Summary
This work deals with the well-posedness of the linearized initial value problem in solid elastoplastodynamics. By well-posedness we understand existence, uniqueness and continuous dependence of the solution with respect to initial and boundary data. The initial value problem is studied in its most general form using a technique from Kreiss [1]. The cases of elastic and elastic-plastic materials with associated and non-associated plastic flow rules are considered. The analysis is carried under the assumption that the material response remains within the same constitutive cone everywhere (i.e.; exclude loading/unloading subregions). A complete answer to the well-posedness question is given without the requirement of major symmetry of the constitutive tensor. First it is shown that in 2-D the necessary and sufficient condition for well-posedness is the strong ellipticity condition. Up to now this condition had been known only to be a sufficient condition.
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References
Kreiss, H. O.: Über sachgemäße Cauchyprobleme. Math. Scand.13, 109–128 (1963).
Hadamard, J.: Lecons sur la propagation des ondes. Paris: Hermann Cie 1903.
Kreiss, H. O.: Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math.23, 277–298 (1970).
Kreiss, H. O., Lorenz, J.: Initial-boundary value problems and the Navier-Stokes equations. San Diego: Academic press 1989.
Drucker, D. C.: A more fundamental approach to plastic stress-strain relations. Proc. U. S. Nat. Congr. Appl. Mech.1, 487–491 (1952).
Il'Iushin, A. A.: On the postulate of plasticity. PMM25, 503–507 (1961) (Bers, L., John, F., Schecter, M., eds.), pp. 503–507. New York: Wiley 1964.
Hill, R.: A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids6, 236–249 (1958).
Rice, J. R., Rudnicki, J. W.: A note on some features of the theory of localization of deformation. Int. J. Solids Struct.16, 597–605 (1980).
Raniecki, B., Bruns, O. T.: Bounds to bifurcation stresses in solids with non-associated plastic flow laws at finite strain. J. Mech. Phys. Solids29, 153–172 (1981).
Mandel, J.: Conditions de stabilité et postulat de Drucker. Proc. IUTAM Symp. on Rheology and Soil Mechanics (Kravtcheko, J., Sirieys, P. M., eds.), pp. 58–68. Heidelberg New York Berlin: Springer 1964.
Browder, F. E.: Strongly elliptic systems of differential equations. Ann. Math. Studies33, 101–159 (1954).
Weyl, H.: The classical groups. Princeton: Princeton University Press 1946.
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Örs, H., Prévost, J.H. On the well-posedness of the initial value problem in elasto-plastodynamics for a linear comparison solid. Acta Mechanica 111, 181–192 (1995). https://doi.org/10.1007/BF01376929
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DOI: https://doi.org/10.1007/BF01376929