Independent Displacement Fields Research Articles

Thermostatics of micropolar elastic thin beams with a circular axis with independent fields of displacements and rotations and the finite element method

In the present paper, on the basis of the previously developed system of hypotheses, the applied (one-dimensional) theory of thermostatic bending of micropolar (with independent fields of displacements and rotations) elastic thin beams with a circular axis is constructed, energy theorems are proved and the corresponding variational principles are established. To solve specific boundary-value problems of the applied theory of thermostatic bending of micropolar thin beams with a circular axis, ways of analytical solutions are considered as well as a variant of the finite element method is developed. On the basis of numerical results and parametric analysis of the problems, it is stated that the micropolar properties of the material, in case of other equal conditions, increase the rigidity of the beams in comparison with the classical case.

3D viscoelastic computational grains with spherical inclusions with or without interphases/coatings for micromechanical modeling of heterogeneous materials

AbstractIn this article, the 3D viscoelastic computational grains (CGs) with spherical inclusions, with and without interphases/coatings are developed to study the viscoelastic behavior of polymer‐based heterogeneous materials. Papkovich–Neuber solutions and spherical harmonics are adopted to develop the independent displacement fields inside the grains, and Wachspress coordinate is used to interpolate compatible displacement fields at inter‐element surfaces. The elemental stiffness matrices are developed with multi‐field boundary variational principles in the Laplace domain, after which Zakian technique is adopted to invert both the homogenized and localized responses back to the time domain with good accuracy and efficiency. With different kinds of models to describe the property of the viscoelastic polymers, the generated homogenized moduli and localized stress distributions are validated against the experimental data, simulations by commercial FE software, and predictions by composite spherical assemblage models. Parametric studies are also carried out to investigate the influence of material and geometric parameters on the behavior of viscoelastic composites. Finally, the viscoelastic CG is also used to study the effect of the negative Young's modulus of particles on the stability and loss tangent of viscoelastic composites.

Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.

N-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

• A model of n -sided polygonal hybrid FE based- topology optimization is presented. • Only line integrals along the element boundary are involved in the computation. • Easy to construct any polygons for adapting complex design domains. • The present model n -sided polygonal hybrid FE is with high accuracy. In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n -sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions . To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n -sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.

Finite Element Analysis of Orthotropic Thin Plates Using Analytical Solution

Despite the high importance of orthotropic bending plate, up to now, only a few studies have been performed based on the analytical solution for analysis of these structures. In this paper, an efficient element is proposed by utilizing the hybrid-Trefftz method to analyze structures. The proposed quadrilateral element (QHT-OR) has four nodes and twelve degrees of freedom. Moreover, two independent displacement fields are defined for inner and boundaries of element. The internal field is achieved by solving the governing equation of orthotropic plates. Furthermore, the boundary field is related to the nodal degrees of freedom by using interpolation functions. It should be mentioned that the interpolation functions of two-node beam elements are applied for boundary field. The responses obtained by using numerical tests show the high accuracy of the QHT-OR element even for coarse meshes.

Uniqueness of the interior plane strain time-harmonic viscoelastic inverse problem

Elasticity imaging has emerged as a promising medical imaging technique with applications in the detection, diagnosis and treatment monitoring of several types of disease. In elasticity imaging measured displacement fields are used to generate images of elastic parameters of tissue by solving an inverse problem. When the tissue excitation, and the resulting tissue motion is time-harmonic, elasticity imaging can be extended to image the viscoelastic properties of the tissue. This leads to an inverse problem for the complex-valued shear modulus at a given frequency. In this manuscript we have considered the uniqueness of this inverse problem for an incompressible, isotropic linear viscoelastic solid in a state of plane strain. For a single measured displacement field we conclude that the solution is infinite dimensional, and the data required to render it unique is determined by the measured strain field. In contrast, for two independent displacement fields such that the principal directions of the resulting strain fields are different, the space of possible solutions is eight dimensional, and given additional data, like the value of the shear modulus at four locations, or over a calibration region, we may determine the shear modulus everywhere. We have also considered simple analytical examples that verify these results and offer additional insights. The results derived in this paper may be used as guidelines by the practitioners of elasticity imaging in designing more robust and accurate imaging protocols.

