The role of non-normality in soil mechanics and some of its mathematical consequences

Transcription

1 Computers and Geotechnics 31 (2004) The role non-normality in soil mechanics and some its mathematical consequences R. Nova * Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, Italy Abstract Experimental data suggest that the normality rule concept, widely used in metal plasticity, is not applicable to geomaterials. The concept plastic potential can be retained, however, provided it is assumed that this is different from the loading function (nonassociate flow rule). The elastic plastic stiffness matrix becomes non-symmetric and the stiffness matrix looses its positive definiteness in the hardening regime. From that point onwards, some its principal minors can become zero. It is proven that, each time this occurs, it is possible to find a particular loading programme for which an infinity solutions exist under a given load increment (homogeneous bifurcation). It is shown that various types instability can be described this way, all occurring in the hardening regime. For example, the onset instability in undrained tests on loose sand, the occurrence drained and undrained shear bands in plane strain conditions, the possibility unlimited pore water pressure generation in undrained conditions can be described. Ó 2004 Elsevier Ltd. All rights reserved. 1. The normality rule and its earlier applications in soil mechanics Taylor and Quinney [1] subjected hollow cylinders various metal types (aluminium, copper, mild steel) to the combined action a tensile force N and a torque T around the axis the cylinder. The experimental data showed two important results: on the one hand, the locus yielding points in the plane N; T was very well fitted by an ellipse, as predicted by the Huber von Mises yield criterion. On the other hand, by plotting on the same plane the vectors whose components are the incremental plastic axial displacement, d _ p, and the incremental plastic rotation, _# p, in such a way that the variables corresponding in the power density equation share the same axes, such vectors were found to be directed orthogonally to the yield locus at the corresponding yield point, Fig. 1. Therefore, if f ðn; T Þ¼0is the expression the yield locus in the plane N; T, the incremental plastic displacement and rotation can be formally derived as follows: * Tel.: ; fax: address: nova@stru.polimi.it (R. Nova). _d p ¼ K _# 0 grad f ¼ K 0 p : ð1þ on om Eq. (1) can be easily rewritten in terms stresses and plastic strain rates _e p _c p ¼ Kgrad f ¼ K or ; ð2þ os and generalised to the full tensorial relation _e p ij ¼ Kgrad f ¼ K : ð3þ or ij Eq. (3) is known as the normality rule the plastic flow, or simply as the normality rule. The normality rule can be assumed as a postulate, or it can be derived as a consequence the so-called Drucker stability postulate. If the normality rule holds true, important theorems can be proven. They concern the uniqueness the collapse load a structure made a stable material and the possibility establishing lower and upper bounds to its value, that can be eventually obtained as the separation element two contiguous classes values (the lower and upper bounds). The normality rule was then assumed as a postulate also for other types materials, such as soils, rocks and X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi: /j.compgeo

2 186 R. Nova / Computers and Geotechnics 31 (2004) M, ϑ. p Yield locus Load path Plastic displacement rate N, δ. p Fig. 1. Yield locus and normality rule concept for metallic hollow cylinders subject to tension and torsion. concretes (geomaterials). Drucker and Prager [2], for instance, defined a convenient failure criterion for such kind materials, whose behaviour depends on the isotropic pressure, p, by generalising the Huber von Mises failure condition f J 2 ðap þ jþ 2 ¼ 0; ð4þ where J 2 is the second invariant the stress deviator. This expression allows the Mohr Coulomb failure condition to be approximated, since the constitutive parameters a and j can be easily linked to the friction angle, /, and the cohesion, c. By means Eq. (3), plastic strain rates can be determined without ambiguities, because the C 1 (i.e. gradient) continuity Eq. (4). The normality concept for frictional materials has a serious drawback, however. Consider for instance the case the retaining wall shown in Fig. 2(a). Imagine the earthfill is made a purely frictional soil, as it is most ten the case in practice. At impending collapse, a wedge soil adjacent to the wall slides down along a shear band where strains tend to localise. If the soil would obey the normality rule, the element Fig. 2(b), belonging to the shear band, would dilate in such a way that the plastic normal strain rate is proportional to the plastic shear strain rate and the proportionality constant is ( tan /). In this way, however, no plastic work would be dissipated during the plastic strain the shear band, Fig. 2(c). The plastic strain rate cannot therefore be normal to the Mohr Coulomb failure condition for thermodynamic reasons. Few years later, Drucker et al. [3] proposed a way, which appeared to be very effective in solving this paradoxical situation. They assumed in fact that yielding soils occurs when the stress