High-order Displacement Functions Research Articles

Free Vibrational Behaviour of Multi-Directional Porous Functionally Graded Structures

The free vibrational frequencies of a multi-directional functionally graded (FG) structure are investigated for the first time in this paper considering the influences of variable grading (power-law, sigmoid, exponential) and porosity distribution (even and uneven type). Also, the structural properties (Young’s modulus, density and Poisson's ratio) varied along with two different directions simultaneously, i.e., the longitudinal and transversional ones, respectively. The present multi-directional grading model of the FG structure is reconstructed mathematically for the numerical analysis by considering adequate state-space deformation kinematics with the help of higher-order displacement functions and shear stress continuity. The general motion equation of a multi-graded structure is expressed using Hamilton’s principle and the finite element method including the necessary porosity effect. Initially, model consistency is verified and the eigenvalues obtained in the analysis are compared with the ones found in the literature. The comparison also includes the directional grading effect on their frequencies. Further, the influence of different parameters, i.e., power exponents (nz and nx), aspect ratio, thickness ratio, geometry, end support conditions, curvature ratio, porosity index, porosity distribution and material grading patterns, on the vibration characteristics of the multi-directional FG structure is computed. The analysis of the numerical results confirms that material grading; porosity distribution pattern and other design parameters have a significant influence on the frequency response characteristics of a single/multi-directional porous FG structure.

Nonlinear deformation and stress responses of a graded carbon nanotube sandwich plate structure under thermoelastic loading

The large deformation and stresses (normal and shear) of the graded nanotube-reinforced sandwich structure are numerically investigated under the influence of mechanical loading and a uniform temperature field. A higher-order nonlinear finite element model in conjunction with the direct iterative technique has been adopted for the solution purpose. Also, the structural distortion was modeled via the full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of higher-order displacement functions. Further, to replicate the actual operational conditions, the temperature-dependent properties of the individual material constituents (i.e., carbon nanotube and polymer) have been implemented in the current material modeling steps. The final deflections and stress values are evaluated via an own developed computer code using the currently proposed nonlinear mathematical formulation. The model accuracy and solution stability are checked by comparing the responses (deflection and stress) with available published results. Lastly, a variety of numerical examples is solved for different design parameters and deliberated in detail.

Taylor dispersion in two-dimensional bacterial turbulence

In this work, single particle dispersion was analyzed for a bacterial turbulence by retrieving the virtual Lagrangian trajectory via numerical integration of the Lagrangian equation. High-order displacement functions were calculated for cases with and without mean velocity effect. The two-regime power-law behavior for short and long time evolutions was identified experimentally, which was separated by the Lagrangian integral time. For the case with the mean velocity effect, the experimental Hurst numbers were determined to be 0.94 and 0.97 for short and long time evolutions, respectively. For the case without the mean velocity effect, the values were 0.88 and 0.58. Moreover, very weak intermittency correction was detected. All measured Hurst numbers were above 1/2, the value of the normal diffusion, which verifies the super-diffusion behavior of living fluid. This behavior increases the efficiency of bacteria to obtain food.

Role of High Order Functions in Analysis of the Creep Behavior of a Short Fiber Composite “Silicon Carbide/Aluminum 6061”

A new analytical approach is proposed to analyze the steady state creep in a short fiber composite SiC/Al6061 (Silicon Carbide/Aluminum 6061) under axial load based on high order displacement functions. The creep behavior of the matrix is described by an exponential law, while the fibers behave elastically. The main originality and purpose of the present paper is to predict the creep behavior in the short fiber composite “Silicon Carbide/Aluminum 6061 using high order functions analytically. The paper is presented in order to prevent undesirable events, as well as controlling the creep deformation rate. Good agreements are found among available experimental methods, FEM and present analytical approach results. As a result, most behaviors have ascending behaviors with smooth gradients, and also are controllable.

Time Domain Spectral Element method to Study Response of a Sandwich Beam with Compliant Core Subjected to Under Water Explosion

A spectral element formulation is developed for a sandwich beam with stiff face sheets and a compliant core. Higher order displacement functions are taken for the core in order to incorporate the core compression effects. Highly efficient spectral element method is used for the study. Accuracy and the computational efficiency of this method are compared with the conventional finite element method in solving transient blast loading case. Using this spectral element method, under water blast loading response of a sandwich beam with composite face sheets and core is studied. A comparison is made between the response of stiff core and compliant core sandwich beam models. The study presents a design approach for an optimal configuration for under water blast resistant structures.

A two-level method for static and dynamic analysis of multilayered composite beam and plate

A computationally efficient method is proposed to construct complicated boundary conditions based on the substructural boundary assumption, which can simplify the complex multilayered composite beam and plate to a reducible structure dimensionally. Moreover, the proposed method can construct various boundary conditions flexibly by the transformation matrix, especially the rational higher-order displacement functions, which can be easily extended into the dynamic analysis by the time integration algorithm. The proposed method shows the consistency with the extended multiscale finite element method (EMsFEM) of the linear boundary condition [40,41]. Finally, some static and dynamic numerical examples are discussed including multilayered beams and the composite lattice structure, which reveal the high accuracy and efficiency of the method.

