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Symmetry is introduced as a basic notion of physics and, in particular, for soil mechanics also. Isotropy and anisotropy are discussed. A special case of isotropy of space is the principle of material frame indifference which plays an eminent role in the development of constitutive equations. The geometric scaling is discussed together with the notion of a simple material, which is – often unconsciously – basic in geotechnical engineering. Invariance with respect to stress and time scales is discussed. Mechanical similarity and the associated Pi theorem is shown to be the basis for the evaluation of so-called physical simulations with model tests.
The notion of collapse and its importance in geotechnical engineering is introduced. The two main approaches are explained: (i) stress fields that fulfil the Mohr–Coulomb limit condition (together with slip line analysis as an application of the method of characteristics) and (ii) analysis of collapse mechanisms consisting of rigid blocks. The harmonisation of codes and the problematic definition of safety on the basis of probability theory are discussed.
As an important application of the theory of elasticity in soil mechanics, the main principles of elastodynamics are introduced. On the basis of waves in 1D-continua the notions of transmission, reflexion and dynamic stiffness are explained, and the body waves are presented as compression and shear waves. Rayleigh waves are presented as an example of surface waves.
The general definition of elasticity is given, and as a special case the linear elasticity with Hooke’s law, is presented together with its derivation on the basis of the Cayley–Hamilton theorem. Some applications of elasticity theory in soil mechanics are presented.
Soil consisting of grains, water and air is an example of a multiphase material (or mixture). The basic concepts for multiphase materials, such as volume fraction, partial quantities and interaction forces are introduced. The Darcy's equation and balance equations for mixtures are introduced.
On their basis, the consolidation theory is presented. The equations describing steady and unsteady groundwater flow are derived. Transport within groundwater by means of convection, diffusion and dispersion is explained. The main principles describing unsaturated soil are presented: capillary and osmotic suction, the function of filters is explained. The soil–water characteristic curve is also introduced. The effective stresses in unsaturated soil are discussed.
The notion of constitutive equations (or laws) is elucidated together with the meaning of related notions such as material constants, calibration and response envelopes. Also discussed is why a comparison of constitutive equations is conceptually difficult.
Several tensors that describe deformation are introduced, as well as stretching and spin. As an example, they are presented for the special case of simple shear. The compatibility equations are discussed together with non-Boltzmann continua.
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.