Hybrid-Trefftz formulation for analysis of anisotropic and symmetric laminated plates

In this paper, two novel and efficient elements are proposed for analysis of anisotropic and symmetric laminated thick plates based on hybrid-Trefftz formulation. In this approach, two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. At the first step, the governing differential equations of anisotropic and symmetric laminated thick plates are derived. Then, the analytical solution of these equations is considered. By using these analytical functions, the internal displacement field of proposed elements is achieved. Also, the boundary field is related to the nodal degrees of freedom by the boundary interpolation functions. For this aim, the interpolation functions of tow-node laminated Timoshenko beam element are calculated for boundary field. By utilizing these displacement fields and hybrid-Trefftz functional, a triangular element THT-7 and quadrilateral element QHT-11 are proposed. These elements have 9 and 12 degrees of freedom, respectively. The results of the numerical testes are proven the absence of the shear locking and high accuracy and efficiency of the proposed elements for analysis of anisotropic and symmetric laminated plates.

Non-linear vibration of Timoshenko beams by finite element method

The paper is concerned with free vibrations of geometrically non-linear elastic Timoshenko beams with immovable supports. The equations of motion are derived by applying the Hamilton principle. The approximate solutions are based on the negligence of longitudinal inertia forces but inclusion of longitudinal deformations. The Ritz method is used to determine non-linear modes and the associated non-linear natural frequencies depending on the vibration amplitude. The beam is discretized into linear elements with independent displacement fields. Consideration of the beams divided into the regular mesh enables one to express the equilibrium conditions for an arbitrary large number of elements in form of one difference equation. Owing to this, it is possible to obtain an analytical solution of the dynamic problem although it has been formulated by the finite element method. Some numerical results are given to show the effects of vibration amplitude, shear deformation, thickness ratio, rotary inertia, mass distribution and boundary conditions on the non-linear natural frequencies of discrete Timoshenko beams.

Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations

A general model of dynamic bending of isotropic micropolar elastic thin plates with independent fields of displacements and rotations is presented. The model has been justified asymptotically based on the solutions for special cases subject to simplifying assumptions. The model incorporates transverse shear deformations. Neglecting transverse shear, a model of the dynamics of micropolar elastic thin plates is also constructed. Then, we study free and forced oscillations and derive the natural frequencies, the amplitudes of the forced oscillations and the resonance conditions for micropolar elastic hinge-supported rectangular and circular plates. Finally, the basic characteristic features of micropolar plates are numerically analysed for different values of various elastic and inertial constants of the micropolar material.

HYBRID TREFFTZ FORMULATION FOR THIN PLATE ANALYSIS

In this paper, two efficient elements are proposed by utilizing hybrid Trefftz method for the analysis of thin plate bending. The triangular element, THT, and the quadrilateral element, QHT, which have 9 and 12 degrees of freedom, respectively. Two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a way that it satisfies the governing equation of the thin plate. Boundary field is dependent on the nodal degrees of freedom via boundary interpolation functions. For deriving boundary interpolation functions, element's edges are assumed to deform like a beam, and the related interpolation functions are used for the boundary fields. The solution accuracies of the famous and hard bench mark problems, such as circular and skew plates, prove the justification of suggested elements. Based on the various test problems, QHT in comparison to other four-sided elements, and THT in comparison to triangular ones, show better results and rapider convergence rate.

Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

A number of hypotheses were formulated using the properties of an asymptotic solution of boundary-value problems of the three-dimensional micropolar (moment asymmetric) theory of elasticity for areas with one geometrical parameter being substantially smaller than the other two (plates and shells). A general theory of bending deformation of micropolar elastic thin plates with independent fields of displacements and rotations is constructed. In the constructed model of a micropolar elastic plate, transverse shear strains are fully taken into account. A problem of determining the stress-strain state in bending deformation of micropolar elastic thin rectangular plates is considered. The numerical analysis reveals that plates made of a micropolar elastic material have high strength and stiffness characteristics.

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.

Two efficient hybrid-trefftz elements for plate bending analysis

This study is devoted to the analysis of the Reissner-Mindlin plate bending. In this paper, the hybrid-Trefftz strategy will be utilized. Two novel and efficient elements are formulated in details. They are a Triangular element (THT) and a quadrilateral element (QHT), which have 9 and 12 degrees of freedom, respectively. In this approach, two independent displacement fields are defined; one within the element and the other on the edges of the element. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. Boundary field is related to the nodal degree of freedoms by the boundary interpolation functions. To calculate these functions, the edges of the element are assumed to behave like a Timoshenko beam. The high accuracy and efficiency of the proposed elements and absence of the shear locking in these formulations are all proven, using various numerical tests.

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

AbstractIn this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.

