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A Journal of Russian Academy of Sciences | | Founded
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| N. G. Bourago and A. N. Kovshov, "A model of fracturing material with dilatancy," Mech. Solids. 36 (5), 89-93 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
5 |
Pages |
89-93 |
| Title |
A model of fracturing material with dilatancy |
| Author(s) |
N. G. Bourago (Moscow)
A. N. Kovshov (Moscow) |
| Abstract |
This work deals with constitutive relations describing the
thermomechanical behavior of geomaterials. A general thermodynamical
method for the construction of constitutive relations for continuous media
with finitely many state parameters is described in [1] and is a
generalization of the investigations started in [2-5]. In the present
paper, the thermodynamical method is systematically applied to modeling
deformation of geomaterials. In contrast to previous publications, our
construction of the model requires only that the free energy and the
dissipation velocity be specified as functions of state parameters. Then,
the dilatancy and other properties of geomaterials become mere
consequences.
Well known experimental studies show that geomaterials possess
some distinctive properties, such as plastic compressibility and
non-monotonicity of the "shear versus tangential stress" loading diagram
which, together with a hardening segment contains
a "dropping" segment corresponding to material softening.
Fracture of geomaterials occurs gradually, with the accumulation of various types of
microdefects; moreover, wave propagation velocities in geomaterials depend on
the damage measure and become lower with its increase.
As shown by numerous experiments, a characteristic feature of geomaterials
is their dilatancy, i.e., the existence of a dependence
of volume strain on shear strains.
The dilatancy of geomaterials can be described in many ways.
As shown in [6], if the deformation of a geomaterial is described by the
equations of plastic flow with the associated law and the yield condition
of the von Mises type with the yield function depending on the
first invariant of the stress tensor, then dilatancy takes place.
Moreover, the mechanical energy dissipation rate and the
volume expansion rate in the case of plane deformation are proportional to
the maximal shear rate, with the coefficient of proportionality equal to
the internal friction coefficient.
This relation implies that there is a dependence between the components of
the plastic strain rate tensor, and this dependence is called the dilatancy condition or
the dilatancy dependence. In [7-9],
the dilatancy condition is taken as an assumption and has the form
of a kinematic constraint on the plastic strain rate components; this constraint involves
the so-called dilatancy rate. As a result, the sign of the dilatancy rate
determines the possibility of loosening or compaction of the dilating
geomaterial under shear. Moreover, the constraint relating the density of
the medium and the pressure p is differential and
this differential constraint is not always integrable.
The function p(ρ) may be different for different deformation paths
of the material elements.
The possibility of constructing models of plastic deformation of
materials (in particular, loose materials) on the basis of the
properties of the dissipation function was considered in [10, 11].
It was shown that for a prescribed dilatancy dependence, the dissipation
function can be defined in such a way that the constitutive relations for
a medium with dilatancy can be introduced by means of the associated law.
The dilatancy dependence on the form of the
stress state is studied in [12], where
the associated plastic flow law is used with a suitably chosen
structure of the yield function and the dilatancy condition is not
prescribed in advance.
Thus, in order to describe the behavior of geomaterials,
various models have been proposed, with the above properties more or less
taken into account. In the so-called models of "gradual fracture",
the constitutive relations rely on the associated elastic-plastic flow law
and the yield surface is specified so as to take into account
the phenomena of hardening, softening, and "residual" strength.
In continuous fracture models, which are currently an object of intensive
studies, fracture of geomaterials is described in terms of a new
structural parameter referred to as the damage parameter. This parameter
pertains to the accumulation of microdefects in a deformed medium and is
responsible for the decrease of wave propagation velocities in geomaterials.
In the models of geomaterials with damage, which seem physically most adequate,
damage accumulation rate is introduced and its backward
influence on the stress-strain state is taken into account in terms of
the dependence of elastic and plastic properties on the damage.
The constitutive relations in damage models are often constructed on
the basis of micromechanical concepts regarding the processes
of deformation and fracture in geomaterials.
The constitutive relations should agree with the general principles of
thermomechanics, i.e., be compatible with the laws of thermodynamics,
the dimensional theory, and the principles of invariance and objectivity.
These requirements can be easily met when constructing the constitutive
relations by the thermodynamical method [1] used below. |
| References |
| 1. | N. G. Bourago, A. I. Glushko, and A. N. Kovshov,
"Thermodynamical method for constructing
constitutive relations for models of continuous media,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 4-15, 2000. |
| 2. | V. I. Kondaurov and V. N. Kukudzhanov,
"On constitutive equations and numerical solution of
the multidimensional problems of the dynamics of
nonisothermic elastic-plastic media
with finite deformations," Arch. Mech., Vol. 31, No. 5, pp. 623-647,
1979. |
| 3. | N. G. Bourago and V. N. Kukudzhanov,
Solution of Elastoplastic Problems by the Finite-Element Method,
Software Package "ASTRA" [in Russian], Preprint 326,
In-t Problem Mekhaniki AN SSSR, Moscow, 1988. |
| 4. | V. N. Kukudzhanov and K. Santaoja,
"Thermodynamics of viscoelastic media with internal parameters,"
Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 115-126, 1997. |
| 5. | V. N. Kukudzhanov, N. G. Bourago, A. N. Kovshov, V. L. Ivanov,
and D. N. Shneiderman,
On the Problem of Damage and Localization of Strains, Preprint 95:11,
Chalmers Univ. Technol., Goetheborg, 1995. |
| 6. | D. C. Drucker and W. Prager,
"Mechanics of solids and plastic analysis or limiting design,"
in Mechanics. New Trends in World Science. Constitutive Laws
in the Mechanics of Soil [Russian translation], pp. 166-177,
Mir, Moscow, 1975. |
| 7. | V. N. Nikolaevskii and N. M. Syrnikov,
"On the limiting flow of a loose dilatant medium,"
Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 159-166, 1970. |
| 8. | V. N. Nikolaevskii,
"Constitutive equations of plastic deformation of a loose
material," PMM [Applied Mathematics and Mechanics], Vol. 35, No. 6, pp. 1070-1082, 1971. |
| 9. | V. N. Nikolaevskii,
Postface to the book. Current Trends in Soil Mechanics [Russian translation], Mir,
Moscow, 1975. |
| 10. | I. A. Berezhnoi, D. D. Ivlev, and V. B. Chadov,
"On the construction of a model of loose media
on the basis of the definition of the dissipation function,"
Doklady AN SSSR, Vol. 213, No. 6, pp. 1270-1273, 1973. |
| 11. | Ya. A. Kamenyarzh, "On the construction of models
of plastic media by means of the dissipation function,"
Doklady AN SSSR, Vol. 215, No. 4, pp. 804-806, 1974. |
| 12. | E. V. Lomakin,
"Plastic flow of a dilatant medium under plane strain conditions,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 58-68, 2000. |
| 13. | H. Ziegler, Extremal Principles of Thermodynamics of
Irreversible Processes in Continuum Mechanics [Russian translation], Mir, Moscow, 1966. |
|
| Received |
15 May 2001 |
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