The main principles of plasticity theory are introduced. The collapse theorems are presented as one of the principal applications of plasticity theory in soil mechanics. Some special plasticity theories for soil (Cam-clay theory and Mohr–Coulomb applications) are also presented.

For civil engineering, calculations are of central importance; they provide a sense of security. However, in soil mechanics the reliability of calculations is reduced for several reasons: aside from incomplete constitutive equations and their calibration based on experiments burdened with errors, some other problems are the spatial scatter of soil properties, missing knowledge of the initial stress field and infinite extending of the considered bodies. Validations reveal the poor reliability of computation results. To cope with such problems increased margins of safety, cautious building (observational method) and reliance upon authorities like experts, computer codes and codes of practice are used.

Barodesy is developed on the basis of the relation between proportional strain paths and proportional stress paths and of the fading memory of soil. This relation is mathematically described by means of a matrix exponential. The fine tuning of barodesy is obtained by consideration of limit states and the dilatancy prevailing there. A new approach to critical states is presented which leaves the hitherto considered critical state line unchanged but introduces an evolution equation for the critical void ratio at non-critical states. It is shown that barodesy includes basic and well-known concepts of soil behaviour. It is shown that the same equation of barodesy holds for sand and clay. Simulations of element tests, oedometric, triaxial drained and undrained ones, show that barodesy is capable of describing them in a satisfactory way, though using equations of outstanding simplicity. In addition, the simulations of cyclic tests exhibiting liquefaction and cyclic mobility are satisfactory.

Brief overview of the importance and peculiarities of granular materials.

Conservation laws (or balance equations) are introduced together with their representation as field and integral equations, jump relations. Some special stress fields are introduced as examples.

Hypoplasticity as an alternative to elastoplasticity theory is introduced. The description of irreversibility with non-linear rate equations is explained, and the incremental non-linearity is elucidated. The emergence and development of hypoplasticity is discussed together with its links to elastoplasticity.

A very short and simple introduction to continuous and discontinuous fields and some related quantities.

The problem of uniqueness in solving initial boundary value problems is considered together with its implications for the simulation of element tests. Some general theorems holding for linear constitutive equations are discussed together with some applications to soil. The formation of shear bands ('faults’ in geology) is described as a special case of loss of uniqueness. Vortices are also introduced as patterns of loss of uniqueness.

Friction is discussed as the main source of strength of granular media. The Mohr–Coulomb strength criterion is introduced. Cohesion as an additional source of strength is discussed together with its controversial physical origin.

This appendix introduces the classification of physical properties according to their behaviour under rotation (scalars, vectors, and in general tensors of a certain rank). Cartesian and spherical representations are presented and explicit expressions for the tensors of second and fourth rank needed when studying liquid crystals are given.

The application of molecular and mesophase symmetries to identify the minimal set of order parameters required to treat single-particle properties for different mesophases and mesogens or solutes is discussed. The use of symmetry to build rotational invariants (Stone S functions) to describe pairwise properties is described and explicit expressions of invariants are provided for uniaxial and biaxial particles.

This chapter introduces the interactions between particles, a key input to the computer simulations described later in the book. Molecular level and fully atomistic interactions are described, having in mind particles forming liquid crystals phases. The empirical level models discussed comprise purely repulsive hard anisotropic particles (ellipsoids, spherocylinders) and attractive-repulsive (uniaxial and biaxial Gay–Berne type) ones. Expressions for electrostatic interactions and in particular charge, dipole and quadrupole ones are derived and typical values for some common mesogens provided. Dispersion interactions, molecular polarizability and chiral interactions are then introduced via quantum mechanical perturbation theory. Since liquid crystals are also formed by colloidal suspensions, dispersive interactions and Hamaker constants are briefly discussed, as well as model potentials for water useful for lyotropic systems, micelles and membranes.

Off-lattice models both based on purely repulsive or attractive-repulsive Gay–Berne models allow us to simulate liquid crystal phases with some positional as well as orientational order. This chapter summarizes simulation results for anisotropic particles of elongated or discotic shape of the two types either pristine or decorated with charges, dipoles and quadrupoles. Beyond showing the effect of key molecular features (e.g. aspect ratios) on morphologies and phase diagrams, applications specific to liquid crystals, like the calculation of elastic constants and the simulation of a TN LCD, are reported. Tapered, bowlic and biaxial GB type single particle systems as well as more complex ones based of multi-particle mesogens (banana phases, polymers, elastomers) are discussed.

Quaternions and their use in treating the orientation of arbitrary rigid particles and their rotations are introduced. Explicit expressions needed to convert between quaternions and Euler angles representations are given.