Ordinary Finite Element Research Articles

Sliding element simulating the response of slip surfaces and its application for the prediction of earthquake-induced landslide movement using one-dimensional dynamic analyses

Earthquake-induced landslides involve excessive movement of slopes, usually along slip surfaces. This seismic movement of slopes may depend crucially on (a) the soil response along the slip surface, which may include strain softening; (b) the rotation of the sliding mass with displacement towards a gentler configuration; and (c) the dynamic response of the soil profile above the underlying bedrock. Ordinary finite element methods cannot be applied to predict large localized movement along slip surfaces. Even though effects (a)–(c) above have been studied in the bibliography, a cost-effective method for simultaneous simulation to predict the seismic displacement along slip surfaces has not been found in the bibliography. The present work proposes such a cost-effective method. For this purpose, first a new sliding element is introduced which simulates effects (a) and (b) above. For effect (b), a new empirical expression is proposed and validated, while effect (a) is simulated by a previously proposed constitutive model. Then, this element replaces the slip-stick constant resistance element at a previously proposed one-dimensional non-linear dynamic model. A numerical solution of the new model is developed and applied at the well-documented Nikawa landslide. The application illustrated that the method is able to predict the displacement of the landslide, as well as the manner that (i) the stiffness of the soil profile, (ii) the shear stress–displacement response along the slip surface, and (iii) the rotation of the sliding mass affect this displacement.

A study on the computational effort of hyper-dual numbers to evaluate derivatives in geometrically nonlinear hyperelastic trusses

We report the use of hyper-dual numbers as a derivative tool for stress and tangent modulus calculation in hyperelastic models. By using the analytical expressions of Ogden’s model as a reference, we confirmed the accuracy of this tool for both first- and second-order derivatives. Moreover, the hyper-dual implementation considering the operator overloading technique corroborated the interesting generality properties of this method; by using virtually the same code syntax, the material behavior was entirely described by the hyper-dual representation of the strain energy function. Therefore, the use of the hyper-dual procedure brings an important advantage in this field since the development of the expressions concerning stress and tangent modulus becomes unnecessary; only the potential function of a given model is implemented. In this context, due to its specific set of arithmetic operations, a detailed study on the efficiency of the hyper-dual scheme in a finite element analysis is demanded. In this research, we proposed a comparative study regarding the effects of the hyper-dual procedure on the analysis processing time. Using a specific mesh generator, we evaluated a wide range of model sizes for a beam structure made of hyperelastic trusses. As a result, when compared to an ordinary finite element execution using analytical expressions for the constitutive model, we found corresponding hyper-dual performance in the larger models. Particularly, we identified the model size from which the hyper-dual approach becomes competitive; considering this element type, this model contains approximately 30,000 elements and 25,000 degrees of freedom. Furthermore, as for the computational time related to the material subroutine and the stiffness matrix, we found that the hyper-dual scheme increased the computational time by a factor of 4 and 2, respectively. We demonstrated that the hyper-dual procedure combines interesting characteristics in terms of accuracy, generality and computational costs. Hence, this numerical strategy for derivative calculation potentially represents an important tool for the development and application of new constitutive models in structural mechanics.

STRESS INTENSITY FACTOR ANALYSIS OF AN EDGE CRACK PERPENDICULAR TO THE INTERFACE IN BI-MATERIALS

The purpose of this paper is to investigate the stress intensity factor in the problem of a crack perpendicular to the interface in bi-materials. Based on the theory of an edge crack perpendicular to the interface between two dissimilar isotropic half-planes, the stress intensity factor was yield out from the stress singularity eigen-equation. The stress intensity factor for a composite beam segment with an edge crack under bending and tension was computed by the ordinary finite element and the singular finite element, and it was found that the singular element method is more accurate and applicable than the ordinary element method. The influence of the distance from the crack tip to the interface and the material’s mechanical properties to the stress intensity factor were analyzed. The results show that the stress intensity factor increases first, and then decreases with decreasing of the distance from the crack tip to the interface, and the stress intensity factor increases with increasing of the elastic modulus of the cracked material.

