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2015 Vol. 36, No. 4
Published: 2015-08-28
277
Nonlinear vibrations of a cantilever subject to a magnetic force of attraction at the free end
The forced vibration of a cantilevered magnetic system is investigated. One small magnet, fixed at the free end, is perpendicularly attracted by the other magnet. The magnetic force is modeled as a fractional function. The nontrivial static equilibrium configuration is derived from the distributed parameter model with the nonlinear boundary. A coordinate transform is introduced only based on the stable nontrivial equilibrium. The effects of system parameters on the natural frequency are shown. The steady-state response of the forced vibration is approximately determined under a small harmonic base excitation. The effect of the initial distance between the two magnets on the frequency-response curve is also presented. The finite difference method is employed to numerically validate these analyzed results. It is concluded that a change in magnitude or direction of the magnetic force has a great effect on the equilibrium, the natural frequency or the steady-state response.
2015 Vol. 36 (4): 277-282 [
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239
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283
THE CHARACTERISTICS OF OPTIMAL TWO-DIMENSIONAL SOLID/SOLID PHONONIC CRYSTALS
Phononic crystals (PnCs) is a kind of periodic composite with elastic band-gap characteristics, the distribution of elastic materials within a unit cell has significant effect on the band-gaps. In this paper, the topology optimization (TOP) techniques are utilized to obtain two-dimensional (2D) square lattice Cu/Epoxy PnCs with maximized relative band-gaps between multiple consecutive bands. The optimization follows two-stage design process which adopts genetic algorithms (GAs) and plane wave expansion method (PWE). The results show that the optimal PnCs with different band-gaps, which can easily be found by the developed method, have different material layout, and the PnCs with the lowest band-gap are simple lattice and have the highest value of application.
2015 Vol. 36 (4): 283-289 [
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300
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290
Two kinds of micro-structures of pentamode metamaterials and their performance analysis
Pentamode materials are new artificial metamaterials called as PM materials. Although PM materials belong to elastic materials, they possess only one load-carrying mode under static forces, and can transmit only one kind of elastic waves under dynamic excitation because of their special micro-structures. In the paper, two kinds of micro-structures of PM materials with different elastic properties are constructed, and one of them can transmit elastic expansion waves while the other transmit elastic shear waves. Then the effective elastic modului of the micro-structures are calculated and analyzed by employing the representative volume element method and the homogenization method, respectively. The results show: (1) The reasonable procedure for analyzing PM materials should be divided into two steps. The first step is to employ the truss model of micro-structures to calculate the effective elastic modului of PM materials and to judge whether the effective elastic modului satisfy the definition of PM materials; the second step is to adopt the solid model to analyze the relationship between the structural parameters of micro-structures and the effective elastic modului and to finish the material design. (2) The representative volume element method is suitable for analyzing the effective elastic modului of PM materials.
2015 Vol. 36 (4): 290-296 [
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263
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297
Coordinate transformation algorithm for pentamode metamaterial design based on non-linear finite element analysis
Pentamode metamaterials called as PM materials belong to a new kind of artificial materials, and their main characteristics are that the five eigenvalues of the effective modulus matrixes of PM materials are zero, which makes PM materials behave like fluid. As a results, PM materials show big potentialities in the design of acoustic cloaks. However, according to A.N.Norris theory[1], the design method for PM materials is different from others for common acoustic metamaterials, because an nonlinear partial differential equation must be satisfied while using the coordinate transformation method to design PM materials. The paper firstly derives the weak form of the partial differential equation, and then the nonlinear finite element formula for solving the partial differential equation are established by using the full Lagrangian methods for finite element analysis; thirdly, an iterative algorithm to solve the partial differential equation is constructed; finally two numerical examples of the coordinate transformation scheme are offered, and one is for 2D round domain, the other is for 3D sphere domain.
