Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 3, 39, 2001 MODELLING OF FRICTION AND DILATANCY EFFECTS AT BRITTLE INTERFACES FOR MONOTONIC AND CYCLIC LOADING Stanisław Stupkiewicz Zenon Mróz Institute of Fundamental Technological Research, Polish Academy of Sciences e-mail: sstupkie@ippt.gov.pl; zmroz@ippt.gov.pl The most important effects related to monotonic and cyclic response of con- tact interfaces of brittle materials are analyzed in the paper. Next, the availa- ble constitutive models are reviewed with respect to their ability to describe these effects. Severalmicro-mechanicalmechanisms are analyzed including de- cohesion, interaction of primary and secondary asperities, asperity wear and damage and formation of a third body granular layer. Finally, we propose new formulations of constitutive models for cyclic interface response. Keywords: constitutive models, contact interfaces, friction, dilatancy 1. Introduction The problem of modelling of material interface response under monotonic and cyclic loading is of fundamental scientific and engineering importance. In fact, such interfaces occur inmost engineering or geotechnical structures such asmasonry structures, fibre-reinforced brittlematrix composites, jointed rock masses, dams, bridges, etc. The structural stiffness and limit load are strongly dependent on inelastic interface response. As the displacement discontinuity resulting from frictional slip along interface occurs, the localized effects of damage and wear develop depending on micro-mechanical effects of asperity interaction. A closely related problem of fluid transport along interfaces (es- sential, for instance, in nuclear waste storage technology), essentially coupled with the mechanical response, will not be discussed in the paper. Thepresentpaper is devoted to the analysis ofmonotonic and cyclic effects at contact interfaces of brittle materials. The class of materials (and interfa- ces) is quite wide and includes: rock joints, artificial and natural joints in civil 708 S.Stupkiewicz, Z.Mróz engineering structures, existing cracks in brittlematerials (e.g. concrete, cera- mics),masonry and other cementitious joints, fibre-matrix interfaces in brittle matrix composites, etc. A special attention is paid to interfacial dilatancy phe- nomena as this aspect does not seem to have been sufficiently analyzed in the literature. Although the emphasis is laid on friction and dilatancy effects, so- meattention is also paid to tensile/compressive behaviour as these phenomena are coupled and cannot be completely separated. Some of the interfaces considered in this work are characterized by initial tensile strength.Typical examples of cohesive interfaces are themasonry joints andfiber-matrix interfaces and also infilled rock joints.Thedecohesion process is understood as a loss of tensile strength along a predefined interface. Thus crack propagation problems in which the crack path is a part of the solution are not considered. This is the case of a weak interface between two dissimilar (or similar) materials. Clearly, the decohesion may occur in tension (mode I), shear (mode II/III) or mixedmodes. In the case of cohesive interfaces, the formation of the actual rough sur- face is a part of the deformation process. As a result, the asperities of one surface match (at least partially) the asperities of the other surface. On the other hand, most of the non-cohesive interfaces studied in this paper (e.g. rock joints) are generated through the prior cracking processes. In such case the asperities of both contacting surfaces also match, depending on the me- chanical and environmental conditions since the time of joint formation. The interaction of interlocked asperities strongly affects the friction and dilatancy response of these interfaces. This, in fact, is a common effect for most of the brittle interfaces. In Section 2 the most important effects observed experimentally are pre- sented, followed by a qualitative discussion of the related micro-mechanical mechanisms. The constitutive models for brittle interfaces are discussed in Section 3. A critical review of existing interface models is provided and some new formulations of constitutive models of cyclic behaviour of interfaces are proposed. 2. Experimental effects of mechanical interface response 2.1. Typical experimental setups Frictional properties of joints/interfaces are usually investigated by perfor- ming shear tests with uniform contact conditions along the interface, Fig.1a. In direct shear tests a constant normal pressure is kept during shearing, thus Modelling of friction and dilatancy effects... 709 allowing for free dilation at the interface. Typically, the friction stress and the relative normal displacement (dilation) are measured as a function of relative tangential displacement (slip). These tests are typically performed for rock joints, masonry joints, etc., cf for example Bandis et al. (1981), Atkinson et al. (1989). Fig. 1. Scheme of uniform shearing (a) and tensile/compressive (b) tests on joints The principle of tensile/compressive tests is similar to that of direct she- aring, Fig.1b. The measured response is the normal pressure and relative normal displacement, cf Bandis et al. (1983), van der Pluijm (1997). The direct shear tests cannot be used for investigation of the properties of fiber-matrix interfaces in brittle matrix composites because of very small dimensions of fibers. Instead, single- or multiple-fiber pulling or pushing tests are usually applied, cf Marshall and Oliver (1990), Marshall et al. (1992). In these tests, however, the contact conditions are not constant along the interface as the debonding zone and slip zone propagate along the interface with increasing load. Unlike in direct shear tests, the normal pressure at the interface cannot be varied, and also due to the matrix surrounding the fiber the interfacial dilation is constrained. Clearly, other types of tests are performeddependingon the joint/interface type and specific requirements. These, for example, include multiply-jointed rock specimens (Bandis et al., 1981), four point bending tests (van derPluijm, 1997), shearing ofmasonrywall panels (Anthoine et al., 1995), etc., containing multiple interface systems. 