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A Saint-Venant principle for a theory of nonlinear plane elasticity. (English) Zbl 0790.73014

A formulation of Saint-Venant’s principle within the context of a restricted theory of nonlinear plane elasticity is described. The theory assumes small displacement gradients while the stress-strain relations are nonlinear. The elastic material occupies a rectangular region. The lateral sides are traction-free while the far end is subjected to a uniformly distributed tensile traction. The near end is subjected to a prescribed normal and shear traction.
If this end loading were also that of uniform tension, then one possible corresponding stress state throughout the rectangle is that of uniform tension. When the near end loading is not uniform, the resulting stress field is expected to approach a uniform tensile state with increasing distance from the near end. This result is established in this article by using the differential inequality techniques for quadratic functionals. It is shown that, under certain constitutive assumptions, an energy-like quadratic functional, defined on the difference between the deformation field and the uniform tensile state, decays exponentially with the distance from the near end. The estimated decay rate (which is a lower bound on the actual rate of of decay) is characterized in terms of the loading level, the domain geometry, and material properties. The results predict a progressively slower decay of end effects with increasing load level. The mathematical issues concerned involve the spatial decay of solution of a fourth-order non-linear elliptic partial differential equation. The paper serves as an incentive to further research in this direction.

MSC:

74G50 Saint-Venant’s principle
74B20 Nonlinear elasticity
74A20 Theory of constitutive functions in solid mechanics
35J60 Nonlinear elliptic equations
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