Skip to main content
Log in

Material constraints, lagrange multipliers, and compatibility. Applications to rod and shell theories

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • S. S. Antman (1972), The Theory of Rods, in Handbuch der Physik, Vol. VIa/2, C. Truesdell, ed., Springer-Verlag, 641–703.

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • S. S. Antman (1982), Material constraints in continuum mechanics, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 70, 256–264.

    Google Scholar 

  • 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.

    Google Scholar 

  • S. S. Antman & J. E. Osborn (1979), The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69, 231–262.

    Google Scholar 

  • S. S. Antman & W. H. Warner (1966), Dynamical theory of hyperelastic rods, Arch. Rational Mech. Anal. 23, 35–352.

    Google Scholar 

  • J. M. Ball (1981), Remarques sur l'existence et la régularité des solutions d'élastostatique nonlinéaire, in Recent Contributions to Nonlinear Partial Differential Equations, H. Berstycki & H. Brezis, eds., Pitman, 50–62.

  • M. F. Beatty & M. A. Hayes (1992), Deformations of an elastic, internally constrained material. Part 1: Homogeneous deformations, J. Elasticity, to appear.

  • J. F. Bell (1985), Contemporary perspectives in finite strain plasticity, Int. J. Plasticity 1, 3–27.

    Google Scholar 

  • F. Brezzi & M. Fortin (1991), Mixed and Hybrid Finite Element Methods, (to appear).

  • P. G. Ciarlet (1990), Plates and Junctions in Elastic Multi-Structures, Masson, Springer-Verlag.

  • P. G. Ciarlet & J. Nečas (1987), Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97, 171–188.

    Google Scholar 

  • H. Cohen (1981), Pseudo-rigid bodies, Util. Math. 20, 221–247.

    Google Scholar 

  • H. Cohen & R. Muncaster (1988), The Theory of Pseudo-Rigid Bodies, Springer-Verlag.

  • F. Davi (1991), The theory of Kirchhoff rods as an exact consequence of three-dimensional elasticity, J. Elasticity, to appear.

  • D. G. Ebin & R. A. Saxton (1986), The initial-value problem for elastodynamics of incompressible bodies, Arch. Rational Mech. Anal. 94, 15–38.

    Google Scholar 

  • J. L. Ericksen (1955), Deformations possible in every compressible, perfectly elastic material, Z. angew. Math. Phys. 34, 126–128.

    Google Scholar 

  • J. L. Ericksen (1986), Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures 22, 951–964.

    Google Scholar 

  • J. L. Ericksen & R. S. Rivlin (1954), Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3, 281–301.

    Google Scholar 

  • A. F. Filippov (1985), Differential Equations with Discontinuous Right-Hand Sides (in Russian), Nauka, English transl., 1988, Kluwer.

  • A. E. Green, N. Laws & P. M. Naghdi (1967), A linear theory of straight elastic rods, Arch. Rational Mech. Anal. 25, 285–298.

    Google Scholar 

  • A. E. Green, N. Laws & P. M. Naghdi (1968), Rods, plates and shells, Proc. Camb. Phil. Soc. 64, 895–913.

    Google Scholar 

  • P. Hartman (1964), Ordinary Differential Equations, Wiley, New York.

    Google Scholar 

  • G. E. Hay (1942), The finite displacement of thin rods, Trans. Amer. Math. Soc. 51, 65–102.

    Google Scholar 

  • M. W. Hirsch & S. Smale (1972), Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press.

  • W. T. Koiter (1970), On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetesch. B 73, 169–195.

    Google Scholar 

  • H. Le Dret (1985), Constitutive laws and existence questions in incompressible nonlinear elasticity, J. Elasticity 15, 369–387.

    Google Scholar 

  • 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.

    Google Scholar 

  • D. G. Luenberger (1969), Optimization by Vector Space Methods, Wiley.

  • L. A. Lyusternik (1934), On constrained extrema of functionals, Mat. Sb. 41, 390–401.

    Google Scholar 

  • R. S. Marlow (1989), On the linearized stress response of an internally constrained elastic material, Doctoral Dissertation, Univ. Illinois, Urbana.

    Google Scholar 

  • A. Mielke (1990), Normal hyperbolicity of center manifolds and Saint-Venant's principle, Arch. Rational Mech. Anal. 110, 353–372.

    Google Scholar 

  • D. Morgenstern & I. Szabó (1961), Vorlesungen über theoretische Mechanik, Springer.

  • J. Moser (1965), On the volume element on a manifold, Trans. Amer. Math. Soc. 120, 286–294.

    Google Scholar 

  • P. M. Naghdi (1972), The Theory of Shells, in Handbuch der Physik, Vol. VIa/2, C. Truesdell, ed., Springer-Verlag, 425–640.

  • 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.

  • V. V. Novozhilov (1948), Foundations of the Nonlinear Theory of Elasticity (in Russian), Gostekhteorizdat, English translation, 1953, Graylock Press.

  • P. Podio-Guidugli (1989), An exact derivation of the thin plate equation, J. Elasticity 22, 121–133.

    Google Scholar 

  • P. Podio-Guidugli (1990), Constrained elasticity, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) 1, 341–350.

    Google Scholar 

  • M. Renardy (1986), Some remarks on the Navier-Stokes equations with a pressuredependent viscosity, Comm. Partial Diff. Eqs. 11, 779–793.

    Google Scholar 

  • G. de Rham (1973), Variétés Différentiables, 3rd edn., Hermann.

  • 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.

  • T. I. Seidman & P. Wolfe (1988), Equilibrium states of an elastic conducting rod in a magnetic field, Arch. Rational Mech. Anal. 102, 307–329.

    Google Scholar 

  • F. Sidoroff (1978), Sur l'équation tensorielle AX+XA=H, C. R. Acad. Sci. Paris A 286, 71–73.

    Google Scholar 

  • R. Temam (1977), Navier-Stokes Equations, North-Holland.

  • 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.

    Google Scholar 

  • C. Truesdell (1977), A First Course in Rational Continuum Mechanics, Vol. 1, Academic Press.

  • C. Truesdell & W. Noll (1965), The Non-linear Field Theories of Mechanics, in Handbuch der Physik III/3, Springer-Verlag.

  • C. Truesdell & R. A. Toupin (1960), The Classical Field Theories, in Handbuch der Physik III/1, Springer-Verlag.

  • E. Volterra (1956), Equations of motion for curved and twisted elastic bars deduced by the “method of internal constraints”, Ing. Arch. 24, 392–400.

    Google Scholar 

  • E. Volterra (1961), Second approximation of the method of internal constraints and its applications, Int. J. Mech. Sci. 3, 47–67.

    Google Scholar 

  • J. Wissmann (1991), Doctoral dissertation, Univ. Maryland.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Antman, S.S., Marlow, R.S. Material constraints, lagrange multipliers, and compatibility. Applications to rod and shell theories. Arch. Rational Mech. Anal. 116, 257–299 (1991). https://doi.org/10.1007/BF00375123

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00375123

Keywords

Navigation