Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions |
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Authors: | D.V. Georgievskii |
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Affiliation: | 1.Department of Mechanics and Mathematics,Moscow State University,Moscow,Russia;2.Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences,Moscow,Russia |
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Abstract: | The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension. |
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