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.

Two new Voronoi cell finite element models for fracture simulation in porous material under inner pressure

Two new elements based on Voronoi cell finite element (VCFEM) model are developed for fracture simulation of material containing massive holes under inner pressure. One is the element with crack initiating from edge of the inner hole. The other one is the element with crack initiating from the edge of cell which generates when cracks extend from adjacent cells. In addition, analytical solution of stress components near crack tip is added to stress function applying to complementary energy functional. When the number of the holes is over a hundred thousand, the ordinary finite element needs about hundreds of millions of meshes which are solved impossibly. However, there is the characteristic of keeping the number of grids with hole number in new Voronoi cell which makes large scale simulation possible. Also grids number in VCFEM remains constant while the number in ordinary element increases greatly with cracks. The results for a large number of numerical experiments which are to calculate stress intensity factor (SIF) have demonstrated that the present models have high precision and capture stress intensity accurately.

A simplified model for high-rate actuation of shape memory alloy torque tubes using induction heating

Shape memory alloy actuators deliver high forces while being compact and reliable, making them ideal for consideration in aerospace applications. Induction heating of shape memory alloy actuators, specifically tubes that twist about the longitudinal axis, has recently been studied experimentally and computationally using finite element analysis. Reduced-order models for the torsional behavior of shape memory alloy tubes and induction heating of general metallic tubes exist, and thus, it is possible for these thermal and mechanical models to be combined and numerically solved. This work develops and implements an engineering model for inductively heated shape memory alloy tubes based on the reduction of the governing partial differential equations in space and time to an ordinary differential equation in time. An example solution is compared to finite element analysis results and agrees well. Finally, the ordinary differential equation is linearized and solved analytically. The linearized model agrees well with the nonlinear ordinary differential equation and finite element model.

Numerical modelling of rock materials with polygonal finite elements

This article presents some preliminary results on numerical modeling of rock ma-terials with polygonal finite elements. A method to describe the rock microstructure based onVoronoi diagrams, representing the rock grain texture, is sketched. In this method, the mineralsconstituting the rock are represented as Voronoi cells which themselves are polygonal finite ele-ments. A three-point bending problem under plane stress linear elasticity condition is solved inorder to compare the performance of polygonal elements to ordinary finite elements. Moreover,it is demonstrated by solving the stress state in uni-axial compression that the heterogeneitydescribed with the present method results in short-range tensile stresses which could initiatemode-I cracks.

Dynamic response analysis of solitary flexible rocking bodies: modeling and behavior under pulse‐like ground excitation

SUMMARYResults obtained for rigid structures suggest that rocking can be used as seismic response modification strategy. However, actual structures are not rigid: structural elements where rocking is expected to occur are often slender and flexible. Modeling of the rocking motion and impact of flexible bodies is a challenging task. A non‐linear elastic viscously damped zero‐length spring rocking model, directly usable in conventional finite element software, is presented in this paper. The flexible rocking body is modeled using a conventional beam‐column element with distributed masses. This model is verified by comparing its pulse excitation response to the corresponding analytical solution and validated by overturning analysis of rocking blocks subjected to a recorded ground motion excitation. The rigid rocking block model provides a good approximation of the seismic response of solitary flexible columns designed to uplift when excited by pulse‐like ground motions. Guidance for development of rocking column models in ordinary finite element software is provided. Copyright © 2014 John Wiley & Sons, Ltd.

