scholarly journals Surface instability of elastic half-spaces by using the energy method

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170854 ◽  
Author(s):  
Yi-chao Chen ◽  
Shengyou Yang ◽  
Lewis Wheeler
Keyword(s):  
Eigenvalue Problem ◽  
Half Space ◽  
Elastic Half Space ◽  
Surface Instability ◽  
Second Variation ◽  
Equilibrium Equations ◽  
Stability And Instability ◽  
Complete Set ◽  
Instability Regions ◽  
The Second Variation

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res. 12 , 168–182 ( doi:10.1007/BF03184638 )) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

2015 ◽  
Vol 7 (3) ◽  
pp. 295-322 ◽  
Author(s):  
Valeria Boccardo ◽  
Eduardo Godoy ◽  
Mario Durán
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Half Space ◽  
Plane Surface ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Quadratic Functional ◽  
Infinite Plane ◽  
Equilibrium Equations ◽  
Finite Element Software

AbstractThis paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Surface Instability and Splitting in Compressed Brittle Elastic Solids Containing Crack Arrays

1982 ◽  
Vol 49 (4) ◽  
pp. 761-767 ◽  
Author(s):  
L. M. Keer ◽  
S. Nemat-Nasser ◽  
A. Oranratnachai
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Special Class ◽  
Compressive Force ◽  
Surface Damage ◽  
Elastic Half Space ◽  
Surface Instability ◽  
Elastic Solids ◽  
Longitudinal Splitting ◽  
Brittle Solids

Surface instability in brittle solids may occur at relatively small values of the inplane compressive force if the solid contains shallow cracks parallel to its free surface. The instability may produce surface damage by spallation. Similarly, the buckling load of a longitudinally compressed strip that contains an array of central cracks is affected to a great extent by the size and the relative spacing of these cracks. The instability in this case may result in longitudinal splitting of the strip. To illustrate these phenomena, the compression of an elastic half space and a layer, each containing an array of coplanar equally spaced cracks, is studied for a special class of hypoelastic materials, and the corresponding weakening due to cracks is analytically estimated.

The Singularity at the Apex of a Bonded Wedge-Shaped Stamp

1979 ◽  
Vol 46 (3) ◽  
pp. 577-580 ◽  
Author(s):  
K. S. Parihar ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Solution ◽  
Integral Equations ◽  
Half Space ◽  
Singular Integral ◽  
Green's Functions ◽  
Green’S Functions ◽  
Singular Integral Equations ◽  
Elastic Half Space

The problem of determining the singularity at the apex of a rigid wedge bonded to an elastic half space is formulated by considerations of Green’s functions for the loaded half space. The eigenvalue problem is reduced to finding the solution of a coupled pair of singular integral equations. A numerical solution for small wedge angles is given.

scholarly journals Surface instability of an elastic half space with material properties varying with depth

2008 ◽  
Vol 56 (3) ◽  
pp. 858-868 ◽  
Author(s):  
Donghee Lee ◽  
N. Triantafyllidis ◽  
J.R. Barber ◽  
M.D. Thouless
Keyword(s):  
Half Space ◽  
Material Properties ◽  
Elastic Half Space ◽  
Surface Instability

Two-Dimensional Contact on an Anisotropic Elastic Half-Space

1994 ◽  
Vol 61 (2) ◽  
pp. 250-255 ◽  
Author(s):  
Hui Fan ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Contact Problem ◽  
Half Space ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Solution Procedure ◽  
Elastic Half Space ◽  
Two Dimensional ◽  
Anisotropic Elastic ◽  
The One

The two-dimensional contact problem for a semi-infinite anisotropic elastic media is reconsidered here by using the formalism of Es he I by et al. (1953) and Stroh (1958). The approach of analytic function continuation is employed to investigate the half-space contact problem with various mixed boundary conditions applied to the half-space. A key point of the solution procedure suggested in the present paper is its dependence on a general eigenvalue problem involving a Hermitian matrix. This eigenvalue problem is analogous to the one encountered when investigating the behavior of an interface crack (Ting, 1986). As an application, the interaction between a dislocation and a contact strip is solved. The compactness of the results shows their potential for utilization to solve the problem of contact of a damaged anisotropic half-space.

Stability of Elastic Half-Spaces by an Energy Method

1999 ◽  
Author(s):  
Yi-chao Chen ◽  
Lewis T. Wheeler
Keyword(s):  
Half Space ◽  
Unbounded Domain ◽  
Minimization Problem ◽  
Energy Method ◽  
Stability Criterion ◽  
Elastic Half Space ◽  
Stability Conditions ◽  
Second Variation ◽  
The Stability ◽  
First And Second Variation

Abstract An energy stability criterion is used to study the stability of deformations of a compressible elastic half-space. A minimization problem is formulated in an unbounded domain, and the first and second variation conditions are derived for this problem. Algebraic stability conditions are derived for general compressible isotropic materials, as well as for neo-Hookean class of Hadamard materials.

On the surface instability of a highly elastic half-space

1974 ◽  
Vol 4 (4) ◽  
pp. 249-263 ◽  
Author(s):  
S. A. Usmani ◽  
M. F. Beatty
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Surface Instability

scholarly journals Lamb's problem at its simplest

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120462 ◽  
Author(s):  
Eduardo Kausel
Keyword(s):  
Half Space ◽  
Full Range ◽  
Classical Problem ◽  
Elastic Half Space ◽  
Lamb’S Problem ◽  
Complete Set ◽  
Explicit Formulae ◽  
Poisson's Ratios ◽  
Poisson’S Ratios ◽  
Single Set

This article revisits the classical problem of horizontal and vertical point loads suddenly applied onto the surface of a homogeneous, elastic half-space, and provides a complete set of exact, explicit formulae which are cast in the most compact format and with the simplest possible structure. The formulae given are valid for the full range of Poisson's ratios from 0 to 0.5, and they treat real and complex poles alike, as a result of which a single set of formulae suffices and also exact formulae for dipoles can be given.

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