2.2. Monotonic loading 2.2.1. Tension of cohesive joints The tensile behaviour of cohesive interfaces resembles that of mode I frac- ture of the quasi-brittle materials (e.g. concrete), where after reaching a peak 710 S.Stupkiewicz, Z.Mróz the strength decreases to zero, cf Fig.2 (we use a notation, inwhich the tensile contact stresses and opening relative displacements are positive). Masonry joints are typical examples of cohesive interfaces. Van der Pluijm (1997) investigated the response of masonry bed joints in tension. The frac- ture occurred at the interface between themortar layer and one of the blocks (bricks). The results were characterized by a large scatter of results in terms of peak stresses, fracture energies and characteristic openingdisplacements for nominally identical specimens. Fig. 2. Typical tensile response of cohesive joints 2.2.2. Compression Under compression the relation between the normal pressure and the nor- mal relative displacement is nonlinear. Typical response curves for rock joints are given in Fig.3, where two cases are shown namely a joint with interloc- ked (fully mated) asperities and with mismatched asperities, cf Bandis et al. (1983), Sun et al. (1985). When the asperities are not interlocked the contact stiffness decreases as the effect of localized deformation at asperity contacts. 2.2.3. Shearing The shearing response under constant normal pressure is usually characte- rized by a peak followed by softening until a residual shear stress is attained, Fig.4a. This type of behaviour is observed for both cohesive (Atkinson et al., 1989; van der Pluijm, 1993; Binda et al., 1994) and non-cohesive joints (Kut- ter and Weissbach, 1980). In the latter case, the response without the peak shear resistancemay also be observed (Bandis et al., 1981; Sun et al., 1985), cf Fig.4b.Generally, the post-peak softening can be attributed to several pheno- mena, often occurring simultaneously, namely to decohesion, configurational Modelling of friction and dilatancy effects... 711 Fig. 3. Typical compressive response of rock joints effects due to dilation, friction softening caused by damage and wear of aspe- rities, etc. These topics are discussed in detail in Section 3 concerned with constitutive modelling of interfaces. Fig. 4. Typical response to direct shear tests: (a) with and (b) without peak The shearingdisplacement is usually accompanied by thedilation of the jo- int.Here several typical dilation curves are observed experimentally depending on the joint type, initial state of the joint andamount of shearingdisplacement applied in the test. Some joints show a nearly linear dilation response as indi- cated in Fig.5a. As the interfacial dilation has to be bounded, such response should be understood as the initial portion of a general nonlinear response with an asymptotic dilation occurring at sliding displacements high enough as compared to the characteristic length of primary asperities, Fig.5b. Finally, some joints are characterized by an increasing dilatancy angle at small she- aring displacements, which then gradually decreases with increasing amount 712 S.Stupkiewicz, Z.Mróz of sliding (provided the test is performed with large enough sliding displace- ments), Fig.5c. Fig. 5. Typical dilation curves in monotonic direct shear tests An important effect is usually observed, namely that dilation is reduced with an increasing normal pressure. This effect is explained by asperity cru- shing at the values of normal pressures relatively high with respect to the unconfined compressive strength of the asperity material. 2.3. Non-monotonic and cyclic effects in shearing of joints Generally, much less experimental results of cyclic tests are available as compared tomonotonic tests. In this section, we only discuss the effects obse- rved in cyclic shearing. The results of cyclic compression of rock joints can be found for example in Bandis et al. (1983) and Sun et al. (1985). 2.3.1. Reversible dilation component As discussed previously, the joint opening (dilation) is commonly observed in themonotonic direct shear tests. However, upon unloading the dilation de- creases until the shear displacement changes its sign and then increases again, Fig.6a,b. The actual shape of the cyclic dilatancy curve depends strongly on the joint type, normal stress and the range of applied shearing displacement. The reversible dilation effect is typical for natural (well-seated) rock joints (Kutter andWeissbach, 1980; Fox et al., 1998) andmasonry joints (Atkinson et al., 1989). The phenomenon of reseating load drop in fiber push-back te- sts is also explained by the reversible dilation component at the fiber-matrix interface, cf Section 2.3.5. 2.3.2. Friction jump at zero relative displacement This effect is commonly observed for many types of interfaces. In the case Modelling of friction and dilatancy effects... 713 Fig. 6. Cyclic dilatancy response: (a) rock joint under low normal pressure, (b) cyclic degradation of dilatancy, (c) masonry bed joint with a dilatancy drop at load reversal of initially mated rough rock joints, the effect may be very strong with the friction stress increasing two-three timeswhen the two contacting surfaces pass through their original position (i.e. at zero relative displacement), cf Hutson and Dowding (1990), Fox et al. (1998). This case is sketched in Fig.7a. On the other hand, the effect is much weaker, but still visible, for other joints, as shown in Fig.7b, cf results of Atkinson et al. (1989) for masonry bed joints. Thedashed line inFig.7b represents the peak followed by softening during the first loading cycle observed both for cohesive joints (such as masonry joints) and for some rock joints. If the normal pressure is high enough to crush andwear the asperities, the jumpdecreases for consecutive loading cycles, cfKutter andWeissbach (1980), Hutson and Dowding (1990), Jing (1990), Atkinson et al. (1989). Otherwise, the cyclic response ishardlyaffectedby thenumberof loadingcycles, cfHutson and Dowding (1990), Fox et al. (1998). 