입자법을 이용한 토사의 대변형 해석법 개발

본 연구에서는 토사 유동과 같은 대변형 해석을 위해 기존 유한요소법이나 유한차분법과 달리 격자를 사용하지 않는 입자법을 사용하였으며, 입자법은 구배, 발산, 라플라시안과 같은 미분연산자에 대응하는 입자간 상호작용모델을 이용하여 연속체의 지배방정식을 이산화하였다. 외부 하중이나 함수비 증가에 따라 고체에서 유체 상태로 변하는 흙의 3차원 대변형 거동을 해석하기 위해 기존 입자법에 흙의 파괴상태를 고려할 수 있는 Mohr-Coulomb 파괴기준을 도입하였으며, 흙의 파괴 이전의 항복이나 경화현상으로 인한 대변형은 흙을 점성 유체로 가정하여 해석하였다. 개발된 입자법은 먼저 구속압이 작용하지 않고 강도가 0인 모래기둥 붕괴실험 결과를 이용하여 검증한 다음, 점착력을 가지고 자립이 가능한 점토기둥 붕괴실험을 실시하여 흙의 항복이나 파괴로 인한 대변형을 시뮬레이션하였다. 개발된 입자법은 모래와 점토의 3차원 대변형 거동을 유사하게 잘 예측하였으며, 파괴 이전에 발생하는 흙의 항복으로 인한 소성변형에 따른 흙기둥 내의 수직 및 전단응력 변화도 계산할 수 있었다. In this study, a particle method without using grid was applied for analysing large deformation problems in soil flows instead of using ordinary finite element or finite difference methods. In the particle method, a continuum equation was discretized by various particle interaction models corresponding to differential operators such as gradient, divergence, and Laplacian. Soil behavior changes from solid to liquid state with increasing water content or external load. The Mohr-Coulomb failure criterion was incorporated into the particle method to analyze such three-dimensional soil behavior. The yielding and hardening behavior of soil before failure was analyzed by treating soil as a viscous liquid. First of all, a sand column test without confining pressure and strength was carried out and then a self-standing clay column test with cohesion was carried out. Large deformation from such column tests due to soil yielding or failure was used for verifying the developed particle method. The developed particle method was able to simulate the three-dimensional plastic deformation of soils due to yielding before failure and calculate the variation of normal and shear stresses both in sand and clay columns.

Flexural analysis of discontinuous tile core sandwich structure

Three-point flexure loading of sandwich beams with a core consisting of discrete ceramic tiles (DTSS) is considered. The tile gaps may be bonded or unbonded (open gaps). The analysis utilizes a layer-wise beam theory approach. The general formulation for the displacements and stresses in the face sheets, face/core adhesive layer, and core is derived. Solutions for stresses and displacements of the beam constituents are obtained from finite element formulation based on analytical solution of the face sheet/tile unit cell. The approach is verified by comparison to stress results obtained from ordinary finite element analysis where each layer is modeled discretely. Effects of load introduction and support conditions on the effective flexural stiffness are examined. It is demonstrated that the face sheets experience substantial stress concentrations at the tile joint locations, especially if the gaps are unfilled. Analysis of beam compliance reveals sensitivity to details of load introduction and support conditions, especially when the span length becomes comparable to the tile length.

3-D Numerical Modelling of Mesoscopic Configuration and Cracking of Concrete

This paper presents a numerical strategy for realistic modelling of the internal material configuration and cracking of concrete on mesoscale. Polyhedron shapes resembling crushed rocks are adopted to represent the aggregate. A packing approach in which the particles are heaped up layer by layer is proposed for 3-D condition. A cubic representative volume element (RVE) is generated using the method. The cohesive approach, in which zero thickness cohesive elements are inserted into the ordinary finite element mesh along the potential cracking path, is adopted to model the cracking developing process. The cracking behaviours of created RVE under uniaxial tension are well simulated using the current approach.

