Particular Solutions, Stress Singularities, and Stress Decay

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Asymptotic Solution ◽  
Numerical Solutions ◽  
Stress Singularities ◽  
Elastic Materials ◽  
Element Scheme ◽  
Regular Elements ◽  
Linear Elastic ◽  
Other Information

Not all boundary value problems are amenable to a simple analytical solution. This is particularly the case when the geometry of the boundary on which the boundary conditions are specified contains a corner. A crack tip is a special corner. When an analytical solution for the entire region is not available, asymptotic solutions near the corner can be obtained which provide useful information on the nature of stress singularities at the corner. They also provide more accurate numerical solutions by a finite element scheme in which the asymptotic solution at the corner is employed in a special element at the corner with regular elements elsewhere (see, for example, Stolarski and Chiang, 1989). Other information that can be obtained by an asymptotic analysis is the decay factor of stress at a large distance from a point at which a self-equilibrated load is applied (Crafter, et al., 1993). For singularities that arise in non-linear elastic materials the reader is referred to the book by Antman (1995). As in Chapter 8 most solutions can be expressed in a real form with the aid of identities presented in Chapters 6 and 7.

Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

2011 ◽  
Vol 3 (6) ◽  
pp. 729-744
Author(s):  
Zhengzheng Hu ◽  
Mansoor A Haider
Keyword(s):  
Finite Element ◽  
Articular Cartilage ◽  
Analytical Solution ◽  
Conjugate Gradient ◽  
Numerical Solutions ◽  
Elastic Inclusion ◽  
Algebraic Multigrid ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Cpu Time

AbstractIn studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.

An Investigation of Dynamic Pulse Buckling of Thick Rings

1992 ◽  
Vol 59 (3) ◽  
pp. 615-621 ◽  
Author(s):  
N. G. Pegg
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Numerical Solutions ◽  
Nonlinear Finite Element ◽  
Finite Element Analyses ◽  
Elastoplastic Behavior ◽  
Impulse Load ◽  
Pulse Buckling ◽  
Comparison Of The Results ◽  
Dynamic Pulse

The occurrence of dynamic buckling of thick rings responding to an impulse load is investigated by analytical and numerical solutions to the equation of motion and by nonlinear finite element analyses. An extension to the linearized analytical solution is made using a finite difference scheme which incorporates a nonlinear moment-curvature relationship to model the effects of elastoplastic behavior and strain-rate reversal on the buckle formation. The finite element solution to the problem is formulated with the nonlinear code, ADINA. A comparison of the results shows that the numerical solutions (and, in particular, the ADINA solution) predict a significant reduction in the amplitude of buckling response and an increase in the predominant wavelength of response with time, in comparison to the linear analytical solution. A limited comparison to published experimental results of dynamic pulse buckling of thick rings is also given.

scholarly journals Solving geometrically nonlinear tasks of the statics of hinged-rod systems basing on finite element method in the form of classical mixed method

Vestnik MGSU ◽  
2016 ◽  
pp. 20-33 ◽  
Author(s):  
Aleksandr Vladimirovich Ignat’ev ◽  
Vladimir Aleksandrovich Ignat’ev ◽  
Ekaterina Valer’evna Onishchenko
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Method ◽  
Numerical Solutions ◽  
Geometrically Nonlinear ◽  
Building Structures ◽  
Engineering Structures ◽  
Linear Elastic ◽  
Rod System ◽  
Element Method

The most widely used numerical method used in linear calculation of building structures is finite element method in traditional form of displacements. Different software is developed on its basis. Though it is only possible to check the certainty of these numerical solutions, especially of non-linear tasks of engineering structures’ deformation by the coincidence of the results obtained by two different methods. The authors solved geometrically nonlinear task of the static deformation of a flat hinged-rod system consisting of five linear elastic rods undergoing great tension-compression strains. The solution was obtained basing on the finite element method in the form of classical mixed method developed by the authors. The set of all equilibrium states of the system, both stable and unstable, and all the limit points were found. The certainty of the solution was approved by the coincidence of the results obtained by other authors basing on traditional finite element method in displacements.

