Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

Abstract

In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces.

Influence of Composite Structure on Temperature Distribution—An Analysis Using the Finite Difference Method

“…The technique, used to average the discontinuous coefficients in Equation (1), is constantly developed and used in the analysis of periodic [ 18 , 19 , 20 , 21 , 22 , 23 ], biperiodic [ 24 , 25 , 26 ] and functionally graded structures [ 27 , 28 ]. The most relevant assumption of the Tolerance Averaging Technique, from the point of view of the carried-out considerations, is the micro–macro decomposition assumption, according to Equation (2): where the total temperature field θ is divided into an averaged part ϑ and an oscillating part in the form of product g a ·ψ a .…”

“…Each cell is composed of two (periodicity) or four (biperiodicity) sub-cells, and the above-mentioned material properties within a given sub-cell are constant. The technique, used to average the discontinuous coefficients in Equation ( 1), is constantly developed and used in the analysis of periodic [18][19][20][21][22][23], biperiodic [24][25][26] and functionally graded structures [27,28]. The most relevant assumption of the Tolerance Averaging Technique, from the point of view of the carried-out considerations, is the micro-macro decomposition assumption, according to Equation (2):…”

A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

“…f 1 , f 2 , h 1 , and h 2 are depicted in Figure 2. By considering the micro-macro decomposition assumption, and the other assumptions and definitions of the tolerance modelling discussed in [23][24][25], Equation (1) describing the heat conduction issue was averaged, leading to the tolerance model equations:…”

“…By considering the micro-macro decomposition assumption, and the other assumptions and definitions of the tolerance modelling discussed in [ 23 , 24 , 25 ], Equation (1) describing the heat conduction issue was averaged, leading to the tolerance model equations: where K stands for the thermal conductivity tensor whose components are k ij , ∇ is a gradient operator defined as (∂ 1 , ∂ 2 , ∂ 3 ), overlined ∇ is a gradient in x 3 direction (0, 0, ∂ 3 ), and ∂ is a gradient operator defined as (∂ 1 , ∂ 2 , 0).…”

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core

“…[ 25 ]. Within the literature, one can find multiple applications of this technique in various mechanical issues, such as stability analysis [ 26 , 27 , 28 , 29 ], dynamics [ 30 , 31 , 32 , 33 ] or even heat conduction issues [ 34 , 35 , 36 , 37 ].…”