NOTES ON THE GENERAL THEORIES OF STRAIN AND ELASTICITY

In this paper, strain problems in a general deformation are treated with methods and observations different from the usual theory of general strain, and elasticity in the consideration of general strain is investigated. In the first part of the paper, the obscurity in the usual definitions for strains is made clear, and it is shown that the usual measure of shear strain is not strictly proper. Then, it is shown that almost all the strain problems, in finite or infinitesimal strain, can be simply treated and expressed by the use of vector quantities, which represent, in amount and direction, a material straight line, the normal between a pair of parallel material planes and the relative displacement of a particle. Among the results thus obtained, these results are most noticeable, that the strain of a material plane and its tangential component, i.e. shear strain, bocome the maximum in a general case at pairs of material planes, the normals of which are perpendicular to the axis of the middle principal strain and have inclinations to the axis of the greatest principal strain, in the unstrained state, shown by cos^<-1>±[{(1+e_1)^2-(1+e_1)^<2/3>(1+e_3)^<2/3>}/{(1+e_1)^2-(1+e_3)^2}]^<1/2> for the maximum strain of a material plane, and by cos^<-1>±((1+e_1)/(2+e_1+e_3))^<1/2> for the maximum shear strain, e_1 and e_2 being the greatest and the least principal strains, while the value of the maximum shear strain is the same as stated in the usual theory. In addition to the analytical treatment, strain problems are illustrated graphically by means of strain ellipsoid (or by reciprocal strain ellipsoid) in a general case and by a plane diagram in a case of small strain, the diagram being drawn after, but with a correction to, Mohr's method. Lastly, it is stated from mathematical reasoning that Hooke's law can not strictly be true, that is to say, stress and strain can not strictly be proportional, and that the law has to be modified to be such that the principal strains are in linear relations to the forces on the principal planes, which have unit areas in the unstrained state.