Solutions to some classical problems in linear elastostatics

This chapter presents and discusses the solution of several classical problems in linear elastostatics, including thick-walled spheres and cylinders under external and internal pressure; bending and torsion of prismatic bars of arbitrary cross section; and the use of Airy’s stress function method to solve several two-dimensional plane strain and plane stress traction boundary value problems, including a demonstration of the extent of the Saint-Venant effect. The discussion also includes an analysis of the asymptotic stress and deformation fields near the tips of sharp cracks, and a discussion of stress intensity factors which are of importance in linear elastic fracture mechanics.

Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances

The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.

On displacement based non-local models for non-linear vibrations of thin nano plates

This paper addresses the formulation of displacement based, non-linear, plate models adopting Eringen's non-local elasticity, to study the modes of vibration of thin, nano plates. Plate models governed by ordinary differential equations of motion with generalized displacements as unknowns have some advantages over mixed type formulations, but difficulties arise in the development of such non-linear models when non-local effects are taken into account. To circumvent those difficulties, approximations of debatable justification can be imposed. Different approximations are discussed here and the accuracy of the best non-local, non-linear displacement based model achieved is put to test, by carrying out comparisons with a model based on Airy’s stress function.

Stress Solutions of some Axisymmetric and Non-Axisymmetric Cases with the Principles of Elasticity: A Review

Stress solutions are reviewed for some typical cases of axisymmetric and non-axisymmetric loads over a structural member with the principles of elasticity. A curved bar is chosen for the analysis. Tangential, radial and shear stress are determined analytically using Airy’s stress function. The curved bar is also modelled by finite element method to obtain numerical values of stress. Analytical and numerical results are in excellent agreement with each other.

The Fault in the Stress Analysis of Pseudo-Stress Function Method

Although Peng Yafei and his co-workers discovered some faults with the pseudo-stress function method suggested by Y. S. Lee in 1987, the authors did not provide convincing arguments. We investigate the crucial assumption in Lee’s method by rewriting it as the form of real part and imaginary part. Through a specific counterexample, we point out that the crucial assumption in Lee’s theory is untenable. Namely, for given Airy’s stress function, it cannot be guaranteed that the pseudo-stress function Λ(x,y) exists. The root cause of the fault with Lee’s method is found in this paper.