WEEK 10 Soil Behaviour at Medium Strains 14. Onset of yielding 14-1. What is yielding: Remember? We have already studied what yielding means; if you don t remember, refer back to Week 2. It is defined by generation of plastic strain. According to the conventional elasto-plastic concept that we studied in Weeks 3-5, soil behaviour was assumed to be elastic within the yield surface. In many realistic cases, however, this is a wild idealisation. If the assumption is true, soil would always exhibit elastic responses against cyclic loadings with a constant stress amplitude. This week we will study real soil behaviour at medium strains, which are larger than those to which the elastic stiffness is relevant, but smaller than those at which the large-scale yielding leads to failure of soil. In the previous weeks, we studied that the stress-strain non-linearity appears against very small strain (and hence stress) increments. Does this signify yielding? Remember, non-linearity per se does not necessarily mean yielding. Isotropic hardening p Inside: Elastic? Stress path Example of E vu ε a curves of London Clay (Addenbrook et al., 1997) 1
14-2. Yielding at relatively small to medium strains The answer to the uestion seems, yes. See how the stress-strain relationships form a loop even for small strain amplitudes (meaning irrecoverability of strains; hysteresis). The τ γdata shown below (from Iwasaki et al., 1978) were obtained with hollow cylinder torsion shear apparatus (refer back to Week 9). A ualitatively same view is obtained for the ε relationship that is obtained with triaxial apparatus, although the τ γand ε relationships may be uantitatively different due to anisotropy. or τ Cyclic loading with gradually increasing stress-strain amplitude or τ p ε or γ Change of hysteresis as strain amplitude increases Definition of damping ratio (Iwasaki et al., 1978) 2
Another effect of plasticity is dependency of the stress-strain relationship on loading histories. Shown here are the (shear) strain contours drawn for loadings from different origins along the K 0 -unloading path, obtained for the Magnus Till (Jardine, 1992). Note how the strain contours align themselves along the unloading path. A change in the loading direction generally leads to stiffer responses. (Jardine, 1992) 3
Another example by Atkinson et al. (1990) on reconstituted and over-consolidated London Clay Influence of stress-path direction changes on subseuent ε relationshps Influence of stress-path direction changes on subseuent p ε p relationshps (Atkinson et al., 1990) 4
14-3. Multiple yield surface and kinematic hardening: Concept To interpret the yield at relatively small to medium strains, it is necessary to prepare more than one yield surface. This concept of multi-surface plasticity is not limited for soils (for example, Iwan, 1967). It is often combined with kinematic hardening rule. The idea is to limit the elastic region to very small size and describe multiple distinct stages of yielding by adopting multiple yield surfaces. Boundary surface (Yield surface) Boundary surface Kinematic yield surface p p Elasticity Plasticity immediately appears for reloading/unloading ε ε End of Critical State elasticity & Onset of large-scale yield End of elasticity Critical State Onset of large-scale yield G 0 G sec G 0 G sec Realistic nonlinearity at small strains may be described logε logε 5
15. Behaviour under cyclic loading and principal stress axis rotations 15-1. Cyclic loading and accumulation of volumetric strain or excess pore water pressure The cyclic and dynamic behaviour of soils will be discussed in detail in the lecture course provided by Professor Miura, so this lecture allocates space for this topic less than it deserves. The soil behaviour discussed under this topic is not necessarily limited to the range of medium strains, however you define it. For example, liuefaction eventually leads to extremely large strains. However, the processes to reach such an ultimate state are dominated by a seuence of medium-scale yielding, so probably it is appropriate to discuss them here. (i) Sands Let us start with sands, with which liuefaction under cyclic loadings is always a great concern (the significance of liuefaction phenomena will be discussed in detail in the other e3 post-graduate course by given the lecturer, Disaster Mitigation Geotechnology ). The data shown here were obtained for the Toyoura Sand in drained cyclic simple shear, performed in hollow cylinder apparatus. See how volumetric strain accumulates over number of loading cycles. How can we interpret, or how can we not interpret this behaviour from what we have learnt previously? Simple shear in hollow cylinder apparatus Shahnazari & Towhata (2002) 6
In undrained cyclic simple shear, the volumetric strain is forced to be zero, but this time the pore water pressure cannot be controlled. As a result, the pore water pressure increases and p decreases. The ultimate state can be liuefaction. How can we interpret the behaviour under drained and undrained conditions in a unified way? Actually, you already know the (or, an ) answer; remember what you studied in Week 3? e Undrained conditions: Because of plastic straining due to the cyclic loading, e wants to decrease, but it cannot ( e must be zero). So p is forced to decrease instead. Drained conditions: e due to plastic straining p Effective stress-paths and shear stress-strain curves of loose and dense Fuji River Sand (Ishihara, 1985; reproduced after Iai et al., 1991) 7
