A model for rockill compressibility

A model for rock®ll compressibility L. A. OLDECOPandE.E.ALONSO  The paper presents a macroscopic constitutive model for rock®ll that includes the effect of water on compressibility and collapse phenomena. Breakage of rock particles and fracture propagation are basic underlying mechanisms controlled by the relative humidity of the air ®lling the rock®ll voids. A conceptual deformation model based on these mechanisms is ®rst proposed and discussed. The results of oedometer tests on a quartzitic slate rock®ll, in which the air relative humidity was controlled, are then presented. A signi®cant ®nding is that bringing the relative humidity within the specimen to its maximum (100% RH) leads to a collapse strain equal to that observed in ¯ooded specimens. An elastoplastic constitutive model, consistent with the basic deformation framework, is developed. Its parameters have a clear physical meaning. Guidelines for parameter determination are given. Model performance is ®nally compared with the results of the experimental programme. KEYWORDS: compaction; compressibility; constitutive relations; gravels; laboratory tests; partial saturation. L'article pre Âsente un mode Ále macroscopique de comportement des mate Âriaux d'enrochement. Il prend en compte l'in¯uence de l'eau sur les phe Ânome Ánes de compressibilite Âet de collapse. Les me Âcanismes fondamentaux sous-jacents conside Âre Âs sont le broyage et la propagation de la fracture a Á l'inte Ârieur des particules, controlle Âs par l'humidite Ârelative de l'air remplissant les vides du mate Âriau d'enrochement. Un mode Âle conceptuel de de Âformation est premie Árement propose Âet discute Âsur la base de ces me Âcanismes. Des re Âsultats d'essais oedome Âtriques a Áhumidite Ârelative contro Ãle Âere Âalise Âs sur des enrochements d'ardoise quartzitique sont ensuite pre Âsente Âs. Un re Âsultat important est le fait que la de Âformation de collapse obtenue par augmentation de l'humidite Ârelative jasqu'a Áune valeur de 100% est la me Ãme que celle observe Âe par inondation de l'e Âchantillon. Un mode Ále e Âlastoplastique cohe Ârent avec le cadre conceptuel de de Âformation propose Âest alors de Âveloppe Â. Ses parame Átres ont une signi®cation physique claire et la de Âmarche pour les obtenir est donne Âe. Les pre Âdictions du mode Ále sont compare Âes avec les re Âsultats du programme expe Ârimental. INTRODUCTION The use of rock®ll as a construction material started during the second half of the nineteenth century with the building of timber-faced mining dams in California (Penman, 1992). Since then, and essentially in connection with the construction and operation of large rock®ll dams in the twentieth century, a large number of ®eld observations concerning rock®ll behaviour have been collected. Rock®ll exhibits a slow accumulation of deformation over many decades (Sowers et al., 1965). When upstream shells of zoned earthdams are submerged, a collapse is often observed. Downstream shells also collapse after rainfall (Marsal et al., 1976; Naylor et al., 1997), a behaviour that is dif®cult to explain with the current understanding of rock®ll deformation mechanisms. Terzaghi (1960) suggested that a possible reason for the large deformations of rock®lls could be the breakage of rock particles in the vicinity of highly stressed contacts and the subsequent rearrangement of the granular structure into a more stable position. During the 1960s and 1970s, much effort was put into testing rock®ll samples in the laboratory. Large-scale testing equipment was developed in some research centres (Sowers et al., 1965; Fumagalli, 1969; Marachi et al., 1969; Marsal, 1973; Penman & Charles, 1976; Veiga Pinto, 1983), and all types of classical soil mechanics tests were carried out. The clear conclusion that arises from all this experimental work is that the main cause of the distinct rock®ll behaviour is particle breakage, a phenomenon that seems to be enhanced by the presence of water. Nobari & Duncan (1972) carried out a systematic investigation of the factors that in¯uence the collapse of rock®ll upon ¯ooding. Their work included one-dimensional and triaxial compression tests on crushed argillite. They found that the initial water content was the most important factor determining the amount of collapse upon ¯ooding: the larger the initial water content, the smaller was the collapse deformation. Sieve analysis carried out before and after ¯ooding the sample showed that during collapse some particle crushing occurs. This observation suggests that the reduction of rock strength due to wetting may be the mechanism that determines the collapse behaviour. Some researchers (Sowers et al., 1965; Marsal, 1973; Clements, 1981) tested isolated rock particles and rock points under compression. In constant-loading tests, Clements (1981) observed time-dependent deformation of contacts. When, after a certain time under constant load, the contact was ¯ooded, additional displacements (also delayed in time) occurred. Clements' results for isolated contacts are compared in Fig. 1 with the results of oedometer tests reported by Sowers et al. (1965). The parallel behaviour between both types of test suggests that time-dependent strains and collapse of rock®lls are both closely related to the