituted or the FPZ, must be surrounded by the singular stress ield o a bimaterial corner tip (except when the laminate thickness is too small, which has been checked not to be the case in practical situations). Thereore, the stress intensity actor K at the interacial crack tip will depend on the stress ield whose magnitude is characterized by the stress intensity actor H o the corner tip. By dimensional analysis, the two stress intensity actors may be related as [10, 15]: iη ' κ +0. 5 iη Kc = Hc c (3) where = dimensionless complex number. For the interacial crack, it is well known that the energy release rate unction or an interacial crack can always be written as [16]: CKK G = (4) E Upon substituting Eqs. 2 and 3 into the oregoing equation, one obtains: 2 2κ σ D 2 G = g (5) 1 2κ Ec where σ = nominal stress = P/bD, and g = C h. Within the LEFM ramework, a crack can propagate once G reaches a certain critical value G, called the racture energy, and this also represents the condition o maximum load 5
P. From Eq. 5, one obtains the LEFM expression o nominal strength σ N (= P/bD) o the bimaterial joint: σ 1 κ 0.5 κ N = g EG c D (6) Eq. 6 represents the large-size asymptote o the size eect law. It is clear that this asymptote ollows a power scaling law, with an exponent directly related to the real part o the strongest stress singularity exponent or the bimaterial corner tip. A general approximate ormula or the size eect on the nominal strength σ N, spanning all the sizes and a range o corner angles, can be obtained through asymptotic matching [13]. A general approximate size eect equation has recently been developed or symmetrically loaded reentrant corners in homogenous materials o various corner angles, which exhibit a real singularity [17]. For the general case with complex singularity, the aorementioned analysis shows that only the real part o singularity exponent matters or the energy release rate. Thereore, an equation o similar type can be used to approximate the general size eect law or the hybrid joint : κ ( β ) 1 D 0 + σ N = σ (7) D0β where σ 0 and D 0β are parameters yet to be calibrated by the size eect tests, κ = real part o the strongest stress singularity exponent at the bimaterial corner, which is a unction o corner angle β. Note that the oregoing equation has been set up to match three essential asymptotic conditions: 1) For D/l 0 0, there must be no size eect since the FPZ occupies the whole structure and what matters is solely the material strength, and not the energy release (the ailure is quasi-plastic). 2) For D/l 0, Eq. 7 must match Eq. 6 as the large-size asymptote o size eect law. 3) For β = π (smooth surace, no corner), the size eect o this type must vanish (there is another type o size eect (Type 1), applicable to ailures at cohesive crack initiation rom a smooth surace). 4. Experimental Investigation on Size Eect Law Based on preliminary computer simulations, 9 specimens were designed and manuactured, see Fig. 5a. To ensure that the axial tensile orce is applied through the center line o the specimen, the specimen is connected to the loading piston through a steel chain at each end (Fig. 5b). Except or the end portion connected by a steel pin, the specimens are geometrically similar, with the scaling ratio 1:4:12. The composite material used in this study is iberglass-epoxy laminates (G-10/FR4 Epoxy Grade procured rom McMaster-Carr, Inc.). The uniaxial tension, compression, and Iosipescu V-notch beam shear tests are conducted to obtain the elastic constants o the composite. The adhesive used to glue the 6