state touches a spherical ÔcapÕ, in the stress space, which evolves according to a hardening rule defined in terms the plastic strains (or the plastic work) experienced. The normality rule is then fulfilled on the cap only, while the Drucker Prager cone gives a limit state condition, associated to zero hardening (perfect plasticity). The plastic strain rate at failure need not to be normal to the limit state condition, therefore, and the thermodynamic requirements are fulfilled. Furthermore, in this way it was possible to model, at least in a qualitative way, the irreversible behaviour shown by soils in unloading reloading tests in which the strain rate components vary proportionally, such as the isotropic or the oedometric test. This irreversibility is in fact at the base the concept overconsolidation traditional Soil Mechanics. Starting from those premises, Roscoe et al. [4] and subsequently [5] modelled in terms effective stresses (indicated by a dash in the following) the behaviour soils as elastic plastic strainhardening (or stening). In order to determine a convenient expression for the loading function, they exploited the normality concept. From experimental data, they knew in fact that in axisymmetric (so-called triaxial) tests on virgin specimens it is possible to establish a one-to-one relationship between the stress ratio g, defined as g q p ¼ 3 r0 1 r0 3 0 r 0 1 þ ; ð5þ 2r0 3 and the plastic dilatancy d, defined in turn as the ratio between the volumetric plastic strain rate and the deviatoric plastic strain rate d _ep m _e p d ¼ 3 2 _e p 1 þ 2_ep 3 _e p 1 : ð6þ _ep 3 In particular they assumed a simple linear relation to fit the data g þ d ¼ M; ð7þ Fig. 2. (a) Sliding wedge behind a wall; (b) dilating shear band element; (c) stress and plastic strain rate at failure following the normality rule (r and _e p are vectors scalar components fr; sg and f_e p ; _c p g, respectively).

3 R. Nova / Computers and Geotechnics 31 (2004) where M is the value the stress ratio at zero dilatancy. This value coincides with the so-called critical state stress ratio, since hardening has been assumed to depend on plastic volumetric strains only: zero dilatancy implies therefore no hardening. With this model, called Cam Clay, it is possible to describe, in a unique conceptual framework, the behaviour normally consolidated (i.e. virgin) and overconsolidated (i.e. preloaded) clays in drained and undrained (i.e. constant volume) conditions, in triaxial tests as well in tests in which the strain rate components vary proportionally. The model is able to capture important features soil behaviour such as the qualitative differences lightly and heavily overconsolidated clays, the former being contractant and ductile, while the latter are dilative and brittle. In particular, Cam Clay is able to model qualitatively well the behaviour in undrained triaxial tests two clay specimens characterised by the same water content, w, defined as the ratio between the weight the water in the pores and the weight the dry specimen, but different preconsolidation history and, consequently, different confining pressures. Fig. 3(a) shows the variation water content for the two specimens during consolidation and undrained shear, Fig. 3(b) illustrates the effective stress path followed and Fig. 3(c) presents the deviatoric stress strain relationship during the shear phase. Loose sand has a behaviour that is conceptually similar to that normally consolidated clay. One would expect that Cam Clay is equally well suited to describe the behaviour such a material. This is not the case, however. In undrained tests, in fact, the typical stress strain relationship is that shown in Fig. 4. The peak in the stress deviator cannot be modelled by an elastoplastic model fulfilling the normality rule, as proven analytically in [6]. This result is intuitive, since, by virtue Eq. (7) the yield locus has a maximum deviator stress at critical state (g ¼ M) and, for virgin soils, the stress path must always be directed out it (occurrence plastic strains). Fig. 4. Loose sand behaviour in undrained compression: comparison between experimental data and backcalculated results with an elastoplastic model with non-associate flow rule: (a) effective stress paths and (b) stress strain relationship (after Di Prisco et al. [7]). Such a behaviour can be easily modelled, instead, if the existence a plastic potential g, different from the loading function f, is postulated, so that _e p ij ¼ Kgrad g ¼ K og : ð8þ or ij For instance, the good agreement between experimental data and numerical results shown in Fig. 4, after Di Prisco et al. [7], was obtained by means the model Di Prisco et al. [8], that, among the rest, uses a plastic potential which is different from the loading function. On the other hand Poorooshasb [9], Tatsuoka and Ishihara [10] and Nova and Wood [11] demonstrated with various experimental techniques combined with theoretical considerations that the flow rule is non-associate for sands, not only for loose but even for dense or medium dense sand. (a) (b) (c) Fig. 3. Cam Clay predictions clay behaviour: (a) variation water content during isotropic consolidation and undrained shear; (b) effective stress paths and (c) stress strain relationship in the undrained phase.