particle breakage phenomenon. The aim of this paper is to present a macroscopic constitutive model for rock®ll stress±strain behaviour, consistent with previous experiences and with more recent experimental results, which are also presented and discussed in the paper. A basic mechanism of rock®ll volumetric deformation that is based on a fracture propagation mechanism is ®rst proposed. This deformation mechanism is able to give a qualitative physical explanation of time-dependent strains and collapse strains of rock®ll, and of their simultaneous dependence on stress and water action. Rock®ll can be considered as having two sets of voids. A set of large voids is formed by the inter-particle spaces. Rock particles have their own natural porosity, and hence a second set of very much smaller voids can be identi®ed within the rock particles. In order to avoid confusion, the following large interparticle voids will be termed rock®ll voids, while the term rock pores will be used to refer to the small pores within the rock particles. In the latter group, only open pores are considered (i.e. those connected with the rock®ll voids). The experimental observations mentioned suggest that the action of water contained within the rock pores has a determinant role on rock®ll deformation. It was thought that tests in which the water content of rock pores could be controlled would provide a better insight into rock®ll behaviour. Such tests were performed on a rock®lltype material using a relative humidity control technique. Their results are interpreted in terms of the proposed conceptual model in the third part of this paper. 127 Oldecop, L. A. & Alonso, E. E. (2001). Ge Âotechnique 51, No. 2, 127±139 Manuscript received 24 May 1999; revised manuscript accepted 3 May 2000 Discussion on this paper to reach editor by 6 September 2001, for further details see inside front cover. Department of Geotechnical Engineering and Geosciences, Universitat Politecnica de Catalunya, Barcelona, Spain. On the basis of the experimental results and the proposed deformation mechanism, a constitutive model for the compressibility of rock®ll that takes into account the in¯uence of water is then proposed. It includes the mechanisms of rock®ll deformation in an elastoplastic framework. In this way, an overall macroscopic constitutive model is developed. The proposed model and the comparison between test results and model performance focus on volumetric deformations. Behaviour under deviatoric stress changes lay outside the scope of the paper. It is believed, however, that the framework developed provides a convenient basis for its extension to generalized stress states. A CONCEPTUAL MODEL FOR ROCKFILL VOLUMETRIC DEFORMATION Among the reported water-induced phenomena leading to particle breakage [loss of cohesion due to mineral super®cial energy reduction (Vutukuri & Lama, 1978); suction reduction (Vutukuri & Lama, 1978); expansion of clay minerals (De Alba & Sesana, 1978; Delgado et al., 1982)], the subcritical crack propagation due to stress corrosion (Atkinson & Meredith, 1987) offers a comprehensive explanation of rock®ll behaviour. Any ¯aw in a piece of rock acts as a stress concentrator, and can initiate a crack that propagates under the action of load and also under the action of water. The stress intensity factor, as de®ned in linear elastic fracture mechanics (LEFM) (Broek, 1985), allows the complete characterization of the stress ®eld in the crack tip vicinity. It embodies the in¯uence of the crack length, the applied load, the geometry and the size of the body containing the crack, and can be computed with the following general expression (Broek, 1985): KLâó  ða p(1) where óis the stress far from the crack in¯uence zone, ais the length of the crack, and âis a dimensionless factor that depends on the geometry of the body: that is, the shape and the ratio a=D, where Dis a generalized size parameter. The subscript Ldenotes the crack load mode: tensile, mode I; shear normal to crack tip, mode II; and shear parallel to crack tip, mode III. For many simple geometries, the factor âcan be obtained from handbooks (Murakami, 1987). The value of KLthat marks the onset of crack propagation for a given material is called the fracture toughness,KLc, and can be obtained for each load mode Lfrom suitable tests. In terms of LEFM, while KL,KLcthe crack does not grow at all, and when KL>KLca catastrophic propagation occurs. Crack propagation mechanisms in pure mode I (tension) are by far the most studied by experimental and theoretical work. In the following discussion, we shall refer to experimental results that were obtained mostly for mode I loaded cracks. However, there is no theoretical reason why the conclusions reached should not be extended to any load mode present in the rock particles. Keeping this in mind, subscript Lwill be omitted in the following. In spite of the LEFM propagation criteria, it was observed that cracks may also grow in situations in which Kis lower than the fracture toughness. This is called subcritical crack growth, and it may be caused by several different mechanisms. The most familiar of these mechanisms is the fatigue due to repeated load application, but subcritical crack propagation may also be due to water-induced corrosion. The theory of stress corrosion (Michalske & Freiman, 1982; Atkinson & Meredith, 1987) states that the strained inter-atomic bonds at the tip of a crack are more vulnerable to the attack of a corrosive agent, such as water, than the unstressed material away from the tip. The corrosion reaction produces a weaker material, which is broken at lower Kvalues than the uncorroded material. A typical relationship between crack propagation velocity, V, and K, as obtained from experiments performed in a constant relative humidity (RH) environment, is shown in Fig. 2 in qualitative terms. The thermodynamic formulation of the reaction rate theory (Laidler, 1987) applied to stress corrosion cracking (Wiederhorn et al., 1980, 1982; Freiman, 1984) yields the following general expression for the crack propagation velocity with water as corrosive agent: VV0(RH)exp[(ÿE{bK)=RT] (2) where RH is the relative humidity in the testing environment, K is the stress intensity factor in the crack, Tis the absolute Crack propagation velocity, V (log scale) Increasing relative humidity 0 K 0 K c Wet Equation (2) Dry Stress intensity factor, K Region ΙRegion ΙΙ Region ΙΙΙ Fig. 2. Schematic subcritical crack growth curves and conceptual model of rock®ll volumetric deformation 0 8 16 24 32 2·0 2·5 3·0 4·0 3·5 0 1000 2000 3000 4000 5000 6000 7000 Settlement of initial height: % Displacement: 10–3 mm 0 30 60 90 120 150 180 Time: min Saturation (a) (b) Time after load applied: min Jetting commenced Jetting commenced Jetting commenced Graywacke: initially dry Sandstone: initially dry Sandstone: initially wet Fig. 1. (a) Effect of saturation in contact test of shaped point; contact angle 169´18(after Clements, 1981). (b) Effect of jetting dry rock®ll under load and of initial wetting on settlement at 16 300 psf in oedometer tests (after Sowers et al., 1965) 128 OLDECOP AND ALONSO temperature, Ris the gas constant, and V0,E{and bare constants usually determined by ®tting experimental data. This type of relationship, implied in the bold part of the curves in Fig. 2, holds for a large body of experimental data, mostly obtained for glasses and ceramics (Wiederhorn et al., 1980, 1982; Freiman, 1984), but also for rocks (Atkinson, 1984; Atkinson & Meredith, 1987). Proportionality between Vand RH was found to hold for RH ranging from 20% to 100% in the case of soda lime±silica glass (Wiederhorn et al., 1982; Freiman, 1984). The use of RH in equation (2) is in fact a convenient way to measure the chemical potential of the corrosive agentÐthat is, waterÐwhich actually governs the rate of the corrosion reaction under constant K(Wiederhorn et al., 1982). Outside the region de®ned by the bold part of the curves in Fig. 2, different mechanisms, other than the reaction rate, control the crack propagation. A plateau in the relationship of V to Kis observed in some materials for high Kvalues (Fig. 2). This is attributed to the fact that Vbecomes controlled by the transport rate of the corrosive agent towards the crack tip and not by the reaction rate. Moreover, when Kapproaches the fracture toughness Kc,Vincreases rapidly (catastrophic rupture) and becomes independent of relative humidity. On the other hand, it is usually assumed (although not yet experimentally con®rmed for most rocks) that a stress corrosion limit, K0, exists below which cracks do not propagate at all. In the case where crack growth occurs in a liquid environmentÐthat is, a solution of water and a non-aqueous solventÐ it was observed that measured values of Vare proportional to the relative humidity of a gas in (thermodynamic) equilibrium with the solution (Freiman, 1984). Thermodynamic equilibrium implies that both gaseous environment and liquid environment will have equal chemical potentials, and hence they would produce the same effect on the stress corrosion reaction: that is, equal reaction rates and equal V(Wiederhorn et al., 1982). This means that it does not matter whether the corrosive agent (water) is in the liquid or the gaseous state. Therefore the sole parameter that controls the in¯uence of the corrosive agent on Vis the RH measured in the environment surrounding the specimen, provided that thermodynamic equilibrium is ensured. Let us assume that we are able to compute the stress intensity factor Kfor every crack or ¯aw existing in the rock particles of a rock®ll sample. Kvalues will depend on the geometry of the particles, the crack lengths, and the load con®guration. Then each crack within the rock®ll sample would occupy a de®ned position along the Kaxis of Fig. 2. Cracks lying in region I (K<K0) will not grow at all. Cracks lying in region II (K0,K,Kc) will grow at a velocity controlled by load and by relative humidity. This crack growth implies in turn an increase in Kfor most crack con®gurations, since it usually increases with the increase in crack length (see equation (1), although K may also decrease with the increase of crack length for `negative geometries'; Bazant & Planas, 1998). Eventually, when Kvalues approach region III (K>Kc) an instantaneous particle breakage will occur. The associated rearrangement of the granular structure will lead to a macroscopic strain increment and a more stable con®guration of contacts and contact loads. By means of this mechanism, a conceptual explanation of the rock®ll behaviour can be given. In