2.3.3. Cyclic degradation of dilatancy For repeated cyclic shearing the dilation decreases with increasing number of cycles. The higher the normal pressure with respect to the joint material strength, the stronger the effect, cf Hutson and Dowding (1990), Fox et al. 714 S.Stupkiewicz, Z.Mróz Fig. 7. Idealized friction jump in the vicinity of original mated joint position: (a) strong, (b) weak effect (1998). A typical cyclic dilatancy curve of a rock joint under relatively high normal pressure is shown in Fig.6b, while for very low pressures the response resembles that in Fig.6a. Cyclic degradation of dilatancy is also observed in the case of masonry bed joints, cf Atkinson et al. (1989). 2.3.4. Drop of dilation angle at load reversals The results of cyclic shearing of masonry bed joints reported by Atkinson et al. (1989) are characterized by a significant drop of dilation angle at each load reversal, Fig.6c. This behaviour is qualitatively different from that of typical rock joints, cf Fig.6a,b. As there is currently no theory explaining this behaviour, it will be discussed in detail in the modelling part of the paper. 2.3.5. Reseating load drop in fiber push-back tests Fiberpushingorpulling tests are commonlyused to examine theproperties of fiber-matrix interfaces in brittlematrix composites (BMC).The importance of interfacial dilation is well illustrated by, so-called, push-back tests in which the fiber is first displaced from its original position (by pushing or pulling) and next pushed in the opposite direction. Once the fiber passes through its initial position, the load required to push the fiber decreases and subsequently increases to the previous value, cf Fig.8. This phenomenon of reseating drop of the load has been observed by several authors, cf Carter et al. (1991), Jero et al. (1991) and Cherouali et al. (1997). The explanation of the phenomenon is that the asperities of the fiber andmatrix perfectlymatch in the original fiber position while after displacing the fiber, the interaction of asperities results in Modelling of friction and dilatancy effects... 715 the dilation of the interface. As the dilation is constrained due to surrounding matrix, the normal pressure (and thus also the friction stress) increases when the fiber is moved from its initial position. Note that in these tests the displa- cement of the fiber is typically much larger than the characteristic asperity length, thus the dilation has a constant asymptotic value, except for a close vicinity of the original position. Fig. 8. Reseating drop of the pushing load in a fibre push-back test 2.4. Interpretation of observed phenomena Before discussing constitutivemodels of interfacial behaviourwe shall brie- fly provide amicro-mechanical explanation of the effects described above. The most important effects are summarized in Table 1. Each effect is assigned a number which will be referred to throughout the rest of the paper. Table 1. Summary of experimentally observed phenomena Monotonic loading A.1 quasi-brittle damage in tension, Fig.2 A.2 decreased normal stiffness of mismatched joints, Fig.3 A.3 peak resistance in direct shear, Fig.4a A.4 shearing response without peak stress, Fig.4b A.5 constant residual shearing resistance, Fig.4a,b A.6 dilation in direct shear, Fig.5 A.7 decreasing dilation with increasing normal pressure Cyclic loading B.1 reversible dilation component, Fig.6a B.2 friction jump at zero relative displacement, Fig.7 B.3 cyclic degradation of dilatancy, Fig.6b B.4 drop of dilation angle at load reversals, Fig.6c B.5 reseating load drop in fiber push-back 716 S.Stupkiewicz, Z.Mróz Now, we shall discuss the main micro-mechanical mechanisms that occur at brittle interfaces. InTable 2we associate thesemechanismswith the effects that are related to them. Note that some effects may be attributed to several mechanisms. Table 2.Micro-mechanisms and related effects Mechanism Related effects decohesion A.1, A.3, A.5, A.6 interaction of well-seated primary asperities A.3, A.4, A.5, A.6, B.1, B.2, B.5 random asperity contact A.2 wear/damage of primary asperities A.7, B.3 wear/damage of secondary asperities A.3, A.5 formation of a third body granular layer B.4 2.4.1. Decohesion Decohesion is a fracture (damage) process which proceeds along a pre- defined, usually weaker interface. In terms of the mechanical response, the decohesion results in some peak strength followed by a softening regime, both in tension and in shear.When the cohesive strength is lost, the tensile strength drops to zero (A.1), while in shear a residual frictional resistance ismaintained (A.3, A.5). In the case of shearing (also under some compressive normal pres- sure), the decohesion process usually proceeds through a distributed damage process associated with volumetric deformation in the process zone, which leads to interfacial dilation (A.6). 2.4.2. Interaction of well-seated primary asperities It is very well known that, regardless of the type of the contact pair, the surfaces in contact are rough, and that asperity interaction is a fundamental mechanism governing contact phenomena. In the context of brittle interfaces two distinct cases may occur, namely 1. well-seated (interlocked) asperities, 2. random asperity contact. Random asperity contact is a usual situation of contact of two non- correlated surfaces.Onthe contrary, the case ofwell-seated asperities is typical for brittle interfaces where the contact surface with its roughness (asperities) is a result of a crack propagation process (decohesion of a cohesive interface or a crack that formed the joint in the past). Modelling of friction and dilatancy effects... 