Bending beams made of a material with an unchangeable volume | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 26 > №2 / 2020 / Pages: 200-211

Bending beams made of a material with an unchangeable volume


Materials that do not change their initial volume under the action of a force load are called incompressible. These are usually low-modulus rubber-like materials, the feature of which is an infinitely large volume module that characterizes the resistance of the medium to changes in the volume of the material. Therefore, of the two undependable physical characteristics (modules) for incompressible materials, only one module remains, which characterizes the resistance of the medium to shape change. There is no Poisson’s ratio equal to 0.5 in the defining relations of the problem. The product of an infinitely large modulus on the deformation of the volume change, equal to zero, is an uncertainty that is replaced by some force function, which is an additional unknown. The term “low-modulus material” does not contradict the property of infinitely large resistance to volume change, since there is no volume module in the defining relations of the mechanics of incompressible media. In all these relationships, the shear modulus appears, which is much smaller than similar modules of widely used construction materials. The additional ratio, which is the absence of volume change, calls into question some classical hypotheses, such as the Kirchhoff hypotheses in plate theory and the Bernoulli hypotheses in beam bending theory. The hypothesis of non-compressibility of fibers in the transverse direction is not of great importance for the construction of determining relations, and the other two hypotheses about the absence of linear deformation in the transverse direction and shear in the plane may lead to an unacceptable solution. Here and further is the longitudinal coordinate that coincides with the neutral line of the beam, and is the transverse coordinate to the neutral line. The origin of coordinates for symmetrical loads and boundary conditions at the ends is located in the middle of the neutral line, and if there is no symmetry at one of the ends of the beam. The problem of bending an incompressible beam is constructed in displacements, although for such a beam the term “in displacements” is conditional, since the physical relations of the incompressible material, known in the scientific literature as “neo-hook” relations, contain a force function that cannot be expressed in terms of deformation or displacement. To define an additional unknown, an incompressibility condition is added to the defining relations of the problem.