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.


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