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

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 x(1)-axis and non-periodic structure on the microlevel. However, along the other two directions, i.e., x(2) and x(3), 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.