Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

Abstract

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.

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

“…• In the framework of the general and standard tolerance models, not only the fundamental lower (ω gtm − ) uni (25), (ω stm − ) uni (29), but also the new additional higher (ω gtm + ) uni (26), (ω stm + ) uni (30) free vibration frequencies can be derived and analysed. The higher free vibration frequencies are caused by a periodic microstructure of the shell strip under consideration, and hence, they depend on a microstructure length parameter λ.…”

“…Hence, the asymptotic model (19) being more simple than the tolerance non-asymptotic ones is sufficient from the point of view of calculations made for the dynamic problem under consideration. From the computational results, it also follows that the differences between values of higher cell-dependent free vibration frequency (26) derived from the general tolerance model and values of higher cell-dependent free vibration frequency (30) obtained from the standard tolerance one are negligibly small. Thus, in order to determine and investigate these celldependent frequencies, the standard tolerance model ( 15)- (17), being more simple than the general tolerance one ( 12)-( 14), can be applied.…”

“…The results were compared with the corresponding results obtained from standard tolerance model ( 15)-( 17) and from asymptotic one (19), which neglects the length-scale effect. It was shown that in the framework of the general and standard tolerance models, not only the fundamental cell-independent lower (25), (29), but also the new additional cell-dependent higher (26), (30) free vibration frequencies can be determined and analysed. The higher free vibration frequencies, caused by a periodic structure of the shell strip, cannot be determined using the asymptotic model.…”

“…We can mention here monograph by Tomczyk [10] and papers by Tomczyk et al [11,12], where the length-scale effect in mechanics of periodic cylindrical shells is investigated; papers by Baron [13], where dynamic problems of medium thickness periodic plates are studied and by Marczak and Je ˛drysiak [14], Marczak [15,16], where dynamics of periodic sandwich plates is analysed; papers by Je ˛drysiak [17][18][19], which deal with stability of thin periodic plates; papers by Łaciński and Woźniak [20], Rychlewska et al [21], Ostrowski and Je ˛drysiak [22], Kubacka and Ostrowski [23], where problems of heat conduction in conductors with periodic structure are analysed. Let us also mention papers by Bagdasaryan et al [24], Tomczyk and Goła ˛bczak [25], Tomczyk et al [26], which deal with coupled thermoelasticity problems, respectively, for multicomponent, multi-layered periodic composites, for thin cylindrical shells with micro-periodic structure in circumferential direction (uniperiodic shells) and for thin cylindrical shells with micro-periodic structure in circumferential and axial directions (biperiodic shells). The extended list of references on this subject can be found in [6][7][8][9][10].…”

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

“…An analysis of the dynamics of periodic thin plates, with the size of the microstructure of an order of the plate thickness, was shown in [73]. Stability, dynamics, and thermoelasticity problems of micro-periodic shells were considered in papers [74][75][76][77][78][79]. Elastostatics of periodic thin plates with large deflections were investigated in [80].…”