717 The surface roughness may be seen as a composition of asperities of dif- ferent length scales. A simplifying assumption is often adopted, namely two scales of asperities are considered: primary asperities (the largest ones) and secondary asperities, cf Mróz and Giambanco (1996), Mróz and Stupkiewicz (1994,1998). This assumption allows to separate the basic mechanisms in a clear way, although in reality the interactions of asperities of different scales are coupled. In this work we assume that the primary asperities are respon- sible for interfacial dilation, while the secondary asperities govern the friction conditions at the inclined contacts of primary asperities, cf Fig.9. Fig. 9. Primary and secondary asperities in an interlocked joint The interaction of well-seated primary asperities is a basic mechanism explaining the dilation effects in joints. Depending on the shape of asperities, different dilatancy curves in direct shear (A.6) may be generated, cf Fig.5. Also the shear resistance response with peak (A.3) or without peak (A.4), as well as a constant residual resistance (A.5), may be predicted depending on the asperity shape and the range of relative displacements considered. Finally, in cyclic shearing the change from downward to upward sliding results in an increase of friction (B.2). These topics are discussed in detail in Section 3.2. 2.4.3. Random asperity contact It is natural to expect that randomly contacting asperities are less stiff and easier tobecrushed than thewell-seated asperities (A.2). In shearing, there are no dilation effects observed for well-seated asperities, however, the wear and damageof asperitiesmay result in compactionat the interface. Letusnote that the randomcontact of asperities occurs after a jointwith interlocked asperities undergoes a shearing displacement larger than the characteristic length of primary asperities (assuming that the asperities are not ideally periodic). 2.4.4. Wear and damage of primary asperities Wear and damage of initially interlocked primary asperities result in a decrease ofdilation for subsequent cycles (B.3).Also thehighcontact pressures 718 S.Stupkiewicz, Z.Mróz accelerate the wear/damage process and lead to a further decrease of the effective dilation angle (A.7). 2.4.5. Wear and damage of secondary asperities The frictional properties at the inclined contacts of primary asperities are governed by the secondary asperities. These also undergo damage and wear in the course of relative sliding, thus the effective friction properties evolve. In particular, a peak followed by a residual friction stress may be associated with wear and damage of secondary asperities (A.3, A.5). 2.4.6. Formation of a third body granular layer Thesurfacedamageassociatedwithasperity crushingandwear leads to the formation of a third body layer. This layer consists of free particles of different sizes andmay be seen as a granularmaterial.While this fact seems to bewell recognized, the respective models are not available. In Section 3.3 we propose a phenomenological model of a granular third body layer. In particular, the effect of thedropof dilation angle at the load reversals (B.4)maybeattributed to the rearrangement of free particles due to the change of shearing direction. 3. Constitutive interface models In this section we review the existing models with respect to their ability to describe the basic effects discussed in the previous section. A new constitu- tive interface model will also be discussed in Section 3.3. The analysis of this section ismostly qualitative aswe concentrate on the effects rather than on re- producing experimental data exactly. For simplicity, we restrict our analysis to two-dimensional problems. Extension to general three-dimensional problems will be presented elsewhere. A general model of friction is formulated using a friction condition F ¬ 0 and a slip potential G = const. The actual forms of functions F and G depend on the specific model types and will be stated for each considered model. In the plasticity theory approach to friction, the total relative velocity u̇= [u̇t, u̇n] ⊤ is decomposed into elastic u̇e and inelastic u̇s components u̇= u̇e+ u̇s (3.1) Modelling of friction and dilatancy effects... 719 and the rate of contact traction ṫ= [ṫt, ṫn] ⊤ is related to the elastic velocity component by the rule ṫ=Du̇e D= [ kt 0 0 kn ] (3.2) where kn,kt are thenormal and tangential contact stiffness parameters,which may generally depend on contact stresses and the respective state variables. The slip potential G= const generates the following slip rule u̇ s = λ̇ ∂G ∂t λ̇­ 0 λ̇F =0 (3.3) where λ̇ is a plastic multiplier. 3.1. Coulomb-type laws Let us first introduce a simple prototype law as a basis for further discus- sion of prediction capabilities and possible enhancements. The limit friction surface F =0 of the Coulomb law is a cone in the space of contact tractions, cf Fig.10, described by the friction condition F(tn, tt)= |tt|+ tn tanΦ¬ 0 (3.4) accompanied by the slip potential G(tn, tt)= |tt|+ tn tanΨ (3.5) Here, Φ and Ψ denote the friction and dilatancy angles, respectively, which are the constitutive parameters of themodel. The slip rule now takes the form u̇ s = λ̇ ∂G ∂t = λ̇m m= [ s tanΨ ] s= tt |tt| (3.6) where s=±1 is the direction of slip. Equation (3.6) predicts sliding inclined at a constant dilatancy angle Ψ with respect to thenominal contact plane.Themonotonic and cyclic dilatancy curves are schematically plotted in Fig.11. Clearly, the predicted response is highly idealized. Firstly, the dilation is not limited as the shear displacement increases. Secondly, the cyclic response is completely different from that obse- rved experimentally, cf Fig.6. Thus the applicability of this law is limited to monotonic processes and small shear displacements. 720 S.Stupkiewicz, Z.Mróz Fig. 10. Limit friction surface of the Coulomb law Fig. 11. Dilatancy response of a simple Coulomb law: (a) monotonic and (b) cyclic shearing The excessive dilation predicted by theCoulomb law at large shear displa- cements may be avoided by assuming an evolution rule for the dilation angle, for example in the following simple form Ψ =Ψ0exp ( − κΨ κΨ ) (3.7) where Ψ0 is an initial dilation angle, κΨ is a state variable governing the evolution of dilation angle and κΨ is amaterial parameter governing the rate of variation of the dilation angle. By analogy to work (or strain) hardening variables in the plasticity theory, the evolution variable κΨ may be defined as follows κ̇Ψ = ttu̇ s t or κ̇Ψ = |u̇ s t| (3.8) The dilation in monotonic shearing is now limited, cf Fig.12a, and thusmore Modelling of friction and dilatancy effects... 