A THREE-DIMENSIONAL PROBLEM FOR HIGHLY ELASTIC MATERIALS SUBJECT TO CONSTRAINTS

1958 ◽  
Vol 11 (1) ◽  
pp. 88-97 ◽  
Author(s):  
J. E. ADKINS
Keyword(s):  
Three Dimensional ◽  
Dimensional Problem ◽  
Elastic Materials ◽  
Highly Elastic Materials

Effects of Type of Rubber, Temperature, and Degree of Crosslinking on the Friction Properties of Elastomers

1960 ◽  
Vol 33 (4) ◽  
pp. 1166-1179 ◽  
Author(s):  
G. M. Bartenev ◽  
Z. E. Styran
Keyword(s):  
Energy Barrier ◽  
Three Dimensional ◽  
Static Friction ◽  
Effective Area ◽  
Elastic Materials ◽  
Static Modulus ◽  
Secondary Bonds ◽  
The Coefficient Of Friction ◽  
Rubbing Pair ◽  
Highly Elastic Materials

Abstract 1. The data on the influence of temperature and rate of slip on the frictional action of elastomers confirms the results of Schallamach for compounds of natural rubber and indicates that the external friction of highly elastic materials is a molecular-kinetic process of transition of chains, which adhere to the other rubbing surface, over an energy barrier under the action of an external force and thermal motion. 2. The energy barrier, which is dependent on the force of adhesion of the rubbing materials (energy of activation), is changed only slightly by changing the temperature, load and crosslink density for compounds from the same rubber. A greater change in the activation energy occurs by going from one polymer to another (from 18 to 32 kcal/mole). It is higher for polar rubbers (SKN) than for nonpolar rubbers (NR, SKBM, SKS). 3. Independently of the nature of the rubbing pair, the friction of elastomers follows Formula (2) over the range of slip velocities of 0.001–10 mm/min. In spite of the use of extremely small slip velocities, such a limit was not reached which might be attributed to the existence of actual static friction with the highly elastic materials. On the contrary, the drop in the coefficient of friction with decreased rate of slip became sharper the smaller the rate. For all elastomers there is observered a drop in the force of friction up to 60–80°. Above this region is an anomolous portion of the curve, the more sharply expressed, the greater the polarity of the rubber. The anomolous change in the force of friction is connected with the increase in the effective area of contact resulting from the destruction of the secondary bonds in the three dimensional network and a decrease in the static modulus of the compound. 5. High modulus compounds give a linear relationship for the force of friction up to 100–120° while low modulus compounds have a deviation in their linearity which appears, the earlier, the lower the equilibrium modulus. The slope of the linear portion of these compounds is greater than that of the lower modulus compounds which is connected with the increase in the effective area of contact at the transition from a hard to a soft compound. 6. At small normal loads (to 3 kg/cm2) the properties of the law of friction may be described by Coulomb's law. The force of friction is a linear function of the effective area of contact.

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Some Remarks on the Behaviour of Surface Source Distributions near the Edge of a Body

1973 ◽  
Vol 24 (1) ◽  
pp. 25-33
Author(s):  
J W Craggs ◽  
K W Mangler ◽  
M Zamir
Keyword(s):  
Flow Direction ◽  
Three Dimensional ◽  
Cross Flow ◽  
Symmetric Case ◽  
The Body ◽  
Dimensional Problem ◽  
Surface Source ◽  
Asymmetric Case ◽  
Main Flow Direction ◽  
Flow Plane

SummaryWhen the incompressible potential flow past a three-dimensional body is represented by source distributions on the body surface, these source distributions have singularities near an edge or corner, for example á trailing edge of a wing or the (unfaired) intersection of a body and a wing. The nature of these singularities is discussed. When assuming slow variations of the geometry in the main flow direction we can consider a two-dimensional problem in the cross-flow plane. Here the tangential velocities and source distributions are proportional to certain powers of the distance from the corner. For example at a convex right-angled corner these powers are − ⅓ in the asymmetric case (the bisector is a potential line) and ⅓ in the symmetric case (the bisector is a streamline) for both sources and tangential velocities. At a concave right-angled corner the corresponding values for the source distributions are ⅓ (asymmetric case) and − ⅓ (symmetric case) whereas they are 1 and 3 respectively for the tangential velocities.

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