4 188 R. Nova / Computers and Geotechnics 31 (2004) This result received recently an important confirmation by Calvetti et al. [12], who simulated the behaviour a granular material as an assembly spherical frictional particles, by means the threedimensional discrete element method. They show unambiguously how the direction the plastic strain rate vector is not orthogonal to the yield locus, which can be precisely determined by virtue the numerical technique adopted. Even normally consolidated clay is characterised by a non-associated flow rule, although the deviation from normality is less marked than in sand. This is not apparent in most axisymmetric cases (and this is the reason why it is ten assumed that the normality rule holds for clay, as in the family the Cam Clay models). In a plane strain test, and particularly in undrained simple shear, the influence non-normality is notable, however, even for normally consolidated clay. For instance, Airey and Wood [13] tested a normally consolidated speswhite kaolin in undrained simple shear. It can be derived from their results that the peak the shear stress corresponds to a mobilised friction angle about 20, while the critical state angle is equal to 24. As for loose sands in undrained axisymmetric tests, such a difference can be reproduced in the framework hardening plasticity, only by admitting that the flow rule is non-associated. Similarly, the occurrence shear bands in biaxial tests before reaching the critical state, see e.g., the experimental results Topolnicki [14] on remoulded clay specimens, cannot be explained if normality rule is assumed to be valid. In the following, we shall assume therefore the validity Eq. (8) and investigate some its mathematical consequences. 2. Bifurcations for elastoplastic models with non-associate flow rule For a material with a non-associate flow rule, when plastic strains occur, the constitutive relationship can be written in vectorial form as _e ¼ C _r 0 ¼ C e þ 1 H mnt _r 0 ð9þ or _r 0 ¼ D_e ¼ D e De mn t D e _e; H þ n t D e m ð10þ where C and C e are the total and the elastic compliance matrices, D and D e are the total and the elastic stiffness matrices, H is the hardening modulus (positive for hardening and negative for stening), while m and n are the gradients the plastic potential and loading function, respectively. A superposed ÔtÕ indicates transposition and a dash effective stress. The limit condition under stress rate control is given by det D ¼ 0: ð11þ In fact, when this occurs, unlimited strain rates, all proportional via an indefinite scalar to the eigenvector matrix D, are possible under zero load increment. It is possible to show via Eq. (10) that such a limit condition is achieved when H ¼ 0. In the hardening regime (H positive), therefore, the determinant D is positive, the stiffness matrix can be inverted and, under full load (equivalent to stress, if strains are small) control, a unique homogeneous solution is possible. It is readily apparent from the Eqs. (9) and (10) that the compliance and the stiffness matrices are not symmetric, however. The Ostrowski and Taussky [15] theorem states then that the loss positive definiteness matrix D (and C) occurs when its determinant is still positive. We have in fact at the onset loss positive definiteness det D P det D s ¼ det C s ¼ 0; ð12þ where D s and C s are the symmetric part D and C, respectively. From this moment onwards, loss uniqueness the incremental response can occur under special loading programmes [16]. For instance, consider a test partly load controlled and partly strain controlled, as it is usual in Soil Mechanics (e.g. a drained triaxial compression test, in which axial strain and cell pressure are controlled). By splitting stress and strain vectors in subvectors, partitioning D accordingly and grouping the control parameters in the l.h.s., the constitutive law can be written _r 0 a _e b ¼ D aa D ab D 1 bb D ba D 1 bb D ba D ab D 1 bb D 1 bb _e a _r 0 : ð13þ b Loss uniqueness under zero increment the control parameters occurs when the determinant the matrix Eq. (13) is nil, which in turn, by the Schur theorem [17], occurs whenever the minor D aa matrix D is zero. It can be shown that when this occurs also the complementary minor C bb is zero. As an example, consider the numerical simulation a true triaxial test by means the model by Di Prisco et al. [8]. In this test, two principal stress increments and one principal strain rate are controlled. Fig. 5, after Imposimato and Nova [18], shows the loci for which the minors C 11, C 22, C 33 are zero. These are also the loci for which a drained shear band is possible [19]. In geotechnical tests, we can control also linear combinations stresses and strains, instead controlling those quantities directly. For instance, in an undrained test the volumetric strain is constrained to be zero, i.e. the sum the principal strains is nil. Together with this condition, the load increase at constant cell

