Hyperelastic Thin Shells: Equilibrium Equations and Boundary Conditions

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
Chi Zhang ◽  
Jian Wu ◽  
Keh-Chih Hwang
Keyword(s):  
Boundary Conditions ◽  
Finite Deformation ◽  
Deformation Analysis ◽  
Thin Shells ◽  
Membrane Stress ◽  
Equilibrium Equations

The equilibrium equations and boundary conditions in terms of the second Piola–Kirchhoff membrane stress and moment are given in this note, which are necessary for the finite deformation analysis of shells.

Large Deformation Analysis of Orthodontic Appliances

1990 ◽  
Vol 112 (1) ◽  
pp. 29-37 ◽  
Author(s):  
A. W. Lipsett ◽  
M. G. Faulkner ◽  
K. El-Rayes
Keyword(s):  
Boundary Conditions ◽  
Local Coordinate System ◽  
Iterative Solution ◽  
Computational Effort ◽  
Deformation Analysis ◽  
Orthodontic Appliances ◽  
Equilibrium Equations ◽  
Nonlinear Approach ◽  
Space Closure ◽  
Time Required

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.

A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential—Part I: Basic Theory

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
J. Wu ◽  
K. C. Hwang ◽  
Y. Huang ◽  
J. Song
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Constitutive Relation ◽  
Finite Deformation ◽  
Shell Theory ◽  
Interatomic Potential ◽  
Bending Moments ◽  
Equilibrium Equations ◽  
Potential Part ◽  
Asymmetric Stress

A finite-deformation shell theory for carbon nanotubes (CNTs) is established directly from the interatomic potential for carbon to account for the effect of bending and curvature. Its constitutive relation accounts for the nonlinear multibody atomistic interactions and therefore can model the important effect of CNT chirality and radius. The equilibrium equations and boundary conditions are obtained for the symmetric stresses and bending moments, which are different from many existing shell theories that involve asymmetric stress and bending moments. The theory is used in Part II of this paper to study the instability of carbon nanotubes subjected to different loadings.

A Variational Approach to Loaded Ply Structures

2003 ◽  
Vol 9 (1-2) ◽  
pp. 175-185 ◽  
Author(s):  
G. H.M. Van Der Heijden ◽  
J. M.T. Thompson ◽  
S. Neukirch
Keyword(s):  
Boundary Conditions ◽  
Energy Method ◽  
Variational Approach ◽  
Energy Analysis ◽  
Numerical Results ◽  
Equilibrium Equations

We show how an energy analysis can be used to derive the equilibrium equations and boundary conditions for an end-loaded variable ply much more efficiently than in previous works. Numerical results are then presented for a clamped balanced ply approaching lock-up. We also use the energy method to derive the equations for a more general ply made of imperfect anisotropic rods and we briefly consider their helical solutions.

A Two-Dimensional Linear Assumed Strain Triangular Element for Finite Deformation Analysis

2005 ◽  
Vol 73 (6) ◽  
pp. 970-976 ◽  
Author(s):  
Fernando G. Flores
Keyword(s):  
Finite Deformation ◽  
Degrees Of Freedom ◽  
Yield Function ◽  
Triangular Element ◽  
Deformation Analysis ◽  
Elastic Model ◽  
Stress Strain ◽  
Metric Tensor ◽  
Assumed Strain ◽  
Non Linear

An assumed strain approach for a linear triangular element able to handle finite deformation problems is presented in this paper. The element is based on a total Lagrangian formulation and its geometry is defined by three nodes with only translational degrees of freedom. The strains are computed from the metric tensor, which is interpolated linearly from the values obtained at the mid-side points of the element. The evaluation of the gradient at each side of the triangle is made resorting to the geometry of the adjacent elements, leading to a four element patch. The approach is then nonconforming, nevertheless the element passes the patch test. To deal with plasticity at finite deformations a logarithmic stress-strain pair is used where an additive decomposition of elastic and plastic strains is adopted. A hyper-elastic model for the elastic linear stress-strain relation and an isotropic quadratic yield function (Mises) for the plastic part are considered. The element has been implemented in two finite element codes: an implicit static/dynamic program for moderately non-linear problems and an explicit dynamic code for problems with strong nonlinearities. Several examples are shown to assess the behavior of the present element in linear plane stress states and non-linear plane strain states as well as in axi-symmetric problems.

Nonlinear Finite Deformation Analysis of Beams and Columns

1994 ◽  
Vol 120 (10) ◽  
pp. 2136-2153 ◽  
Author(s):  
Ronald Y. S. Pak ◽  
Eric J. Stauffer
Keyword(s):  
Finite Deformation ◽  
Deformation Analysis

Equilibrium equations and boundary conditions of strain gradient theory in arbitrary curvilinear coordinates

2014 ◽  
Vol 23 (5-6) ◽  
pp. 169-176
Author(s):  
Mikhail Guzev ◽  
Chengzhi Qi ◽  
Jiping Bai ◽  
Kairui Li
Keyword(s):  
Boundary Conditions ◽  
Plane Strain ◽  
Strain Gradient ◽  
Special Form ◽  
Curvilinear Coordinates ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Equilibrium Equations ◽  
Plane Strain Problem

AbstractEquilibrium equations and boundary conditions of the strain gradient theory in arbitrary curvilinear coordinates have been obtained. Their special form for an axisymmetric plane strain problem is also given.

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