Stress distribution over plates vibrating at large amplitudes
Introduction
The plates are one of the major types of structural elements. During the operation they are often subjected to periodic loads, which lead to their vibrations. Linear theory does not always accurately describe such vibrations, so various nonlinear theories are used to describe them. Most commonly in such problems the theories taking into account the geometrical nonlinearity (the nonlinearity of strain–deflection relations) are applied [1]. Physical nonlinearity, i.e. nonlinearity of stress–strain relations, in the problems of vibrations of plates and shallow shells is used much rarely. Most of these studies are devoted to the thermomechanical behavior of structural elements [2], [3].
Chinh [4] studied oscillations of simply supported spherical and cylindrical shallow shells, taking into account softening-type physical nonlinearity. The linear approximation for the transverse normal strain is used. The bifurcation analysis of viscoelastic panel with physical nonlinearity is carried out by Han and Hu [5]. The bifurcation diagrams are presented and the stability of regimes, which exist due to the presence of internal resonances, is studied.
The following investigations that take into account both the geometrical and physical nonlinearities should be cited here. Pastushikhin [6] derived the equations of free vibrations of very thin shallow shells and studied equilibrium positions of these shells. The method for studying nonlinear dynamics of plates and shallow shells, both with and without taking into account temperature change, is presented by Awrejcewicz and Krys'ko [2], [7]. Models with different types of nonlinearities are analyzed, the comparison of obtained results is presented.
Most of research papers concerning the dynamical stress distribution are devoted to the study of plates with holes and cracks or composite, functionally graded and thick plates. As a rule, works from the first group focus on the stress state around a notch [8]. Works from the second group pay more attention to the stress distribution through the thickness. Thus in [9] the distribution through the thickness of stress components for thick rectangular plates and plates with laminate is presented. Data obtained by using fifth-order plate theory are in good correspondence with analytical solutions. Linear oscillations of thick rectangular plates with various boundary conditions are considered by Batra and Aimmanee [10]. The higher order shear, normal deformable plate theory and finite element method are used to calculate through the thickness stress distribution.
The distributions of twisting moments, bending moment and shear forces over the plate vibrating at small amplitudes are computed in [11]. Results obtained by using the Rayleigh–Ritz method with four different kinds of basic functions are compared.
Physically nonlinear vibrations are studied in [12], [13]. In [12] the results obtained with single-mode and four-mode deflection approximations are compared. It is mentioned that frequency responses, as well as the maximum dynamic stresses, are close for both approximations. Unfortunately, only the maximum dynamic stresses, but not the stress distribution, are presented in this paper.
The flexural stress distribution for rectangular plates with two simply supported and two clamped edges vibrating at large amplitudes is shown in [14]. The results for linear and geometrically nonlinear vibrations of plates with different aspect ratios are presented. This study is extended to plates with other boundary conditions in [15].
The effect of large vibration amplitudes on the nonlinear natural frequencies and mode shapes of clamped and simply supported circular plates is studied by Haterbouch and Benamar [16], [17], [18]. The solution of free geometrically nonlinear vibration problem is reduced to a solution of a set of nonlinear algebraic equations. The purely bending vibrations are considered in [16]. In [17], [18] the analysis is extended to multimode oscillations, which also include in-plane displacements. It is concluded that the stress distribution, even taking into account only the geometrical nonlinearity, differs significantly from that predicted by linear theory. It is found that the resonant nonlinear frequencies predicted by the single-mode approach are in good agreement with the iterative solution for a wide range of vibration amplitudes. Also quite accurate membrane stresses are obtained using single-mode approximation. It is pointed out that the only drawback of single-mode approximation is its inability to describe a change in the vibration mode shape for large amplitudes vibrations.
The present study focuses on the stress intensity distribution over thin plate vibrating at large amplitudes. Both the geometrical and physical nonlinearities are taken into account. It is assumed that the stresses do not exceed the yield point stress, i.e. vibrations remain elastic. The physical nonlinearity is taken into account because for many materials (some non-ferrous metals, polymers) stress–strain diagram deviate from Hooke's law, while loading and unloading stress–strain curves coincide [19]. This allows one to call such a deformation elastic. Physical nonlinearity is assumed to be small, so corresponding equation for the transverse normal strain is solved by using the small parameter method. The dynamic deflections are expanded into truncated series of eigenmodes. The comparison of results obtained taking into account geometrical nonlinearity, physical nonlinearity and both together is made. The stress intensity distribution patterns over the plate surface as well as through the thickness stress distribution are shown.
