## On the Modelling of State-Dilatancy and Mechanical Behaviour of Frictional Materials

##### Abstract

Understanding and characterizing the mechanical behaviour of soils is an important area of engineering for geotechnical engineers. This is for a good reason. Soils, as engineering materials, carry structures of different sorts where safety depends on the stability of the soil mass the various structures rest on. Soils are also used as building materials.
Soils, as aggregate materials, are able to sustain loads of various kinds through friction, cohesion, and interlocking. Soils that mainly sustain loads through friction and interlocking are referred to as frictional materials in this thesis and the particular focus is on sand-like materials. The load carrying capacity and the deformation characteristics of such materials depend not only on the mineralogy and grain characteristics of the aggregants but also on the condition at which the particles are arranged. The arrangement depends on the configuration of particles, void proportions, and the effective stress level.
The stress-strain behaviour of soils, and hence frictional materials, is highly non-linear and depends on the state of deposition (void-solid proportion and arrangement or fabric), effective stress state, and the history of loading conditions they have been subjected to. The constitutive behaviour of soils in general is thus complex, as the variables that affect it are complex and random. However, through careful observations, important frameworks have been deciphered. Two of the most successful frameworks are the critical state/ steady state theory and the stress-dilatancy theory.
According to the critical state/steady state theory, there exists an ultimate state that can be described independently of the configuration state. The evolution tendency of various state variables during continuous monotonic deformation is towards this ultimate state which is a target reference state. Such an approach relies on the following four key ingredients. First, state variables must be defined. This is to some extent arbitrary. Nevertheless, the void ratio and the effective stress are the main ones in the list. Fabric and intergranular strain may also be considered to obtain a more complete picture. Second, the initial values of each state must be specified. Third the evolution of the various state variables must be established. Fourth, the ultimate state must be investigated and defined.
The stress-dilatancy theory considers the coupling between shear and volume changes in several materials, especially in soils, rocks and concrete. Such materials undergo volumetric changes when sheared. This behaviour is best described by Osborne Reynolds who initiated the scientific enquiry of shear-volume coupling in granular materials, “distortion necessitates volume change”. These volumetric changes give rise to deformation and energy dissipation behaviour that is characteristic for such materials. Mathematical formalisms that describe volumetric changes are generally known as stress-dilatancy theories. The stress-dilatancy formalisms can be established in different ways. Some theoretical abstractions consider certain principles in the energy dissipation mechanism while in others arbitrary empirical equations are defined. From the stress-dilatancy formalisms, the so-called plastic potential functions which define the “flow” of direction of plastic strains can be constructed. Plastic potential functions may also be arbitrarily defined. Often it is possible to decipher the implied stress-dilatancy relation. Plastic potential functions are generally distinguished into associated and nonassociated. Flow rules are defined associated when the plastic potential is the same as the yield function(s), a (set of) constraint(s) that a stress state is assumed to obey. If the plastic potential function is different from the yield function(s) then the flow condition is called nonassociated. Such is often the case with the deformation behaviour of several geomaterials and thus frictional materials.
In this thesis, state variables and important states of soils are investigated. Some empirical and theoretical foundations of the resistance of soils are investigated. A flow rule which accommodates non-associativity is established. Several yield functions that were proposed for frictional materials are presented and discussed. Empirical equations that relate plastic distortional strain to mobilization of the yield functions are proposed. Both loading and unloading conditions are considered. Employing the concepts of plasticity theory, a tangent stiffness tensor is composed. Then, an extended stress-dilatancy and plastic flow theory called adaptable cyclic stress-dilatancy and non-associated flow theory is proposed. Both loading and unloading are considered such that the proposed theory can be applied for cyclic loading conditions. The proposed theory can be adapted to describe the stress-dilatancy behaviour of a wide range of geomaterials. This is demonstrated by establishing novel stress-dilatancy formalism for Hoek-Brown materials. The theory is further improved by considering the noncoaxiality between the axes of principal stresses and principal plastic strain rates. Novel inequalities are proposed. Furthermore, a novel evolution rule is established for the degree of non-coaxiality. Finally, a new constitutive framework named C-StaD is proposed and evaluated in simple element tests. The model is developed by hierarchically building from a simple elastoplastic model. The model development targets both the small and the large strain ranges. The model framework is organized such that further extension is easier.