A finite element method for the Saint-Venant torsion and bending problems for prismatic beams

In this work, we present a finite element formulation for the Saint-Venant torsion and bending problems for prismatic beams. The torsion problem formulation is based on the warping function, and can handle multiply-connected regions (including thin-walled structures), compound and anisotropic bars. Similarly, the bending formulation, which is based on linearized elasticity theory, can handle multiply-connected domains including thin-walled sections. The torsional rigidity and shear centers can be found as special cases of these formulations. Numerical results are presented to show the good coarse-mesh accuracy of both the formulations for both the displacement and stress fields. The stiffness matrices and load vectors (which are similar to those for a variable body force in a conventional structural mechanics problem) in both formulations involve only domain integrals, which makes them simple to implement and computationally efficient. (C) 2014 Elsevier Ltd. All rights reserved.

Switching and Release Time Analysis of Electrostatically Actuated Capacitive RF MEMS Switches

This paper explains the reason behind pull-in time being more than pull-up time of many Radio Frequency Micro-Electro-Mechanical Systems (RF MEMS) switches at actuation voltages comparable to the pull-in voltage. Analytical expressions for pull-in and pull-up time are also presented. Experimental data as well as finite element simulations of electrostatically actuated beams used in RF-MEMS switches show that the pull-in time is generally more than the pull-up time. Pull-in time being more than pull-up time is somewhat counter-intuitive because there is a much larger electrostatic force during pull-in than the restoring mechanical force during the release. We investigated this issue analytically and numerically using a 1D model for various applied voltages and attribute this to energetics, the rate at which the forces change with time, and softening of the overall effective stiffness of the electromechanical system. 3D finite element analysis is also done to support the 1D model-based analyses.

A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

A residual based a posteriori error estimator is derived for a quadratic finite element method (FEM) for the elliptic obstacle problem. The error estimator involves various residuals consisting of the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. The choice of the discrete Lagrange multiplier yields an error estimator that is comparable with the error estimator in the case of linear FEM. Further, an a priori error estimate is derived to show that the discrete Lagrange multiplier converges at the same rate as that of the discrete solution of the obstacle problem. The numerical experiments of adaptive FEM show optimal order convergence. This demonstrates that the quadratic FEM for obstacle problem exhibits optimal performance.

The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems

Schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems. Hence, the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging. In this work, we show that time finite element methods because of their inherent energy momentum conserving property (in the case of linear and nonlinear elastodynamics), provide a robust time-stepping method for nonlinear dynamic equations (including chaotic systems). We also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework; thus, the time finite element provides a unification of time-stepping schemes used in diverse disciplines. We demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic. (C) 2015 Elsevier Inc. All rights reserved.

The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems

Schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems. Hence, the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging. In this work, we show that time finite element methods because of their inherent energy momentum conserving property (in the case of linear and nonlinear elastodynamics), provide a robust time-stepping method for nonlinear dynamic equations (including chaotic systems). We also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework; thus, the time finite element provides a unification of time-stepping schemes used in diverse disciplines. We demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic. (C) 2015 Elsevier Inc. All rights reserved.

Further analysis of indentation loading curves: Effects of tip rounding on mechanical property measurements

The effects of indenter tip rounding on the shape of indentation loading curves have been analyzed using dimensional and finite element analysis for conical indentation in elastic-perfectly plastic solids. A method for obtaining mechanical properties from indentation loading curves is then proposed. The validity of this method is examined using finite element analysis. Finally, the method is used to determine the yield strength of several materials for which the indentation loading curves are available in the literature.

Determination of Proper Processing Conditions of Material Surface Heat Treatment Through Finite Element Method

A two-dimensional model has been developed based on the experimental results of stainless steel remelting with the laminar plasma technology to investigate the transient thermo-physical characteristics of the melt pool liquids. The influence of the temperature field, temperature gradient, solidification rate and cooling rate on the processing conditions has been investigated numerically. Not only have the appropriate processing conditions been determined according to the calculations, but also they have been predicted with a criterion established based on the concept of equivalent temperature area density (ETAD) that is actually a function of the processing parameters and material properties. The comparison between the resulting conditions shows that the ETAD method can better predict the optimum condition.