A new hybrid-Trefftz triangular and quadrilateral plate elements

We develop 9 DOF triangular and 12 DOF quadrilateral plate bending elements based on the hybrid-Trefftz method. Among the two independent displacement fields, i.e. the internal and the boundary displacements, we use the Mindlin–Reissner’s thick plate solution with a particular solution under pressure load as the internal displacement field. For the boundary displacement field, we approximate transverse displacement w ˜ and rotations β ˜ x and β ˜ y by cubic and quadratic hierarchical shape functions, respectively. We derive the transverse shear strains from constitutive equations and equilibrium equations, respectively, and then, remove additional degrees of freedom of hierarchical shape functions using the relations between these two shear strains. Results of numerical tests reveal that the new triangular ( T 32 - 7 ) and quadrilateral ( Q 32 - 11 ) elements are robust, accurate and free of shear locking in the thin limit.

Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverseproblem in two dimensions

We examine the uniqueness of an N-field generalization of a 2D inverse problem associated with elastic modulus imaging:given N linearly independent displacement fields in an incompressible elasticmaterial, determine the shear modulus. We show that for the standard case,N=1, the general solution contains two arbitrary functions whichmust be prescribed to make the solution unique. In practice, the datarequired to evaluate the necessary functions are impossible to obtain. ForN=2, on the other hand, the general solution contains at most four arbitraryconstants, and so very few data are required to find the unique solution. ForN=4, the general solution contains only one arbitrary constant. Our results apply to bothquasistatic and dynamic deformations.

Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach

Abstract A refined higher order shear deformation theory is used to investigate the dynamic instability associated with composite plates with delamination that are subject to dynamic compressive loads. Both transverse shear and rotary inertia effects are taken into account. The theory is capable of modeling the independent displacement field above and below the delamination. All stress free boundary conditions at free surfaces as well as delamination interfaces are satisfied by this theory. The procedure is implemented using the finite element method. Delamination is modeled through the multi-point constraint approach using the transformation matrix technique. For validation purposes, the natural frequencies and the critical buckling loads are computed and compared with three-dimensional NASTRAN results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions and plate thickness. As expected, the natural frequencies and the critical buckling load of the plates with delaminations decrease with increase in delamination length. Increase in delamination length also causes instability regions to be shifted to lower parametric resonance frequencies. The effect of edge delamination on the static and dynamic stability as well as of delamination growth is investigated.

Unified Approach to Modeling Matrix Cracking and Delamination in Laminated Composite Structures

A higher-order theory is developed to model the behavior of composite laminates with delamination and transverse matrix cracking. The higher-order displacement field is used in the sublaminates for accurate representation of the effects of transverse shear. This also allows description of the independent displacement fields above and below the delamination. A refined displacement field is obtained through the satisfaction of stress-free boundary conditions at all free surfaces, including delaminated interfaces. The effect of matrix cracking is determined using a separate finite element model of a representative crack and implemented into the structural model by means of reduced laminate stiffnesses. Matrix crack closure creates a bimodularity where stiffness under compression is greater than stiffness under tension. This bimodularity is addressed using an iterative process. The effect of this bimodularity on mode shapes is shown to be small, and a method is presented for determining natural frequencies in the presence of matrix cracks. Results show that this model provides a consistent level of accuracy for a variety of laminate materials and configurations, with various combinations of delaminations and matrix cracks.

Finite elements satisfying all governing equations inside the element

We introduce a formulation of the finite element method for the solution of elasticity problems in which all the governing equations will be satisfied inside the element. The displacements inside the elements are approximated by polynomials, whose coefficients are given from equilibrium equations. Another independent displacement field, which is common to adjacent elements, is defined on the element boundaries only. The element stiffness equations are derived using two modified variational principles. The element can have arbitrary shape and can be connected to isoparametric displacement elements and/or can be used as a transition element. Although the solution is shown for two-dimensional elasticity, the extension on three-dimensional and/or other field problems is straightforward.

Special finite elements including stress concentration effects of a hole

Special finite elements are developed for efficient evaluation of stress concentration around a hole in complex structures. The complex variable formulation is used to derive a special set of stress functions which embody the stress concentration effects of a hole. The stress functions in combination with an independent displacement field assumed along the element boundary are used to construct the special elements with the hybrid displacement finite element method. Several numerical examples are presented to show that the used of special finite elements to model critical regions around a hole, together with conventional finite elements to model other regions away from the hole, is not only very convenient but also highly accurate.