IV International Workshop on Modern Trends in Geomechanics
G. Viggiani, D. Salciarini, F. Silvestri, C. Tamagnini, G.M.B. Viggiani(Eds) Assisi, May 16–18, 2016

SOME ASPECTS OF BARODESY
Dimitrios Kolymbas
Division of Geotechnical and Tunnel Engineering
Faculty of Technical Sciences
Universität Innsbruck, Austria
e-mail: dimitrios.kolymbas@uibk.ac.at

Keywords: Constitutive relations, sand, asymptotic states.
Summary. Barodesy is a mathematical frame for constitutive modelling that makes possible to derive a constitutive relation from basic properties of soil. As such are considered those ones that are asymptotically obtained with monotonousdeformation. The latter erase any initial disturbances imposed erratically by the sample preparation. The obtained equations have a simple mathematical structure which facilitates understanding of the underlying processes and numericalimplementation. In this way, barodesy could serve as a basis for further developments in a field with many open questions
and which is erroneously considered as completed.

1

DIRECTIONS OF PROPORTIONAL PATHS

G OLDSCHEIDER observed that proportional strain paths starting from the stress-free state lead to proportional stresspaths. A proportional strain path is characterized by a constant stretching, D = const, hence the normalized stretching
D0 := D/|D| indicates its direction. The direction of the pertinent proportional stress path is given by a tensor R. We seeka relation R(D0 ). We know that all contractant proportional strain paths should produce proportional stress paths that lie
within the compression octant of the principal stress space. This means that all principal values of R must be negative if
trD < 0. The condition detR < 0 iff trD0 < 0 suggests to use the following function for the relation R(D0 ):R(D0 ) = − exp(aD0 )(1)

The collection E of all proportional strain paths with trD0 = 0 consists of volume preserving (’undrained’) proportionalstrain paths and defines the limit between dilatant and contractant paths. The relation (1) maps E into the collection S ofthe corresponding proportional stress paths. Clearly, S is the critical state surface in stress space. Interestingly, the crosssection of S , as obtained with equ. (1), with a deviatoric plane trT = const is a curve that practically concides with theM ATSUOKA -NAKAI limit curve.
With the abbreviations σ := |T| and ε̇ := |D| a constitutive equation of the rate type can be formulated that holds for
proportional strain paths: Ṫ = hR0 ε̇. Herein, h is a function of σ and expresses the fact that the stiffness increases withincreasing σ .
2ASYMPTOTIC PATHS

G OLDSCHEIDER observed also that starting from a non-vanishing stress state and applying a proportional strain path leads
asymptotically to the proportional stress path that would be obtained starting from the stress-free state. This behaviour
can be modelled if we add to the aforementioned constitutive equation a second term which is proportional to the stress
T. This leads to the following final form of the barodetic constitutive equation:Ṫ = h( f R0 + gT0 )ε̇,(2)

which allows to describe many effects of soil behaviour in an elegant way. Of course, to accomplish the equation, thescalar quantities f and g have to be specified.

D. Kolymbas

Limit states, defined by vanishing stiffness Ṫ = 0, are manifested either as peak or residual (critical) limit states. Inbarodesy, Ṫ = 0 implies f R0 + gT0 = 0. This tensorial equation implies the ’flow rule’ R0 = T0 and the ’yield condition’f + g = 0. Critical limit states are obtained with δ := tr D0 = 0 and e = ec , whereas peak limit states are obtained with
δ > 0 and e < ec . A simple way to model limit states of either type is to set f + g = δ + c3 (ec − e) with ec being thecritical void ratio. It can be shown that the critical state line, i.e. a relation ec (σ ) can be derived from the above formalism.
3LIMIT CYCLES

Proportional paths are not the only attractors in barodesy. Being an ordinary differential equation, it exhibits also limit
cycles or cyclic orbits as further attractors. In mechanics, the existence of such attractors is known as ’shake-down’ andimplies that stress cycles lead asymptotically to cyclic changes of void ratio. For oedometric deformation, this implies
also cyclic strain. This phenomenon is related to vibrocompaction. However, the original barodetic equation needssome modification to realistically model cyclic loading and small strain stiffness. This modification can be achieved byincorporating the acceleration of deformation or, more precisely, the time rate of normalized stretching.
4BENEFITS OF BARODESY

As hypoplasticity, barodesy avoids any whatsoever defined surfaces in stress space to describe the material behaviour.
Even if some surfaces can be a posteriori derived from the original equations, they are not necessary for the calculation,and this is an advantage, as it renders any return mapping superfluous. In addition, barodesy constitutes a frame ofamazing simplicity to describe material behaviour including barotropy and pyknotropy. And last but not least, it offers a
simple theoretical platform to understand and describe the behaviour of soil.