The Cosserat Spectrum Theory in Thermoelasticity and Application to the Problem of Heat Flow Past a Rigid Spherical Inclusion

1998 ◽  
Vol 65 (3) ◽  
pp. 614-618 ◽  
Author(s):  
Wensen Liu ◽  
X. Markenscoff ◽  
M. Paukshto
Keyword(s):  
Variational Principle ◽  
Heat Flow ◽  
Displacement Field ◽  
Elastic Energy ◽  
Rigid Inclusion ◽  
Spherical Inclusion ◽  
Three Dimensional ◽  
Practical Method ◽  
Cosserat Spectrum ◽  
Thermoelastic Displacement

We apply the Cosserat Spectrum theory to boundary value problems in thermoelasticity and show the advantages of this method. The thermoelastic displacement field caused by a general heat flow around a spherical rigid inclusion is calculatedand the results show that the discrete Cosserat eigenfunctions converge fast and thus provide a practical method for solving three-dimensional problems in thermoelasticity. In the case of uniform heat flow, the solution is obtained analytically in closed form and a variational principle within the frame of the Cosserat Spectrum theory shows that the solution maximizes the elastic energy.

A Generalized Form Of Integral-Finite Element Formulations For Diffusion Equation

1984 ◽  
pp. 1-6
Author(s):  
Wan Mokhtar Nawang
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Variational Principle ◽  
Heat Flow ◽  
Three Dimensional ◽  
Integral Form ◽  
Physical Problem ◽  
Integral Formulation ◽  
Governing Differential Equation ◽  
Finite Element Formulations

A physical problem such as diffusion can be described mathematically in two ways, i.e. by Differential Equation Formulation or Integral Formulation. An integral form is derived from its governing differential equation using the method of Variational Principle for a three-dimensional heat flow equation.The equivalent Integral Formulation will be a very useful and an inevitable tool in the formulation of finite element equatipn.

Stress Concentrations in Three-Dimensional Electrostriction

1967 ◽  
Vol 34 (2) ◽  
pp. 431-436 ◽  
Author(s):  
T. E. Smith
Keyword(s):  
Displacement Vector ◽  
Spherical Inclusion ◽  
Crack Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Electric Field ◽  
Inclusion Problem ◽  
Stress Concentrations ◽  
Displacement Potential ◽  
Penny Shaped Crack

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.

Modified Mixed Variational Principle and the State-Vector Equation for Elastic Bodies and Shells of Revolution

1992 ◽  
Vol 59 (3) ◽  
pp. 587-595 ◽  
Author(s):  
Charles R. Steele ◽  
Yoon Young Kim
Keyword(s):  
Variational Principle ◽  
Equations Of State ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Shells Of Revolution ◽  
Elastic Bodies ◽  
Mixed Variational Principle ◽  
Stress Components ◽  
Independent Variable ◽  
Nonsymmetric Deformation

A modified mixed variational principle is established for a class of problems with one spatial variable as the independent variable. The specific applications are on the three-dimensional deformation of elastic bodies and the nonsymmetric deformation of shells of revolution. The possibly novel feature is the elimination in the variational formulation of the stress components which cannot be prescribed on the boundaries. The result is a form exactly analogous to classical mechanics of a dynamic system, with the equations of state exactly in the form of the canonical equations of Hamilton. With the present approach, the correct scale factors of the field variables to make the system self-adjoint are readily identified, and anisotropic materials including composites can be handled effectively. The analysis for shells of revolution is given with and without the transverse shear deformation considered.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

scholarly journals Determination of the Optimal Syndesmosis Fixation Angle Using Mortise View of The Ankle Joint

2021 ◽  
Author(s):  
Oğuzhan Tanoğlu ◽  
İzzet Özay Subaşı ◽  
Mehmet Burak Gökgöz
Keyword(s):  
Ankle Joint ◽  
Articular Surface ◽  
Three Dimensional ◽  
Vertical Plane ◽  
Functional Results ◽  
Practical Method ◽  
Medial Malleolus ◽  
Significant Difference ◽  
Horizontal View ◽  
Mortise View

Background: Syndesmosis is an important soft tissue component supporting the ankle stability and commonly injured accompanying with ankle fractures. The accurate reduction and fixation of syndesmosis is essential to obtain better functional results. Therefore, we aimed to find a practical method using the mortise view of ankle to determine the optimal syndesmosis fixation angle intraoperatively. Methods: We randomly selected 200 adults (100 women and 100 men) between 18 - 60 years of age. Three-dimensional anatomical models of tibia and fibula were created using Materialise MIMICS 21. We created a best fit plane on articular surface of medial malleolus and a ninety degrees vertical plane to medial malleolus plane. We determined two splines on cortical borders of tibia and fibula distant from the most superior point of ankle joint in horizontal view. We created two spheres that fit to the predefined splines. The optimal syndesmosis fixation angle was determined measuring the angle between the line connecting the center points of spheres, and the ninety degrees vertical plane to medial malleolus plane. Results: We observed no statistically significant difference between gender groups in terms of optimal syndesmosis fixation angles. The mean age of our study population was 47.1 {plus minus} 10.5. The optimal syndesmosis fixation angle according to mortise view was found as 21 {plus minus} 4.3 degrees. Conclusions: We determined the optimal syndesmosis fixation angle as 21 {plus minus} 4.3 degrees in accordance with the mortise view of ankle. The surgeon could evaluate the whole articular surface of ankle joint with the medial and lateral syndesmotic space in mortise view accurately and at the same position syndesmosis fixation could be performed at 21 {plus minus} 4.3 degrees.

Lee waves in three-dimensional stratified flow

1984 ◽  
Vol 148 ◽  
pp. 97-108 ◽  
Author(s):  
G. S. Janowitz
Keyword(s):  
Stream Function ◽  
Displacement Field ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Vertical Displacement ◽  
Function Solution ◽  
Dispersion Surface ◽  
Phase Calculation ◽  
Lee Waves ◽  
The Green Function

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.

Consistent Asymptotic Expansion Multiscale Formulation for Heterogeneous Column Structure

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Dongdong Wang ◽  
Pinkang Xie ◽  
Lingming Fang
Keyword(s):  
Asymptotic Expansion ◽  
Displacement Field ◽  
Three Dimensional ◽  
Axial Direction ◽  
Unit Cell ◽  
Local Unit ◽  
Cell Problem ◽  
Element Discretization ◽  
Free Condition ◽  
Column Structure

A consistent asymptotic expansion multiscale formulation is presented for analysis of the heterogeneous column structure, which has three dimensional periodic reinforcements along the axial direction. The proposed formulation is based upon a new asymptotic expansion of the displacement field. This new multiscale displacement expansion has a three dimensional form, more specifically, it takes into account the axial periodic property but simultaneously keeps the cross section dimensions in the global scale. Thus, this formulation inherently reflects the characteristics of the column structure, i.e., the traction free condition on the circumferential surfaces. Subsequently, the global equilibrium problem and the local unit cell problem are consistently derived based upon the proposed asymptotic displacement field. It turns out that the global homogenized problem is the standard axial equilibrium equation, while the local unit cell problem is completely three dimensional which is subjected to the periodic boundary condition on axial surfaces as well as the traction free condition on circumferential surfaces of the unit cell. Thereafter, the variational formulation and finite element discretization of the unit cell problem are discussed. The effectiveness of the present formulation is illustrated by several numerical examples.

Sign in / Sign up

Export Citation Format

Share Document