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Material constraints, Lagrange multipliers, and compatibility. Applications to rod and shell theories. (English) Zbl 0769.73012

The authors deal with the theory of material (internal) constraints in continuum mechanics, formulating a global constraint principle, having a physically and mathematically natural interpretation. An appropriate general role for the Lagrange multipliers as reactive stresses is put into evidence; it is shown for which kind of constraints the global constraint principle reduces to the classic local principle. Applications to rods, shells and discrete models are presented.

MSC:

74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K15 Membranes
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[1] S. S. Antman (1972), The Theory of Rods, in Handbuch der Physik, Vol. VIa/2, C. Truesdell, ed., Springer-Verlag, 641-703.
[2] S. S. Antman (1976a), Ordinary differental equations of one-dimensional nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells, Arch. Rational Mech. Anal. 61, 307-351. · Zbl 0354.73046
[3] S. S. Antman (1976b), Ordinary differental equations of one-dimensional nonlinear elasticity II: Existence and regularity theory for conservative problems, Arch. Rational Mech. Anal. 61, 353-393. · Zbl 0354.73047
[4] S. S. Antman (1982), Material constraints in continuum mechanics, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 70, 256-264. · Zbl 0508.73002
[5] S. S. Antman (1983), Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders, Arch. Rational Mech. Anal. 83, 1-52, Corrigenda, ibid. 95 (1986) 391-393. · Zbl 0524.73044
[6] S. S. Antman & J. E. Osborn (1979), The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69, 231-262. · Zbl 0403.73003
[7] S. S. Antman & W. H. Warner (1966), Dynamical theory of hyperelastic rods, Arch. Rational Mech. Anal. 23, 35-352.
[8] J. M. Ball (1981), Remarques sur l’existence et la r?gularit? des solutions d’?lastostatique nonlin?aire, in Recent Contributions to Nonlinear Partial Differential Equations, H. Berstycki & H. Brezis, eds., Pitman, 50-62.
[9] M. F. Beatty & M. A. Hayes (1992), Deformations of an elastic, internally constrained material. Part 1: Homogeneous deformations, J. Elasticity, to appear. · Zbl 0786.73018
[10] J. F. Bell (1985), Contemporary perspectives in finite strain plasticity, Int. J. Plasticity 1, 3-27. · Zbl 0612.73049
[11] F. Brezzi & M. Fortin (1991), Mixed and Hybrid Finite Element Methods, (to appear). · Zbl 0788.73002
[12] P. G. Ciarlet (1990), Plates and Junctions in Elastic Multi-Structures, Masson, Springer-Verlag. · Zbl 0706.73046
[13] P. G. Ciarlet & J. Ne?as (1987), Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97, 171-188. · Zbl 0628.73043
[14] H. Cohen (1981), Pseudo-rigid bodies, Util. Math. 20, 221-247. · Zbl 0482.70002
[15] H. Cohen & R. Muncaster (1988), The Theory of Pseudo-Rigid Bodies, Springer-Verlag. · Zbl 0687.70001
[16] F. Davi (1991), The theory of Kirchhoff rods as an exact consequence of three-dimensional elasticity, J. Elasticity, to appear.
[17] D. G. Ebin & R. A. Saxton (1986), The initial-value problem for elastodynamics of incompressible bodies, Arch. Rational Mech. Anal. 94, 15-38. · Zbl 0599.73012
[18] J. L. Ericksen (1955), Deformations possible in every compressible, perfectly elastic material, Z. angew. Math. Phys. 34, 126-128. · Zbl 0064.42105
[19] J. L. Ericksen (1986), Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures 22, 951-964. · Zbl 0595.73001
[20] J. L. Ericksen & R. S. Rivlin (1954), Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3, 281-301. · Zbl 0055.18103
[21] A. F. Filippov (1985), Differential Equations with Discontinuous Right-Hand Sides (in Russian), Nauka, English transl., 1988, Kluwer.
[22] A. E. Green, N. Laws & P. M. Naghdi (1967), A linear theory of straight elastic rods, Arch. Rational Mech. Anal. 25, 285-298. · Zbl 0146.46204
[23] A. E. Green, N. Laws & P. M. Naghdi (1968), Rods, plates and shells, Proc. Camb. Phil. Soc. 64, 895-913. · Zbl 0172.50403
