Nonlinear Shell Deflections With Thickness Updating

1994 ◽  
Author(s):  
C. W. S. To ◽  
M. L. Liu
Keyword(s):  
Finite Strain ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Hybrid Strain ◽  
Finite Rotations ◽  
Stiffness Matrices ◽  
Strain Formulation ◽  
Updated Lagrangian Formulation ◽  
Updated Lagrangian ◽  
Locking Phenomena

Abstract In the present investigation the incremental Hellinger-Reissner variational principle, a hybrid-strain based formulation and an updated Lagrangian formulation are adopted to derive explicit expressions for element stiffness matrices of various flat triangular shell finite elements. These elements are developed for application to the analysis of thin and thick shell structures undergoing large geometrically nonlinear deformation at finite strain. Correct representations of finite rotations and the specifically chosen strain field maintain appropriate rigid body motions and prevent shear locking phenomena. Consideration of thickness updating and a finite strain formulation relaxes any plane stress assumptions and enables the analyst to deal with three-dimensional constitutive models. Two numerical examples are presented to demonstrate the performance of the elements.

Lower Order Tetrahedral Finite Elements With Rotational Degrees of Freedom for Solid Modelling

2007 ◽  
Author(s):  
Cho W. S. To ◽  
Xiao Hua
Keyword(s):  
Finite Elements ◽  
Lower Order ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Benchmark Problems ◽  
Solid Modelling ◽  
Hybrid Strain ◽  
True Solution ◽  
Stiffness Matrices ◽  
Strain Formulation

The investigation reported in this paper is concerned with the development of efficient, conceptually simpler, mathematically rigorous and physically meaningful three-dimensional (3D) finite elements for solid modelling. A mixed variational principle based on the hybrid strain formulation has been adopted for the derivation of element stiffness matrices of two lower order tetrahedral finite elements. Explicit expressions for element matrices have been derived with a combination of hand manipulation and computer algebraic package, MAPLE. Each of the two finite elements has four nodes. Every one of the latter has six degrees of freedom (DOF). These include three translational and three rotational DOF. Element performance is evaluated with benchmark problems. For brevity, only two benchmark problems are included in this paper. It is shown numerically that the results converge to the true solution and have superior accuracy compared with those previously published in the literature. Mathematically, the elements being proposed are simple and frame invariant. Computationally, they are very efficient compared with higher order tetrahedral finite elements and other lower order tetrahedral finite elements.

Non-Conservative and Conservative Loads in Geometrically Nonlinear Shells

2003 ◽  
Author(s):  
Cho W. S. To ◽  
Meilan L. Liu
Keyword(s):  
Finite Element ◽  
Large Strain ◽  
Small Strain ◽  
Shell Structures ◽  
Element Formulation ◽  
Geometrically Nonlinear ◽  
Hybrid Strain ◽  
Shell Elements ◽  
Updated Lagrangian Formulation ◽  
Updated Lagrangian

Responses of geometrically nonlinear shell structures under combined conservative and non-conservative loads are investigated and presented in this paper. The shell structures are discretized by the finite element method and represented by the hybrid strain based three node flat triangular shell elements that were developed previously by the authors. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are employed. Features such as large or small strain deformation, finite rotation, updated thickness so as to account for the “thinning effect” due to large strain deformation, and inclusion or exclusion of the mid-surface director field are incorporated in the finite element formulation. Representative results of two examples are included to demonstrate the capability, accuracy and efficiency of the computational strategy proposed.