721 realistic. However, the cyclic response is still qualitatively different from that observed experimentally, Fig.12b. Fig. 12. Dilatancy response of a Coulomb lawwith decreasing dilation angle: (a) monotonic and (b) cyclic shearing The simple Coulomb law can bemodified to include the effects of decohe- sion and softening response of the interface. Assume the yield surface in the form F(tn, tt,η)= |tt|+µtn− (1−η)τu ¬ 0 (3.9) where 0¬ η¬ 1 is a damage-like variable. The evolution of η is governed by η̇= |u̇st| ucrt for η< 1 (3.10) where ucrt = 2Gf/τu. When η = 1, the cohesive strength is lost and a pure Coulomb friction model is obtained. The parameters of the model are the ultimate shear strength τu, specific fracture energy Gf and residual friction coefficient µ. The response of the model is depicted in Fig.13. As the model is very simple, it does not properly describe tensile and mixed tensile-shear decohesion modes. Several interface models have been derived from the Coulomb law by ad- ding cohesionwith hardening/softening effects and evolution rules for dilation angle, cf Gens et al. (1989), Lotfi and Shing (1994), Gambarotta and Lago- marsino (1997), Lourenco and Rots (1997), Raous et al. (1999). The simple Coulomb’s cone has been replaced by a hyperbolic limit surface (with a hyper- bolic sliding potential) to improve the description of mixed mode decohesion and the effect of normal stress on friction and dilation by Lotfi and Shing (1994). A model with additional surfaces related to tensile and compressive damage has been proposed by Lourenco and Rots (1997). 722 S.Stupkiewicz, Z.Mróz Fig. 13. The response of the cohesive Coulomb law in direct shear: (a) monotonic loading; (b) unloading response In these Coulomb-type interface laws much effort has been put to capture several important effects of the friction (peak and residual resistance, decohe- sion) and dilation response in monotonic loading processes. As a result, the monotonic behaviour of different types of joints is well described by these models. However, the cyclic effects, cf Tables 1 and 2, which originate from the interaction of interlocked asperities are completely ignored in these mo- dels. Particularly, the cyclic dilation behaviour, Fig.12b, is unacceptable as compared to the actual typical response, cf Fig.6. 3.2. Asperity interaction models 3.2.1. Configurational dilatancy model Let us first introduce a simple model of fully reversible configurational dilation. The dilation is assumed to solely result from the interaction of inter- locked asperities. Thewear and damage of asperities aswell as initial cohesion of the joint are neglected at this stage, thus no irreversible dilatancy effects occur. The interaction of initiallywell-seated asperities is schematically illustrated in Fig.14. Since wear and damage of asperities are neglected, we can assume that dilation depends only on the shearing displacement of the joint, cf the concept of the relative displacement surface of sliding introduced by Gerrard (1986). A physically acceptable and consistent model requires the dilation to be limited for increasing shearing displacement. The following function is Modelling of friction and dilatancy effects... 723 adopted to describe the dilation function δ(ust), cf Stupkiewicz (1996), usn = δ(u s t)= δAδu (ust uA ) (3.11) where δA and uA denote the characteristic height and length of primary asperities. The dimensionless function δu defines the actual shape of the di- lation response curve andmust satisfy two conditions, namely δu(0)= 0 (i.e. no dilation in the absence of shearing displacement) and δu(±∞) = 1 (i.e. asymptotic dilation equal to δA). Fig. 14. Configurational dilation resulting from interaction of initially matching primary asperities: (a) ust =0, u s n =0; (b) 0< |u s t|uA, u s n = δA Some insight into possible forms of the function δu can be gained by studying the interaction of periodic asperities. Periodic saw-tooth and sine- shaped asperities are shown together with respective dilation curves inFig.15. Note that the periodicity assumption leads to a non-physical dilation decrease once the asperity peaks enter neighbouring valleys, cf dashed lines in Fig.15. As the real asperities are not periodic, it is reasonable to assume that after the asperities leave their initial valleys, the random contact of asperity peaks occurs resulting in a constant dilation for ust >uA. The local asperity contacts are inclinedwith respect to thenominal contact plane.We shall assume that friction at the inclined contacts of primary aspe- rities is governed by the Coulomb law (with no dilation). Thus the effective friction angle for sliding in an upward direction is Φ+ |Ψc| and in a downward direction is Φ− |Ψc|, cf Fig.16. The angle of configurational dilatancy Ψc follows from (3.11) Ψc = dδ dust (3.12) The limit friction condition can be formulated using global contact stresses tt, tn or, in a rotated coordinate system,using local stresses t ′ t, t ′ n, cfFig.16. In the space of global stresses the limit friction condition and the slidingpotential have the form 724 S.Stupkiewicz, Z.Mróz Fig. 15. Dilation curves resulting from the assumption of periodic saw-tooth (a) and sine-shaped (b) asperities F(tn, tt)= stt+ tn tan(Φ+sΨc)¬ 0 (3.13) G(tn, tt)= stt+ tn tan(Ψ+sΨc) where Ψ is the dilation angle at the inclined contact of primary asperities (for Ψ = 0 only the dilatancy resulting from the interaction of primary asperities is modeled, cf (3.11)). Note that here the sliding direction factor s = ±1 is explicitly involved in the description. In the case of Φ< |Ψc| the direction of sliding is different from the direction of friction stress, thus s= sgnu̇t should be used in Eqs (3.13) rather than s= sgntt as in Eqs (3.6). Alternatively, the limit condition can be formulated in terms of local stres- ses t′t, t ′ n, namely F ′(t′n, t ′ t)= |t ′ t|+ t ′ n tanΦ¬ 0 (3.14) G′(t′n, t ′ t)= |t ′ t|+ t ′ n tanΨ While the limit friction condition and sliding potential are exactly the same as in case of the Coulomb law (3.4) and (3.5), the coordinate system rotates as the dilation angle changes t ′ =Qt u̇′ =Qu̇ (3.15) where Q= [ cosΨc sinΨc −sinΨc cosΨc ] Q ⊤ =Q−1 (3.16) Modelling of friction and dilatancy effects... 