A SHEAR TRANSFER MODEL FOR ROUGH JOINTS BASED ON CONTACT AND FRACTURE MECHANICS

VIII International Conerence on Fracture Mechanics o Concrete and Concrete Structures FraMCoS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) A SHEAR TRANSFER MODEL FOR ROUGH JOINTS BASED ON CONTACT AND FRACTURE MECHANICS MARCO PAGGI, ALBERTO CARPINTERI Politecnico di Torino Department o Structural, Geotechnical and Building Engineering Corso Duca degli Abruzzi 24, 1129 Torino, Italy e-mail: marco.paggi@polito.it, http://sta.polito.it/marco.paggi Key words: Rough joints, Contact mechanics, Fracture mechanics, Fractality, Size-scale eects Abstract: This article addresses the undamental issue o shear transer across rough joints, a problem o paramount importance in several civil engineering applications. Starting rom the examination o the experimental results obtained rom shear tests on concrete-concrete joints, rockrock joints and concrete-soil interaces, the main eatures o the physical phenomenon are presented. The experimental results are then interpreted according to a novel ormulation based on the integration o contact mechanics, suitable or the description o the pre-peak response o the joint, and nonlinear racture mechanics, or the description o the post-peak sotening branch. The proposed treatment ully takes into account the size-scale eects on the shear response o the joints due to interace roughness, providing a signiicant step orward with respect to the state-o-the-art modelling based on elasto-plasticity with a history dependent riction coeicient. 1 INTRODUCTION The treatment o shear transer across rough joints arises in several civil engineering problems. They range rom retroitting o damaged concrete structures with new concrete overlays [1] or with iber-reinorced polymers [2] to the analysis o the shear response o rock joints [3] and concrete-soil interaces in pile oundations [4]. Moreover, it is an aspect o paramount importance or the undamental understanding o the behaviour o rough interaces subjected to Mode II deormation. In these problems, although the joint is not glued or soldered as in composites, roughness induces similar eects, as we will quantitatively show in the sequel. Starting rom the experimental evidence, shear tests on rock joints [3], concrete-soil interaces [4], concrete-concrete cold joints [5] and bolt-astened concrete layers [1] present common eatures and show that several mechanisms contribute to shear transer. The most important are riction, cohesion due to asperity interlocking and the dowel action in the presence o bolts. The shear strength, which is as the peak shear traction evaluated rom a shear test, is an increasing unction o the normal pressure. This trend is shown in Fig. 1 or shear tests on concrete-soil interaces [4], where the shear traction q is plotted vs. the sliding displacement g T. The amplitude o roughness is also increasing the shear resistance o the joint, as experimentally observed in [3]. The specimen size, on the other hand, leads to an opposite trend, which is diicult to be explained according to classical Euclidean geometry. Large specimens have a lower strength than the small ones [3], as shown in Fig. 2. This size-scale eect on the shear strength has been interpreted in [6-8] by 1

considering the ractality o rough contact interaces. q (kpa) 14 12 1 8 6 4 2 p=25 kpa p=1 kpa p=2 kpa 5 1 15 2 25 Figure 1: Shear test results or concrete-soil contact [4] showing the eect o the normal pressure p. q (kpa) 6 5 4 3 2 Δ=4 cm Δ=2 cm 1 Δ=1 cm Δ=5 cm 2 4 6 8 Figure 2: Shear test results or rock-rock contact [3] showing the eect o the specimen size Δ. The post-peak branch o the curve obtained rom a shear test is also very complex and may present a transition rom sotening or small specimen sizes or or high normal pressure to a plastic plateau or large specimens or or low normal pressure, see Figs. 1 and 2. When steel bolts are included, the post-peak branch usually presents hardening. This phenomenon has been investigated in [1] by perorming shear tests on concrete-concrete joints astened using 212 steel bolts, or dierent amplitudes o roughness. The non planarity o the interace is originated by surace blasting with high pressure water (Fig. 3(a)), or with sand (Fig. 3(b)). In both cases, the total tangential orce, F T, is an increasing unction o the sliding displacement g T up to a peak value (thick solid curves in Fig. 3). Aterwards, a sotening branch takes place, which is ollowed by hardening or very large sliding displacements. The dowel action exerted by bolts was also measured in [1] and it is superimposed to the previous igures using a dashed-dotted line. The dierence between the curve corresponding to the global response and the curve related to the dowel action is numerically computed here and it represents the shear contribution due to roughness (see the dashed curves in Fig. 3). This contribution shows a sotening branch, in close analogy with the previous results displayed in Figs. 1 and 2. Sand blasting induces a roughness amplitude lower than high pressure water blasting and thereore the peak shear strength is lower in Fig. 3(b) than in Fig. 3(a). This conirms that roughness has a decisive role on the rictional response o cold joints. F T (kn) F T (kn) 2 15 1 5 Dowel action Global response Roughness contribution 5 1 15 2 (a) High pressure water blasted suraces. 14 12 1 8 6 4 2 Dowel action Global response Roghness contribution 5 1 15 2 (b) Sand blasted suraces. Figure 3: Shear test results or concrete-concrete contact [1] showing the eect o the dowel action and o roughness, or two dierent surace treatments. 2

