Prediction of breakage-induced couplings in unsaturated granular soils

Transcription

1 Zhang, Y. D. & uscarnera, G. Géotechnique [ TECHNICAL NOTE Prediction of breakage-induced couplings in unsaturated granular soils Y. D. ZHANG and G. USCARNERA This note addresses the interplay between particle breakage, water-retention cure and deformation processes. For this purpose, a continuum theory for unsaturated granular soils is used to capture the effects of loading and/or wetting in the grain-crushing regime. The theory proides a relation to express the air-entry alue of the retention cure as a function of the degree of particle breakage, thus generating a coupled hydro-mechanical formulation. These features are exploited by deriing the incremental relations that goern the hydro-mechanical response. It is then shown that breakagedependent retention cures generate arious coupling effects, such as the eolution of the degree of saturation along constant-suction paths and the pressure dependence of the wetting-collapse strains. Since these capabilities are an outcome of coupling terms deried from grain-scale considerations, these results suggest that microstructure-inspired models are important tools to capture the feedbacks between the macro-scale response and the eolution of micro-scale attributes, as well as to minimise the recourse to phenomenological assumptions. KEYWORDS: constitutie relations; partial saturation; particle crushing/crushability; sands; suction Manuscript receied 8 May 214; reised manuscript accepted 5 January 215. Discussion on this paper is welcomed by the editor. Northwestern Uniersity, Department of Ciil and Enironmental Engineering, Eanston, IL, USA. INTRODUCTION Recent studies hae shown that changes of the hydraulic state are a source of crushing and compaction in coarsegrained soils subjected to high pressures (Oldecop & Alonso, 23; Alonso et al., 25; Oalle et al., 213). Similarly, new eidence has shown that crushing may alter the soil water retention cure (SWRC) of particulate media (Jamei et al., 211). Such mechanisms are a challenge for constitutie models, as they suggest that the SWRC of crushable soils must co-eole with the grain size distribution (GSD), making coupled hydro-mechanical formulations necessary to simulate their response. Although these couplings are controlled by specific microscopic mechanisms (i.e. grain breakage), it is arguable that the mathematical structure of models for unsaturated crushable soils must share similar features with those typically used for fine-grained materials, in which such macroscopic couplings are often obsered. It is therefore readily apparent that the formulation of models for unsaturated crushable soils is a challenging proing ground, as well as a tremendous opportunity to deelop a better understanding of the microscopic origin of hydro-mechanical couplings. The purpose of this note is to tackle such challenges by using the microstructure-inspired model formulated by uscarnera & Eina (212), in which the co-eolution of GSD and capillary solid fluid interactions is captured by way of statistical homogenisation. The purpose of the study is two-fold. On the one hand, the aim is to discuss the rich set of hydro-mechanical couplings that may emerge in crushable soils as a result of the grain-size dependence of their water-retention properties. On the other, the aim is to comment on some interesting mathematical features of the model that derie from the incorporation of microstructural ariables. Indeed, some of the peculiar features displayed by this specific microstructure-inspired formulation hae the potential to inspire new strategies to model hydro-mechanical coupling also in other classes of unsaturated soils. COUPLING ETWEEN SWRC AND PARTICLE FRAGMENTATION Comminution alters the GSD of a granular soil, affecting its capacity to retain water. Seeral empirical or semiempirical models aailable in the literature recognise the link between GSD and SWRC (Arya & Paris, 1981; Aubertin et al., 23; Likos & Jaafar, 213). One of the most widely used models of this kind has been proposed by Arya & Paris (1981) (hereafter referred to as the AP model), who hae used the GSD and the oid ratio to estimate the water retention cure of a granular soil. Although none of these models is combined with a strategy to assess how physical and mechanical agents alter the particle grading, they all imply that any alteration of the GSD impacts the SWRC. The unsaturated breakage model (UM) (uscarnera & Eina, 212) shares similar features with the aboe-mentioned approaches, in that it predicts the eolution of the SWRC as a function of the alterations of the