laminate to the steel block is E-60 HP metal-plastic bonder with high shear strength and high peel resistance. The specimens are loaded under displacement control by an MTS machine. To eliminate dierences due to the rate eect, it is essential to ensure that, or all the specimens, the FPZ is ully ormed within approximately the same time duration. In all the tests, the maximum load was reached in about 5 to 6 minutes. It is observed that all the specimens ailed by interacial racture. By careul investigation o the damage pattern, one inds the damage o the laminate to be concentrated in the regions near the interior corners o the joint. Thereore, the crack in the interace must have initiated rom the inner corner, as predicted by both the computational and analytical studies. A strong size eect can be seen in the plot o the nominal strength σ N versus the specimen size D. By least-square statistical regression, one can calibrate the analytical size eect expression derived in the preceding, as shown in Fig. 6. Note that the asymptotic slope o the theoretical size eect curve agrees well with the test data. 5. Conclusion The analytical study indicates that the large size asymptote o size eect law ollows a power law with an exponent related to the order o the stress singularity. The generalized size eect law, which is developed through asymptotic matching, is shown to it the experimental results very well. The numerical, analytical and experimental studies all show a strong size eect on the strength o the hybrid joint. Acknowledgement Financial support under Grant N00014-07-0313 rom the Oice o Naval Research, monitored by Dr. Roshdy Barsoum, is grateully acknowledged. Reerences: [1] Barsoum, R.S., The best o both worlds : Hybrid ship hulls use composite & steel. The AMPTIAC Quarterly 7 (3), 55-61. [2] Bahei-El-Din, Y.A. and Dvorak, G.J. New designs o adhensive joints or thick composite laminates. Composites Sci. and Tech. 61, 2001, 19-40. [3] Dvorak, G.J., Zhang, J., and Canyurt, O. Adhesive tongue-and-groove joints or thick composite laminates. Composites Sci. and Tech. 61, 2001, 1123-1142. 7
[4] Bažant, Z.P., Zhou, Y., Daniel, I.M., Caner, F.C., and Yu, Q. Size eect on strength o laminate-oam sandwich plates. J. Eng. Mater. Tech. ASME 128 (3), 2006, 366-374. [5] Bažant, Z.P., Zhou, Y., Novak, D., and Daniel, I.M. Size eect on lexural strength o iber-composite laminate. ASME J. Eng. Mater. Tech. 126, 2004, 29-37. [6] Bažant, Z.P., Daniel, I.M., and Li, Z. Size eect and racture characteristics o composite laminates. J. Eng. Mater. Tech. ASME 118 (3), 1996, 317-324. [7] Bažant, Z.P., Kim, J.-J.H. Kim, Daniel, I.M., Becq-Giraudon, E., and Zi, G. Size eect on compression strength o iber composites ailing by kink band propagation. Int. J. Fract. 95, 1999, 103-141. [8] Caner, F.C. and Bažant Z.P. Size eect on strength o laminate-oam sandwich plates: Finite element analysis with interace racture, Int. J. o Solids and Structures. 2007, (submitted) [9] Desmorat, R. and Leckie, F.A. Singularities in bi-materials: parametric study o an isotropic/anisotropic joint. Eur. J. Mech., A/Solids 17, 1998, 33-52. [10] Grenestedt, J.L. and Hallstrom, S. Crack initiation rom homogeneous and bi-material corners. J. Appl. Mech. ASME 64 (December), 1997, 811-818. [11] Leguillon, D. Strength or toughness? A criterion or crack onset at a notch. Eur. J. Mech., A/Solids 21, 2002, 61-72. [12] Bažant, Z.P. and Planas, J. Fracture and Size Eect in Concrete and Other Quasibrittle Materials, CRC Press, 1997. [13] Bažant, Z.P. Scaling o Structural Strength. Hermes Penton Science, London; 2nd updated ed., Elsevier 2005. [14] Barenblatt, G.I. Scaling, Sel-Similarity and Intermediate Asymptotics. Cambridge University Press, Cambridge, UK, 1996. [15] Liu, D. and Fleck, N.A. Scale eect in the initiation o cracking o a scar joint. Int. J. Fract. 95, 1999, 66-88. [16] Hutchinson, J.W. and Suo, Z. Mixed mode cracking in layered materials. Adv. Appl. Mech. 29, 1992, 64-191. [17] Bažant, Z.P. and Yu, Q. Size eect on strength o quasibrittle structures with reentrant corners symmetrically loaded in tension. J. Eng. Mech. ASCE 129 (11), 2006, 1168--1176. 8
(a) (b) Figure 1. Geometry o double-lap hybrid joint = s 2 s + β 2 t Figure 2. Cohesive law or interacial ratcure Figure 3. Similated size eect curve 9
Figure 4. Exponent o displacement singularity a) b) Figure 5 a) test specimens; b) test set-up Figure 6 Calibration o size eect law 10