The present author (Nakaza), introduced a new elastic theory in 2005. According to this theory, the internal stress of an isotropic elastic material consists of an elastic stress governed by Hooke‘s law and an internal pressure governed by the state equation. In conventional theory, the internal stress consists of the two elastic stresses corresponding to the two elastic moduli and thus the force causes the Poisson effect cannot be explained, even if we observe the occurrence of the lateral strains under a uniaxial loading. According to Nakaza's elastic theory, Hooke‘s law has only one elastic modulus and in this sense, the new theory is consistent with the assertion of Navier who used only one elastic constant to derive his fundamental elastic theory. The new elastic theory may change the stress evaluation from the existing theory.
Even if we observe deformations of a material (i.e., a distribution of strain), we cannot immediately judge whether the distribution corresponds to the distribution of the internal stress. A strain distribution is converted into a stress distribution based on Hooke‘s law, which is characterized as having two elastic moduli. For example, when we observe the deformation of the material under longitudinal (vertical) uniaxial loading, not only longitudinal strain, but also lateral strains are generally observed in the material. In such a case, non-zero internal stress exists only in the longitudinal direction in the material, i.e., the stresses in the lateral directions are evaluated to be zero.
In this case, the distortions in the transverse directions are caused by the Poisson effect. By relying on the traditional elastic theory, the stress that causes these cannot be explained. Then, by further loading the material, the material shows cracks or splitting in the longitudinal direction, leading to failure. At this time, the fracture surface is in the longitudinal direction. Why did the material crack run vertically? The conventional theory cannot answer this simple question because two elastic moduli are used in the traditional theory