On dynamics and stability of thin periodic cylindrical shells

Proc Appl Math and Mech

2008

Self Cite

Free vibrations of thin linear-elastic Kirchhoff-Love cylindrical shells, having a periodic structure along one direction tangent to the shell midsurface, is considered. In order to take into account the effect of the periodicity cell size in this problem, a new averaged non-asymptotic model of such shells, proposed by Tomczyk (2006), is applied. The new additional higher-order free vibration frequencies dependent on the microstructure size will be derived and discussed. PreliminariesThe new non-asymptotic mathematical model for problems of dynamics and dynamical stability of thin linear-elastic KirchhoffLove-type cylindrical shells with periodically variable thickness as well as periodically variable inertial and elastic material properties along one direction tangent to the shell midsurface has been proposed in [2]. In order to derive this model the tolerance averaging technique, presented in [3], was applied. Shells under consideration are called uniperiodic.The aim of this contribution is to apply the model equations derived in [2] to investigate the effect of a cell size on free vibrations of a circular periodically stiffened shell as shown in Fig.1. The length-scale effect is neglected in commonly used homogenized models of such shells, derived by means of asymptotic methods, [1].Definestand for a Cartesian orthogonal coordinate system in the physical space E 3 . A cylindrical shell midsurface M will be given by its parametric representation M ≡ {x ∈ E 3 : x = r(ξ), (ξ) ∈ Ω}, where r(·) is the smooth function such that ∂ 1 r · ∂ 2 r = 0, ∂ 1 r · ∂ 1 r = ∂ 2 r · ∂ 2 r = 1. It means that on M we have introduced the orthonormal parametrization and hence L 1 , L 2 are length dimensions of M . Let the shell, considered here, be closed and circular. It means that now L 1 = 2πρ, where ρ is the midsurface curvature radius. Let δ stand for the constant shell thickness. The shell is reinforced by two families of ribs, which are periodically and densely distributed in circumferential direction, cf. Fig. 1. The stiffeners of both kinds are assumed to have constant cross-sections with A 1 , A 2 as their areas and with I 1 , I 2 as their moments of inertia. Let a 1 , a 2 be the widths of the ribs. It is assumed that both the shell and stiffeners are made of homogeneous isotropic materials and let us denote by E, ν Young's modulus and Poisson's ratio of the shell material, respectively, and by E 1 , E 2 Young's moduli of the rib materials. Let µ 0 and µ 1 , µ 2 stand for the shell mass density and for the mass densities of the stiffeners, respectively, cf. Fig. 1. On 0ξ1 ξ 2 -plane we define λ as the period of the stiffened shell structure in ξ 1 -direction, cf. Fig. 1. The period λ satisfies the conditions: λ/δ max 1, λ/ρ min 1 and λ/L 1 1. We also assume that L 2 > L 1 and hence λ/L 2 1. The basic cell ≡ [−λ/2, λ/2] × {0} has a symmetry axis for η 1 = 0, where η 1 ∈ [−λ/2, λ/2]. It is assumed that the shell is simply supported on the edges ξ 2 = 0, ξ 2 = L 2 . The tensile and bending rigidities of the stiffeners are ...

Section: Preliminariesmentioning

confidence: 99%

Section: On 0ξmentioning

confidence: 99%

“…The system of governing equations of the tolerance model given in [2] is separated now into two independent systems…”

Section: On 0ξmentioning

confidence: 99%

“…The second expression represents the approximate form of ω * obtained by neglecting some terms, which under assumption λ/L 2 1 are small in comparison with unity.…”

Section: On 0ξmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Vibrations of thin cylindrical shells with a periodic structure

Proc Appl Math and Mech

2008

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Length-scale effect in stability problems for thin biperiodic cylindrical shells: extended tolerance modelling

Continuum Mech. Thermodyn.

Gołąbczak

Litawska

et al. 2020

Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells of periodically micro-inhomogeneous structure in circumferential and axial directions (biperiodic shells) are investigated. The aim of this contribution is to formulate and discuss a new averaged nonasymptotic model for the analysis of selected stability problems for these shells. This, so-called, general nonasymptotic tolerance model is derived by applying a certain extended version of the known tolerance modelling procedure. Contrary to the starting exact shell equations with highly oscillating, noncontinuous and periodic coefficients, governing equations of the tolerance model have constant coefficients depending also on a cell size. Hence, the model makes it possible to investigate the effect of a microstructure size on the global shell stability (the length-scale effect).

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

Continuum Mech. Thermodyn.

Gołąbczak

Litawska

et al. 2021

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.