GSD induced by comminution. This model relies on a thermo-mechanical formulation in which the stored energy is assumed to be gien by the sum of a strain-dependent term and a saturation-dependent term, both deried from microscopic scaling laws through statistical homogenisation. The total Helmholtz free energy potential of this model can be expressed as follows Ø(å e,, ) ¼ (1 W M )ł M r (åe ) þ (1 þ W H )ł H r () (1) where W M and W H are grading indices; Ø and ł r are the total and reference Helmholtz free energy functions, respectiely; the superscripts and subscripts M and H indicate mechanical and hydraulic components, respectiely; å e is the elastic strain; is the degree of saturation; and <, 1 is the breakage ariable quantifying the degree of comminution 1

2 2 ZHANG AND USCARNERA (Eina, 27). Within such formalism, the SWRC can be expressed as follows ns ¼ (1 þ W H r where n is the porosity and ns the smeared suction. The SWRC in equation (2) is factorised by a term (1 þ W H ) that eoles with the GSD through. As a result, it suggests that the eolution of the SWRC is inherently coupled with the mechanical processes. In particular, as breakage takes place, the SWRC shifts towards larger alues of suction. The magnitude of such moement is controlled by the grading index W H, which is computed on the basis of an inerse grain-size scaling inspired by the capillary theory. A schematic description of such alterations of the SWRC is illustrated in Fig. 1 with reference to AP and UM models (where, for the latter, a simple hyperbolic SWRC cure has been used for the unbroken assembly (uscarnera & Eina, 212)). oth models predict significant changes of the SWRC as comminution takes place. It can be noticed that the AP model predicts dramatic changes in the shape of the SWRC (Fig. 1), while the inflection point associated with the air-entry alue (AEV) is fixed. y contrast, the UM retention cure captures the increase in suction for an assigned degree of saturation through an increase of the AEV, which is predicted to occur for a constant shape of the SWRC. oth predictions hae their merits. In fact, as the GSD becomes broader, the shape of the SWRC is expected to change, as indicated by the AP model. On the other hand, as crushing generates more fines, it is expected to hae large pores occupied by small particles, haing an increase in the AEV similar to that captured by the UM model. Grain crushing is therefore likely to alter both the shape and the air-entry point of the SWRC of a soil. This aspect is confirmed by recent experiments on crushable materials done by Jamei et al. (211), as well as by empirical models, in which the shape of the SWRC and its AEV depend on both the particle diameter at 1% of the cumulatie GSD, D 1, and the coefficient of uniformity, C u (e.g. Aubertin et al., 23). Assessing which of the two effects is more important in a crushable soil would require an extensie set of experiments, and it is therefore beyond the scope of this note. Neertheless, in either of the aboe-mentioned scenarios grain crushing causes significant changes of the water-retention capacity, which are inherently coupled with the stress strain response. As a result, gien the unique ability of the UM model to capture the interplay between microstructure eolution, mechanical behaiour and hydrologic properties, here its simplifying assumptions are accepted, with the purpose of studying the hydro-mechanical implications of a breakage-dependent SWRC. Finer by mass: % s: kpa s: kpa d: mm (c) Fig. 1. Comparison of predicted changes in SWRC from different GSD-dependent models: GSDs at arying alues of ; AP retention model (Arya & Paris, 1981); (c) UM retention model (uscarnera & Eina, 212) INCREMENTAL CONSTITUTIVE FORMULATION Following uscarnera & Eina (212), the yield condition in the presence of coupled breakage frictional dissipation is y(e,, ó9) ¼ E E c (1 ) 2 þ q 2 1 < (3) Mp9 where E is the breakage energy; E c is the critical breakage energy at the onset of comminution; ó9 ¼ ó u a ä þ s ä is the aerage skeleton stress, where u a is the air pressure and ä is the Kronecker s delta; p9, q are the mean and deiatoric stresses; and M is the stress ratio at frictional failure. Thus, the consistency condition can be written as d þ de : dó9 @ó9 Using hyperelastic relations, the increment of the ariables E and ó9 can be deried dó9 e de M r ¼ 2 ł M r ¼ (1 W M e M dåe r W e d : dåe W H d (5a) (5b)

