Tractions, balances, and boundary conditions for
nonsimple materials
Eliot Fried
Department of Mechanical and Aerospace Engineering
Washington University in St. Louis
St. Louis, MO 63130-4899, USA
Abstract
In two monumental works, Toupin (1962, 1964) derived general balance equations and associated
traction boundary conditions for an elastic body whose strain energy depends on first and second gradients
of the deformation. Toupin’s derivation is based on a virtual work principle asserting that the variation
of the total elastic energy be equal to the virtual work exerted on the body by tractions and body forces.
A central consequence of Toupin’s work is the observation that Cauchy’s hypothesis — that the surface
traction at a point x on a surface S depend on S through its normal field at x — is not valid in a theory
involving second gradients of the deformation, because in Toupin’s theory the traction depends also on
the curvature of S at x.
Unfortunately, because it assumes that the material is elastic and the body is in equilibrium, Toupin’s
derivation of the balance equations and associated traction boundary conditions cannot be applied within
a general dynamical framework that includes dissipation.
Our goal here is to derive Toupin’s results within a framework that is independent of constitutive
equations. To do so we use a nonstandard form of the principle of virtual power (Gurtin 2001). Conven-
tional versions of this principle are formulated for the body B as a whole rather than for control volumes
and as such generally involve particular boundary conditions applied to the boundary ∂B of B. Such
formulations allow for a weak statement of the basic force balances and when combined with constitutive
equations result in weak statements of the resulting boundary-value problems. Here the principle of
virtual power is used instead as a basic tool in determining the structure of the tractions and of the local
force balances. As such, conditions on ∂B play a role no different from those on the boundary of any
control volume. Basic to this view is the premise, central to all of continuum mechanics, that any basic
law for the body should hold also for all subregions of the body.
References
• Gurtin, M. E., 2001. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary
dislocations, J. Mech. Phys. Solids 84, 809–819.
• Toupin, R. A., 1962. Elastic materials with couple stresses, Arch. Ration. Mech. Anal. 11, 385–414.
• Toupin, R. A., 1964. Theory of elasticity with couple stress, Arch. Ration. Mech. Anal. 17, 85–112.
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