As regards modelling, an attempt to interpret these shear phenomena has been made in the ramework o the inite element method and elasto-plasticity. A history dependent riction coeicient according to a semi-empirical expression and itted on experimental results was proposed in [1] to capture the pre- and post-peak branches o the curves in Fig. 1. Although successul or these tests, the use o seven parameters reduces the possibility to generalize the results o this approach to other types o joints. In act, all the model parameters are just best-itting coeicients not related to measurable roughness properties. In the present contribution, a new model based on the integration o contact mechanics and racture mechanics is proposed. The novelty o the approach consists in modelling the pre-peak and the post-peak branches o the shear response by using and integrating two dierent methodologies, a modelling strategy not yet explored in the literature. The nonlinear pre-peak response is considered as the result o the phenomenon o micro-slipping o the asperities in contact. This problem is solved using the contact mechanics approach or rough suraces proposed in [9] and generalized in [1] or cyclic loading. Ater achieving the peak shear stress, the post-peak regime is considered to be the result o a phenomenon o cohesion exerted by asperity interlocking. This stage will be modelled by introducing a new Mode II cohesive zone model. The micromechanical contact ormulation and the nonlinear racture mechanics model are integrated to provide a complete description o the joint behaviour. In case o bolt-astened concrete blocks, the dowel action can also by added to the previous contributions to simulate the post-peak hardening, ollowing the methodology presented in [1]. The main achievement o the proposed model is the ability to take into account the size-scale eect observed in experiments. Moreover, it involves parameters solely dependent on the topology o rough suraces that can be measured according to tribology standards [11]. 2 MODELLING THE PRE-PEAK RESPONSE BY USING CONTACT MECHANICS 3.1 The normal contact problem Dimensional analysis considerations and incomplete similarity arguments presented in [12] have demonstrated that the normal contact stiness o a rough joint is a power-law unction o the normal pressure. Experimental results and numerical simulations have also conirmed this power-law trend, see [12]. This is due to the non Euclidean (ractal) character o roughness. In dimensionless orm, the power-law relation between pressure and mean plane separation is: dp β p dd (1) where the dimensionless variables p = pδ /( Eσ ) and d = d / σ have been introduced. The variable d is schematically shown in Fig. 4 and represents the distance between the average plane o the composite topography and a rigid lat indenting plane. The variable Δ denotes the upper cut-o length to the power-law behaviour o the power spectral density unction o the rough ractal surace [11]. In the present case, it can be related to the lateral size o the sample. The remaining parameters, E and σ, are the Young's modulus and the r.m.s. roughness o the surace. Figure 4: Sketch o the normal contact problem between a equivalent composite rough surace and a smooth rigid plane. Integration o the ordinary dierential equation (1) relating the normal stiness to the pressure gives the pressure-separation relation. 3

