Summary
We consider the process of compression between rigid plates of a multilayer consisting of materials with different yield limits. The model of a rigid perfectly plastic body is assumed and theLagrangian description of the motion is employed. Closed formulae are derived for the deformation, state of stress and the force required for the process to take place. The conditions of a joint uniform deformation of the multilayer are examined.
Zusammenfassung
Wir betrachten die Kompression eines Vielschichtstoffes bestehend aus Materialen mit verschiedenen Fließgrenzen zwischen starren Platten. Das Modell eines starr-ideal plastischen Körpers wird zugrunde gelegt.Lagrangesche Darstellung der Bewegung wird verwendet. Für die Deformation, den Spannungszustand und die erforderliche Kraft zur Einleitung des Prozesses werden geschlossene Formeln hergeleitet. Die Bedingungen für eine gemeinsame homogene Deformation der Vielschicht werden untersucht.
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Abbreviations
- X, Y :
-
Coordinates of a particle in the material Lagrangian coordinate system
- x(X, Y, t), y(X, Y, t) :
-
Coordinates of the particleX, Y at instantt in a fixed Cartesian Eulerian coordinate system
- χ:
-
Angle between thex-axis and the bisectrix of the angle between the tangents to the material linesX (Y=const),Y (X=const)
- ϑ:
-
Angle between the above bisectrix and the tangent to theX-line
- h X ,h Y :
-
Lame parameters of theX, Y lines
- θ:
-
Angle between thex-axis and the tangent to the slip line α (which makes angle −π/4 with the direction of the algebraically greater principal stress)
- σ:
-
Mean pressure
- σ xx , σ yy , σ xy :
-
Components of theCauchy stress tensor
- τ:
-
Tangent stress on the line of constact of two materials
- τ r :
-
The shear strength of the joint of two materials
- (0) (n):
-
Friction coefficients on the contacts with the plates
- K :
-
Yield limit
- H, h :
-
Thickness of the layer in the initial and current states, respectively
- 2 L, 2 l :
-
Length of the layer in the initial and current states, respectively
- S, s :
-
Y-coordinate of the contact line of two materials in the initial and current states, respectively
- a, b, c, d, e, f, g :
-
Auxiliary functions defined in the text
- A, B :
-
Auxiliary functions defined in the text
- ω, Ω:
-
Auxiliary functions defined in the text
References
Prandtl, L.: Anwendungsbeispiele zu einem Henckyschen Satz über das plastische Gleichgewicht. ZAMM3, 401–406 (1923).
Arkulis, G. E.: A joint plastic deformation of different metals (in Russian). Moscow: Izdatelstvo Metalurgyia, 1964.
Engineering methods for computing technological processes of working of metals by pressure (in Russian). Edited byI. Y. Tarnovski. Moscow: Metallurgizdat, 1964.
Rychlewski, J.: Plane plastic strain for jump non-homogeneity. Int. J. Nonlinear Mech.1, 57–78 (1966).
Kuznetsov, A. I.: The problem on the inhomogeneous layer. Arch. Mech. Stos.12, 163–172 (1960).
Ilyushin, A. A.: Some problems of the plastic flow (in Russian). Izv. AN SSSR, OTN,2, (1958).
Arcisz, M. andJ. Rychlewski: Lagrangian description of plane plastic flow. Bull. Acad. Polon. Sci., Série Sci. Techn.17, 445–451 (1969).
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Arcisz, M. Joint deformation of a metal multilayer. Acta Mechanica 9, 67–77 (1970). https://doi.org/10.1007/BF01176610
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DOI: https://doi.org/10.1007/BF01176610