Chapter 1 Introduction

1.1 Background

In geotechnical engineering,the non-coaxiality is defined as the non-coincidence of the principal stress direction and the principal strain increment direction.It has been widely recognized as an important feature in engineering practice which has been observed and recognized in soil tests using both simple shear and hollow cylinder apparatuses.Numerical analysis carried out by Yu and Yuan(2005,2006),Yang and Yu(2006)showed that the non-coaxiality of a granular soil has very important consequences in geotechnical design.They concluded that the design of shallow foundations without considering the effects of the non-coaxiality might be unsafe.The importance of introducing non-coaxiality into the design of geotechnical structures has been approbated for a more secure project(Yu and Yuan,2005).Models with the consideration of non-coaxiality have been built to simulate the soil behaviour by several researchers(Yatomi et al.,1989;Gutierrez et al.,1993;Li and Dafalias,2004;Lashkari and Latifi,2007;Jiang et al.,2005a and b;Yang and Yu,2006a and b;Yu and Yuan,2006;Yu,2006).

The evidence of the non-coaxial behaviour in granular materials has been observed in both numerical and experimental studies.In simple shear tests,Roscoe et al.(1967)and Roscoe(1970)reported the non-coincidence between principal stress direction and principal strain rate direction.Based on the experimental micro-mechanical study using a photoelastic disc assembly as a two-dimensional analogue of granular media,Drescher and de Josselin de Jong(1972)reported further evidence of non-coaxiality.Using direct shear testing,Wong and Arthur(1986)showed that the deviation between the principal stress and the principal strain incremental directions can be larger than 30° in sand specimens subjected to continuous rotation of the principal stresses axes.Tests using a hollow cylinder apparatus have shown the fact that the behaviour of granular materials is non-coaxial when specimens were subjected to the rotation of principal stress axes(Symes et al.,1982;Ishihara and Towhata,1983;Miura et al.,1986;Pradel et al.,1990;Gutierrez et al.,1991).Non-coaxiality was observed by Alonso-Marroquín et al.(2005)from their 2D simulations with a model consisting of randomly generated convex polygons.Thornton and Zhang(2006)have reported non-coaxial behaviour similar to the results of Roscoe's study(1970)by a 2D numerical simulation using the discrete element method.More recently,Li and Yu(2009)carried out 2D DEM experiments to investigate the directional dependence of the behaviour of granular material under monotonic loading.The non-coaxiality was found to be dependent on the material anisotropy,as well as the loading history.

Figure 1.1 shows an anisotropic specimen.In Figure 1.1(a),if the loading direction is normal to the bedding plane,then the directions of principal stress and principal strain increment will be coaxial even if the specimen fabric is anisotropic.However,as shown in Figure 1.1(b),when the loading direction and bedding plane is not normal to each other,the strain increment axis will deviate from the principal stress axis,thus non-coaxiality is induced.

Figure 1.1 The interrelation of anisotropy and non-coaxiality.

(a)loading direction is normal to the bedding plane;(b)loading direction is not normal to the bedding plane

The theoretical origin of non-coaxiality can be found in the kinematic models for the flow of granular materials developed by de Josselin de Jong(1958).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 has been exhausted.Spencer(1964)used the same concept of double sliding to establish a set of kinematic equations termed 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 justification for the non-coaxiality of principal stress and principal strain rate directions.These "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 them to account for dilat ant,elasto-plastic and pre-peak strain hardening response(Mehrabadi and Cowin,1978;Anand,1983;de Josselin de Jong,1988;Teunissen and Vermeer,1988;Yu and Yuan,2006).Rudnicki and Rice(1975)also reported that non-coaxial behaviour plays an important role in shear band formation in sands.In some pre-failure plasticity models that have been proposed for granular materials,such as a hypoplastic model(Wang et al.,1990;Kolymbas,1991)and a multi-laminate model(Iai et al.,1992),non-coaxiality was also evident.