3 Using the elastic plastic strain decomposition (då e ¼ då då p ) and the flow rules (då p ¼ º(@y =@ó9) and d ¼ º(@y =@E )), it is possible to derie the plastic multiplier, º, for a hydro-mechanical plastic-breakage process, as follows º ¼ 1 (î K M : då î H d ) (6) P where K :î is the plastic-breakage modulus, y is the yield function in the dissipatie space and the terms î M, î H and î are gien by î M r e þ (1 2 ł M e e, REAKAGE-INDUCED COUPLINGS IN UNSATURATED GRANULAR SOILS 3 î W H (7) % pass: by mass d: mm 8 7 Data Model Ultimate GSD Intermediate GSDs 8, σ 65 9 MPa 5, σ 31 6 MPa 2, σ 21 9 MPa 1 8 The incremental constitutie relations can be written in matrix form as follows dó ¼ [D e (ˆ b þ ˆ p)]de (8) where Ó ¼[ ó9 ns ] T and E¼[ å ( )] T are generalised stress and strain ectors; D e is an elastic matrix, while the matrices ˆ b and ˆ p reflect the effect of breakage and plasticity, respectiely 2 3 D e ¼ (1 4 e 5 (1 þ W H ł (9a) H 2 r 2 3 ˆ b W M î e M W M r 4 î e H 5 (9b) W H r î î H " # ˆ p ¼ 1 M 2 ł M K 2 ł M r î H (9c) P 6 e σ : MPa Data Model compression Model breakage where the symbols indicate zero matrices. Equation (8) shows that the constitutie response is the sum of an elastic contribution and two inelastic terms. It can be noticed that, while the mechanical and hydraulic properties are decoupled in the elastic regime (D e in equation (9a)), cross-coupling terms are introduced by the comminution matrix ˆ b: These terms depend on the eolution of the breakage ariable, and can therefore be interpreted as a direct outcome of the eoling microstructure. Similarly, the onset of plastic strains introduces a further off-diagonal term in the constitutie equations through ˆ p, which is also inherently coupled with the growth of through the flow rules. The constitutie properties of such particular coupled formulation will be explored hereafter by simulating the effects of loading and/or wetting paths in the crushing regime. HYDRO-MECHANICAL COUPLING IN CRUSHALE SOILS The constitutie relations can be defined by assigning specific expressions to the potentials in equation (1). In the following, a hyperbolic retention cure and a pressuredependent elastic potential hae been used, calibrating their parameters on the basis of aailable data for Toyoura sand (Fig. 2; Table 1). Gien the simplicity of the selected s: kpa Data Model (c) Fig. 2. Calibration of: initial grain size distribution; elastic properties and yielding threshold for Toyoura sand (data after Nakata et al., 21); and (c) water retention parameters for Toyoura sand (data after Unno et al., 28) Table 1. Model parameters for Toyoura sand W M.757 W H K 55 G 522 M 1.2 ø: degrees 64 E c : MPa.25 K w :kpa 4

4 4 ZHANG AND USCARNERA retention model, the simulations can only approximate the sharp kink of the data at the AEV of the sand. This problem, howeer, can be remoed by using more sophisticated retention functions. Further details about the implications of different elastic potentials and retention models can be found in Zhang & uscarnera (214). Seeral oedometric tests hae been simulated at different alues of initial suction (Fig. 3). The model predicts that the yielding stress associated with the maximum rate of comminution (and, hence, with the maximum rate of compaction) tends to increase with suction, that is, when drier conditions are approached. The breakage eolution cures eentually conerge at high stress leels, suggesting the effect of the initial hydraulic state becomes less important at adanced stages of crushing. During the early stages of loading, the compressibility of unsaturated samples is predicted to decrease compared to saturated conditions, with larger alues of suction magnifying such a stiffening effect. y contrast, unsaturated states are predicted to be characterised by larger alues of compressibility when loaded in the high-pressure regime, where dry states inole a larger rate of compaction compared to saturated states. Fig. 3 illustrates the effect of wetting paths, using the two simulations at s ¼ and s ¼ 1 MPa as a reference (marked as tests a and b, respectiely). The tests c, d, e and f are simulated wetting paths imposed at different alues of ertical net stress. After suction is decreased to zero, all the simulated tests exhibit a compression breakage response similar to that associated with the saturated case (test a). Figure 4 presents the simulated stress paths for tests a (s ¼ ) and b (s ¼ 1 MPa). It is readily apparent that the 7 6 e 5 4 Compression response Toyoura sand Test a, s MPa Test b, s 1 MPa s 1 MPa s 3 MPa reakage eolution σ net : MPa 7 6 Compression response Toyoura sand 6 Test a, s MPa e 5 Test b, s 1 MPa Test c f, s 1 MPa 4 Wetting at different σ net 4 2 reakage eolution σ net : MPa c d e f Fig. 3. Predicted compression response and breakage eolution for: simulated suction controlled oedometric compression at different leels