725 Fig. 16. Friction condition at an inclined contact of primary asperities Although the description is simpler, the respective incremental relations invo- lve the rotation velocity terms since Ψ̇c 6=0. The slip rule takes now the form u̇ ′s = λ̇′ ∂G′ ∂t′ = λ̇′m′ m′ = [ s′ tanΨ ] s′ = t′t |t′t| (3.17) The slip direction in the global coordinate system is obtained by rotating u̇′s, thus u̇ s =Q⊤u̇′s = λ̇′Q⊤m′ = λ̇′m (3.18) and the resulting effective dilatancy angle is u̇sn/u̇ s t =tan(Ψc+s ′Ψ), in agre- ement with (3.13). By putting Ψ = 0 a model with reversible configurational dilatancy is obtained. Themodel is fully defined by the friction angle Φ and the dilatancy curve δ(ust). Some typical dilatancy curves and the resulting friction response curves in cyclic direct shear are sketched in Fig.17. The curves in Fig.17a,b predict an unlimited dilation with increasing shear displacement and are thus only suitable for small relative displacements. Note, however, that similar re- sponse may be obtained by assuming the dilation curves of Fig.17c,d with a small shear displacement amplitude. The curves in Fig.17a,c have a singular point at ust =0.While this is close to the behaviour observed experimentally, it would lead to numerical problems when applying the model in structural analysis. Assuming a dilatancy curve with a non-zero radius at ust =0 is also justified in view of asperity wear and damage during relative sliding. 726 S.Stupkiewicz, Z.Mróz Fig. 17. Configurational dilation: typical dilation curves and resulting cyclic shear response Modelling of friction and dilatancy effects... 727 The specific form of the configurational dilation curve used in the current study is given by usn = δ(u s t)= δA tanh [√ (ust tanΨ0) 2+g20 −g0 δA ] (3.19) where δA is the asperity height, g0 is a parameter defining the radius of the dilation curve at ust = 0 and Ψ0 is the initial dilation angle in case of g0 =0. For small values of g0 a nearly singular curve is obtained, while with increasing g0 the radius also increases, cf Fig.18. Let us note that by choosing proper values of model parameters, all the typical dilation curves depicted in Fig.17 may be described by Eq. (3.19). Fig. 18. The effect of g0 on the dilation curve (3.19) for g0/δA =0.01, 0.2, 1 3.2.2. Asperity wear and damage The effect of asperity wear and damage can be accounted for by assuming the dilation curve to depend additionally on the frictionwork.A simplemodel of this formhasbeenproposedbyStupkiewicz (1996), byassumingtheasperity height δA, cfEqs (3.11) and (3.19), to decrease due to the accumulated friction work. Let us consider the slip rule resulting from the model of configurational dilationwithwear anddamage effects.We assumean explicit relation between thedilation usn, relative slidingdistance u s t and the accumulated frictionwork κδ, where κ̇δ = ttu̇ s t (3.20) 728 S.Stupkiewicz, Z.Mróz Note that other variables governing the wear effects may also be used, e.g. κ̇δ = ttu̇ s t + tnu̇ s n or κ̇δ = t ′ tu̇ ′ t s. By taking the time derivative of the relation usn = δ(u s t,κδ) we have u̇sn = ∂δ ∂ust u̇st + ∂δ ∂κδ κ̇δ =tanΨcu̇ s t + ∂δ ∂κδ ttu̇ s t = (3.21) = [ tanΨc−s ∂δ ∂κδ tn tan(Φ+sΨc) ] u̇st where the limit friction condition (3.13)1 has also been used. Thus in addition to the configurational dilation there is a compaction term resulting fromwear. With increasingnormalpressure this additional term increases andthedilation is reduced, cf effect A.7 in Table 1. Adopting the dilation curve (3.19), thewear effect can be accounted for by assuming that the asperity height decreases due to accumulated friction work κδ, thus usn = δ(u s t,κδ)= δAe − κ δ κ δ tanh [√ (ust tanΨ0) 2+g2+g20 −g0 δA ] (3.22) We also assume that the deposition of wear debris results in a decrease of curvature of the dilation curve at ust =0. The latter effect is governed by the term g = gf[1− exp(−κδ/κg)] in Eq. (3.22). The effect of increasing g on the dilation curve (3.22) is illustrated in Fig.19. The parameters of themodel related to wear effects are κδ, κg and gf. Two examples of cyclic response with wear effects are shown in Fig.20. The response in Fig.20a is obtained with 1/κg = 0 so that the wear effects are only due to decreasing asperity height. Friction and dilation models based on the interaction of interlocked aspe- rities have been studied by several authors and start from theworks of Patton (1966) and Jaeger (1971) who considered wedge-like asperities and a constant dilation angle.Models with dilation angle decreasing with increasing shear di- splacement have been proposed by Plesha (1987), Snyman andMartin (1992) andMróz andGiambanco (1996). Additional effects, such as the effect of wear on asperity slope (and dilation angle), as well as independent evolution of ”ri- ght” and ”left” asperity slopes, have also been accounted for byPlesha (1987). Mróz andGiambanco (1996) additionally considered themicro-slip phenome- na, andGiambancoandDiGati (1997) extended theasperity interactionmode by adding cohesion and softening effects. Modelling of friction and dilatancy effects... 