Engineering Transactions, 16, 3, pp. 385-409, 1968

### Płaskie płyniecie ustalone ośrodka Coulomba z uwzględnieniem sił bezwładności i sił masowych

C. Szymański
Zakład Mechaniki Ośrodków Ciągłych Instytutu Podstawowych Problemów Techniki
Poland

The problem under consideration is that of steady plane fow of a Coulombian body using equations of motion in which the inertia and mass forces are involved. The body is assumed to be incompressible. The constitutive equation is the coaxiality condition of the stress and strain rate deviator tensors.
The partial differential equations of the problem with four unknown functions (two functions describing the state of stress, the remaining two describing the displacement rate field )in an arbitrary system of curvilinear coordinates (in the plane of the motion) are derived. The set of equations above is found to be hyperbolic. They have four families of characteristics in the plane of the motion one pair of which are Coulomb slip-lines, the other representing the trajectories of maximum shear rate. It is found that the differential relations along the slip lines are transformed equations of motion and the differential relations along the trajectories of
maximum shear rate are identical with those of H. Geiringer for the quasi-static problem (in which the inertia terms in the equations of motion have been rejected). The essential point of the present paper is to give a method of investigation of quasi- linear partial differential equations and derivation of the equations of the characteristic lines of these equations. This method is used for the analysis of the differential equations resulting from the mechanical model assumed and transformed to an appropriate system of curvilinear coordinates normalized to the natural form. The derivation of the equations of the characteristics (characteristic directions and the relevant differential relations along the characteristic lines in the plane of independent variables) reduces to the obtainment of some simple algebraic relations between the angles of relative inclination of the coordinate lines and the inclination angles of the principal direction σ1 to the
coordinate lines.
This method enables us to avoid toilsome manipulation connected with the derivation of the equations of the characteristics by the method of eigenfunctions and eigenvectors of the relevant matrices of the differential equations of the problem.

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#### References

R. COURANT, D. HILBERT, Partial Differential Equations, Vol. II, New York, London 1962.

A. DRESCHER, K. KWASZCZYNSKA, Z. MRÓZ, Statics and kinematics of the granular medium in the case of wedge indentation, Arch. Mech, Stos., 1, 19 (1967).

D. C. DRUCKER, W. PRAGER, Soil mechanics and plastic analysis or limit design, Quart, Appl. Math., 10 (1952), 157-165.

[in Russian]

[in Russian]

[in Russian]

[in Russian]

A. W. JENIKE, R. T. SHIELD, On the plastic flow of Coulomb solids beyond original failure, J. Appl. Mech., 27 (1959), 599-602.

G. DE JOSSELIN DE JONG, Statics and Kinematics in the Faible Zone of a Granular Material, Waltman, Delft 1959.

J. MANDEL, Sur les équations d'écoulement des sols idéaux en déformation plane et le concept

du double glissement, J. Mech. Phys. Solids, 14 (1966), 303-308.

J. NAJAR, Inertia effects in the problem of compression of a perfectly plastic layer between two rigid plates, Arch. Mech. Stos., 1, 19 (1967).

W. OLSZAK, P. PERZYNA, The constitutive equations for elastic|viscoplastic soils, IUTAM Conference in Grenoble, 1964.

A. J. M. SPENCER, The dynamic plane deformation of an ideal plastic-rigid solid, J. Mech. Phys. Solids, 4, 8 (1960), 262-279.

A. J. M. SPENCER, A theory of the kinematics of ideal soils under plane strain conditions, J. Mech. Phys, Solids, 5, 12 (1964), 337-351,

W. SZCZEPINSKI, Dynamic expansion of an rotating solid cylinder of mild steel, Arch. Mech. Stos., 1, 19 (1967), 75-88.

P. P. TEODORESCU, Le problème pian de la théorie de l'élasticité en coordonnées curvilignes arbitraires, Bull, Acad. Polon. Sci., Série Sci. Techn., 7 10, (1962), 269 (403)-299(433),

[in Russian]

J. ZAWIDZKI, Płaskie stany ośrodków plastycznych w nieortogonalnych układach współrzędnych krzywoliniowych, Rozpr. Inzyn., 4, 15 (1967).