Abstract
This paper deals with the transformation of the inelastic continuum problem from the differential to the variational form. Due to the lack of symmetry of the constitutive law, the variational formulation of the problem has been, above all, obtained using techniques of symmetrization of the overall problem operator (as with Tonti’s techniques). In the present paper, instead, the adopted method is based on the symmetrization of the constitutive law only. On this basis, taking into account the consequent transformations to be done on the equilibrium and on the compatibility boundary condition operators, a symmetric global operator for the original problem is achieved. This leads to a new ‘enlarged’ formulation and then, by the choice of a suitable operator, to a new wide class of variational principles and, by specialization, to classical and recent variational principles.
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References
Alliney, S. and Tralli, A., ‘Extended variational formulations and F.E. methods in the stability analysis of non-conservative mechanical problems’, Comp. Meth. Appl. Mech. Eng. 51 (1986) 209–219.
Auchmuty, G., ‘Variational principle for operator equations and initial value problems’, Nonlinear Anal. Theor. Meth. Appl. 12(5) (1988) 531–564.
Carini, A., ‘Colonnetti's minimum principle extension to generally nonlinear materials’, Int. J. Solids Struct. 33 (1996) 121–144.
Carini, A., ‘Saddle-point principles for general nonlinear material continua’, ASME J. Appl. Mech. 64 (1997) 1010–1014.
Carini, A. and De Donato, O., ‘A comprehensive energy formulation for general nonlinear material continua’, ASME J. Appl. Mech. 64 (1997) 353–360.
Carini, A. and Genna, F., ‘Some variational formulations for continuum nonlinear dynamics’, J. Mech. Phys. Solids 46 (1998) 1253–1277.
Carini, A. and Genna, F., ‘Variational formulations in the non-associated flow theory of plasticity’, Int. J. Eng. Sci. 39 (2001) 1765–1784.
Carini, A., Gelfi, P. and Marchina, E., ‘An energetic formulation for the linear viscoelastic problem. Part I: Theoretical results and first calculations’, Int. J. Numeric. Meth. Eng. 38 (1995) 37–62.
Dall'Asta, A. and Menditto, G., ‘A variational formulation of the perturbed motion problem for a viscoelastic body’, Int. J. Solids Struct. 31 (1994) 247–256.
Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.
Gurtin, M.E., ‘Variational principles for linear initial-value problems’, Quart. Appl. Math. 22 (1964) 252–256.
Huet, C., ‘Bounds for the overall properties of viscoelastic heterogeneous and composite materials’, Arch. Mech. 47 (1995) 1125–1155.
Leipholz, H.H.E., ‘Variational principles for non-conservative problems: a foundation for a finite element approach’, Comput. Meth. Appl. Mech. Eng. 17/18 (1979) 609–617.
Magri, F., ‘Variational formulation for every linear equation’, Int. J. Eng. Sci. 12 (1974) 537–549.
Morse, P.M. and Feshbach, H., Methods of Theoretical Physics, Vol. 1, McGraw-Hill, New York, 1953.
Ortiz, M., ‘A variational formulation for convection-diffusion problems’, Int. J. Eng. Sci. 23(7) (1985) 717–731.
Rafalski, P., ‘Orthogonal projection method. I: Heat conduction boundary problem’, Bull. Acad. Polon. Sci. Ser. Sci. Tech. 17 (1969a) 63–67.
Rafalski, P., ‘Orthogonal projection method. II: Thermoelastic problems’, Bull. Acad. Polon. Sci. Ser. Sci. Tech. 17 (1969b) 69–74.
Reiss, R. and Haug, E.J., ‘Extremal principles for linear initial value problems of mathematical physics’, Int. J. Eng. Sci. 16 (1978) 231–251.
Telega, J.J., ‘Dual extremum principles in rate boundary value problems of non-associated plasticity’, Int. J. Eng. Sci. 17 (1979) 215–226.
Telega, J.J., ‘Derivation of variational principles for rigid-plastic solids obeying non-associated flow laws — II. Unprescribed jumps’, Int. J. Eng. Sci. 20 (1982) 935–946.
Tonti, E., ‘Variational formulations for every nonlinear problem’, Int. J. Eng. Sci. 22(11–12) (1984) 1343–1371.
Vainberg, M.M., Variational Methods for the Study of Nonlinear Operators, Holden Day, 1964.
Yosida, K., Functional Analysis, Springer-Verlag, Berlin and New York, 1965.
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Carini, A., De Donato, O. Inelastic Analysis by Symmetrization of the Constitutive Law. Meccanica 39, 297–312 (2004). https://doi.org/10.1023/B:MECC.0000029361.81361.c5
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DOI: https://doi.org/10.1023/B:MECC.0000029361.81361.c5