a steady situation, all cracks will lie in region I and hence the rock®ll does not deform. If a load increment is then applied, some cracks will move to region III, producing an instantaneous deformation increment. Some other cracks will fall in region II and cause the time-dependent component of deformation. Under constant load, the number of source cracks lying in region II will decrease in time owing to the breakage process, and this implies a reduction of strain rate with time. If relative humidity increases at a given time, cracks in region II will increase their propagation velocity, causing a sudden increase in the strain rate. Additionally, some cracks previously lying in region I will move to region II owing to the decrease in K0implied in Fig. 2, causing an additional amount of strain. This is, indeed, the nature of collapse: an increment of rock®ll strain not related to an increment in load. ROCKFILL DEFORMATION TESTS The tested material is a slate obtained from the Pancrudo River outcrop (AragoÂn, Spain). The slate belongs to the Almunia formation, which has a Cambric origin. Its mineralogical composition, obtained from X-ray diffraction tests, includes quartz, muscovite, clinochlore, calcite and dolomite. The outcrop is planned to serve as quarry for the construction of the shells for the Jiloca River Regulation Dam, a zoned earth and rock®ll dam. The engineering properties of the rock are summarized in Table 1. The water retention curve for rockÐthat is, the Pancrudo slateÐis shown in Fig. 3. The retention curve characterizes the water storage features of the rock, and should serve as a reference material property when interpreting the collapse tests with suction control. For the range of water content values covered in this retention curve, water is essentially stored within the rock particles: that is, in the rock pores. As suction approaches zero, rock pores tend to become saturated. However, such a limiting condition is dif®cult to achieve in the retention curve experiments, although it can be de®ned by extending to very low suctions the interpolated retention curves shown. It is worth noting that the theoretical saturation water content of rock particles, computed from the rock porosity value given in Table 1, is 3´15%, which is close to the limiting water content for zero suction derived from Fig. 3. The ®gure shows the results of several tests (equilibrium points connected by broken lines). For high suctions, a rock sample was stored in an Table 1. Properties of Pancrudo slate Mean value Range Uniaxial compression strength 20´5 MPa 14´2±31´9 MPa Solid speci®c gravity (ASTM C97-90) 2´754 Ð Water absorption (ASTM C97-90) 1´356% Ð Porosity (from 84 mm core specimens) 8% 6´3±11´8% 1000 100 10 1 0·1 0·01 0·001 Suction: MPa 0 0·5 1·0 1·5 2·0 2·5 3·0 Water content: % Tensiometer technique Axis translation technique Vapour equilibrium technique Wetting path Drying path All tests. Fitted limiting curves for drying and wetting paths Fig. 3. Retention curve for the Pancrudo slate rock A MODEL FOR ROCKFILL COMPRESSIBILITY 129 isolated container where the RH was controlled by means of saturated saline solutions (`vapour equilibrium technique' in the ®gure). In the low-suction range, a ceramic suction plate was used to apply initially a negative water pressure to the rock pores (`tensiometer technique') and later an air overpressure (`axis translation technique') to the rock sample. A drying curve and a wetting curve (solid lines in Fig. 3) were ®tted to all the experimental data. A sample of crushed rock was obtained from the quarry with the aid of a digger. Recovered blocks having 200±400 mm maximum dimensions were broken down with the aid of a hammer. The maximum particle size for tests was ®xed at 40 mm, and the adopted grain size curve is shown in Fig. 4. An oedometer test program, including ®ve tests, was carried out in a Rowe-type cell, 300 mm sample diameter and an approximate sample height of 200 mm. Specimen compaction was carried out directly in the oedometer ring by means of a handy compaction hammer speci®ed for the Marshall test. This hammer delivers the same energy per blow as the modi®ed Proctor hammer, but the compaction effort is transmitted to the material by means of a ®xed base. This base prevents the hammer from striking the rock particles directly, thus preventing the grain breakage that would otherwise occur in the surface of each compaction layer. Specimens were compacted in four layers with a compaction energy in the range 600±700 J=l, which is close to the compaction energy used in the Standard Proctor Test (584´3 J=l). All specimens tested reached a void ratio e0:55 0:03 after compaction. All specimens were allowed to reach thermodynamic equilibrium with the laboratory environment (approximately 50% RH and 228C), resulting in a low initial water content (under 1%). Test procedure The test programme included three classic oedometer tests with specimen ¯ooding at some particular con®ning stress (tests 1, 2 and 3) and two oedometer tests with relative humidity control (tests 4 and 5). For tests 4 and 5 a relative humidity control system was added to the oedometer. The aim of this system is to produce a gradual variation in the specimen water content by adding controlled quantities of water in a uniform manner