2015 Vol. 36 (4): 297-318 [
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249
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319
EFFECT OF INTERFACE YIELDING/DEBONDING ON CRACK PROPAGATION IN ANISOTROPY COMPOSITES
The influence of interface yielding/debonding on stress redistribution ahead of the crack tip is performed in laminated composites. By method of the superposition principle, a continuous distribution of sliding dislocation density is used to represent the interface yielding/debonding. This procedure reduces the problem to a singular integral equation which can be solved numerically by using of Chebyshev polynomials. The distribution of dislocations at the interface is numerically obtained. The role of the interface debonding which redistributes the stress ahead of the crack tip in the neighboring uncracked layer is also presented. It is concluded that the interface debonding or yielding weakens the stress singularity near the crack tip and has a shielding effect on the crack growth in anisotropic composites.
2015 Vol. 36 (4): 319-328 [
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353
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329
A 2-DIMENSIONAL HYBRID STRESS ELEMENT FOR STRESS ANALYSES OF ORIFICE PLATE
A novel 2-dimensional hybrid stress finite element is proposed for stress analyses of plate with holes. The new element, which is a 4-node quadrilateral plane element, is denoted as P-HS4-8β. Firstly, a stress in polar coordinates is obtained by physical equations and geometric equations. Secondly, the obtained stress is introduced into energy functional of the Hellinger-Reissner variational principle for plane stress problem in polar coordinates. In the new energy functional, the number of the stress variables is 2. According to this, the hybrid stress finite element formation is established. A matrix for stress interpolation is obtained by the traction-free condition along the edge of hole and the compatibility equation, and is introduced into the hybrid stress finite element formation. Numerical results show that, the new method gives a good accuracy of the stress near the holes.
2015 Vol. 36 (4): 329-336 [
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217
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337
Nonlinear dynamics of initial geometrical imperfection FGM circular cylindrical shells
A research on the nonlinear dynamics of initial geometrical imperfection clamped-clamped FGM circular cylindrical shell with different volume fractions is presented in this paper. Suppose the effective properties of FGM circular cylindrical shell are geometrical changed of gradient in thickness direction. Based on Classical Shear deformation theory and von-Karman type nonlinear strain-displacement relationship combined with Hamilton’s principle, the clamped-clamped FGM circular cylindrical shell nonlinear partial differential governing equations of motion are obtained. Considering of symmetric mode of clamped-clamped circular cylindrical shell in this paper,Galerkin’s method is utilized to discretize the governing partial equations，the differential form of nonlinear dynamics equation is obtained. Runge-Kutta method is applied for numerical simulation, and plotted its maximum lyapunov index. Numerical results are presented to show the influences of plane loads on the nonlinear dynamics,and the comparison of the influences of different volume fractions on nonlinear dynamics is given.
2015 Vol. 36 (4): 337-345 [
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434
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346
Buckling Analysis of Composite Laminates Using Hierarchical Finite Strip Mehtod
Hierarchical Finite Strip Method (HFSM) can be obtained by hybridizing hierarchical and finite strip method, the shape functions of which are polynomial series. Then the buckling of composite laminates under complex boundary and load conditions are analyzed by HFSM. The numerical results illustrate that the HFSM combines the merits of both hierarchical and finite strip, and can be used for analyzing the buckling problems of composite laminates quickly and precisely.
2015 Vol. 36 (4): 346-359 [
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275
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360
Buckling Analysis of Composite Sandwich Structures Based on Layerwise Plate Theory
Transverse shear deformation at each layer must be considered accurately when buckling loading is analyzed for composite sandwich structures. The reason is that there is much difference of mechanical properties between each layer. A four-node plate element based on Reddy’s Layerwise plate theory is developed in this paper for buckling analysis of composite sandwich structures. The first-order shear deformation theory is applied at each layer. Mixed interpolation of tensorial component is used to fix the problem of shear locking. Three numerical examples were analyzed and compared to the results in open literatures. The results show that the proposed element can discretely consider the mechanical properties of each layer; the accuracy of the proposed finite element is good; the results fit well with the results based on three dimensional elastic theory or high-order plate theory when the plate is modelled by several layers elements and the results fit well with the results based on first-order shear deformation plate theory when the plate is modelled by single layer elements
2015 Vol. 36 (4): 360-366 [
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370
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