5 R. Nova / Computers and Geotechnics 31 (2004) Fig. 5. Loci loss uniqueness the incremental response when two principal stress increments and one principal strain rate are controlled (after Imposimato and Nova [18]). Fig. 6. Loci loss uniqueness the incremental response in undrained tests in which one principal strain is kept constant and the two orthogonal principal stresses are increased the same amount (after Imposimato and Nova [18]). pressure is controlled. This latter condition implies that, for small strains at least, what is actually controlled is the deviator stress, i.e. the difference between the maximum and minimum principal stress. In general, it is possible then to define a vector generalised stress variables n and a vector generalised strain variables g linked to the vectors stress and strain as follows: _n ¼ T r _r 0 ; ð14þ _g ¼ T e _e ¼ðT t r Þ 1 _e; where T r and T e are matrices constants. The constitutive relationship can be written as _n ¼ T r DT t r _g ¼ D_g: ð15þ A way reasoning similar to that leading to Eq. (13) can be repeated for the generalised variables. For instance, consider an undrained test in which also the vertical strain, e 1, is kept constant whilst the two horizontal principal stresses are increased at the same time, the same quantity. The trivial solution is that the pore water pressure increases exactly the amount which the two horizontal stresses were increased and no incremental strain occur. However, it can be shown that, for the stress state such that a particular minor matrix D is nil, unlimited pore pressures and horizontal strains (one opposite to the other) may take place. Fig. 6, after [18], shows the calculated loci for which this can occur in the deviatoric plane (with permutation indices). Furthermore, it can be shown that such loci coincide with those for which the formation an undrained shear band is possible [19]. The spontaneous generation pore water pressures and associate unlimited strains can occur even in axisymmetric conditions, if convenient variables are controlled. In such conditions, the stress strain relationship can be written in terms the variables defined in Eqs. (5) and (6) _e m _e d ¼ Cpp C pq C qp C qq _p 0 _q 0 : ð16þ Since the volumetric strain rate is nil in undrained conditions, the deviatoric strain rate can be related to the stress deviator increment via the following equation: _e d ¼ C ppc qq C pq C qp C pp _q ¼ det C C pp _q: ð17þ Since the determinant matrix C is positive in the hardening regime, the condition for the occurrence a peak the deviator stress in an undrained test in which the deviator strain (i.e. the axial one) is monotonically increased is C pp ¼ 0: ð18þ It is possible to show [19] that the locus for which this condition is met is a straight line passing through the origin axes in the plane p 0 ; q. This line is known as instability line [20]. The mobilised friction angle at which this occurs for a loose sand is the order 16, while the friction angle at limit state for the same sand in drained and undrained conditions is roughly 30. The specimen instability in undrained conditions occurs therefore much earlier than ordinary (drained) failure.

6 190 R. Nova / Computers and Geotechnics 31 (2004) On the other hand, in an undrained load controlled test, the stress strain relationship becomes " _e m ¼ C CpqC pp qp C pq # C qq C qq _p 0 : ð19þ _q _e d Cqp C qq 1 C qq For the stress state for which Eq. (18) is fulfilled, the determinant the matrix above becomes zero. There are possible therefore eigensolutions the type _p 0 _e d ¼ k 1 C qp ; ð20þ where k is an arbitrary scalar. Since the external isotropic pressure does not change, it follows that the pore water pressure increment is opposite to k, and can be unlimitedly large. Deviatoric strains can also increase accordingly. Imposimato and Nova [19] demonstrated experimentally that this is actually the case. They performed a drained compression test on a specimen loose Hostun R.F. sand. At a stress level close to the value for which, in an undrained test, the deviator stress reaches a peak, they kept constant the axial loading for 24 h. Then the drainage valve was closed. The small creep the sand specimen under constant loading caused a small increase the pore water pressure, until the instability line was reached. At that point, a marked increase pore water pressures was recorded and the collapse the specimen took place, Fig. 7. It is quite interesting to note that the response the material under the same loading conditions (axisymmetric undrained compression), but different control parameters (fully strain controlled or partly strain and partly stress controlled) changes dramatically. In the former case, the test is fully controllable (unique solution for a given increment controlling parameters), while in the latter a homogeneous bifurcation takes place for the stress state corresponding to a stress peak in the former one. Finally, Fig. 8 shows a generalisation the instability line for general stress conditions. Three loci are depicted in the deviatoric plane. The outer locus is the limit state locus for which the determinant the stiffness matrix is zero (ordinary failure). The inner one is that for which the determinant the symmetric part the stiffness matrix is zero (loss positive definiteness). The intermediate one is that for which Eq. (18) is fulfilled. It is shown in Imposimato and Nova [18] that such a condition is equivalent to the possibility homogeneous bifurcations when the controlling variables are the following: Fig. 8. Generalisation the Ôinstability lineõ concept. The outer and inner loci correspond to the nullity the determinants the stiffness matrix and its symmetric part, respectively. The intermediate locus is that for which C 33 ¼ C pp ¼ 0 (after Imposimato and Nova [18]). Fig. 7. Spontaneous generation pore water pressures in undrained conditions: (a) stress path and (b) pore water generation as a function time after closure drainage (after Imposimato and Nova [19]).