This relation is exponential only in case o β = 1 [12]. For the most general case β 1, we have: p = K( A Bd ) γ (2) which is a power-law i and only i A=. The exponent γ is related to β via γ = 1/(1 β ). The coeicient K is a scaling actor taking into account the nonlinear size-scale eect on the shear strength. A qualitative sketch o Eq. (2) is shown in Fig. 5. Considering a loading scenario where two rough suraces are progressively brought into contact rom a very large separation, the pressure is increasing by reducing d, up to a inal coniguration denoted by the index in Fig. 5. p~ p~ O Figure 5: Dimensionless pressure-separation relation. 3.2 The tangential contact problem Following the approach in [9,1], the solution o the tangential contact problem between rough suraces can be derived rom the solution o the normal contact problem. In ormulae we have: g~ ( ~ ~ T = μ g N, g N ) q~ = ( ~ p ~ (3) μ p ), Final state d ~ d ~ where μ is the iction coeicient, g = g / σ T T is the dimensionless sliding displacement and q = qδ/( Eσ ) is the dimensionless tangential traction. Hence, the slip and the tangential tractions can be obtained rom the solution o the normal contact problem at the inal state o a normal contact simulation, minus a corrective term related to a lower pressure level. Equation (3) is sets up in terms o the dimensionless contact compliance, g ~ N = g N /σ, which should be measured rom the irst point in contact (see Fig. 4). Although a tallest asperity always exists in numerical realization o rough suraces, its elevation is aected by large variations governed by the extreme value theory. Hence, it is preerable to rewrite Eq. (3) in terms o the mean plane separation, a quantity much more stable rom the statistical point o view. Noting that g~ ~ ~ N = d max d and introducing this relation into Eq. (3), we obtain: ~ ~ g~ T = μ( d d ) q~ ( ~ p ~ (4) = μ p ) ~ Note that Eq. (4) does not depend on d max. The pressure range that can be explored is rom up to p~. Correspondingly, the tangential tractions vary rom q = μ p down to. Intermediate values o tractions between and the value corresponding to ull slip, q, will be denoted as q. The obtained dimensionless tangential traction vs. sliding relation is qualitatively shown in Fig. 6. It can be considered as a regularization o the Mohr- Coulomb riction law, which would imply zero sliding until ull slip takes place or q = q. This sharp situation would happen only in the limit case o two ideally lat planes in contact. Physically speaking, the reason or this regularization is due to the non constant normal contact pressures supported by the asperities and induced by roughness. Using Eq. (2), the ollowing explicit relation between q ~ and ~g T is obtained: γ KB q = μ p KA g T KBd μ (5) 4

q 6 5 Δ=5 cm Δ=1 cm q = μ p q q (kpa) 4 3 2 Δ=2 cm Δ=4 cm 1 O g T g T, Figure 6: Dimensionless tangential traction-sliding relation. 1 2 3 4 ~g T Figure 7: Tangential traction vs. sliding displacement: numerical predictions (solid lines) and experimental curves (dashed lines). 3.4 Interpretation o the pre-peak experimental results In this subsection, the experimental shear tests related to the curves in Fig. 2 are numerically simulated and the predicted prepeak branches are compared with the experimental ones. To do so, the Young's modulus and the r.m.s. roughness are chosen as to represent the rock material analyzed in [3]. The parameters A and B in Eq. (2) are set equal to 2.4 and.41, respectively. Four dierent values o the specimen size, Δ = 5 cm, 1 cm, 2 cm and 4 cm, are considered. The range o dimensionless separations explored is rom 5 (large separations, very weak interaction) to (high pressure level). This is also the inal state or the application o Eq. (3) and it leads to normal pressure levels consistent with the experimental ones in [3]. Considering K=1, the solid curve in Fig. 7 corresponding to Δ = 5 cm is obtained. All the other curves corresponding to dierent specimen sizes are obtained by selecting a dierent Δ in the omulae and applying the coeicient K reported in Tab. 1. Table 1: Value o the coeicient K in Eq. (2). Δ (cm) K.5 1..1 1.5.2 2.1.4 3.2 It is remarkable to note that, without the use o the coeicient K, the size-scale eect predicted by the application o Eq. (5) would be much higher. The coeicient K re-scales the vertical coordinates and it is related to the size-scale eect on the shear strength o the joint. Hence, the knowledge o this scaling law allows an easy computation o the parameter K by matching the peak traction given by Eq. (5) with K=1 and the shear strength given by the scaling law. For rocks, the expression o the scaling law or the shear strength has been determined in [7,8] by considering a wide range o length scales ranging rom the size o typical laboratory specimens to the size o a natural d ault, obtaining q = q * Δ q. Determining q * and dq rom best-itting o experimental results, in logarithmic orm we have: log1 q = 4.33.34 log1 Δ (6) 5