across the specimen. In order to achieve this goal, water has to be transported in vapour state. The system includes a closed-loop air circulation circuit, as shown schematically in Fig. 5. The air, impelled by an electric pump, bubbles in a solution vessel and then ¯ows across the specimen from the upper to the lower drainage plate. Saline solutions were used in order to impose a controlled relative humidity to the air ¯ow. A hygrometer inserted in the air ¯ow coming out of the specimen allowed the progress of the test to be monitored. As the test proceeds, the air ¯ows through the rock®ll voids, impelled by the pump. In a wetting path, the RH imposed by the solution in the air ¯ow is larger than the current RH within the rock pores. Water vapour is transported from the vessel to the rock®ll voids by advection. Further transport of water vapour occurs from rock®ll voids to rock pores by molecular diffusion owing to the RH gradient created between rock®ll voids and rock pores. Water vapour will condense within the rock pores wherever the width of the pore is smaller than approximately twice the radius of curvature at equilibrium of the gas±liquid interface (see detail in Fig. 5). The curvature depends on the surface tension of water and the difference between the gas pressure and the pressure in liquid water. This difference is usually termed the matric suction,spgÿpl. The absorption of water by the rock®ll specimen during the test was measured by recording the loss of weight of the solution vessel. Under constant imposed RH, the rate of water absorption decreased gradually until an equilibrium state (nil absorption rate) was reached. When the whole system (specimen and RH control device) reaches thermodynamic equilibrium, the RH is equal at any point of the system's gas phase, and no further transport of water vapour can occur. In such an equilibrium state, the RH in the gas phase and the matric suction in the rock pore water are related by the psychrometric relationship (Coussy, 1995): 100 90 80 70 60 50 40 30 20 10 0 Percent finer by weight 100 10 1 Grain size: mm Fig. 4. Grain size distribution of the tested material Oedometer (dia. = 300 mm) σ σ 200 mm Rockfill specimen NaCI solution Air pump Hygrometer Liquid water ( P L) Rock pore Rockfill void Vapour diffusion Air flow (RH%, P g ) Fig. 5. Rock®ll oedometer test set-up with a relative humidity control system 130 OLDECOP AND ALONSO RH exp ÿv(sð) RT  (3) where íis the molar volume of liquid water and ðis called the osmotic suction, which is due to the presence of solutes in the rock pore water. The sum øsðis usually called the total suction. Total and matric suction would be equal in the case where rock pores contained only pure water with no solutes. Test results Figure 6 shows plots of vertical strain against vertical stress in log and natural scales for the ®ve oedometer tests carried out. The load was applied in increments, allowing the sample to deform for a maximum of 24 h before the application of the next load increment. The stress±strain curves in Fig. 6 correspond to a constant time interval of 1000 min. Fig. 7 shows the paths followed by the four tests in the vertical stress±total suction space. Total suction, derived from the RH measurements using equation (3), has here the physical meaning given in the previous section: it is the sum of matric suction plus osmotic suction existing in the liquid water contained in the rock pores, and it is assumed to be in thermodynamic equilibrium with the specimen's gas phase (that is, the air ®lling the rock®ll voids). On the other hand, it seems reasonable to assume that, in a rock®ll, suction does not introduce relevant inter-particle capillary forces, as is the case for soils. It is worth noting that if stresses are represented in a natural scale, the normal compression lines corresponding to the `dry' stage (initial water content) of tests 2, 3 and 4 (Fig. 6(b)) plot nearly as straight lines passing through the origin. Fig. 8 shows unloading±reloading compression curves for the dry stage of tests 2 and 3 (which were omitted in the previous ®gures for simplicity). It is apparent that permanent deformations occur from the very beginning of the test (under very low stresses), suggesting that rock®ll has no initial elastic region during ®rst loading. In test 1, the specimen was ¯ooded at a very low load (0´01 MPa). During ¯ooding, and for an additional period of 24 h, a moderate swelling of specimen 1 was recorded (Fig. 6(b)). The total measured swell was 0´27%. Under further load increase, the normal compression for this saturated specimen also approaches a straight line but only beyond a threshold stress (close to 0´4 MPa), and shows a softer behaviour than the dry specimens. Collapse due to specimen ¯ooding was observed in test 3 under 0´6 MPa constant vertical stress. The amount of collapse strain was very close to the strain difference between the normal compression lines corresponding to the initial water content (dry state) and the saturated state. The subsequent loading after collapse leads to a stress±strain curve that follows the NCL for the saturated material obtained in tests 1 and 2. Fig. 9 shows a photograph of specimen 3 after test completion. The behaviour of specimen 2 during ¯ooding contrasts with the collapse observed