Natural vibrations of prestressed thin functionally graded plates with dense system of ribs in two directions

Dynamic problems of elastic non-periodically laminated solids are considered in this paper. It is assumed that these laminates have a functionally graded structure on the macrolevel along the x1-axis and non-periodic structure on the microlevel. However, along the other two directions, i.e., x2 and x3, their properties are constant. The effects of the size of a microstructure (the microstructure effect) on the behaviour of the composites can play a significant role. This effect can be described using the tolerance modelling method. This method allows us to derive model equations with slowly varying coefficients. Some of these terms can depend on the size of the microstructure. These governing equations of the tolerance model make it possible to determine formulas describing not only fundamental lower-order vibrations related to the macrostructure of these composite solids, but also higher-order vibrations related to the microstructure. Here, the application of the tolerance modelling procedure is shown to lead to equations of the tolerance model that can be used for non-periodically laminated solids. Then, these model equations are mainly used to analyse a simple example of vibrations for functionally graded composites with non-periodically laminated microstructure (FGL). Similar problems were investigated in the framework of the homogenised (macrostructural) model (Jędrysiak et al. 2006); the resulting equations neglect the microstructure effect.

Section: Introductionmentioning

confidence: 99%

Dynamics of Functionally Graded Laminated (FGL) Media—Theoretical Tolerance Modelling

Jędrysiak

2023

“…For instance, dynamics for longitudinally graded plates was investigated in [40,41], but stability of similar plates in [42]. Modelling of thin-walled structures with dense system of ribs was shown in [43,44]. Heat conduction of composite cylindrical conductors with non-uniform distribution of constituents was analysed in [45,46].…”

Section: Introductionmentioning

confidence: 99%

The Effect of the Material Periodic Structure on Free Vibrations of Thin Plates with Different Boundary Conditions

Jędrysiak

2022

Thin elastic periodic plates are considered in this paper. Since the plates have a microstructure, the effect of its size on behaviour of the plates can play a crucial role. To take into account this effect, the tolerance modelling method is applied. This method allows us to obtain model equations with constant coefficients, which involve terms dependent of the microstructure size. Using the model equations, not only can formulas of fundamental lower-order vibration frequencies be obtained, but also formulas of higher-order vibration frequencies related to the microstructure. In this paper, the effect of the material periodic microstructure on free vibration frequencies for various boundary conditions of the plates was analysed. To obtain proper formulas of frequencies, the Ritz method is applied. Moreover, some results are compared to the results calculated using the FEM.

“…The tolerance modelling approach can be also successfully applied to consider various problems of thermomechanics for functionally graded structures. For instance: investigations of dynamics for longitudinally graded plates were presented in References [ 75 , 76 , 77 ]; vibrations of thin functionally graded plates with the size of the microstructure of an order of the thickness of the plate were analysed in Reference [ 78 ]; vibrations of thin transversally graded plates with the thickness smaller than the microstructure size were shown in Reference [ 2 , 79 , 80 ]; dynamics of a thin-walled structure with a system of ribs was presented in Reference [ 81 , 82 ]; heat distribution of composite cylindrical conductors having non-uniform microstructure was analysed in Reference [ 83 , 84 ]; problems of thermoelasticity in transversally graded laminates were considered in Reference [ 85 ]; dynamics of thin functionally graded microstructured shells was presented in Reference [ 86 , 87 , 88 ].…”

Section: Introductionmentioning

confidence: 99%

“…It seems that most of them show models based on the tolerance modelling method, cf. References [ 2 , 60 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 86 , 87 , 88 ], and between them References [ 76 , 88 ] present stability problems of functionally graded plates or shells.…”

Section: Introductionmentioning

confidence: 99%

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

Jędrysiak

Kaźmierczak-Sobińska

2020

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.