Relaxation of Thick-Walled Cylinders and Spheres

1982 ◽  
Vol 49 (3) ◽  
pp. 487-491 ◽  
Author(s):  
N. S. Ottosen
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Approximate Solutions ◽  
Nonlinear Creep ◽  
Closed Form Solutions ◽  
Cylinders And Spheres

Using the nonlinear creep law proposed by Soderberg, closed-form solutions are derived for the relaxation of incompressible thick-walled spheres and cylinders in plane strain. These solutions involve series expressions which, however, converge very quickly. By simply ignoring these series expressions, extremely simple approximate solutions are obtained. Despite their simplicity these approximations possess an accuracy that is superior to approximations currently in use. Finally, several physical aspects related to the relaxation of cylinders and spheres are discussed.

On the Prediction of the Strain Rate Intensity Factor in Metal Forming Processes

2008 ◽  
Author(s):  
Elena Lyamina
Keyword(s):  
Intensity Factor ◽  
Strain Rate ◽  
Closed Form ◽  
Upper Bound ◽  
Approximate Solutions ◽  
Closed Form Solutions ◽  
Perfectly Plastic ◽  
Strain Rate Intensity Factor ◽  
Rate Intensity ◽  
Strain Rate Intensity

The strain rate intensity factor is the coefficient of the principal singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. Such singular behaviour occurs in the case of several rigid plastic models (rigid perfectly plastic solids, the double-shearing model, the double slip and rotation model, some of viscoplastic models). Since it is only possible to introduce the strain rate intensity factor for singular velocity fields, it is obvious that standard finite element codes cannot be used to calculate it. The currently available distributions of the strain rate intensity factor have been found from closed form solutions or with the use of simple approximate solutions (for instance upper bound solutions). Closed form solutions are available for boundary value problems with simple geometry (flow through infinite rough channels, compression of infinite layers between rough plates and so on) and, therefore, are mostly of academic interest. Simple approximate solutions can predict general tendencies in the distribution of the strain rate intensity factor but cannot predict its distribution with a sufficient accuracy for industrial applications. For, the strain rate intensity factor reflects a very local effect inherent in the velocity field whereas simple approximate methods, such as the upper bound method, estimate global parameters, such as the limit load. The purpose of the present research is to propose a special numerical technique for calculating the strain rate intensity factor in the case of plane strain deformation of rigid perfectly plastic materials and to verify it by means of comparison with an analytical solution. The technique is based on the method of Riemann.

scholarly journals Closed-form solutions for the effective conductivity of two-phase periodic composites with spherical inclusions

2013 ◽  
Vol 469 (2151) ◽  
pp. 20120339 ◽  
Author(s):  
Q. D. To ◽  
G. Bonnet ◽  
V. T. To
Keyword(s):  
Closed Form ◽  
Effective Conductivity ◽  
Approximate Solutions ◽  
Static Structure Factor ◽  
Static Structure ◽  
Two Phase ◽  
Closed Form Solutions ◽  
Periodic Composites ◽  
Spherical Inclusions ◽  
Volume Limit

In this paper, we use approximate solutions of Nemat-Nasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.

POSTBUCKLING OF COMPRESSIVELY LOADED IMPERFECT COMPOSITE PLATES: CLOSED-FORM APPROXIMATE SOLUTIONS

2010 ◽  
Vol 10 (04) ◽  
pp. 761-778 ◽  
Author(s):  
CHRISTIAN MITTELSTEDT ◽  
KAI-UWE SCHRÖDER
Keyword(s):  
Closed Form ◽  
Composite Plates ◽  
Approximate Solutions ◽  
Laminated Plates ◽  
Engineering Practice ◽  
Nonlinear Behaviour ◽  
State Variables ◽  
Shape Functions ◽  
Closed Form Solutions ◽  
Out Of Plane

In this paper, closed-form approximate solutions for the geometrically nonlinear behaviour of rectangular laminated plates with flexural orthotropy under longitudinal compression are presented. Based on the governing Marguerre-type differential equations postulated for imperfect plates, two plate configurations are discussed in detail, representing important application cases in practical engineering work. The first configuration is a laminated plate that is simply supported at all four edges (the so-called SSSS plate), while for the second configuration clamped unloaded longitudinal edges are considered (denoted as the SSCC plate). For both plate configurations, rather simple closed-form approximations in the form of trigonometric shape functions are employed for the description of the out-of-plane postbuckling plate deflections. Based on the chosen shape functions, the compatibility condition with respect to the in-plane strains is fulfilled exactly, while the out-of-plane equilibrium condition for a deflected plate element is not, but is solved using a Galerkin-type formulation instead. Eventually, very simple closed-form solutions for all postbuckling state variables (deflections, in-plane edge displacements, and effective widths) are derived that can be used very conveniently in engineering practice. The high accuracy of the presented analysis methods is established by comparison with the results of other authors.

scholarly journals A point source in the presence of spherical material inhomogenity: Analysis of two approximate closed form solutions for electrical scalar potential

2009 ◽  
Vol 22 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Predrag Rancic ◽  
Miodrag Stojanovic ◽  
Milica Rancic ◽  
Nenad Cvetkovic
Keyword(s):  
Closed Form ◽  
Numerical Experiments ◽  
Scalar Potential ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Green Functions ◽  
Closed Form Solutions ◽  
Very Low Frequency ◽  
Periodic Current ◽  
Approximate Expressions

A brief review of derivation of two groups of approximate closed form expressions for the electrical scalar potential (ESP) Green functions that originates from the current of the point ground electrode (PGE) in the presence of a spherical ground inhomogenity, are presented in this paper. The PGE is fed by a very low frequency periodic current through a thin isolated conductor. One of approximate solutions is proposed in this paper. Known exact solutions that have parts in a form of infinite series sums are also given in this paper. Here, the exact solution is solely reorganized in order to facilitate comparison to the closed form solutions, and to estimate the error introduced by the approximate solutions. Finally, error estimation is performed comparing the results for the electrical scalar potential obtained applying the approximate expressions and accurate calculations. This is illustrated by numerous numerical experiments. .