Study on the Thermo-Elastic-Plastic Cutting Model for 3-D Tool with Chip Breaker

1998 ◽  
Vol 120 (4) ◽  
pp. 265-274 ◽  
Author(s):  
Zone-Ching Lin ◽  
Yan-Liang Zheng
Keyword(s):  
Simulation Model ◽  
Cutting Force ◽  
Three Dimensional ◽  
Initial Contact ◽  
Chip Breaker ◽  
Elastic Plastic ◽  
Updated Lagrangian Formulation ◽  
Updated Lagrangian ◽  
Chip Separation ◽  
Cutting Model

This paper used large deformation finite element theory, updated Lagrangian formulation, finite difference method, and incremental theory to develop a three-dimensional thermo-elastic-plastic simulation model for a tool with chip breaker. Both the critical strain energy density theory and the tool feed geometrical location were introduced as the chip separation criterion for cutting. The algorithm of tool movement geometrical limitations was used to examine and correctly the node so as to conform to real cutting conditions. In this model, the tool moved step by step in the simulation, which ran from the initial contact between tool and workpiece to the formation of steady cutting force. Finally, the numerical simulation model proposed in this paper was used to analyze the changes in workpiece and chip shapes, stress, strain rate, residual stress, temperature and cutting force of mild steel workpiece under different chip breaker lengths. The results were also compared with those from tools without chip breaker. The findings indicate that the chip breaker length affects the shorter the chip breaker length, the better the effects of chip breaker, and the lower the values of the aforementioned physical properties.

Finite Strain Viscoplasticity Based on Unified Constitutive Equations and Decomposition of Logarithmic Strain: Applications to Shells

1997 ◽  
Vol 50 (11S) ◽  
pp. S184-S192 ◽  
Author(s):  
C. Sansour ◽  
F. G. Kollmann
Keyword(s):  
Finite Strain ◽  
Large Strain ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Strain Range ◽  
Basic Feature ◽  
Point Of View ◽  
Logarithmic Strain ◽  
Additive Structure

The paper is concerned with a formulation of large strain viscoplasticity based on the concept of unified constitutive models as well as on an additive decomposition of a logarithmic strain tensor. The constitutive model due to Bodner and Partom is modified as to fit within the theoretical framework presented. A basic feature of the formulation is the fact that the additive structure of the infinitesimal theory is preserved in the finite strain range. Based on an essential result, a closed form of the tangent operator is derived which is very efficient from the numerical point of view. As an application, finite shell deformations are considered. The shell theory used allows for the application of three-dimensional constitutive laws and is geometrically exact. The computations are based on an enhanced strain functional where the right Cauchy-Green tensor is enhanced. Two examples of large shell deformations including loading-unloading cycles are presented.

Finite-Element Analysis of the Double Lateral Compression of Clad Tube into a Symmetric Square-Tube

2011 ◽  
Vol 337 ◽  
pp. 332-335 ◽  
Author(s):  
Tsung Chia Chen ◽  
Jiun Ming Ye
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Strain Hardening Exponent ◽  
Element Analysis ◽  
Circular Tubes ◽  
Geometric Ratio ◽  
Updated Lagrangian Formulation ◽  
Symmetric Square ◽  
Clad Tube ◽  
Updated Lagrangian

The squaring process to shape a circular tube into a symmetric square clad tube is examined by a three-dimensional incremental elastic-plastic finite-element method based on an updated Lagrangian formulation. The effects of various parameters, such as geometric ratio R/t, strain hardening exponent n, friction coefficient μ, and the length of tube L, on the occurrence of collapse in the squaring process are discussed and interpreted in a theoretical manner. The findings show that geometric ratio is the major factor in the process of squaring circular tubes. When R/t=25, serious collapse is likely to appear. Aiming at circular tubes with geometric ratio R/t=25, this study proposes six analysis configurations for clad tubes to discuss the possibility of clad tubes avoiding collapse. The findings showed that clad tubes could effectively reduce the collapse ratio.

Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zdeněk Dostál ◽  
Tomáš Brzobohatý ◽  
Oldřich Vlach
Keyword(s):  
Condition Number ◽  
Three Dimensional ◽  
Schur Complement ◽  
Decomposition Methods ◽  
Stiffness Matrices ◽  
Spectral Bounds ◽  
Schur Complements ◽  
Linear Problems ◽  
Primal Method ◽  
Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

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