Equation (6) its very well the shear strength results o the tests in Fig. 2, as shown in the comparison o Fig. 8. log 1 q max (kpa) 1.9 1.8 1.7 1.6 1.5 1.4 1.3 Experimental data Scaling law in Eq. (6) -1.5-1 -.5 log 1 Δ (m) Figure 8: Size-scale eects on the shear strength: experimental data rom Fig. 2 vs. scaling law in Eq. (6). 4 MODELLING THE POST-PEAK RESPONSE BY USING FRACTURE MECHANICS The post-peak branches o the curves in Fig. 2 are modelled in this section according to a cohesive zone approach or Mode II deormation. The cohesion resistance to sliding is due to interlocking between the asperities, an aspect that takes place or large sliding displacements ater the achievement o the peak strength. For this analysis, we ollow the methodology proposed in [13,14] already adopted or the interpretation o uniaxial tensile tests and compression tests (Mode I problems). As a preliminary operation, the inelastic contribution to the deormation in the postpeak branch is isolated by removing rom the total sliding displacement the elastic deormation o the pre-peak branch. Ater this operation, the post-peak inelastic traction vs. sliding curves o Fig. 2 are transormed in the curves plotted in Fig. 9. They show a signiicant size-scale eect, much more evident than in Mode I problems [13]. Figure 9: Inelastic post-peak branches obtained rom Fig. 2, depending on the specimen size Δ. In order to obtain a single size-independent cohesive zone model in Mode II, a proper renormalization o the vertical and horizontal coordinates has to be made. Regarding the shear tractions, the scaling law in Eq. (6) suggests to divide q by Δ d q, i.e., plotting the scaling invariant material property q *. The horizontal coordinates can also be renormalized in a similar way, considering the scaling law typical or the displacements which is inspired by the Cantor set [13], i.e., ( 1 d g ) gt = gt Δ. In this case, the ractal exponent d g can range rom to.61, considering the equality relating the sum o the ractal exponents or the strength, the strain and the racture energy (reer to [13] or more details about the description o the ractal domains). For the present problem, the best dimensionless representation with the minimum scatter between the curves is obtained or d =, which leads g to g T = gt Δ. The collapse o the curves in Fig. 1 shows that the post-peak branch o the dierent experimental shear tests can be eectively modelled according to a single scale-invariant Mode II cohesive zone model. 6

q*=q Δ d q 2 15 1 5 Δ=2 cm Δ=4 cm Δ=1 cm Δ=5 cm peak branches, a single scale-invariant Mode II cohesive zone model has been proposed. This methodology is expected to have an impact in the design ormulae or the estimation o the shear strength o cold joints and it overcomes all the main disadvantages o the purely numerical elasto-plastic approach proposed so ar in the literature. 2 4 6 8 1 * g =gt /Δ T Figure 1: Scale-invariant Mode II cohesive zone model. The various curves correspond to dierent specimen sizes Δ. 5 CONCLUSIONS In this work, the undamental problem o shear transer between rough suraces has been analyzed in reerence to the existing experimental results available in the literature. Shear tests on rock-rock joints, concreteconcrete interaces, and concrete-soil oundations present several common aspects. In particular, the pre-peak response o the shear traction vs. sliding displacement curve is highly nonlinear, due to progressive sliding o micro-contacts. Ater the achievement o the shear strength, the post-peak branch presents sotening and plastic plateau, when steel bolts are not included. Finally, the shear strength exhibits a strong size-scale eect, as also typically observed in previous studies regarding Mode I problems [15-17]. In order to interpret both regimes, a hybrid ormulation based on contact mechanics or the pre-peak stage and on nonlinear racture mechanics or the post-peak one has been proposed. This modelling strategy is consistent with the physics o the phenomenon, since the pre-peak response is mainly related to the progressive slippage o micro-contacts, whereas the post-peak behaviour is governed by cohesion originated by asperity interlocking and rictional eects. As a result, we have been able to predict with a reasonable accuracy the pre-peak regime and the related size-scale eects, proposing the simple Eq. (5) and an algorithm easy to use or the simulation o shear tests rom the knowledge o the normal contact stiness. Regarding the post- ACKNOWLEDGEMENTS The support o the Italian Ministry o Education, University and Research to the Project FIRB 21 Future in Research Structural mechanics models or renewable energy applications (RBFR17AKG) is grateully acknowledged. REFERENCES [1] Randl, N. and Wicke, M., 2. Schubübertragung zwischen Alt- und Neubeton. Beton und Stahlbetonbau 95: 461-473. [2] Cottone, A. and Giambanco, G., 29. Minimum bond length and size eects in FRP substrate bonded joints. Engng. Fract. Mech. 76:1957-1976. [3] Bandis, S., Lumsden, A.C. and Barton N.R., 1981. Experimental studies o scale eects on the shear behaviour o rock joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18:1-21. [4] Reul, O., 2. In-situ-Messungen und numerische Studien zum Tragverhalten der kombinierten Pahl-Plattengründung, Tech. Report 53, Institut und Versuchsanstalt ür Geotechnik, TU Darmstadt, Darmstadt, Germany. [5] Červenka, J., Chandra Kishen, J.M., and Saouma, V.E., 1998. Mixed mode racture o cementitious bimaterial interaces; part II: numerical simulation. Engng. Fract. Mech. 6:95-17. [6] Carpinteri, A. and Paggi, M., 25. Sizescale eects on the riction coeicient. Int. J. Sol. and Struct. 42:291-291. 7

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