of suction; simulated suction controlled wetting stages at different alues of ertical net stress ns: MPa q: MPa s MPa, S 1 r s 1 MPa, S 1 1 r p : MPa s 1, S 1 r % passing by mass d: mm predicted expansion of the elastic domain under unsaturated conditions (test b) produces a steeper stress path at low stress leels compared to the saturated one (test a). Neertheless, once a considerable amount of breakage has occurred (e.g. when ¼. 55) the two paths tend to conerge. These effects are a direct consequence of the assumed shape of the hydraulic potential, which influences the predicted changes in size and shape of the elastic domain (uscarnera & Eina, 212). Such changes of the yield surface are the main cause of the stiffer compression response in Fig. 3, which is eentually suppressed in the post-yielding regime when the two simulated stress paths conerge. Fig. 4 illustrates the eolution of SWRC and GSD at selected alues of corresponding to the stress paths in Fig. 4. It is readily apparent that the SWRC eoles with the accumulation of breakage, thus promoting higher suction AEVs and an increase of the degree of saturation. As illustrated in Fig. 5, the model predicts the stressdependence of the wetting-induced compaction process. This feature has been pointed out by simulating a series of wetting paths under oedometric conditions for different alues of initial suction and net stress. Each leel of suction is predicted to be associated with a specific alue of ertical net stress at maximum collapse, as well as with the prediction of larger collapse strains for increasing alues of initial suction. Such features of the wetting response (most notably the existence of a stress leel at which compaction is maximised), hae been obsered by many researchers on , 1, 32, 55, 78 Fig. 4. Stress paths during simulated oedometric compression (dotted lines indicate yield surfaces at selected states for saturated conditions (path with circular symbols) and unsaturated conditions (path with square symbols); eoling hydraulic state and SWRCs for the simulated test b

5 Collapse olumetric strain: % s s s REAKAGE-INDUCED COUPLINGS IN UNSATURATED GRANULAR SOILS 5 1 MPa 3 MPa 1 MPa : MPa σ net CONCLUSIONS The authors hae used the breakage mechanics framework for unsaturated granular media to inestigate the coupled hydro-mechanical behaiour of crushable soils, giing emphasis to the dependence of the water-retention cure on the degree of particle breakage. A notable feature of the selected approach is that the coupled effects hae not been assumed a priori, but hae rather been deduced from micromechanical considerations based on the capillary theory and a procedure of statistical homogenisation. To study the implications of a water-retention cure that eoles during comminution, the incremental hydro-mechanical relations of the model hae been deried. Such deriation has shown that particle breakage introduces crosscoupling terms able to link the hydrologic response and the compression behaiour of a crushable granular soil. Using this formulation, a series of oedometric tests hae been simulated, showing that breakage-dependent functions enable the simulation of a broad range of coupled effects, such as suction-dependent stress thresholds at the onset of comminution, stiffer compression response for increasing alues of suction and wetting-induced breakage. The simulated breakage eents, either caused by loading or wetting, are always predicted to be accompanied by a simultaneous increase in the suction air-entry point. Furthermore, it has been shown that the mathematical structure of the model proides the possibility to reproduce the pressure-dependence of the wetting-induced strains, thus capturing the existence of a stress leel associated with the maximum collapse. Such hydro-mechanical feedbacks hae been captured by a mathematical formulation considerably different from those typically used to mimic similar patterns in other classes of unsaturated soils. One key aspect that has led to this result is that hydro-mechanical coupling is not directly injected into the constitutie functions through phenomenology, but is rather predicted by using scaling relations that link the microstructural attributes (e.g. the particle size) to the continuum properties. Hence, these results suggest that microstructure-inspired arguments can lead to mathematical formulations capable of minimising the number of phenomenological assumptions required to capture complex macroscale effects. It is the authors opinion that this approach to constitutie modelling is susceptible to future generalisations based on the incorporation of other micro-scale attributes (e.g. pore size, inter-particle bonds), as well as on a more systematic use of micro-scale characterisation techniques. ACKNOWLEDGEMENT This work has been suppoted by grant no. CMMI awarded by the Geomechanics and Geomaterials program of the U. S. National Science Foundation. Fig. 5. Predicted wetting-induced olumetric strains as a function of the imposed alue of constant ertical net stress (oedometric conditions) other classes of unsaturated soils (Sun et al., 27; Munoz- Castelblanco et al., 211), and usually imply considerable efforts to be incorporated satisfactorily in phenomenological models (Josa et al., 1992; Georgiadis et al., 25). As illustrated by the example in Fig. 5, the links between microstructure eolution and macroscopic response embedded in the breakage mechanics model enable this aspect to be reproduced without specific phenomenological assumptions. NOTATION breakage index C u coefficient of uniformity D 1 particle diameter at 1% of the cumulatie GSD D e elastic stiffness matrix d grain diameter E breakage energy E c critical breakage energy e oid ratio K, G elastic stiffness constants K P breakage-plastic modulus K w retention cure parameter M stress ratio at failure n porosity degree of saturation s suction y, y yield function in true and dissipatie stress space ˆb, ˆp stiffness reduction terms due to breakage and plasticity ä Kronecker s delta å, å e, å P total, plastic and elastic strain tensors W M, W H mechanical and hydraulic grading indices º plastic multiplier î M, î, î H consistency coefficients Ó, E generalised stress and strain ectors ó9, p9, q skeleton, mean skeleton and deiatoric stresses ó total ertical stress ó net, ó net net stress tensor, ertical net stress Ø total Helmholtz free energy ł M r, łh r mechanical and hydraulic reference Helmholtz free energy ø breakage-plasticity coupling angle REFERENCES Alonso, E. E., Oliella, S. & Hugas, J. (25). Modelling the behaiour of an earth and rockfill dam during construction and impoundment. In Unsaturated soils: Numerical and theoretical applications (ed. T. Schanz), pp erlin, Germany: Springer. Arya, L. M. & Paris, J. F. (1981). A physico-empirical model to predict the soil moisture characteristic from particle size distribution and bulk density data. Soil Sci. Soc. Am. J. 45, No. 6, Aubertin, M., Mbonimpa, M., ussière,. & Chapuis, R. P. (23). A model to predict the water retention cure from basic geotechnical properties. Can. Geotech. J. 4, No. 6, uscarnera, G. & Eina, I. (212). The yielding of brittle unsaturated granular soils. Géotechnique 62, No. 2, , dx.doi.org/1.168/geot.1.p.118. Eina, I. (27). reakage mechanics. Part I. Theory. J. Mech. Phys. Solids 55, No. 6, Georgiadis, K., Potts, D. M. & Zdrakoic, L. (25). Threedimensional constitutie model for partially and fully saturated soils. Int. J. Geomech. 5, No. 3, Jamei, M., Guiras, H., Chtourou, Y., Kallel, A., Romero, E. & Georgopoulos, I. (211). Water retention properties of perlite as a material with crushable soft particles. Engng Geol. 122, No. 3,

6 6 ZHANG AND USCARNERA Josa, A., almaceda, A., Gens, A. & Alonso, E. E. (1992). An elasto-plastic model for partially saturated soils exhibiting a maximum of collapse. Proceedings of the 3rd international conference on computational plasticity, arcelona, ol. 1, pp Swansea, UK: Pineridge. Likos, W. J. & Jaafar, R. (213). Pore-scale model for water retention and fluid partitioning of partially saturated granular soil. J. Geotech. Geoeniron. Engng 139, No. 5, Munoz-Castelblanco, J., Delage, P., Pereira, J. M. & Cui, Y. J. (211). Some aspects of the compression and collapse behaiour of an unsaturated natural loess. Géotechnique Lett. 1, No. 2, 17 22, Nakata, Y., Kato, Y., Hyodo, M., Hyde, A. F. & Murata, H. (21). One-dimensional compression behaiour of uniformly graded sand related to single particle crushing strength. Soils Found. 41, No. 2, Oldecop, L. A. & Alonso, E. E. (23). Suction effects on rockfill compressibility. Géotechnique 53, No. 2, , dx.doi.org/1.168/geot Oalle, C., Dano, C. & Hicher, P. Y. (213). Experimental data highlighting the role of surface fracture energy in quasi-static confined comminution. Int. J. Fracture 182, No. 1, Sun, D. A., Sheng, D. & Xu, Y. (27). Collapse behaiour of unsaturated compacted soil with different initial densities. Can. Geotech. J. 44, No. 6, Unno, T., Kazama, M., Uzuoka, R. & Sento, N. (28). Liquefaction of unsaturated sand considering the pore air pressure and olume compressibility of the soil particle skelton. Soils Found. 48, No. 1, Zhang, Y. D. & uscarnera, G. (214). Grainsize dependence of clastic yielding in unsaturated granular soils. Granular Matter, 16, No. 4,