Application of fictitious discrete models with variable characteristic dimensions in calculations for the strength of composite bodies

To analyze the stress-strain state of homogeneous and composite bodies (CB), the method of multigrid finite elements (MMFE) is effectively used, which uses multigrid finite elements (MgFE). MMFE generates multigrid discrete models of small dimension, in which the inhomogeneous structure of bodies is taken into account within the framework of a micro-approach using MgFE. Basic discrete models (BM), taking into account the heterogeneous structure of bodies, have a high dimension. To reduce the dimensionality of discrete models of bodies, MMFE is used. However, there are BM CB (for example, BM bodies with a microhomogeneous structure), which have such a high dimension that the implementation of MMFE for such BM, due to limited computer resources, is difficult. In addition, for multigrid discrete models of high dimension, the MMFE generates numerically unstable solutions, which is associated with the error of computer calculations. To solve these problems, it is proposed here to use fictitious discrete models in calculations, the peculiarity of which is that their dimensions are smaller than the dimensions of BM CB.

In this paper, we propose a method of fictitious discrete models (MFDM) for calculating the static strength of elastic composite bodies with an inhomogeneous, micro-homogeneous molecular structure. MFDM is implemented using MMFE using adjusted strength conditions that take into account the error of approximate solutions. The MFDM is based on the position that the solutions that meet the BM CB differ little from the exact ones, i. e. we consider these solutions to be accurate.

The calculation of CB by MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FM), which have the following properties. FM reflect: the shape, characteristic dimensions, fastening, loading and type of inhomogeneous structure of the CB, and the distribution of elastic modulus corresponding to BM CB. The dimensions of FM are smaller than the dimensions of BM CB. The sequence consisting of FM converges to BM, i. e. the limiting FM coincides with BM. Calculations show that the convergence of such a sequence ensures uniform convergence of the maximum equivalent stresses of the FM to the maximum equivalent stress of the BM CB, which allows the use of such FM in the calculations of elastic bodies for strength.

Two types of FM are considered. The first type is scaled FM, the second type is FM with variable characteristic sizes. In this paper, the FM of the second type is considered in detail. Calculations show that the implementation of MMFE for FM with one, two or three variable characteristic sizes leads to a large saving of computer resources, which allows the use of MFDM for bodies with a micro-homogeneous regular structure. Calculations for the strength of CB according to MFDM require several times less computer memory than a similar calculation using BM CB, and does not contain a procedure for grinding BM. The given example of calculating the strength of a three-dimensional composite beam according to MFDM using FM with three variable characteristic dimensions shows its high efficiency.