Basic Analysis of Microstructural Fracture Behavior in Structural Materials by Using FEM with Interface Element

A new finite element method with the interface element was developed for examining the microstructural fracture behavior, where the anisotropy of grain was modeled by the ordinary finite element while both the opening and shear deformations was demonstrated by the interface element. From the serial computations using two-dimensional virtual polycrystalline models obtained through Voronoi tessellations, it was found that the interaction between the interaction between opening and shear deformations would be a dominant factor of the fracture processes. Also, it can be concluded that this method would be a useful tool for examining microstructural fracture behavior. © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11

Preliminary Numerical Research of Microstructural Fracture Behavior in Metal by Using Interface Element

In order to demonstrate not only the deformation of grain but also the opening and/or sliding at grain boundary, the interface element was introduced into the ordinary finite element method, and this numerical method was applied for examining the microstructural fracture behavior in two-dimensional ideal microstructure obtained through Voronoi tessellations. As for the grain, the anisotropy in elastic modulus due to the grain orientation was taken into account, while the fracture strength at grain boundary was assumed to be related to the boundary energy which could be determined by the atomic disorder at the boundary. From the serial computational results for examining the influences of elastic properties in grain (isotropy and anisotropy), mechanical property at grain boundary (interaction between opening and sliding deformation), and grain configurations, it was revealed that all the factors varied in this research might affect the microstructural fracture behavior. Also, it can be concluded that this numerical method with the interface element can be useful for demonstrating the microstructural fracture behavior including the deformation at grain boundary.

An implicit stress gradient plasticity model for describing mechanical behavior of planar fiber networks on a macroscopic scale

The plasticity behavior of fiber networks is governed by complex mechanisms. This study examines the effect of microstructure on the macroscopic plastic behavior of two-dimensional random fiber networks such as strong-bonded paper. Remote load is a pure macroscopic mode I opening field, applied via a boundary layer assuming small scale yielding on the macroscopic scale. It is shown that using a macroscopic classical homogeneous continuum approach to describe plasticity effects due to (macroscopic) singular-dominated strain fields in planar fiber networks leads to erroneous results. The classical continuum description is too simple to capture the essential mechanical behavior of a network material since a structural effect, that alters the macroscopic stress field, becomes pronounced and introduces long-ranging microstructural effects that have to be accounted for. Because of this, it is necessary to include a nonlocal theory that bridges the gap between microscopic and macroscopic scales to describe the material response in homogeneous continuum models. An implicit stress gradient small deformation plasticity model, which is based on a strong nonlocal continuum formulation, is presented here that has the potential to describe the plasticity behavior of fiber networks on a macroscopic scale. The theory is derived by including nonlocal stress terms in the classical associated J2-theory of plasticity. The nonlocal stress tensor is found by scaling the local Cauchy stress tensor by the ratio of nonlocal and local von Mises equivalent stresses. The model is relatively easy to implement in ordinary finite element algorithms for small deformation theory. Fairly good agreements are obtained between discrete micromechanical network models and the derived homogeneous nonlocal continuum model.

Three dimensional FEM of moving coordinates for the analysis of transient vibrations due to moving loads

Abstract This paper is concerned with the transient vibration analysis of railway-ground system under fast moving loads. A 3D finite element method in a convected coordinate system moving with the load is formulated, together with viscous-elastic transmitting boundary conditions in order to limit the finite element mesh. A method is proposed to introduce Rayleigh type material damping in the finite element formulation in the moving coordinate system, while measures have also been taken to improve the numerical stability of the solution procedure. The performance of the transmitting boundary and the entire solution procedure are assessed via comparison with the ordinary finite element solution of some relatively simple problems and through a comparison with field measurements. The reasonable agreement found from these comparisons demonstrates the validity of the proposed method.

Improved finite element for 3D laminate frame analysis including warping for any cross-section

