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As it is well known now, the asymmetric theory of elasticity was first conceived by
Voigt (1887) and later expanded by Cosserat brothers (1909). In this theory, a
majority of the work is concerned with the formulation and derivation of
fundamental relationships (Truesdell and Toupin, 1960; Grioli, 1960; Aero and
Kuvshinskii, 1961; Grioli, 1962; Toupin, 1962; and Mindlin and Tiersten, 1962).
A concise form of field equations derived by Cosserat (1909) was developed by
Truesdell and Toupin (1960). Associated constitutive equations for finite
deformation of elastic solids were derived by Grioli (1960 and 1962) and Toupin
(1962). Grioli establishes six boundary conditions to be satisfied at the surface
of the body, whereas the actual number of meaningful boundary conditions that
can be specified is five. The authors Truesdell and Toupin (1960) admit the same
error in their description of the earlier work by Voigt and Cosserat brothers.
Mindlin and Tiersten (1962) linearized Toupin’s constitutive equations and
applied them to a number of problems, such as the effects of couple stress on
vibration, wave motion and stress concentration. As far as the general theory is
concerned, the work of Mindlin and Tiersten (1962) is no doubt elegant and
concise, but it suffers from the use of a consistent dyadic notation which is most
uncommon to many dedicated readers. Aero and Kuvshinskii (1961)
independently studied and derived the stress equilibrium equations and
constitutive equations for the general anisotropic case appropriate to the corresponding linearized theory in 1960 and its Russian and English versions were available in 1961. Bulygin and Kuvshinskii (1967) considered the twodimensional theory for plain strain case. |
