2.2 Non-coaxiality of Soil Behaviour

In geomechanics,non-coaxiality is defined as the non-coincidence between the principal stress axis and the principal strain rates axis.In contrast,the assumption of coaxiality was postulated by Saint Venant when he applied Tresca's yield criterion to a problem in metal plasticity(de Saint Venant,1870).Therefore it has also been termed Saint Venant's postulate.However it has long been recognized that coaxiality cannot be satisfied in the case of anisotropy.

2.2.1 Definition of Non-coaxiality

As the non-coaxiality is between principal stress and principal strain increment directions,tensors σ_ij and dε_ij,which describe the state of stress and strain rate at a point,are considered to define the non-coaxiality.The two ten sors use the same reference axis x_i=x,y,z which can be easily chosen.

If the principal stress and principal strain increment are used,the tensors σ_ij and dε_ijcontain only the principal components and have zero off-diagonal components,then

Tensors σ_ij and,dε_ij and dtogether with their respective reference axes are equivalent representations of the state of stress and strain increment at a point in a continuum.

The tensor σ_ij and dε_ij can be obtained from σ_ijand dε_ij by suitable transformations:

where A_ij and B_ij are transformation tensors giving the directions of the principal stresses and principal strain increments from the reference axis x_i to which the tensors σ_ij and dε_ij are referred.Ifis denoted as the principal stress axis,then A_ij may be expressed as A_ij=.If A_ij≠ B_ij,the non-coaxiality ofandis obtained.

2.2.2 Previous Studies on Non-coaxiality

From numerous studies on granular materials,it was found that the coaxiality assumption is only valid for isotropic media.When the strain rate depends not only on the principal stress but also on other vectors and tensors,for the requirement of anisotropic behaviour,new theories have been developed and applied to represent the non-coaxiality for granular material behaviour.

The theoretical origin of non-coaxiality can be found in some pre-failure plasticity models that have been proposed for granular materials,such as hypoplastic models(Wang et al.,1990;Kolymbas,1991)and multi-laminate models(Iai et al.,1992).Rudnicki and Rice(1975)focused on the strain localization of materials and reported that non-coaxiality plays an important role in shear band formation in sands.Moreover,by introducing vertices into the yield surface,in contrast to a smooth and continuous yield surface,the flow becomes dependent on the directions of the stress and stress increment,and then the flow becomes non-coaxial for non-straight ahead loading.As obtained in experiments,this may facilitate strain localization in the strain hardening region(Vardoulakis,1980).

Non-coaxiality has been a feature of a number of physically established plasticity models that describe "fully developed" plane plastic flow of granular materials by means of kinematic theories.The earliest kinematic models for granular material flow were developed by de Josselin de Jong(1958)with graphical methods.The so-called "double sliding,free rotating model" for planar flow was based on the assumption of shear flow occurring along two surfaces where the available shear resistance had been exhausted.Then Spencer(1964)used the same concept of double sliding to establish a set of kinematic equations termed as the "double shearing model" with a different rotation term from the de Josselin de Jong model.A similar model was also proposed independently by Mandel(1966).Further analysis of the double sliding model was made by Mandel and Fernandez(1970)with further justifications for the non-coaxiality between principal stress and principal strain increment directions.These original "double-sliding free rotation" and "double-shearing" models were developed for non-dilatant,rigid-plastic and post-peak flow of granular materials.Several researchers have extended those models to account additionally for dilatant,elasto-plastic and pre-peak strain hardening response(Mehrabadi and Cowin,1978;Anand,1983;de Josselin de Jong,1988;Teunissen and Vermeer,1988;Harris,1993;Joer et al.,1998).Therefore the models become determinate,but they have not been shown to be able to reproduce the non-coaxial behaviour as observed in Roscoe's simple shear tests(Roscoe,1970).

Figure 2.1 shows the experimental results reported by Roscoe(1970).Before this,Roscoe et al.(1967)had shown that the principal axis of strain rate and of stress were not coincident before reaching peak shear stress during a simple shear test of sand.From Figure 2.1,the rotations of the principal stress and the principal plastic strain are non-coaxial,particularly at the early stage of loading.Then the axes tend to become coincident at large shear strains.

Figure 2.1 Experimental curves showing principal stress and strain increment rotations against shear strain during simple shear tests.