729 Fig. 19. The effect of g on the dilation curve (3.22) with g0/δA =0.01 and for g/δA =0, 0.1, 0.5 As thewear effects are considered, themain difference between the current approach (cf Stupkiewicz, 1996) and the works mentioned above is that we postulate dilation as an explicit functionof relative displacement, frictionwork and possibly other state variables. In other models (Plesha, 1987; Mróz and Giambanco, 1996) it is the dilation angle that is specified as a function of relative displacement and friction work. In our approach the current dilation angle follows fromEq. (3.21) and is different from the configurational dilation angle Ψc which defines the rotated contact tractions t ′ t and t ′ n. 3.3. Phenomenological model of a third body granular layer In this section we propose a phenomenological description of a third body granular layer. Asperity wear, damage and, more generally, also surface wear at the contact interface lead to formation of a third body layer consisting of wear debris and particles of crushedmaterial of contacting bodies. In the case of brittlematerials this layer can be assumed to be a layer of granularmaterial with particles of different sizes.While the fact of formation of such third body layer seems to bewell-recognized, there are nomodels describing its effects on the interface response. Further, as alreadymentioned in Section 2.4, the effect of dilation angle drop (cf Section 2.3.4 andFig.6c) cannot be explained by any of the knownmicro-mechanical interactionmodes.We propose an explanation of this effect by considering the rearrangement of free third body particles at the interface. 730 S.Stupkiewicz, Z.Mróz Fig. 20. Configurational dilation with wear effects: friction and dilation response for (a) 1/κg =0 and (b) 1/κg > 0 Consider two bodies in contact separated by a layer of granular material. Upon shearing inonedirection, a steady state configurationof particles occurs. This is schematically illustrated in Fig.21a. If now a change of sliding direc- tion takes place, the configuration of particles is rearranged,Fig.21b, andafter some shear displacement related to the transient state, a steady state configu- ration associatedwith sliding in an opposite direction is obtained, Fig.21c. As schematically illustrated in Fig.21, the change of sliding direction results in compaction and subsequentdilation of the interface duringa transient particle rearrangement process. Fig. 21. Configuration rearrangement of particles after the change of sliding direction Modelling of friction and dilatancy effects... 731 In order to describe thememory associated with configuration of particles in the layer we shall introduce a rotating yield surface F =0 moving within the steady state surface Fs = 0 as illustrated in Fig.22. The steady state surface Fs =0 is characterized by a friction angle Φs (dashed lines in Fig.22) Fs(tn, tt)= |tt|+ tn tanΦs =0 (3.23) and dilation angle Ψs so that the steady state slip potential is Gs(tn, tt)= |tt|+ tn tanΨs (3.24) Fig. 22. Rotating yield surface model for granular layer Theactual yield surface F =0 is presented in a coordinate system rotated by angle Θ with respect to the nominal contact plane F(t′n, t ′ t)= |t ′ t|+ t ′ n tanΦ=0 (3.25) where Φ<Φs and t ′ =Qt Q= [ cosΘ sinΘ −sinΘ cosΘ ] (3.26) The concept of a rotating cone adopted in the present model is similar to the kinematic hardeningmodel for sandproposedbyGajo andMuirWood (1999). The slip rule is generated by the slip potential G(t′n, t ′ t)= |t ′ t|+ t ′ n tanΨ (3.27) 732 S.Stupkiewicz, Z.Mróz so that u̇ ′s = λ̇′ ∂G′ ∂t′ = λ̇′m′ m′ = [ s′ tanΨ ] s′ = t′t |t′t| (3.28) where u̇′s =Qu̇s is the (inelastic) relative velocity in the rotated coordinate system. Note that the effective dilation angle (i.e. the dilation angle in the global coordinate system) is Ψeff =arctan u̇sn |u̇st| = s′Θ+Ψ (3.29) where Ψeff > 0 for u̇ s n > 0 (dilation) and Ψeff < 0 for u̇ s n < 0 (compaction). In the steady state the stress point lies on the steady state surface Fs =0 so that the rotation angle Θ equals Θs = s ′(Φs −Φ). Thus we assume a simple law to describe the evolution of Θ in the transient state Θ̇= [s′(Φs−Φ)−Θ] κ̇Θ κΘ (3.30) where thehardeningvariable κΘ is definedby (other forms including frictional work are also possible) κ̇Θ = |u̇ s t| (3.31) and κΘ is amodel parameter (i.e. a characteristic sliding displacement related to the rotation of the yield surface F =0). We also assume that in the steady state the effective dilation angle Ψeff is equal to the steady state dilation angle Ψs. This condition provides the following relation between the dilation angles Ψ and Ψs Ψ =Ψs−s ′Θs =Ψs− (Φs−Φ) (3.32) which, importantly, is assumed to hold also in the transient states. As the dilation behaviour is the most important part of this model, let us illustrate the assumptions (3.30) and (3.32) in the case of shearing at constant normal pressure. Assume that the steady state dilation angle is constant and equal to Ψs = 0 and consider a steady state sliding situation with u̇ s t > 0. Since Ψs = 0, the normal relative velocity is u̇ s n = 0, cf point 1 in Fig.23. When the friction stress is reduced, sliding in an opposite direction occurs after some elastic unloading stage – point 2 in Fig.23. Note that the effective dilation angle at point 2 is Ψeff = −2Θ = −2(Φs −Φ) thus sliding in an Modelling of friction and dilatancy effects... 733 opposite direction is associated with non-zero compaction velocity, Fig.23b. We can integrate (3.30) along the path 2-3-4 analytically, so that Θ=(Φs−Φ) ( 2e u s t −ũ s t κ Θ −1 ) (3.33) where ũst ¬ 0 is the sliding displacement at load reversal. Fig. 23. Rotating conemodel (with Ψs =0): (a) evolution of the yield surface upon load reversal; (b) dilatancy and frictional response Assuming that the effective dilation angle Ψeff is small, for s ′ = −1 we can write dusn =Ψeff|du s t|=(s ′Θ+Ψ)s′ dust =(Θ−Ψ)du s t = (3.34) = 2(Φs−Φ)e u s t −ũ s t κΘ dust where we have also used (3.32) and (3.33). Integrating (3.34) along the path 2-3-4 we finally obtain the dilation response, namely usn− ũ s n κΘ =2(Φs−Φ) ( e u s t −ũ s t κ Θ −1 ) (3.35) where ũsn is the relative normal displacement at load reversal. The dilatancy and frictional response resulting fromthe solution (3.33) and (3.35) is depicted in Fig.23b. 734 S.Stupkiewicz, Z.Mróz Let us note that the assumptions adopted in the above example result