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.
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.
S. S. Antman (1982), Material constraints in continuum mechanics, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 70, 256–264.
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.
S. S. Antman & J. E. Osborn (1979), The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69, 231–262.
S. S. Antman & W. H. Warner (1966), Dynamical theory of hyperelastic rods, Arch. Rational Mech. Anal. 23, 35–352.
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.
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.
H. Cohen (1981), Pseudo-rigid bodies, Util. Math. 20, 221–247.
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.
J. L. Ericksen (1955), Deformations possible in every compressible, perfectly elastic material, Z. angew. Math. Phys. 34, 126–128.
J. L. Ericksen (1986), Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures 22, 951–964.
J. L. Ericksen & R. S. Rivlin (1954), Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3, 281–301.
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.
A. E. Green, N. Laws & P. M. Naghdi (1968), Rods, plates and shells, Proc. Camb. Phil. Soc. 64, 895–913.
P. Hartman (1964), Ordinary Differential Equations, Wiley, New York.
G. E. Hay (1942), The finite displacement of thin rods, Trans. Amer. Math. Soc. 51, 65–102.
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.
H. Le Dret (1985), Constitutive laws and existence questions in incompressible nonlinear elasticity, J. Elasticity 15, 369–387.
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.
D. G. Luenberger (1969), Optimization by Vector Space Methods, Wiley.
L. A. Lyusternik (1934), On constrained extrema of functionals, Mat. Sb. 41, 390–401.
R. S. Marlow (1989), On the linearized stress response of an internally constrained elastic material, Doctoral Dissertation, Univ. Illinois, Urbana.
A. Mielke (1990), Normal hyperbolicity of center manifolds and Saint-Venant's principle, Arch. Rational Mech. Anal. 110, 353–372.
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.
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.
P. Podio-Guidugli (1990), Constrained elasticity, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) 1, 341–350.
M. Renardy (1986), Some remarks on the Navier-Stokes equations with a pressuredependent viscosity, Comm. Partial Diff. Eqs. 11, 779–793.
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.
F. Sidoroff (1978), Sur l'équation tensorielle AX+XA=H, C. R. Acad. Sci. Paris A 286, 71–73.
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.
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.
E. Volterra (1961), Second approximation of the method of internal constraints and its applications, Int. J. Mech. Sci. 3, 47–67.
J. Wissmann (1991), Doctoral dissertation, Univ. Maryland.
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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
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DOI: https://doi.org/10.1007/BF00375123