(ii) Clays In Clays, cyclic loading also leads to excess pore water generation and hence reduction in p. It is not common, however, for p to reach zero and attain a state of liuefaction. Literature does not cite liuefaction in clays. Towhata (2008) notes some similarity between behaviour of dense sands and clays. Pore water pressure changes during cyclic simple shear of Kaoline (Ohara & Matsuda,1988) Effective stress-path and shear stress-strain relationships of Eastern Osaka Clay (Adachi et al., 1995) 8
15-2. Soil behaviour under rotation of principal stress axes First of all, get familiar with rotations of principal stress axes in 2D. σ 1 σ y τ P D (Pole with regard to direction) ( σ x, τ xy ) τ xy y σ x σ 3 σ 1 σ x σ 1 ( σ y, τ xy ) σ y τ σ 1 τ xy ( σ x, τ xy ) y x σ x σ 1 σ 3 ( σ y, τ xy ) σ 1 σ If the stress state (i.e. σ x, σ y and τ xy ) is changed in such a way that the Mohr s stress circle s centre (= (σ x + σ y )/2) and radius (= [[(σ x -σ y )/2] 2 +[τ xy ] 2 ] 0.5 ) are not changed, a pure rotation of the principal stress axes occurs. It means that only the directions of σ 1 and σ 3 change, but not their magnitudes. If σ 1 and σ 3 do not change,, t, p, s, etc. do not change either (ignore σ 2 for the moment). So, according to the models based on these invariant uantities would not predict any change in state. You stay where you are. Is this realistic? or t p or s 9
In what situations do rotations of the principal stress axes matter particularly? In uite many a situation, actually. Examples: (Bjerrum,1973) Rotation of principal stress directions due to embankment construction (Jardine & Smith., 1991) The rotation of principal stress directions is accompanied by increases in p and. Cyclic rotation of principal stress directions in seabed due to wave loading (Ishihara & Towhata., 1983) The rotation of principal stress directions occurs almost with constant p and. 10
15-2. Soil behaviour under rotation of principal stress axes If we assume elastic behaviour, ε x ε y 1/ G 0 ( σ x σ y)/ 2 = γ xy 0 1/ G τ xy (see p.11, Week 2) X = ( z θ σ σ )/ 2 Y =τ zθ / 2 z θ or, ε x ε y 1 ( σ x σ y)/ 2 = γ xy G τ xy A strain increment is parallel to the stress path. However, the drained hollow cylinder test results by Gutierrez et al. (1991) indicate significant plastic strains for any rotational stress path. Towhata and Ishihara (1985) demonstrated that even liuefaction can be triggered by pure rotation of the principal stresses. Increase of, with p and principal stress directions fixed Rotation of principal stress directions, with p and fixed (Gutierrez et al.,1993) 11
References Adachi, T., Oka, F., Hirata, T., Hashimoto, T., Nagaya, J., Mimura, M. and Pradhan, T.B.S. (1995) Stress-strain behavior and yielding characteristics of Eastern Osaka Clay, Soils and Foundations, 35(3) 1-13. Addenbrooke, T.I., Potts, D.M. and Puzrin, A.M. (1997) The influence of pre-failure soil stiffness on the numerical analysis of tunnel construction, Geotechniue 47(3) 693-712. Atkinson, J.H., Richardson, D. and Stallebrass, S.E. (1990): Effect of recent stress history on the stiffness of overconsolidated soil, Geotechniue 40(4) 531-540. Bjerrum, L. (1973) Problems of soil mechanics and construction on soft clays and structurally unstable soils (collapsible, expansive and others), Proceedings of 8th International Conference on Soil Mechanics and Foundation Engineering, Moscow, Vol.3, 111-159. Gutierrez, M., Ishihara, K. and Towhata, I. (1991): Flow theory for sand during rotation of principal stress direction, Soils and Foundations 31(4) 121-132. Iai, S., Matsunaga, Y. and Kameoka, T. (1992): Strain space plasticity model for cyclic mobility, Soils and Foundations 32(2) 1-15. Ishihara, K. (1985) Stability of natural deposits during earthuakes, Proceedings of 11th International Conference on Soil Mechanics and Foundation Engineering, San Francisco, Vol.1, 327-376. Ishihara, K. and Towhata, I. (1983) Sand response to cyclic rotation of principal stress directions as induced by wave loads, Soils and Foundations 23(4) 11-26. Iwan, W.D. (1967): On a class of models for the yielding behavior of continuous and composite systems, Journal of Applied Mechanics 34(E3) 612-617. Iwasaki, T., Tatsuoka, F. and Takagi, Y. (1978): Shear modulus of sands under cyclic torsional shear loading, Soils and Foundations 18(1) 39-56. Jardine, R.J. (1992): Some observations on the kinematic nature of soil stiffness, Soils and Foundations 32(2) 111-124. Jardine, R.J. and Smith, P.R. (1991) Evaluating design parameters for multi-stage construction, Proceedings of the International Conference on Geotechnical Engineering for Coastal Development, Geo-coast 91, Vol.1, 197-202. Ohara, S. and Matsuda, H. (1988) Study on the settlement of saturated clay layer induced by cyclic shear, Soils and Foundations, 28(3) 103-113. Shahnazari, H. and Towhata, I. (2002) Torsion shear tests on cyclic stress-dilatancy relationship of sand, Soils and Foundations 42(1) 105-119. Towhata, I. (2008) Geotechnical earthuake engineering, Springer-Verlag Berlin Heidelberg. Towhata, I. and Ishihara, K. (1985) Undrained strength of sand undergoing cyclic rotation of principal stress axes, Soils and Foundations 25(2) 135-147. 12