in contact compaction at each load reversal (∆usn =−2(Φs−Φ)κΘ between steady sliding states, cf Eq. (3.35)). Since this is a non-physical result,we shall no longer assume that the steady state dilation angle Ψs is constant. Instead we assume that there is an asymptotic thickness of the granular layer which depends on the normal pressure. Thus even for the points of the steady state surface some dilation may occur until the asymptotic thickness is attained. We thus propose the following evolution law for Ψs Ψ̇s =− [usn−u as n κΨ +2ηΨs ]κ̇Ψ κΨ κ̇Ψ = |u̇ s t| (3.36) where uasn = u as n (tn) is a relative normal displacement related to the asymp- totic thickness of the layer and κΨ is a parameter providing the displacement scale for evolution of Ψs. Since in view of (3.30) and (3.32), equation (3.36) is in fact a second order differential equation for usn(u s t), we introduce a critical damping term 2ηΨs with η > 1 in order to avoid oscillatory solutions. The dilatancy response to cyclic shearing is schematically shown in Fig.24. Fig. 24. Rotating conemodel for granular layer: dilatancy response to cyclic shearing Concluding, the model of the third body granular layer involves the yield surface (3.25) and the slip rule (3.28) accompanied by the evolution laws for rotation angle Θ and steady state dilation angle Ψc, Eqs (3.30) and (3.36). Theparameters of themodel are: Φs, Φ,κΘ and κΨ.Also the function u as n (tn) must be specified (as the first approximation wemay assume uasn =0). 4. Discussion and conclusions The friction and dilatancy effects at brittle interfaces have been studied in thepaper.As alreadydiscussed, the term brittle interfacesdenotes awide class Modelling of friction and dilatancy effects... 735 of both cohesive and non-cohesive interfaces where the contacting bodies are brittle. This, however, implies another common characteristic feature, namely the wear debris and thematerial of crushed asperities form a third body layer which can be regarded a granular material. Also, even more importantly, in practice most of the brittle interfaces are characterized by interlocked aspe- rities and related friction and dilatancy effects. The latter effects are crucial for constitutive modelling, especially in the case of non-monotonic and cyclic loading programs. We have shown that some of the effects observed in the experiments may result from more than one mechanism. This mostly regards the monotonic response effects, such as peak friction and dilation in direct shear (effects A.3 and A.6 in Table 2). Thus the response to monotonic shearing may not be sufficient to distinguish between the mechanisms and, quite obviously, to predict the non-monotonic or cyclic response. Themost importantmicro-mechanisms occurring at brittle interfaces have been identified, namely decohesion, interaction of interlocked primary asperi- ties including wear and damage and formation of a third body granular layer. Simple constitutive models have also been proposed providing description of each of these mechanisms separately, and through a qualitative analysis the predicting capabilities of the models have been assessed. The analysis of existing models for cohesive interfaces (Section 3.1) leads to the conclusion that thesemodels aremostly suitable formonotonic loading processes. Particularly, the predictions of cyclic dilation response, cf Fig.12b, are qualitatively different from the typical response types observed in experi- ments, cfFig.6.The reason is thatmost of thesemodelsneglect the interlocked asperity interaction mode. Themodel of Giambanco andDi Gati (1997) is an exception as it provides a transition from the cohesive mode to the asperity interaction mode. InSection3.2, amodel of configurational dilatancyhasbeenproposed.This model generalizes the concepts of interaction of interlocked primary asperities (well-known in the field of rock joints). Themodel provides a transition from the interlocked to the random asperity contact modewith increasing shearing displacement and describes the related friction anddilatancy effects. A refined model might also include the description of the evolution of elastic stiffness properties of the interface coupled with the transition from the interlocked to the random asperity contact, cf Fig.3. Finally, in Section 3.3, a model for a granular third body layer has been proposed based on the concept of a rotating cone representing rearrangement of particle configurations associated with shearing in different directions. The 736 S.Stupkiewicz, Z.Mróz model predicts a drop of the dilation angle at each load reversal as well as transient and hysteretic effects. Fig. 25. 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Raous M., Cangemi L., Cocu M., 1999, AConsistentModel Coupling Ad- hesion, Friction and unilateralContact,Comp.Meth. Appl. Mech. Engrg., 177, 383-399 31. Snyman M.F., Martin J.B., 1992, A Consistent Formulation of a Dilatant Interface Element, Int. J. Num. Anal. Meth. Geomech., 16, 493-527 Modelling of friction and dilatancy effects... 739 32. Stupkiewicz S., 1996, Fiber Sliding Model Accounting for Interfacial Micro- Dilatancy,Mech. Mater., 22, 65-84 33. SunZ.,GerrardC., StephansonO., 1985,RockJointComplianceTests for Compression and Shear Loads, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 4, 197-213 Modelowanie tarcia i dylatacji w kruchych warstwach kontaktowych przy obciążeniach monotonicznych i cyklicznych Streszczenie Praca zawiera analizę najważniejszych efektów powstających w procesach defor- macji kruchych warstw kontaktowych przy obciążeniach monotonicznych i cyklicz- nych.Dostępnemodele konstytutywne poddano krytycznej analizie pod kątemmożli- wości opisania tych efektów. Rozważono licznemechanizmymikromechaniczne, takie jak dekohezja, oddziaływanie pierwotnych iwtórnychnierówności, zużycie i uszkodze- nie nierówności oraz powstawanie warstwy trzeciego ciała. Zaproponowano również nowe sformułowaniamodeli konstytutywnych dla stanów cyklicznych. Manuscript received December 20, 2000; accepted for print January 8, 2001