Defining equations of deformation of materials with double anisotropy

A. A. Treschev ; Yu. A. Zavyalova ; M. A. Lapshina ; A. E. Gvozdev ; O. V. Kuzovleva ; E. S. Krupitsyn

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 370-384

Résumé

The mechanical properties of composite and polymer materials widely used in engineering are analyzed. It is confirmed that the absolute majority of them have structural anisotropy of different classes. In addition, it is shown that these structural materials often exhibit a sensitivity of the deformation characteristics to the type of stress state. Due to the fact that classical mathematical models describing the states of such materials lead to gross errors in the calculation of structural elements, and the well-known, specially developed theories for them are quite contradictory and have significant drawbacks, the authors propose an energy model of the determining relations for media with structural and deformation anisotropies. This model is based on the use of the normalized stress tensor space, which has an undoubted advantage over the singular functions and parameters having an infinite interval of change, which are used in the known versions of the theories of deformation of materials with double anisotropy. As a specific class of structural anisotropy, orthotropic materials are accepted, for which the strain potential defined in the main structural axes is postulated. By differentiating the formulated potential according to the recommendations of the Castigliano rules, the equations of connection of two tensors of the second rank - strains and stresses - are established. It is shown that these equations have a nonlinear form, which aggravates the problem of uniqueness of solutions to boundary value problems. To identify the resulting model of the defining equations, we recommend an experimental program that includes mechanical tests for uniaxial tension and compression along the main axes of the anisotropy of the material, as well as for a net shift in the three planes of orthotropy. The main technical constants of a number of composite and polymer materials widely used in engineering are given. On the basis of the use of the postulate about the positive certainty of the energy surface, the consistency of the proposed strain potential is verified. Using this test, we prove the uniqueness theorem for solving boundary value problems in the mechanics of a deformable solid. Taking into account the rules of transformation of the components of the second-rank tensors when the axes of the selected coordinate system are rotated, it is shown that the stresses calculated in the main axes of orthotropy are recalculated in the new system according to traditional formulas.

Détail

Keywords: deformation anisotropy, structural orthotropy, strain potential, second-rank tensors, uniqueness theorem, Drucker postulate, principal axes of orthotropy.