A New 3D Analytical Method for Calculating the Longitudinal Stress-Strain of Bones, as ‘N’ Different Asymmetrical Linear Elastic Materials, Loaded Together Eccentrically

2000 ◽  
Author(s):  
Nader Arafati
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Analytical Method ◽  
Hip Prosthesis ◽  
Three Dimensional ◽  
Elastic Materials ◽  
Stress Strain ◽  
Element Analysis ◽  
Linear Elastic ◽  
The Finite Element Analysis

Abstract We describe a three-dimensional analytical method to evaluate the stress-strain of « n » different asymmetrical linear elastic materials, loaded together by eccentric forces and moments. If one assumes that each bone is composed of a number of different linear elastic materials, this method could be used to determine the stress-strain values of any bone. Mesh generation is unnecessary with this method, the only requirement being to determine the contours of each material. We used this method to analyze the longitudinal stresses imposed on the femur, after implantation of cemented hip prosthesis. The results were compared with those of finite element analysis without any contact elements. Considerable differences were found, particularly for exterior prosthesis contours. When the same error was entered into a cylindrical analytical model, the similar errors were found, indicating that not using contact elements may result in unacceptable errors with the finite element analysis method.

scholarly journals Fatigue Crack Growth Analysis with Extended Finite Element for 3D Linear Elastic Material

Metals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 397
Author(s):  
Yahya Ali Fageehi
Keyword(s):  
Finite Element ◽  
Fatigue Crack ◽  
Fatigue Life ◽  
Crack Propagation ◽  
Crack Growth ◽  
Mixed Mode ◽  
Propagation Path ◽  
Loading Conditions ◽  
Extended Finite Element ◽  
Linear Elastic

This paper presents computational modeling of a crack growth path under mixed-mode loadings in linear elastic materials and investigates the influence of a hole on both fatigue crack propagation and fatigue life when subjected to constant amplitude loading conditions. Though the crack propagation is inevitable, the simulation specified the crack propagation path such that the critical structure domain was not exceeded. ANSYS Mechanical APDL 19.2 was introduced with the aid of a new feature in ANSYS: Smart Crack growth technology. It predicts the propagation direction and subsequent fatigue life for structural components using the extended finite element method (XFEM). The Paris law model was used to evaluate the mixed-mode fatigue life for both a modified four-point bending beam and a cracked plate with three holes under the linear elastic fracture mechanics (LEFM) assumption. Precise estimates of the stress intensity factors (SIFs), the trajectory of crack growth, and the fatigue life by an incremental crack propagation analysis were recorded. The findings of this analysis are confirmed in published works in terms of crack propagation trajectories under mixed-mode loading conditions.

scholarly journals Theoretical analysis and experimental research on multi-layer elastic damping track structure

2021 ◽  
Vol 13 (2) ◽  
pp. 168781402199497
Author(s):  
Guanghui Xu ◽  
Shengkai Su ◽  
Anbin Wang ◽  
Ruolin Hu
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Finite Element Simulation ◽  
Reaction Force ◽  
Vibration Reduction ◽  
Vertical Direction ◽  
Propagation Path ◽  
Track Structure ◽  
Test Results ◽  
Element Simulation

The increase of axle load and train speed would cause intense wheelrail interactions, and lead to potential vibration related problems in train operation. For the low-frequency vibration reduction of a track system, a multi-layer track structure was proposed and analyzed theoretically and experimentally. Firstly, the analytical solution was derived theoretically, and followed by a parametric analysis to verify the vibration reduction performance. Then, a finite element simulation is carried out to highlight the influence of the tuned slab damper. Finally, the vibration and noise tests are performed to verify the results of the analytical solution and finite element simulation. As the finite element simulation indicates, after installation of the tuned slab damper, the peak reaction force of the foundation can be reduced by 60%, and the peak value of the vertical vibration acceleration would decrease by 50%. The vibration test results show that the insertion losses for the total vibration levels are 13.3 dB in the vertical direction and 21.7 dB in the transverse direction. The noise test results show that the data of each measurement point is smoother and smaller, and the noise in the generating position and propagation path can be reduced by 1.9 dB–5.5 dB.

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