Almost-everywhere injectivity in nonlinear elasticity

1988 ◽  
Vol 109 (1-2) ◽  
pp. 79-95 ◽  
Author(s):  
Tang Qi
Keyword(s):  
Weak Convergence ◽  
Nonlinear Elasticity ◽  
Partial Regularity ◽  
Three Dimensional ◽  
Inverse Function ◽  
Sufficient Condition ◽  
Almost Everywhere ◽  
Elasticity Problems ◽  
Image Position ◽  
Three Dimensional Elasticity

SynopsisThis paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

An almost everywhere central limit theorem

1988 ◽  
Vol 104 (3) ◽  
pp. 561-574 ◽  
Author(s):  
Gunnar A. Brosamler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Finite Measure ◽  
Standard Normal Distribution ◽  
Almost Everywhere ◽  
Real Functions ◽  
Standard Normal ◽  
Image Position

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean thatfor all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

The method of caustics for the determination of normal loads acting on surfaces of elastic bodies

1980 ◽  
Vol 15 (1) ◽  
pp. 37-41 ◽  
Author(s):  
P S Theocaris ◽  
N I Ioakimidis
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Elastic Bodies ◽  
Plate And Shell ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Quantities Of Interest

The optical method of caustics constitutes an efficient experimental technique for the determination of quantities of interest in elasticity problems. Up to now, this method has been applied only to two-dimensional elasticity problems (including plate and shell problems). In this paper, the method of caustics is extended to the case of three-dimensional elasticity problems. The particular problems of a concentrated force and a uniformly distributed loading acting normally on a half-space (on a circular region) are treated in detail. Experimentally obtained caustics for the first of these problems were seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of various, more complicated, three-dimensional elasticity problems, including contact problems, by the method of caustics is also possible.

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