7 R. Nova / Computers and Geotechnics 31 (2004) < n 1 ¼ r 0 1 r0 2 ; n 2 ¼ r 0 3 r0 2 ; ð21þ : g 3 ¼ e 1 þ e 2 þ e 3 ; and corresponds to the nullity C 33, C being the generalised compliance matrix for this loading programme. 3. Conclusion Experimental data and theoretical considerations suggest that the normality rule concept, widely used in metal plasticity, is not applicable to geomaterials. The concept plastic potential can be retained, however, provided it is assumed that this is different from the loading function (non-associate flow rule). The elastoplastic stiffness matrix becomes non-symmetric, therefore, and the stiffness matrix looses its positive definiteness in the hardening regime. From that point onwards, some its principal minors can become zero. It is proven that each time this occurs, it is possible to find a particular loading programme for which an infinity solutions exist under a given load increment (homogeneous bifurcation). It is shown that, various types instability can be described in this way, all occurring in the hardening regime. For instance, the onset instability in undrained tests on loose sand, the occurrence drained and undrained shear bands in plane strain conditions, the possibility unlimited pore water pressure generation in undrained conditions can be calculated. However, homogeneous bifurcations can occur only if appropriate control variables are chosen. References [1] Taylor GI, Quinney H. The plastic distortion metals. Trans R Soc Lond A 1931;230: [2] Drucker DC, Prager W. Soil mechanics and plastic analysis or limit design. Quart Appl Mech 1952;10: [3] Drucker DC, Gibson RE, Henckel DJ. Soil mechanics and workhardening theories plasticity. Trans ASCE 1957;122: [4] Roscoe KH, Schield AN, Wroth CP. On the yielding soils. Geotechnique 1958;8: [5] Schield AN, Wroth CP. Critical state soil mechanics. Chichester: McGraw-Hill; [6] Nova R. A note on sand liquefaction and soil stability. In: Proceedings the 3rd International Conference on Constitutive Laws Engineering Materials. Tucson; p [7] Di Prisco C, Matiotti R, Nova R. Theoretical investigation the undrained stability shallow submerged slopes. Geotechnique 1995;45(3): [8] Di Prisco C, Nova R, Lanier J. A mixed isotropic kinematic hardening constitutive law for sand. In: Kolymbas D, editor. Modern approaches to plasticity. Amsterdam: Elsevier; p [9] Poorooshasb HB. Deformation sand in triaxial compression. In: Proceedings the 4th Asian Regional Conference on Soil Mechnaics Foundation Engineering, vol. 1. Bangkok; p [10] Tatsuoka F, Ishihara K. Yielding sand in triaxial compression. Soils and Foundations 1974;12: [11] Nova R, Wood DM. An experimental programme to define the yield function for sand. Soils and Foundations 1978;18: [12] Calvetti F, Viggiani G, Tamagnini C. A numerical investigation the incremental behavior granular soils. Rivista Italiana di Geotecnica 2003;37(3): [13] Airey DW, Wood DM. An evaluation direct simple shear tests on clay. Geotechnique 1987;37(1): [14] Topolnicki M. Observed stress strain behaviour remoulded saturated clay and examination two constitutive models. PhD thesis, University Karlsruhe, [15] Ostrowski A, Taussky O. On the variation the determinant a positive definite matrix. Neder Akadem Wet Proc 1951;A54: [16] Nova R. Controllability the incremental response soil specimens subjected to arbitrary loading programmes. J Mech Behav Mater 1994;5: [17] Schur I. Uber Potenzreihen, die im Innern des Einheitskreises beschraenkt sind. Reine Angew Math 1917;147: [18] Imposimato S, Nova R. An investigation the uniqueness the incremental response elastoplastic models for virgin sand. Mech Cohes-Frict Mater 1998;3: [19] Imposimato S, Nova R. Instability loose sand specimens in undrained tests. In: Adachi T, Oka F, Yashima A, editors. Proceedings the 4th International Workshop on Localisation and Bifurcation Theory for Soils and Rocks p [20] Lade PV. Static instability and liquefaction loose fine sandy slopes. J Geotech Eng ASCE 1992;118:51 71.