in test 3. It was ¯ooded after reaching a vertical stress of 1 MPa, unloading to 0´03 MPa and reloading up to 0´2 MPa (Figs 6(b) and 7). No collapse was observed –0·5 0 0·5 1·0 1·5 2·0 2·5 3·0 3·5 4·0 4·5 5·0 5·5 6·0 6·5 Vertical strain: MPa 0·01 0·1 1 Vertical stress: MPa (a) Test 1 Test 2 Test 3 Test 4 Test 5 Fig. 6. Results of one-dimensional compression test on rock®ll: (a) vertical stress in logarithmic scale; (b) (overleaf) vertical stress in natural scale. Broken lines indicate the saturated condition of rock pores (either by specimen ¯ooding or by RH increase). Open circles indicate specimen ¯ooding A MODEL FOR ROCKFILL COMPRESSIBILITY 131 –0·5 0 0·5 1·0 1·5 2·0 2·5 3·0 3·5 4·0 4·5 5·0 5·5 6·0 6·5 Vertical strain: MPa 0 0·2 0·4 0·6 0·8 1·0 Vertical stress: MPa (b) Test 1 Test 2 Test 3 Test 4 Test 5 Fig. 6. (continued) 0 0·2 0·4 0·6 0·8 1·0 1·2 Vertical stress: MPa Test 1 Test 2 Test 3 Test 4 Test 5 1000 100 10 1 0·1 0·01 Total suction + p atm: MPa Fig. 7. Vertical stress±total suction paths followed by tests 0 0·5 1·0 1·5 2·0 2·5 Vertical strain: % 0 0·2 0·4 0·6 0·8 1·0 Vertical stress: MPa Test 2 Test 3 Fig. 8. NCL and URL for `very dry' rock®ll. Vertical stress in natural scale 132 OLDECOP AND ALONSO during saturation. Instead, a small swelling (0´016% axial strain) was measured. Once saturated, the strain±stress curve under a new load increase approaches the NCL for the saturated material obtained in test 1. Test 4 has a stress path identical to that of test 3, but during the collapse process the relative humidity was controlled by means of the previously described set-up. The aim of this test was to induce collapse in a gradual manner as water content increases. Figs 10 and 11 show plots of collapse strain against water content and collapse strain against total suction for the data recorded during test 4. Note that specimen 4 was never ¯ooded, and the `bulk degree of saturation' (referred to the saturation of both rock pores and rock®ll voids) remained quite low during the whole test. As can be seen in Fig. 10, the ®nal water content of specimen 4 at the end of collapse was 3´2%. By contrast, sample 3 was fully saturated during ¯ooding, and it reached a water content of nearly 20%. Nevertheless, the amount of collapse strain that occurred in both specimens is almost the same. Test 5 was carried out at a later date with the objective of serving as a veri®cation test for the constitutive model developed. It followed the stress±total suction path shown in Fig. 7. The specimen response will later be compared with model predictions. Discussion and interpretation On the basis of the reported tests, it may be accepted that a unique normal compression line exists for a given total suction or water content. The stiffness during normal compression depends on suction: the material stiffens as total suction increases. An elastic domain may also be de®ned if small hysteresis effects are neglected. The effect of wetting (decreasing suction) depends on the applied con®ning stress. For low stress levels (small) swelling strains are measured, whereas collapse strains occur beyond a certain con®ning stress value. This is also a feature implied in the previously presented rock®ll conceptual model (Fig. 2). In fact, if rock®ll is wetted under low stress (low Kvalue), the material will not deform at all (no collapse). However, if stress is increased, Kvalues will move towards region II and the associated cracks will react with a rapid increase in their propagation velocity if an increase in relative humidity occurs. Test 4, in which collapse was produced by gradually raising the relative humidity, showed that the water content at the end of the wetting path was a small fraction (15%) of the saturation water content of the rock®ll (given by the water weight necessary to ®ll both rock pores and rock®ll voids). If the specimen water content at the end of the wetting path (w3:2%, see Fig. 10) is referred to the water retention curve of the rock (Fig. 3), it appears that during test 4 the major part of the liquid water was essentially stored within the rock particles (that is, in the rock pores), and that only a very small amount of liquid water was present in the rock®ll voids. This fact was con®rmed by visual inspection at the end of test 4. Hence saturation of the rock particles seems suf®cient to produce the same collapse deformation as a full ¯ooding of the rock®ll specimen. An additional analysis of the observed behaviour was carried out by distinguishing the instantaneous strain increments caused by each load increment from the subsequent time-dependent strains in tests 1±4. The incremental instantaneous compliance and the incremental time-dependent compliance are plotted in Figs 12(a) and 12(b) against the applied stress. Incremental compliance is de®ned as the strain increment, either instantaneous or time dependent (a time period of 1000 minutes was considered), divided by the stress increment. Instantaneous strains are the major component of strain for low con®ning stresses: they tend to decrease when con®ning stress increases, and they are only slightly affected by sample ¯ooding. On the other hand, time-dependent strains are relatively low for low con®ning stresses and dry states. Beyond a certain threshold stress value the time-dependent