(a)σ_yy=135kPa;(b)σ_yy=396kPa(after Roscoe,1970)

One of the first evidences for non-coaxiality was reported by Drescher and de Josselin de Jong(1972)on the base of the experimental micro-mechanical study of a photoelastic disc assembly as a two-dimensional analogue of granular media.Besides,non-coaxiality has been observed in experimental studies on sands using the hollow cylinder apparatus(HCA),which allows full rotation of the principal stresses(Ishihara and Towhata,1983;Symes et al.,1982,1984,1988;Miura et al.,1986;Pradel et al.,1990;Gutierrez et al.,1991,1993;Gutierrez and Ishihara,2000;Lade et al.,2009).The studies included drained and undrained tests using different types of sand.Deviation between principal stress direction and principal strain increment direction was noticed when specimens were subjected to monotonic shearing at a fixed principal stress direction or subjected to pure rotation of principal stress axis at constant deviator stress.These experimental studies using HCA will be introduced in detail in sintion 2.4.Similar experimental evidence has also been shown by Wong and Arthur(1986)in both dense and loose sands during cyclic rotation of principal stresses using the directional shear cell apparatus.These studies showed that the deviation between the principal stress and the principal strain incremental directions could be more than 30° in sand during continuous rotation of the principal stress axes.

2.2.3 Previous Numerical Studies on Non-coaxiality

Due to the limitation of laboratory method to explore the underlying mechanisms and particle scale information,numerical techniques like discrete element method(DEM)can be a useful method of study of soil behaviour.DEM is a numerical method proposed by Cundall(1971)for computing the motion of a large number of particles like molecules or grains of sand.Alonso-Marroquín et al.(2005)combined the continuous and the discrete method to investigate the effect of the induced anisotropy on the elastoplastic response of a two dimensional model.The 2-D discrete element model consisted of randomly generated convex polygons which had adjusted shapes and no voids between particles.The authors concluded that the incremental response of the plastic response was unidirectional.Thornton and Zhang(2006)carried out a series of two dimensional numerical simulations to study the shear banding and simple shear non-coaxial flow rules.5000 elastic spheres with seven different sizes were simulated using DEM.From Figure 2.2 we can see that the non-coaxial behav iour agrees with the results of Roscoe's study in Figure 2.1(Roscoe,1970).When the specimens have approached critical state,the directions of principal and principal strain increment were coaxial.

Figure 2.2 Evolutions of the angle of non-coaxiality

(after Thornton and Zhang,2006)

Li and Yu(2009)used a two dimensional DEM model to simulate granular material behaviour under monotonic loading at fixed strain increment directions.The study was focused on the effect of anisotropy on non-coaxiality.To investigate the initial anisotropy produced during specimen preparation,a specimen was generated using a controlled deposition method.Another specimen was prepared by preloading initial anisotropic specimen along the deposition direction and then unloading it to an isotropic stress state.The specimens were tested in a number of loading directions varying from vertical to horizontal at 15° intervals.Figure 2.3 shows the curves of the directions of principal strain increments and principal stress versus the stress ratio.In Figure 2.3(a),very limited deviation angles were observed between the calculated principal stress directions and the principal strain increments directions,the greatest value was under 5°.In this case the soil behaviour could be approximately considered as coaxial.This conclusion agrees well with the result of Miura et al.(1986)and Gutierrez et al.(1991).In Figure 2.3(b),much more significant non-coincidence between the axes of principal stress and strain increment was observed.The deviations were especially significant when the loading direction was close to the normal direction of the previous loading,with the exception of α=0°,where the symmetrical axis of the specimen coincides with the loading direction.The deviation between directions of principal stress and strain increment diminished gradually as shearing progressed to higher shear strain and larger stress ratio.

For a safer geotechnical design,a precise prediction of magnitude and direction of deformation in soil is required.Although the effects of non-coaxiality have been studied widely by a number of researchers,the theory should be efficiently used to develop advanced plasticity models by introducing non-coaxial flow rules into analytical models.Yu and Yuan(2005)published their opinion of the importance of accounting for non-coaxial behaviour in modeling soil-structure interaction.Design might be unreliable due to the lack of consideration of non-coaxiality.Non-coaxiality has been applied into new models in geotechnical engineering.Yatomi et al.(1989)used a non-coaxial cam-clay model to simulate the formation of localized shear bands.Based on experimental studies,Gutierrez et al.(1993)proposed an elastoplastic constitutive model for the de formation of sand during rotational loading.In their model,the plastic principal strain increment direction was defined based on the current stress and the effects of inherent fabric anisotropy on non-coaxiality.Motivated by the observations of non-coaxial behaviour,Li and Dafalias(2004)introduced an extended platform model for anisotropic sand.The model treated the tangent loading as additional loading,called the rotational loading,which produced the non-coaxial and volumetric deformation components.Recently,Lashkari and Latifi(2007)focused on the simulation of non-coaxiality and presented a constitutive model to predict the anisotropic behaviour of granular soils under different stress paths.Yu and his co-workers have been doing lots of work on the non-coaxiality of granular materials(Jiang et al.,2005a and b;Yang and Yu,2006a and b;Yu and Yuan,2006;Yu,2006).This project is part of the research study in the Nottingham Centre of Geomechanics(NCG)to gain experimental support for the numerical models.

Figure 2.3 Stress and strain increment directions.

(a)initially anisotropic specimens;(b)preloaded

specimens

(after Li and Yu,2009)