Section snippets
Mathematical model
Consider a thin isotropic plate vibrating at large amplitudes. The Kirchhoff–Love theory is used. The dynamics of the plate is described by the Lagrange equations of the second kind. In this paper, rotary inertia is neglected. As follows from the results of Abe et al. [20], its effect on geometrically nonlinear vibrations is insignificant. Also the neglect of rotary inertia is the widely used assumption for physically nonlinear problems [4], [5], [13].
The plate kinetic energy has the following
Problem formulation and linear vibrations analysis
For numerical investigation of dynamic stress distribution the simplest type of a plate – simply supported rectangular plate – is chosen. The plate's edges are assumed to be restricted from in-plane motions. The plate occupies the region Λ={(x,y)∈[0;a]×[0;b]}. The boundary conditions have formwhere is the plate boundary; n is the outer normal to the boundary; τ is the tangential vector to ∂Λ.
The geometrical parameters of the plate are the following: a=1 m, b=1.2
Conclusions
The method of analysis of stress intensity distribution over vibrating plates is presented. Both the geometrical and physical nonlinearities are taken into account. The following conclusions based on the study of vibrations of a supported rectangular plate are drawn:
- •
For a small physical nonlinearity the solution of equation for the transverse normal strain can be obtained in a convenient manner for analytical transformation form by using the small parameter method.
- •
For an accurate account of the
References (33)
- et al.
Non-linear vibrations of shell-type structures: a review with bibliography
Journal of Sound and Vibration
(2002) Dynamics of shallow shells taking into account physical non-linearities
International Journal of Mechanical Sciences
(2000)- et al.
Bifurcation analysis of a nonlinear viscoelastic panel
European Journal of Mechanics—A/Solids
(2001) - et al.
Nonlinear coupled problems in dynamics of shells
International Journal of Engineering Science
(2003) - et al.
Vibration analysis of rectangular plates with side cracks via the Ritz method
Journal of Sound and Vibration
(2009) - et al.
An exact analysis for vibration of simply supported homogeneous and laminate thick rectangular plates
Journal of Sound and Vibration
(1970) - et al.
Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories
Computers and Structures
(2005) - et al.
Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape
Computers and Structures
(2003) - et al.
Geometrically non-linear transverse vibrations of C–S–S–S and C–S–C–S rectangular plates
International Journal of Non-Linear Mechanics
(2006) - et al.
The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations
Journal of Sound and Vibration
(2003)
The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part II: iterative and explicit analytical solution for non-linear coupled transverse and in-plane vibrations
Journal of Sound and Vibration
Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates
Journal of Sound and Vibration
Non-linear vibration characteristics of clamped laminated shallow shells
Journal of Sound and Vibration
Normal modes for nonlinear vibratory systems
Journal of Sound and Vibrations
The construction of non-linear normal modes for systems with internal resonance
International Journal of Non-Linear Mechanics
The structural intensities of composite plates with a hole
Composite Structures
Cited by (10)
Modelling of bidirectional functionally graded plates with geometric nonlinearity: A comparative dynamic study using whole domain and finite element method
2024, Communications in Nonlinear Science and Numerical SimulationUnderstanding moisture effect on nonlinear vibrations of epoxy thin film via a multiscale simulation
2023, Journal of Sound and VibrationOn 3D and 1D mathematical modeling of physically nonlinear beams
2021, International Journal of Non-Linear MechanicsCitation Excerpt :Birger [3] proposed a modification of the latter method to solve physically nonlinear problems in the theory of plasticity and creep. Breslavsky [4], based on the method of small parameters, investigated the stress intensity distribution of vibrating plates taking into account geometric and physical nonlinearity. The regularities of the stress intensity distribution in the plate were analyzed.
Forced vibration response of axially functionally graded non-uniform plates considering geometric nonlinearity
2017, International Journal of Mechanical SciencesCitation Excerpt :Lewandowski [7,8] in a two part paper presented a detailed study on geometrically nonlinear behaviour of plates using Galerkin, Ritz and harmonic balance method. Breslavsky and co. [9,10] investigated both physical and geometric nonlinearities of plates using different models. Xiao et al. [11] studied nonlinear forced vibration of plate with four free edges on elastic foundations using Hamilton's principle in conjunction with Galerkin method and harmonic balance method.
Physically and geometrically non-linear vibrations of thin rectangular plates
2014, International Journal of Non-Linear MechanicsCitation Excerpt :A simple phenomenological model was developed by Kauderer, yet ignoring the framework of the widely accepted strain energy formulation [5]. This model is suitable for the description of non-linear elasticity of some metals, like copper for instance, and recent associated applications for dynamical problems of shell-type structures can be found in [6,7]. Nevertheless, models of hyperelasticity are preferred [5] for the description of rubber-like materials and soft biological tissues.
Experimental and Analytical Approach to Study the Effect of Large Vibration Amplitude of Rectangular Plates
2022, Journal of Vibration Engineering and Technologies