Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

    When dealing with mechanical constraints, it is usual in continuum mechanics to enforce ideas that stem from the seminal work of Bernoulli and D'Alembert and require that internal constraints do no work. The usual procedure is to split the stress into a constraint response and a constitutively determined response that does not depend upon the variables that appear in the constraint response (i.e. the Lagrange multiplier), and further requiring that the constraint response does no work. While this is adequate for hyperelastic materials, it is too restrictive in the sense that it does not permit a large class of useful models such as incompressible fluids whose viscosity depends upon the pressure—a model that is widely used in elastohydrodynamics.

    The purpose of this short paper is to develop a purely mechanical theory of continua with an internal constraint that does not appeal to the requirement of worklessness. We exploit a geometrical idea of normality of the constraint response to a surface (defined by the equation of constraint) in a six-dimensional Euclidean space to obtain (i) a unique decomposition of the stress into a determinate and a constraint part such that their inner product is zero, (ii) a completely general constraint response—even constraint equations that are nonlinear in the symmetric part of the velocity gradient D as well as when the coefficients in the determinate part depend upon the constraint response and (iii) a second order partial differential equation for the determination of the constraint response. The geometric approach presented here is in keeping with the ideas of Gauss concerning constraints in classical particle mechanics.

    References

    • Andrade E.C. 1930 Viscosity of liquids. Nature. 125, 309–310. CrossrefGoogle Scholar
    • Antman S.S. 1982 Material constraints in continuum mechanics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Math. Nat. 70, 256–264. Google Scholar
    • Antman S.S& Marlow R.S. 1991 Material constraints, Lagrange multipliers and compatibility, application to rod and shell theories. Arch. Ration. Mech. Anal. 116, 257–299. Crossref, Web of ScienceGoogle Scholar
    • Bridgman P.W The physics of high pressure. 1931 New York:The Macmillan Company. Google Scholar
    • Carlson D.E, Fried E& Tortorelli D.A. 2003 Geometrically-based consequences of internal constraints. J. Elasticity. 70, 101–109. Crossref, Web of ScienceGoogle Scholar
    • Dettmann C.P& Morriss G.P. 1996 Hamiltonian formulation of the gaussian isokinetic themostat. Phys. Rev. E. 54, 2495–2500. Crossref, Web of ScienceGoogle Scholar
    • Evans D.J& Morriss G.P Statistical mechanics of nonequilibrium liquids. 1990 London:Academic Press. Google Scholar
    • Gauss C.F. 1830 On a new general principle of mechanics. Phil. Mag. 8, 137–140.[Translation of 1829 Ueber ein neues allegemeines Grundgesetz der Mechanik. J. Reine Angew. Math. 4, 232–235.]. Google Scholar
    • Goldstein H Classical mechanics. 2nd edn. 1980 Boston:Addison-Wesley Publishing Company. Google Scholar
    • Hron J, Málek J& Rajagopal K.R. 2001 Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. A. 457, 1603–1622.doi:10.1098/rspa.2000.0723. . LinkGoogle Scholar
    • Lagrange J.L Mechanique analytique. 1787 Paris:Mme Ve Courcier. Google Scholar
    • Málek J, Necas J& Rajagopal K.R Global analysis of flows of fluids with pressure dependent viscosities. Arch. Ration. Mech. Anal. 165, 2002a 243–269. Crossref, Web of ScienceGoogle Scholar
    • Málek J, Necas J& Rajagopal K.R Global existence of solutions for flows of fluids with pressure and shear dependent viscosities. Appl. Math. Lett. 15, 2002b 961–996. Crossref, Web of ScienceGoogle Scholar
    • Murali Krishnan J& Rajagopal K.R. 2003 Review of the uses and modeling of bitumen from ancient to modern times. Appl. Mech. Rev. 56, 149–214. CrossrefGoogle Scholar
    • O'Reilly O.M& Srinivasa A.R. 2001 On the nature of constraint forces in dynamics. Proc. R. Soc. A. 457, 1307–1313.doi:10.1098/rspa.2000.0717. . Google Scholar
    • Rajagopal K.R. 2003 On implicit constitutive equations. Appl. Math. 48, 279–319. CrossrefGoogle Scholar
    • Stokes G.G. 1845 On the theories of internal friction of fluids in motion and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc. 8, 287–305. Google Scholar
    • Szeri A.Z Fluid film lubrication: theory and design. 1998 Cambridge:Cambridge University Press. Google Scholar
    • Truesdell C A first course in rational continuum mechanics. 1977 New York:Academic Press. Google Scholar
    • Truesdell C& Noll W The nonlinear field theories of mechanics. 2nd edn. 1992 Berlin:Springer. Google Scholar