Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

Construction of piezoelectric and flexoelectric models of composites by asymptotic homogenization and application to laminates

The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment.

Dynamic asymptotic model of rolling bearings with a pitting fault based on fractional damping

PurposeThe purpose of this paper is to discuss the asymptotic models of different parts with a pitting fault in rolling bearings.Design/methodology/approachFor rolling bearings with a pitting fault, the displacement deviation between raceways and rolling elements is usually considered to vary instantaneously. However, the deviation should change gradually. Based on this shortcoming, the variation rule and calculation method of the displacement deviation are explored. Asymptotic models of different parts with a pitting fault are discussed, respectively. Besides, rolling bearing systems have prominent fractional characteristics unconsidered in the traditional models. Therefore, fractional calculus is introduced into the modeling of rolling bearings. New dynamic asymptotic models of different parts with a pitting fault are proposed based on fractional damping. The numerical simulation is performed based on the proposed model, and the dynamic characteristics are analyzed through the bifurcation diagrams, trajectory diagrams and frequency spectrograms.FindingsCompared with the model based on integral calculus, the proposed model can better reflect the periodic characteristics and fault characteristics of rolling bearings. Finally, the proposed model is verified by the experiment. The dynamic characteristics of rolling bearings at different rotating speeds are analyzed. The experimental results are consistent with the simulation results. Therefore, the proposed model is effective.Originality/value(1) The above models are idealized, i.e. the local pitting fault is treated as a rectangle. When a component comes into contact with the fault, the displacement deviation between the component and the fault component immediately releases if the component enters the fault area and restores if the component leaves. However, the displacement deviation should change gradually. Only when the component touches the fault bottom, the displacement deviation reaches the maximum. (2) Due to the material's memory and fluid viscoelasticity, rolling bearing systems exhibit significant fractional characteristics. However, the above models are all proposed based on integral calculus. Integral calculus has some local characteristics and is not suitable for describing historical dependent processes. Fractional calculus can better describe the essential characteristics of the system.

Asymptotic Shallow Models Arising in Magnetohydrodynamics

In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green–Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in Castro and Lannes (2014) to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model.

Support of Dynamic Measurements Through Similitude Formulations

Up to now, similitude methods have been used in order to overcome the typical drawbacks of experimental testing and numerical simulations by reconstructing the full-scale model behavior from that of the scaled model. The novelty of this work is the application of similitude theory not as a tool for predicting the prototype dynamic response, but for supporting, and eventually validating, experimental measurements polluted by noise. Two Aluminium Foam Sandwich (AFS) plates are analyzed with Digital Image Correlation (DIC) cameras. First, an algorithm for blind source separation problems is used to extract information about the excitation; then, SAMSARA (Similitude and Asymptotic Models for Structural-Acoustic Research Applications) similitude method is applied to both the force spectra and velocity responses of prototype and model. The reconstruction of force and velocity curves demonstrates that the similitude results are coherent with the quality of the experimental measurements: when the spatial pattern in resonance is recognizable, then the curves overlap. Instead, when the displacement field of just one model is not well identified, the reconstruction exhibits discrepancies. Therefore, similitude methods reveal to be an interesting tool for understanding if a set of measurements is reliable or not and their application should not be underestimated, especially in the light of the expanding range of approaches which can extract important information from noisy observations.

Micro-Vibrations and Wave Propagation in Biperiodic Cylindrical Shells

The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to study a certain long wave propagation problem related to micro-fluctuations of displacement field caused by a periodic structure of the shells. This micro-dynamic problem will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling applied here includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Both these procedures are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. It will be shown that the micro-periodic heterogeneity of the shells leads to exponential micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.

On some model equations for pulsatile flow in viscoelastic vessels

© 2019 Considered here is the derivation of partial differential equations arising in pulsatile flow in pipes with viscoelastic walls. The equations are asymptotic models describing the propagation of long-crested pulses in pipes with cylindrical symmetry. Additional effects due to viscous stresses in bio-fluids are also taken into account. The effects of viscoelasticity of the vessels on the propagation of solitary and periodic waves in a vessel of constant radius are being explored numerically.

On some model equations for pulsatile flow in viscoelastic vessels

© 2019 Considered here is the derivation of partial differential equations arising in pulsatile flow in pipes with viscoelastic walls. The equations are asymptotic models describing the propagation of long-crested pulses in pipes with cylindrical symmetry. Additional effects due to viscous stresses in bio-fluids are also taken into account. The effects of viscoelasticity of the vessels on the propagation of solitary and periodic waves in a vessel of constant radius are being explored numerically.

Asymptotic Models for Tropical Intraseasonal Oscillations and Geostrophic Balance

In the tropics, rainfall is coupled with waves in the form of, for example, convectively coupled equatorial waves (CCEWs) and the Madden–Julian oscillation (MJO). In perhaps the simplest viewpoint of CCEWs, the effects of moisture and convective adjustment can predict the basic aspects of their propagation and structure: reduced propagation speeds and reduced meridional length scales. Here, a similar simple viewpoint is investigated for the MJO’s propagation and structure. To do this investigation, budget analyses of a model MJO are first presented to illustrate and motivate the asymptotic scaling assumptions. Asymptotic models are then derived for the MJO. In brief, the structure of the asymptotic MJO is described by a tropical geostrophic balance, and the slow propagation arises from the dynamics of moist static energy. To be specific, if the moist static energy has a background vertical gradient that is asymptotically weak (i.e., a moist stability that is nearly neutral), then it supports a slowly propagating wave. Beyond these main aspects, other processes also have an influence, such as eddy diffusion of moisture. In comparing the simple viewpoints of CCEWs and the MJO, one main difference is in the propagation speeds: relative to a dry wave speed of 50 m s−1, the MJO has a speed of 5 m s−1, resulting from a reduction factor of 0.1 related to moist stability, whereas the basic CCEW speed is 15 m s−1, resulting from a reduction factor of the square root of 0.1, related to the square root of the moist stability.