Bernoulli hypotheses in the problem of bending a mechanically incompressible beam

The incompressibility condition for an isotropic linearly elastic material seriously restricts the application of the classical hypotheses of the beam bending theory formulated by Bernoulli for small deformations and displacements. At the same time, it is assumed that such a strong kinematic condition as the condition of immutability of volume must be unconditionally fulfilled. The term “mechanical incompressibility” implies the impact on the beam exclusively of a force load, but with thermal action on it, the deformation of the volume change is a function of temperature. Nevertheless, in both of these cases, the condition of mechanical incompressibility may conflict with the classical hypotheses of beam bending, which may lead to the problem degeneration. Therefore, before solving any problem for mechanically incompressible materials, it is necessary to check all hypotheses used and sufficiently justified for conventional materials for compliance with the kinematic condition of volume immutability. In case of inconsistency, it is necessary to build a calculation model based on other hypotheses that do not contradict incompressibility, which will not lead to a serious complication of the tasks being solved. For a bent beam, the Bernoulli model is used, the basis of which is the kinematic hypothesis of a straight normal (the transverse segment after deformation remains straight, orthogonal to the curved axis of the beam and the distances between the points of the segment remain unchanged) and the force hypothesis of the non-compressibility of the beam fibers in the transverse direction. Each of the above hypotheses should be checked for compliance with the condition of immutability of the beam volume when exposed to the surface force bending load. The consideration of transverse deformations is relevant for low-modulus materials and especially for materials with a low shear modulus in the transverse direction. Incompressible materials, as a rule, belong to low-modulus, but this is not their property that is decisive in the analysis of Bernoulli hypotheses.