On Natural Boundary Conditions in Linear 2nd-Grade Elasticity
This work aims at drawing the attention of mechanicians interested in the development of extended continuum theories on the unresolved issue of the physical interpretation of the additional boundary conditions introduced by 2nd-grade models. We discuss this issue in the context of the linearized theory of elasticity as an appropriate platform for discussion. Apart from lineal densities of edge-forces, 2nd-grade models allow for the prescription of force-like quantities energy-conjugated to the gradient of the velocity field on the boundary. Previous works proposed reductionistic interpretations, treating 2nd-grade models as particular cases of continua with affine microstructure; from the latter one can deduce field equations reminiscent of 2nd-grade models, either in the “low-frequency, medium wavelength” limit or by constraining the microstructural degrees of freedom to the gradient of the velocity field. The interpretation we propose here is based on the concept of ortho-fiber, and has the merit of achieving a simple physical interpretation of the boundary conditions without recourse to extensive algebra or the need to invoke microstructure.
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