Complex variable methods and closed form solutions to dynamic crack and punch problems in the classical theory of elasticity

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated force in the plane of the plate and in a direction tangential to the circle. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

Download Full-text

Stiffness of Annular Bonded Rubber Flanged Bushes

Rubber Chemistry and Technology ◽

10.5254/1.3547935 ◽

2006 ◽

Vol 79 (2) ◽

pp. 233-248 ◽

Cited By ~ 4

Author(s):

J. M. Horton ◽

G. E. Tupholme

Keyword(s):

Closed Form ◽

Finite Length ◽

Classical Theory ◽

Theory Of Elasticity ◽

Numerical Results ◽

Torsional Stiffness ◽

Radial Stiffness ◽

Physical Data

Abstract Closed-form expressions are derived for the torsional stiffness, radial stiffness and tilting stiffness of annular rubber flanged bushes of finite length in three principal modes of deformation, based upon the classical theory of elasticity. Illustrative numerical results are deduced with realistic physical data of typical flanged bushes.

Download Full-text

Axial Loading of Annular Bonded Rubber Blocks

Rubber Chemistry and Technology ◽

10.5254/1.3547797 ◽

2003 ◽

Vol 76 (5) ◽

pp. 1194-1211 ◽

Cited By ~ 4

Author(s):

J. M. Horton ◽

G. E. Tupholme ◽

M. J. C. Gover

Keyword(s):

Experimental Data ◽

Stress Distribution ◽

Closed Form ◽

Cross Section ◽

Classical Theory ◽

Cross Sections ◽

Axial Loading ◽

Theory Of Elasticity ◽

Governing Equations ◽

Annular Cross Section

Abstract Closed-form expressions are derived using a superposition approach for the axial deflection and stress distribution of axially loaded rubber blocks of annular cross-section, whose ends are bonded to rigid plates. These satisfy exactly the governing equations and conditions based upon the classical theory of elasticity. Readily calculable relationships are derived for the corresponding apparent Young's modulus, Ea, and the modified modulus, Ea′, and their numerical values are compared with the available experimental data. Elementary expressions for evaluating Ea and Ea′ approximately are deduced from these, in forms which are closely analogous to those given previously for blocks of circular and long, thin rectangular cross-sections. The profiles of the deformed lateral surfaces of the block are discussed and it is confirmed that the assumption of parabolic lateral profiles is not valid generally.

Download Full-text

Elastohydrodynamic Effects in a Clearance Seal

Journal of Lubrication Technology ◽

10.1115/1.3451400 ◽

1970 ◽

Vol 92 (2) ◽

pp. 310-313 ◽

Cited By ~ 3

Author(s):

N. M. Wang ◽

M. M. Kamal

Keyword(s):

High Pressure ◽

Fluid Flow ◽

Closed Form ◽

Linear Function ◽

Reynolds Equation ◽

Hydrodynamic Lubrication ◽

Theory Of Elasticity ◽

Exponential Dependence ◽

Closed Form Solutions ◽

Metal Seal

An elastohydrodynamic solution for a high-pressure, low-clearance metal seal is presented. The fluid flow is assumed to satisfy Reynolds equation of hydrodynamic lubrication, and the deformation of the shaft and the seal is governed by the linear theory of elasticity. The viscosity of the fluid is assumed to have an exponential dependence on the pressure, while the density of the fluid is a linear function of the pressure. Closed-form solutions are obtained for two asymptotic limiting cases: (i) when the length of the seal is much greater than the radius of the shaft, and (ii) when it is much less. For intermediate ratios of the seal length to shaft radius, solutions are obtained numerically and examples are given to show the effect of seal length on the rate of mass flow.

Download Full-text

The Elastic Plane With a Circular Insert, Loaded by a Radial Force

Journal of Applied Mechanics ◽

10.1115/1.3640419 ◽

1961 ◽

Vol 28 (1) ◽

pp. 103-111 ◽

Cited By ~ 54

Author(s):

J. Dundurs ◽

M. Hete´nyi

Keyword(s):

Closed Form ◽

Elastic Properties ◽

Classical Theory ◽

Radial Force ◽

Theory Of Elasticity ◽

Elastic Plane ◽

Elementary Functions ◽

Explicit Formulas

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated radial force in the plane of the plate. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

Download Full-text