The most ordinary finite element formulations for 3D frame analysis do not consider the warping of cross-sections as part of their kinematics. So the stiffness, regarding torsion, should be directly introduced by the user into the computational software and the bar is treated as it is working under no warping hypothesis. This approach does not give good results for general structural elements applied in engineering. Both displacement and stress calculation reveal sensible deficiencies for both linear and non-linear applications. For linear analysis, displacements can be corrected by assuming a stiffness that results in acceptable global displacements of the analyzed structure. However, the stress calculation will be far from reality. For nonlinear analysis the deficiencies are even worse. In the past forty years, some special structural matrix analysis and finite element formulations have been proposed in literature to include warping and the bending-torsion effects for 3D general frame analysis considering both linear and non-linear situations. In this work, using a kinematics improvement technique, the degree of freedom “warping intensity” is introduced following a new approach for 3D frame elements. This degree of freedom is associated with the warping basic mode, a geometric characteristic of the cross-section. It does not have a direct relation with the rate of twist rotation along the longitudinal axis, as in existent formulations. Moreover, a linear strain variation mode is provided for the geometric non-linear approach, for which complete 3D constitutive relation (Saint–Venant Kirchhoff) is adopted. The proposed technique allows the consideration of inhomogeneous cross-sections with any geometry. Various examples are shown to demonstrate the accuracy and applicability of the proposed formulation.

Mechanical analysis of functionally graded plates based on the input data of metallographic diagrams

A microelement method for scale-span analysis of material microstructure and member macro response has been proposed. Instead of material parameter input for traditional mechanical analysis, the method is based on the input of metallographic diagram information. In order to express the material microstructure, this method arranges concentrated microelements in ordinary finite element before calculation, and transfers the node degrees of freedom of each microelement into the ones of the same finite element via compatibility conditions. This method can realize direct transition analysis from material microstructure to macro responses of members, while computation elements and degrees of freedom are equal to those of ordinary FEM. Based on the complicated microstructure diagrams proposed by material metallographic diagram, the mechanics responses of functionally graded plates are calculated and 3-D distributions of macro-mechanical variables and identical stress lines tendency on microstructure are given.

On the influence of singular-type finite elements on the critical force in studying the buckling of a circular plate with a crack

The axisymmetric buckling (delamination) of a circular disk (plate) with a penny-shaped crack is analyzed using a continuum model, piecewise-homogeneous model, and the three-dimensional linearized theory of stability. The FEM is used. The analysis is carried out using various singular and ordinary finite elements. The numerical results obtained indicate that it is not necessary to use singular finite elements to solve the problem

Finite elements using absolute nodal coordinates for large-deformation flexible multibody dynamics

A family of structural finite elements using a modern absolute nodal coordinate formulation (ANCF) is discussed in the paper with many applications. This approach has been initiated in 1996 by A. Shabana. It introduces large displacements of 2D/3D finite elements relative to the global reference frame without using any local frame. The elements employ finite slopes as nodal variables and can be considered as generalizations of ordinary finite elements that use infinitesimal slopes. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices and generalized gravity forces as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This approach allows applying known abstractions of real elastic bodies: Euler–Bernoulli beams, Timoshenko beams and more general models as well as Kirchhoff and Mindlin plate theories. Shabana et al. proposed a sub-family of thick beam and plate finite elements with large deformations and employ the 3D theory of continuum mechanics. Despite the universality of such approach it has to use extra degrees of freedom when simulating thin beams and plates, which case is most important. In our research, we propose another sub-family of thin beams as well as rectangular and triangle plates. We use Kirchhoff plate theory with nonlinear strain–displacement relationships to obtain elastic forces. A number of static and dynamic simulation examples of problems with 2D/3D very elastic beams and plate underwent large displacements and/or deformations will be shown in the presentation.

Magnetic field analysis of laminated core by using homogenization method

This paper describes an approach to efficient modeling of the microscopic structure of magnetic materials in the nonlinear magnetostatic field analysis of a laminated core. When the ordinary finite element method is applied, large memory and computational cost are required due to the huge number of elements generated by detailed modeling of a laminated core. In order to overcome the above difficulties, we introduce the homogenization method, which is widely used in continuum mechanics, into the magnetic field computation and expand the method into the nonlinear analysis. Furthermore, the specialized homogenization method for a laminated core is carried out to verify the above homogenization method in the electromagnetism. Some numerical results are also presented to demonstrate the effectiveness of the proposed method.