strains are strongly affected by water: that is, the time-dependent compliance rapidly increases with increasing water content. Unlike conventional granular soil behaviour, the rock®ll time-dependent strain component is not Fig. 9. Photograph of specimen 3 after test completion 0 0·2 0·4 0·6 0·8 1·0 1·4 1·2 Collapse strain: % 1100·1 0·45% w 0= 3·2% Collapse strain due to flooding = 1·347% χw = 6·842 × 10–3 Water content: % Fig. 10. Collapse strain plotted against gravimetric water content in oedometer test 4. Vertical stress =0´6 MPa 0 0·2 0·4 0·6 0·8 1·0 1·4 1·2 Collapse strain: % 1 10 1000·1 67 MPa χψ = 2·047 × 10–3 ψ + p atm: MPa Test 3 Test 4 Fig. 11. Collapse strain plotted against total suction under 0´6 MPa vertical stress (tests 3 and 4) A MODEL FOR ROCKFILL COMPRESSIBILITY 133 negligible, as Fig.12(b) demonstrates. Furthermore, these timedependent deformations incorporate the effect of water and explain the rock®ll collapse behaviour. These features will be taken into account to formulate a constitutive model for rock®ll compressibility. COMPRESSION MODEL FOR ROCKFILL McDowell & Bolton (1998) distinguish four successive deformation mechanisms of granular materials in compression as the con®ning stress is increased. Under low stresses, deformation is due to particle rearrangement. The second mechanism is called clastic yielding, and it is attained when the applied stress causes the onset of particle crushing. This stage is marked by a strongly curved strain±log stress relationship, with concavity directed towards the strain axis. However, when plotted in natural scale the particle rearrangement and clastic yielding stages appear as a nearly linear strain±stress relationship for many previously published experimental data for granular materials (Nobari & Duncan, 1972; Marsal, 1973; Clements, 1981; Pestana & Whittle, 1995; Yamamuro et al., 1996). If load is further increased, particle crushing continues and a granular structure arises in which the grain size distribution is a fractal and neighbouring particles are never of equal size (Sammis, 1997). McDowell & Bolton (1998) showed that, in such a granular structure, only the smallest particles would undergo crushing under the successive stress increments. Successive reduction of the smallest particle size is called clastic hardening and it was shown that the strain±log stress relationship during this stage should be linear, as observed in many experimental results. Finally, for very high stresses the strain±log stress relationship has an upward-directed concavity. This is attributed to the comminution limit of small particles (McDowell & Bolton, 1998). Deformations due to particle rearrangement under the ®rst load increments can be identi®ed in the strain±stress curves shown in Fig. 8 (upward concavity) for stresses varying from 0 to 0´4 MPa approximately. Clastic yielding seems to occur beyond this threshold value, and it leads to a nearly linear strain±stress relationship. Neither the clastic hardening mechanism nor the comminution limit are attained within the stress range used in the test programme. This interpretation, which is based on the shape of the strain±stress curves, is also consistent with the proposed conceptual model and the analysis of instantaneous and timedependent strains discussed before. In fact, clastic yielding initiates beyond a threshold stress value since it is associated with crack propagation, which requires a minimum stress intensity factor K0to develop (Fig. 2). Since time-dependent deformations are associated with crack propagation it also follows from Fig. 2 that for a very dry state (low vapour pressure) time-dependent strains vanish since K0tends to Kc and the region II disappears. For a wet state of the material, both instantaneous and time-dependent strains should occur within the clastic yielding stage. Two deformation mechanisms producing plastic strains are included in the proposed model: the instantaneous deformation mechanism (IDM) and the time-dependent deformation mechanism (TDM). However, no time dependence is explicitly taken into account by the model presented here. In view of Fig. 12(a) it will be assumed that the IDM is present at any stress level and stress increment. It is considered independent of water action. On the other hand, the TDM is active beyond a threshold total stress value, óy, and depends strongly on the water action. It is the source of the collapse phenomenon, and vanishes when the material is in `very dry state'. A more precise de®nition of such a `very dry state' will arise from the model formulation. The effect of water will be introduced in two alternative ways: either as the gravimetric water content, w, or by means of the total suction, ø. Suction is used here as a parameter and not as a true stress variable. Both formulations (water content and suction based) are similar, and if the uniqueness of the retention curve is assumed, they are completely equivalent. Linear normal compression lines are assumed for both IDM and TDM, and also a linear strain±stress relationship is assumed for unloading±reloading paths. The incremental strain± stress relationship along the normal compression line results from the addition of the strains increments produced by both mechanisms: for ó<óydådåiëidó(4) for ó.