Transverse compression of a thin elastic disc Alzaidi, Ahmed S. M.; Kaplunov, Julius; Nikonov, Anatolij ...
Zeitschrift für angewandte Mathematik und Physik,
06/2024, Volume:
75, Issue:
3
Journal Article
Peer reviewed
Open access
The mathematical formulations for transverse compression of a thin elastic disc are considered, including various boundary conditions along the faces of the disc. The mixed boundary conditions ...corresponding to the loading by normal stresses in absence of sliding are studied in detail. These conditions support an explicit solution in a Fourier series for the boundary layers localised near the edge of the disc and also do not assume making use of the Saint-Venant principle underlying the traditional asymptotic theory for thin elastic structures. As an example, an axisymmetric problem is studied. Along with the leading order solution for a plane boundary layer, a two-term outer expansion is derived. The latter is expressed through the derivatives of the prescribed stresses. Generalisations of the developed approach are addressed.
Plane-strain motion of a flexural seismic metasurface in the form of a regular array of thin Kirchhoff plates attached to the surface of an elastic half-space is analysed. Two types of contact ...conditions, including simply supported plates and plates moving along horizontal rails are studied. Dispersion of time harmonic waves is investigated both asymptotically and numerically. A major effect of the contact conditions on metasurface behaviour is discovered. In particular, it is shown that frequency band gaps are not the feature of the array composed of simply supported plates. It is also demonstrated that the scaling laws, expressed through geometric and material problem parameters, drastically differ from each other for two considered setups.
For polymeric materials, the relaxation spectrum contains complete information about the time-dependent part of the material response. The contribution of different numerical schemes, i.e., different ...methods of reconstructing the dynamic relaxation modulus, to the accuracy of the approximation by calculating the corresponding relaxation spectra is investigated by analysing experimental data for four types of polysaccharides. It was found that there is no unique mathematical approach for the calculation of relaxation spectra that ensures a satisfactory approximation of the experimentally determined dynamic moduli for the selected types of polymeric materials. It is recommended to combine different numerical methods in parallel to achieve a reasonable approximation of the material functions.
The limitations of the dynamic theories for the thin layered elastic structures, which very often have different physical and geometrical contrast properties of the layers in today's high-tech ...applications, bring a number of challenges in the numerical computation of the dynamic response. This is a strong motivation to develop a suitable computational methodology for accurate evaluation and implementation of the numerical results, aiming at accurate interpretation of the vibration spectra and the associated displacement and stress fields.
In this paper, a newly developed numerical engineering approach is presented for the study of elastic wave dispersion in composite plates (sandwich plates) with high-contrast properties of the layers using modal finite element method (FEM) analysis implemented in commercial software. The obtained results are compared with the iterative numerical solution of the Rayleigh-Lamb dispersion equation for the fundamental flexural wave and the first shear harmonic.
It is shown that the complexity of the dispersion phenomena, including the cut-off frequencies of higher order vibrational modes, has been captured very accurately and that the developed computational methodology provides a valuable insight into the frequency range in which the respective mode can be activated. This perspective shows a great potential of the approach to be employed in many engineering applications involving multi-layered structures with arbitrary number of layers.
•Lamb wave dispersion in high-contrast three-layered composite plates is studied.•Numerical engineering approach to analyze wave dispersion phenomena is developed.•Frequency ranges in which the respective mode can be activated are determined.•The dispersion curves and the cut-off frequencies are captured very accurately.•The approach enables to separate fundamental flexural mode and first shear mode.
A 3D problem in linear elasticity is considered for a thin inhomogeneous layer subject to transverse compression. For the first time, the effect of arbitrary vertical inhomogeneity is elucidated. Two ...sets of boundary conditions along the faces of the layer are adapted for modelling transverse compression. Robust asymptotic formulae involving repeated integrals across the thickness are derived for displacements and stresses. As an illustration, numerical results are presented for the elastic moduli having a transverse parabolic variation. The obtained results have a potential to be implemented in modern technology, including manufacturing and design of functionally graded materials.
We study theoretically the biaxial bending of symmetric, flat layered ceramic composites (laminates) due to external loading. We focus on three-layered alumina/zirconia laminates. We compare the ...principal stresses in the samples in the case of static and harmonic dynamic loading. The dynamic equation within the Kirchhoff theory for thin homogeneous plates is first generalized to the case of multilayered plates. It is solved numerically with the relaxation method, which we have developed for this purpose.
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