Partition Energy Absorption of Axially Crushed Aluminum Foam-Filled Hat Sections

The "interaction effect" between aluminum foam and metal column that takes place when foam-filled hat sections (top-hats and double-hats) are axially crushed was investigated in this paper. Based on experimental examination, numerical simulation and analytical models, a systemic approach was developed to partition the energy absorption quantitatively into the foam filler component and the hat section component, and the relative contribution of each component to the overall interaction effect was therefore evaluated. Careful observation of the collapse profile found that the crushed foam filler could be further divided into two main energy-dissipation regions: densified region and extremely densified region. The volume reduction and volumetric strain of each region were empirically estimated. An analytical model pertinent to the collapse profile was thereafter proposed to find the more precise relationship between the volume reduction and volumetric strain of the foam filler. Combined the superfolding element model for hat sections with the current model according to the coupled method, each component energy absorption was subsequently derived, and the influence of some controlling factors was discussed. According to the finite element analysis and the theoretical modeling, when filled with foam, energy absorption was found to be increased both in the hat section and the foam filler, whereas the latter contributes predominantly to the interaction effect. The formation of the extremely densified region in the foam filler accounts for this effect.

A finite element method for cracked components of structures

In this paper, a method is developed for determining the effective stiffness of the cracked component. The stiffness matrix of the cracked component is integrated into the global stiffness matrix of the finite element model of the global platform for the FE calculation of the structure in any environmental conditions. The stiffness matrix equation of the cracked component is derived by use of the finite variation principle and fracture mechanics. The equivalent parameters defining the element that simulates the cracked component are mathematically presented, and can be easily used for the FE calculation of large scale cracked structures together with any finite element program. The theories developed are validated by both lab tests and numerical calculations, and applied to the evaluation of crack effect on the strength of a fixed platform and a self-elevating drilling rig.

Finite Element Solutions for Plane Strain Mode I Crack with Strain Grading Effect

In this paper, a new phenomenological theory with strain gradient effects is proposed to account for the size dependence of plastic deformation at micro- and submicro-length scales. The theory fits within the framework of general couple stress theory and three rotational degrees of freedom omega(i) are introduced in addition to the conventional three translational degrees of freedom mu(i). omega(i) is called micro-rotation and is the sum of material rotation plus the particles' relative rotation. While the new theory is used to analyze the crack tip field or the indentation problems, the stretch gradient is considered through a new hardening law. The key features of the theory are that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the term of stretch gradient is represented as an internal variable to increase the tangent modulus. In fact the present new strain gradient theory is the combination of the strain gradient theory proposed by Chen and Wang (Int. J. Plast., in press) and the hardening law given by Chen and Wang (Acta Mater. 48 (2000a) 3997). In this paper we focus on the finite element method to investigate material fracture for an elastic-power law hardening solid. With remotely imposed classical K fields, the full field solutions are obtained numerically. It is found that the size of the strain gradient dominance zone is characterized by the intrinsic material length l(1). Outside the strain gradient dominance zone, the computed stress field tends to be a classical plasticity field and then K field. The singularity of stresses ahead of the crack tip is higher than that of the classical field and tends to the square root singularity, which has important consequences for crack growth in materials by decohesion at the atomic scale. (C) 2002 Elsevier Science Ltd. All rights reserved.

Effect of dentin tubules on the mechanical properties of dentin - part III: Numerical, analysis

Imitating a real tooth and the periodontal supporting tissues, we have established a 2D finite element model and carried out a numerical analysis based on the inhomogeneous and anisotropic (IA) stress-strain relation and strength model of dentin proposed in the preceding Parts I and II, and the conventional homogeneous and isotropic (III) model, respectively. Quite a few cases of loadings for a non-defected and a defected tooth are considered. The numerical results show that the stress level predicted by the IA model is remarkably higher than that by the III model, revealing that the effect of the dentin tubules should be taken into a serious consideration from the viewpoint of biomechanics.

Computation of stress intensity factors by the sub-region mixed finite element method of lines

Based on the sub-region generalized variational principle, a sub-region mixed version of the newly-developed semi-analytical 'finite element method of lines' (FEMOL) is proposed in this paper for accurate and efficient computation of stress intensity factors (SIFs) of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations (ODEs) and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.