óydådåidåd[ëiëd(w)] dó(5a) or dådåidåd[ëiëd(ø)] dó(5b) where óis the vertical stress applied in the oedometer, óyis the clastic yield stress (that is, the stress threshold value that marks the onset of particle breakage), åis the total vertical strain, åiand ådare respectively the IDM strain component and the TDM strain component, ëiis the slope of the normal compression line when only the IDM is active, and (ëiëd)is the slope of the normal compression line when both plastic deformation mechanisms are active. Overall rock®ll water content may be split in two parts: the signi®cant water content (which varies from zero to the saturation water content of rock particles, w0) and the dead water content that will ®ll the rock®ll voids and will have no effect on the collapse behaviour. Suction, ø, is related (through the retention curve) to the signi®cant water content. Fig. 12. Axial compliance computed from: (a) the instantaneous strain increments; (b) the time-deferred strain increments. Broken lines indicate the saturated condition of rock pores (either by specimen ¯ooding or by RH increase). Open circles indicate specimen ¯ooding 0·05 0·04 0·03 0·02 0·01 0 Instantaneous axial compliance: MPa–1 0 0·2 0·4 0·6 0·8 1·0 Vertical stress: MPa (a) Test 1 Test 3 Test 4 Test 2 (before flooding) Test 2 (after flooding) 0·05 0·06 0·07 0·08 0·04 0·03 0·02 0·01 0 Delayed axial compliance: MPa–1 0 0·2 0·4 0·6 0·8 1·0 Vertical stress: MPa (b) Test 1 Test 3 Test 4 Test 2 (before flooding) Test 2 (after flooding) 134 OLDECOP AND ALONSO The elastic strain increments within unloading±reloading (URL) paths are given by dåekdó(6) where the slope kis assumed to be independent of the water action. The observed moderate swelling behaviour caused by the increase in water content is assumed to be a reversible (elastic) strain. A linear relationship is assumed to exist between the swelling strain and the logarithm of water content, w: dåwÿkw dw w(7a) or ø, dåøkø dø (øpatm)(7b) where patm is the atmospheric pressure and kwand køare the water content and suction based expansion/compression indexes. For simplicity, kwand køare assumed to be independent of the stress level. Yield surface, hardening law and ¯ow rule for the water content formulation Consider in Fig. 13(b) the behaviour of two identical rock®ll samples that are initially in a very dry state (water content wi close to zero; point O). Water is added to one of them in an unstressed state until a water content w, smaller than the rock saturation water content, is reached (w,w0). As observed in the experiments, the increase in water content will cause a moderate swelling up to point A. The `wet' sample is then loaded up to a vertical stress ó0larger than óy(point B). The `dry' sample is loaded up to a stress ó 0larger than ó0(point C). If the `stress states' B and C belong to the same yield surface, then it should be possible to carry the `wet' sample from state B to state C by means of a virtual elastic path (Fig. 13(a)). This involves a constant stress drying from B to D and a constant water content loading from D to C. The following equation expresses the deformation at point C computed along two alternative paths OABDC and OC: ÿkwln(w=wi)ëióy[ëiëd(w)](ó0ÿóy) kwln(w=wi)k(ó 0ÿó0)ëió 0(8) Rearranging this expression, the yield surface in the ó±wspace is derived as F(ó,w)ó0[ëiëd(w)ÿk]ÿóyëd(w)ÿó 0(ëiÿk)0 (9a) where ó 0, the yield stress of the dry state rock®ll, is appropriately selected as the hardening parameter. Equation (9a) is the yield surface for ó.óy. If the stress state does not exceed the clastic yield stress, óy, the yield surface simply reads F(ó,w)ó0ÿó 00 (9b) A simple plastic volumetric strain hardening law is proposed: dó 0dåp ëiÿk(10) Applying the consistency condition, the ¯ow rule becomes for ó.óy:dåp[ëiëd(w)ÿk]dó0 (ó0ÿóy)@ë(w) @wdw(11a) for ó<óy:dåp[ëiÿk]dó0(11b) A linear relationship between collapse strains and the logarithm of rock water content was observed in the experiments carried out (Fig. 10). It is therefore suggested that the compressibility coef®cient ëd(w) accounting for clastic yielding is linearly related to the logarithm of water content through a material parameter áw: ëd(w)ëd 0ÿáwln w0 w  (12) where ëd 0is the maximum clastic compressibility index (for w>w0). Therefore ëd(w) is bounded as follows: 0<ëd(w)<ëd 0(13) The lower bound condition in equation (13) allows the `very dry state' to be de®ned. For the case of the Pancrudo slate tested in this work, this state is attained for w<0:45%, as implied in Fig. 10. A relationship between áwand the experimental coef®cient ÷wde®ned in Fig. 10 may be derived if the collapse strains measured in experiments are compared with predicted strain as given by the model. For an incremental wetting path at constant stress ó0(ó0.óy), the total strain increment is given by the sum of elastic and plastic components (equations 7(a) and 11(a) respectively): dåcollapse ÿkw dw w(ó0ÿóy)áw dw w(14) The measured collapse strain increment will be given by (Fig. 10) Dry Dry Wet Wet Elastic domain Elastic path O O A A B B DC Yield focus Total suction, ψ Water content, w w i w 0σ0* σ0* σ0 σ0σ σy σy Axial stress, σ (a) (b) DC λi + λd λi κ ε Fig. 13. (a) Stress paths considered to derive the yield surface. (b) Idealized model response A MODEL FOR ROCKFILL COMPRESSIBILITY 135