General Solutions for Three-Dimensional Thermoelasticity of Two-Dimensional Hexagonal Quasicrystals and an Application

By introducing four displacement functions, the governing equations of the three-dimensional thermoelasticity of two-dimensional hexagonal quasicrystals are decoupled into two uncorrelated problems. Two higher-order displacement functions are introduced to represent the general solutions, which are eighth-order and fourth-order, respectively, for the two problems. By taking a decomposition and superposition procedure, the general solutions are further simplified in seven cases in terms of six quasi-harmonic displacement functions. To show the application of the general solutions obtained, a closed form solution is obtained for an infinite space containing a penny-shaped crack, subjected to a uniformly distributed temperature at the crack surface.

Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads

This paper presents an efficient Rayleigh–Ritz numerical model to predict the deformed shape and the multistable behaviour of free-standing square and rectangular 0/90 unsymmetric composite plates subjected to thermal and environmental loads. The out-of-plane displacement functions are generated through proper development of the analytical solution presented by Ashwell [34] for the nonlinear pure bending of isotropic plates; the in-plane displacement functions are formulated following a procedure similar to that employed by Galletly and Guest [32], starting from geometrical considerations about the actual deformations of the distorted structure. The resulting model is characterised by high-order displacement functions and by few unknown terms, it is appropriate to achieve sufficient accuracy, with high efficiency. The performance of the proposed approach is compared to that of some alternative models available in the literature, to FE simulations and to experimental results.

General solutions of three-dimensional problems for two-dimensional quasicrystals

Three-dimensional problems are systematically investigated for the coupled equations in two-dimensional hexagonal quasicrystals, and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) solutions and generalized Elliott–Lodge (E–L) solutions, are presented, respectively. By introducing two higher-order displacement functions, an operator analysis technique is applied in a novel way to obtain generalized LHN solutions. For further simplification, a decomposition and superposition procedure is taken to replace the higher-order displacement functions with five quasi-harmonic displacement functions, and then generalized E–L solutions are simplified in terms of these functions. In consideration of different cases of characteristic roots, generalized E–L solutions take different forms, but all are in simple forms that are conveniently applied. To illustrate the application of the general solutions obtained, the closed form solution is obtained for an infinite quasicrystal medium subjected to a point force at an arbitrary point.

A theory of general solutions of 3D problems in 1D hexagonal quasicrystals

A theory of general solutions of three-dimensional (3D) problems is developed for the coupled equilibrium equations in 1D hexagonal quasicrystals (QCs), and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) and Elliott–Lodge (E–L) solutions, respectively, are presented based on three theorems. As a special case, the generalized LHN solution is obtained from our previous general solution by introducing three high-order displacement functions. For further simplification, considering three cases in which three characteristic roots are distinct or possibly equal to each other, the generalized E–L solution shall take different forms, and be expressed in terms of four quasi-harmonic functions which are very simple and useful. It is proved that the general solution presented by Peng and Fan is consistent with one case of the generalized E–L solution, while does not include the other two cases. It is important to note that generalized LHN and E–L solutions are complete in z-convex domains, while incomplete in the usual non-z-convex domains.

Development of high-order manifold method

The Manifold Method of Material Analysis (MM) with high-order displacement functions has been derived based on triangular meshes for the requirement of high accurate calculations from practical applications. The matrices of equilibrium equations for the second-order MM have been given in detail for program coding. The derivation of the method is made by means of approximation theory and very few new mathematical concepts are used in this paper. So, it may be understood by most engineering researchers. By close comparison with widely used Finite Element Method, the advantages of MM can be seen very clearly in the following aspects: (1) the capability of processing large deformation and handing discontinuities like block oriented Discontinuous Deformation Analysis method; (2) making element meshes easily and (3) using high-order displacement functions easily. The C program codes for the second-order MM has been developed, and it has been applied to the example of a beam bending under a central point loading. The calculated results are quite good in agreement with theoretical solutions. By contrast, the results calculated for the same model by use of the original first-order MM are far from the theoretical solutions. Therefore, it is important and necessary to develop high-order Manifold Method for the complicated deformation problems. © 1998 John Wiley & Sons, Ltd.

Thermal buckling analysis of antisymmetric laminated cylindrical shell panels

The thermal buckling analysis of antisymmetric angle-ply laminated cylindrical shells that are simply supported and subjected to a uniform temperature rise is analyzed by a finite element method based on the higher order displacement functions. Comparisons to the first order displacement theory are made. Both theories allow transverse shear deformation, but only the higher order one takes into account the transverse normal strain. The numerical results show that first order theory overestimates the thermal buckling temperature of the shell panel, which suggests that the higher order displacement fields should be used in the analysis of thermal buckling for a laminated shell. Effects of important parameters are also studied.