On Solutions of Plane Strain Consolidation Problems by Finite Element Methods

1971 ◽  
Vol 8 (1) ◽  
pp. 109-118 ◽  
Author(s):  
C. T. Hwang ◽  
N. R. Morgenstern ◽  
D. W. Murray
Keyword(s):  
Finite Element ◽  
Numerical Analysis ◽  
Closed Form ◽  
Plane Strain ◽  
Excellent Agreement ◽  
Finite Element Methods ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Algebraic Equations ◽  
Closed Form Solutions

The formulation is presented of the equations governing the solution of true three-dimensional consolidation problems by means of finite element methods. The numerical procedure for handling the resulting algebraic equations is described briefly and detailed results from the solution of several problems are given. Excellent agreement is found between the numerical analysis and existing closed form solutions.

An Approach to Masonry Structural Analysis by the No-Tension Assumption—Part I: Material Modeling, Theoretical Setup, and Closed Form Solutions

2010 ◽  
Vol 63 (4) ◽  
Author(s):  
Alessandro Baratta ◽  
Ottavia Corbi
Keyword(s):  
Structural Analysis ◽  
Closed Form ◽  
Solution Process ◽  
Steel Structures ◽  
Approximate Solutions ◽  
Material Model ◽  
Masonry Structures ◽  
Closed Form Solutions ◽  
Structural Problems ◽  
Material Theory

The prevalent feature that characterizes masonry structures and makes them dissimilar from modern reinforced concrete and steel structures is quite definitely their inability to resist tensile stresses. Therefore, it is natural that the material model that is intended to be an “analog” of real masonry cannot resist tensile stress but possibly behaves elastically under pure compression, opening a perspective on the adoption of the no-tension (NT) material constitutive assumption. Founded on the NT theory, the basic direction and aim of the present work is to propose a unified treatment of masonry structures presenting an overall and comprehensive up-to-date insight in the analysis of masonry constructions while providing basic and advanced concepts and tools already available or original. The note is intended to lead to several notable results including, just to mention some of the outcomes, that the St. Venant’s postulate does not hold in NT solids, no energy is dissipated by fracture, special accommodations for discontinuous loads are needed, and the relevant developments are provided, among other significant outcomes such as the identification of operative procedures for engineering solutions of structural problems. In the first part of the paper, the basics for the foundation of a NT material theory are illustrated, and the relevant principles for structural analysis, mainly identified in the classic energy theorems and suitably adapted to the material at hand, are formulated. In (apparently) simple cases, closed-form solutions can be obtained, or, at least, the solution process can be prepared after a preliminary screening of the equilibrium scenario. The application to two-dimensional no-tension elasticity is then illustrated in the last section, with reference to two sample cases. The first example proves that the solution in a NT panel acted on by vertical loads on the top is clearly identified in terms of stress, but if discontinuities are present in the load pattern, these reflect in some strong singularities in the deformation field that requires to deepen the problem. The second example is concerned with a NT-elastic half-plane, and a technique to find approximate solutions is outlined and carried on in detail. In Part II of the note, it is demonstrated that discontinuous or lumped load patterns can be looked at as frontier cases: They are not in contrast, but in some sense they are not natural to NT solids. Still in Part II, it is shown how the equilibrium problem can be managed both through the strain and the stress approach.

A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
L. Rubio ◽  
J. Fernández-Sáez
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Direct Method ◽  
Approximate Solutions ◽  
Cracked Beam ◽  
Bending Vibrations ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
A Direct Method ◽  
Cracked Beams

The main goal of this note is to discuss the applicability of approximate closed-form solutions to evaluate the natural frequencies for bending vibrations of simply supported Euler–Bernoulli cracked beams. From the well-known model, which considers the cracked beam as two beams connected by a rotational spring, different approximate solutions are revisited and compared with those found by a direct method, which has been chosen as reference.

scholarly journals On the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutions

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
I. L. Ferreira ◽  
A. Garcia ◽  
A. L. S. Moreira
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Single Phase ◽  
Moving Boundary ◽  
Error Function ◽  
Solid State Diffusion ◽  
Two Phase ◽  
Closed Form Solutions ◽  
State Diffusion ◽  
Cylinders And Spheres

Analytical solutions for the transient single-phase and two-phase inward solid-state diffusion and solidification applied to the radial cylindrical and spherical geometries are proposed. Both solutions are developed from the differential equation that treats these phenomena in Cartesian coordinates, which are modified by geometric correlations and suitable changes of variables to achieve closed-form solutions. The modified differential equations are solved by using two well-known closed-form solutions based on the error function, and then equations are obtained to analyze the diffusion interface position as a function of time and position in cylinders and spheres. These exact correlations are inserted into the closed-form solutions for the slab and are used to update the roots for each radial position of the moving boundary interface. The predictions provided by the proposed analytical solutions are validated against the results of a numerical approach. Finally, a comparative study of diffusion in slabs, cylinders, and spheres is also presented for single-phase and two-phase solid-state diffusion and solidification, which shows the importance of the effects imposed by the radial cylindrical and spherical curvatures with respect to the Cartesian coordinate system in the process kinetics. The analytical model is proved to be general, as far as, a semi-infinite solution for diffusion problems with phase change exists in the form of the error function, which enables exact closed-form analytical solutions to be derived, by updating the root at each radial position the moving boundary interface.

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