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On the Theory of Finite Deformations of Elastic Crystals

R. Furth

1942
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Proceedings of the Royal Society A
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Imperfections in the structure of cohalt 2 85 and the lines with l odd. It seems therefore that the assumption of faults in the cobalt structure, distributed at random along the hexagonal axis, explains satisfactorily both the line-broadening and the difference in breadth between the lines with l even and with l odd. The general theory of finite deformation of cubic crystals at zero tem perature is developed to a second-order approximation, and the cases of (1) a uniform hydrostatic pressure,
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... ) a tension in the direction of one of the axes, (3) a shear along the (0, 1, 0) planes, and (4) a shear along the (0, 1, 1) planes of the lattice, are worked out in detail. A number of 'secondorder effects' (deviations from Hooke's law) are predicted which in case (1) have been observed and measured by Bridgman, and in the remaining cases certainly can be detected and measured by suitable experimental arrangements. Assuming the particular force law between the particles of the lattice which was first introduced by Mie and Gruneisen, and later used in the investigations of Lennard-Jones and of B om and his collaborators, and using some of the results of the latter authors, the constants governing the above-mentioned second-order effects are expressed in terms of the con stants governing the force law, and calculated numerically for a number of special values of these constants. Thus by comparing the calculated values of these constants with the results of measurements at low tempera ture the unknown force law could probably be determined. Vol. 180. A. on July 20, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from 286 R. Furth 1, The classical theory of elasticity is based upon two assumptions: that the potential energy of elastic deformation is a quadratic form in the strain components, and that the deformations are infinitely small. An immediate consequence of these assumptions is the validity of Hooke's law, namely, the linear connexion between strain and stress. Actually the deformations are always finite, and therefore also the first assumption, which holds foi* any reasonable form of potential energy function in the case of infinitely small deformations, is no longer valid and has to be replaced by a more general assumption about this function. Although within the range of purely elastic behaviour of real solids these deformations are in fact gener ally very small, there are certain cases, for instance, compression of solids under very high uniform pressure, where the deformations are large enough to produce marked deviations from Hooke's law, and also in other cases the accuracy of measurement could undoubtedly be increased so as to detect deviations from Hooke's law within the range of elastic deformations, even if they are comparatively small. The problem of developing a theory of finite elastic deformations of solids is therefore of some physical interest. The theory of finite deformations of isotropic elastic solids has been developed independently by Mumaghan (1937) and Brillouin (1938). These theories give the general relation between strain and stress for any supposed form of the potential energy function consistent w ith the assump tion of an isotropic medium. If, for example, this function is developed with respect to the strain components up to the third order, instead of up to the second order as in the classical theory, the elastic properties of the medium are determined by five instead of two elastic constants, and the stress com ponents are not linear but quadratic functions of the strain components. B y comparison with experimental measurements it should be possible to test whether these relations are satisfied, and to determine the values of the elastic constants. Although this is perhaps interesting from the technical point of view, it is not interesting for the physicist unless the knowledge thus gained can be used to reveal new facts about the molecular structure of solids and the laws governing the forces between their particles. This, however, can only be achieved by developing a theory of finite elastic deformations of crystals. Indeed, for a crystal with given lattice structure and given force law between the particles, the potential energy is a known function of the deformation and the constants appearing in the force law, and therefore the elastic constants of the first and of the second order can be expressed in terms of those constants. Thus by measuring the elastic constants the unknown constants of the force law could be determined. In the following investigation the theory for cubic crystals at zero tem perature is developed, for which a certain amount of work has already been done in a paper by Bom & Misra (1940) (further quoted as B.M.). The second-order effects for three particular cases, namely, compression under on

doi:10.1098/rspa.1942.0041
fatcat:ct5fmwuupbdynayljhbb5tfiw4