Accurate rigid-body modes representation for a nonlinear curved thin-shell element

For certain highly curved shells, such as bellows, the formulation of a curved-shell finite element with curvilinear displacement components may fail to properly model some rigid body modes, even with either the explicit inclusion of rigid-body terms or the use of high-order displacement functions. It is suggested in this paper that the rigid-body modes can be properly included if the Cartesian displacement components are used. A 48-degree-of-freedom (DOF) curved thin-shell element is formulated, and both the curvilinear and the Cartesian forms are used for this investigation. Examples of the nonlinear analyses of a bellows shell and a spherical cap are given to demonstrate the advantage of using the Cartesian formulation. Curved elements may also suffer from membrane locking, which is caused by the inability of an element to bend without stretching. Numerical data are presented to show that the present element does not membrane lock. Some nonlinear examples are also presented to further demonstrate the versatility of this element.

A new 48 d.o.f. quadrilateral shell element with variable‐order polynomial and rational B‐spline geometries with rigid body modes

AbstractA 48 degrees‐of‐freedom (d.o.f.) quadrilateral thin elastic shell finite element using variable‐order polynomial functions, B‐spline functions and rational B‐spline functions to model the shell surface is developed. This development may allow the stiffness formulation of the shell element to be linked to the geometry data bases created by computer aided design systems. The displacement functions are that of bicubic Hermitian polynomials. The displacement functions and d.o.f. are expressed and investigated in both the curvilinear and Cartesian forms. The cuivilinear form is simpler and can provide the proper solution for a certain class of shell problems. For certain highly curved shells such as bellows, however, the curvilinear form fails to properly model some rigid body modes even with either the explicit inclusion of rigid body terms or the high order displacement functions. It is suggested in this study that such difficulty can be circumvented and the rigid body modes can be properly included if a Cartesian form is used for displacement functions. The strain–displacement equations are expressed in curvilinear co‐ordinates. Thus, the Cartesian displacement functions require a transformation to curvilinear displacement at each numerical integration point. Examples include a pinched cylinder, a translational shell under central load, a uniformly loaded hypar shell, a pressurized ovel shell, a semi‐toroidal bellows and a U‐shaped bellows. For the first four examples, geometric modellings consist of polynomials of second‐order (subparametric), third‐order (isoparametric), and fourth and fifth‐order (both superparametric) as well as B‐spline functions of fourth‐ and fifth‐order. The geometries of the pinched cylinder, the semi‐toroidal bellows, and the U‐shaped bellows were modelled exactly using rational B‐spline functions. All the results obtained are in good agreement with alternative existing solutions.

Use of higher order displacement functions in the free vibration analysis of shells of revolution having meridional singularities

A finite element technique in which higher order displacement functions are used in the analysis of vibration of shells of revolution with joints and poles is described. The joints typically arise where a meridional curve is made up from a number of different curves, as in the case of a cone-toroid cooling tower or a cone-cylinder chimney. A technique for overcoming the difficulties associated with the representation of the stress discontinuities at the joints when higher order displacement functions are used is discussed. A number of typical shells with joints are analysed by using this technique, and the results are found to compare very well with existing theoretical and experimental data. A pole of a shell of revolution is a point where the meridian intersects the axis of revolution, such as the apex of a spherical dome or the apex of a conical shell. It is shown that, in analysing such shells, it is not necessary to exclude the pole by introducing a small hole around it or to develop special elements. A cone-cylinder combination with an apex is analysed by using a combination of the techniques for analysing jointed shells and shells with poles. The calculated frequencies and mode shapes are shown to agree well with experimental results. It is demonstrated that the cone-cylinder combination with the apex can be idealized accurately by using a curved shell element with four nodal points. The “non-classical modes” of vibration reported in the experiment could not be predicted theoretically, even though the curved shell element was used and the value of the circumferential wave number n was not prescribed.

Finite element analysis of translational shells

A finite element procedure is presented for solving thin shell problems of arbitrary geometry and boundary conditions. The actual smoothly curved shell surface is approximated by the assemblage of flat triangular plate elements. A quadratic displacement function for the tangential displacements in the triangular element is assumed and the membrane stiffness is derived accordingly. A complete fourth order displacement function is used to derive the flexural stiffness of the triangular element. These higher order displacement functions, resulting in 27 generalized displacements, are used for the purpose of obtaining a more accurate solution to the stress conditions especially in the regions near supporting edge members for various doubly curved shells. The paper includes shells which are supported by edge beams or is stiffened by ribs. A beam type member is idealized as an assemblage of segments of straight beam elements. The twisting and axial stiffnesses, the eccentricity and the bending stiffness of the beam element are all considered.