Articles techniques et présentations

An approach to the development of the higher order theories of beams, rods, plates and shells have been proposed in our previous publications. It is based on series expansion of the corresponding field equations in term of the cross-section coordinates. The proposed approach is general and can be applied to the wide range of problems in sciences and engineering. For example, in our publications higher order theories of beams, rods, plates and shells for the cases elasticity, thermoelasticity, electro-magneto-elasticity, as well as micropolar, couple stress and nonlocal field theories have been considered.

The approach is based on a variational principle of virtual power (or more general) and the expansion of the 3-D field equations into generalized series in terms of cross-section coordinates. For example, in the case of micropolar elasticity the stress and strain tensors, as well as vectors of displacements and rotation, have been expanded into series in terms of cross-section coordinates. Thereby, all equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the series of cross-section coordinates. Then, a system of differential equations in terms of displacements and rotation with corresponding boundary conditions for the coefficients of the series of cross-section coordinates have been obtained. The obtained system of different equation is infinite dimensional. Taking into account finite number (one, two, etc.) of members in the series expansion one can obtain first, second, …, N order micropolar theory of the reduced dimension. In the same way, reduced dimension differential equations for any field theory can be developed. Obtained equations can be solved analytically, using appropriate analytical tools or numerically, using for example Finite Element Method (FEM). In the last case, it is not necessary to find a corresponding system of differential equations. Instead, based on the variational formulation, one can obtain corresponding matrixes of the FEM and solve obtained system of the finite dimensional algebraic equations.

We propose to apply above approach to the general system of different equations presented in the COMSOL Multiphysics^® simulation software, and obtain reduced dimension differential equations of the general form. As a result we will get equations of the higher order theories of beams, rods, plated and shells for wide range of multiphysics and multiscale problems in the COMSOL Multiphysics^® environment. It is important to mention that any classical theory is included in the proposed theory as special case. Obtained in such way equations can be easy solved by advanced users of the Mathematics Module of COMSOL Multiphysics^®. In order to simplify application of the developed higher order theories we propose to create corresponding modules in which users will just input order of the theory, physical and geometrical characteristics of structures and get solution as COMSOL Multiphysics^®

output.

Finally, we have to mention that proposed approach can be extended to wide range of nonlinear problems.