[24] P. Hartman (1964), Ordinary Differential Equations, Wiley, New York. · Zbl 0125.32102
[25] G. E. Hay (1942), The finite displacement of thin rods, Trans. Amer. Math. Soc. 51, 65-102. · Zbl 0061.42206
[26] M. W. Hirsch & S. Smale (1972), Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press. · Zbl 0309.34001
[27] W. T. Koiter (1970), On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetesch. B 73, 169-195. · Zbl 0213.27002
[28] H. Le Dret (1985), Constitutive laws and existence questions in incompressible nonlinear elasticity, J. Elasticity 15, 369-387. · Zbl 0648.73013
[29] P. Le Tallec & J. T. Oden (1981), Existence and characterization of hydrostatic pressure in finite deformations of incompressible elastic bodies, J Elasticity 11, 341-357. · Zbl 0483.73035
[30] D. G. Luenberger (1969), Optimization by Vector Space Methods, Wiley. · Zbl 0176.12701
[31] L. A. Lyusternik (1934), On constrained extrema of functionals, Mat. Sb. 41, 390-401.
[32] R. S. Marlow (1989), On the linearized stress response of an internally constrained elastic material, Doctoral Dissertation, Univ. Illinois, Urbana.
[33] A. Mielke (1990), Normal hyperbolicity of center manifolds and Saint-Venant’s principle, Arch. Rational Mech. Anal. 110, 353-372. · Zbl 0706.73016
[34] D. Morgenstern & I. Szab? (1961), Vorlesungen ?ber theoretische Mechanik, Springer. · Zbl 0097.16404
[35] J. Moser (1965), On the volume element on a manifold, Trans. Amer. Math. Soc. 120, 286-294. · Zbl 0141.19407
[36] P. M. Naghdi (1972), The Theory of Shells, in Handbuch der Physik, Vol. VIa/2, C. Truesdell, ed., Springer-Verlag, 425-640.
[37] W. Noll (1966), The foundations of mechanics, in Non-Linear Continuum Theories (C.I.M.E. Conference), G. Grioli & C. Truesdell, eds., Cremonese, 159-200. · Zbl 0202.56101
[38] V. V. Novozhilov (1948), Foundations of the Nonlinear Theory of Elasticity (in Russian), Gostekhteorizdat, English translation, 1953, Graylock Press.
[39] P. Podio-Guidugli (1989), An exact derivation of the thin plate equation, J. Elasticity 22, 121-133. · Zbl 0692.73049
[40] P. Podio-Guidugli (1990), Constrained elasticity, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) 1, 341-350. · Zbl 0726.73030
[41] M. Renardy (1986), Some remarks on the Navier-Stokes equations with a pressuredependent viscosity, Comm. Partial Diff. Eqs. 11, 779-793. · Zbl 0597.35097
[42] G. de Rham (1973), Vari?t?s Diff?rentiables, 3rd edn., Hermann. · Zbl 0284.58001
[43] O. Richmond & W. A. Spitzig (1980), Pressure dependence and dilatancy of plastic flow, in Theoretical and Applied Mechanics, Proc. XV Intl. Cong., F. P. J. Rimrott & B. Tabarrok, eds., North Holland, 377-386.
[44] T. I. Seidman & P. Wolfe (1988), Equilibrium states of an elastic conducting rod in a magnetic field, Arch. Rational Mech. Anal. 102, 307-329. · Zbl 0668.73071
[45] F. Sidoroff (1978), Sur l’?quation tensorielle AX+XA=H, C. R. Acad. Sci. Paris A 286, 71-73. · Zbl 0372.73082
[46] R. Temam (1977), Navier-Stokes Equations, North-Holland. · Zbl 0383.35057
[47] T. C. T. Ting (1985), Determination of C 1/2, C ?1/2 and more general isotropic tensor functions of C, J. Elasticity 15, 319-323. · Zbl 0587.73002
[48] C. Truesdell (1977), A First Course in Rational Continuum Mechanics, Vol. 1, Academic Press.
[49] C. Truesdell & W. Noll (1965), The Non-linear Field Theories of Mechanics, in Handbuch der Physik III/3, Springer-Verlag. · Zbl 0137.19501
[50] C. Truesdell & R. A. Toupin (1960), The Classical Field Theories, in Handbuch der Physik III/1, Springer-Verlag.
[51] E. Volterra (1956), Equations of motion for curved and twisted elastic bars deduced by the ?method of internal constraints?, Ing. Arch. 24, 392-400. · Zbl 0072.19501
[52] E. Volterra (1961), Second approximation of the method of internal constraints and its applications, Int. J. Mech. Sci. 3, 47-67.
[53] J. Wissmann (1991), Doctoral dissertation, Univ. Maryland.
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