A necessary and sufficient boundary integral formulation for plane elasticity problems

Abstract

SUMMARYWith respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem.

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

“…For a degenerate-scale problem containing an elliptical boundary, two different degenerate scales are given in equation (15) and equation (16). The exact solution of displacement field is given by equation (4) and equation (5), subjected to the boundary conditions of equation (6), where the angle of rotation ( r  ), the Lamé constants G and  are given as 30°, 1.0 and 0.25, respectively.…”

“…According to the above analytical result, it is found that only the coefficients of the constant term could not be determined due to the zero denominator when the size of the domain is a degenerate scale of equation (15) and equation (16). Range deficiency by a constant term for the solution space appears.…”

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

“…For a degenerate-scale problem containing an elliptical boundary, two different degenerate scales are given in equation (15) and equation (16). The exact solution of displacement field is given by equation (4) and equation (5), subjected to the boundary conditions of equation (6), where the angle of rotation ( r  ), the Lamé constants G and  are given as 30°, 1.0 and 0.25, respectively.…”

“…According to the above analytical result, it is found that only the coefficients of the constant term could not be determined due to the zero denominator when the size of the domain is a degenerate scale of equation (15) and equation (16). Range deficiency by a constant term for the solution space appears.…”

Boundary elements with equilibrium satisfaction: a consistent formulation for plate bending analysis considering Reissner’s theory

“…This subject has already called the attention of other researchers who also discuss alternative procedures to achieve this goal, mainly to combine boundary and finite elements-the latter naturally satisfies the equilibrium. Here, the works by Khun [2], Dumont [3], Elleithy et al [4] and He et al [5], to name but a few, can be mentioned.…”

Some Aspects of the BEM Research in China

“…He found that the conventional boundary integral equations for 2D potential and elasticity problems are not equivalent to the boundary value problem of corresponding partial differential equation. He and his co-authors at Peking University and Zhejiang University, presented the sufficient and necessary boundary integral equation equivalent to the partial differential equation for the boundary value problems [34,35] .…”

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

“…For a degenerate-scale problem containing an elliptical boundary, two different degenerate scales are given in equation (15) and equation (16). The exact solution of displacement field is given by equation (4) and equation (5), subjected to the boundary conditions of equation (6), where the angle of rotation ( r  ), the Lamé constants G and  are given as 30°, 1.0 and 0.25, respectively.…”

“…According to the above analytical result, it is found that only the coefficients of the constant term could not be determined due to the zero denominator when the size of the domain is a degenerate scale of equation (15) and equation (16). Range deficiency by a constant term for the solution space appears.…”

Boundary elements with equilibrium satisfaction: a consistent formulation for plate bending analysis considering Reissner’s theory

“…This subject has already called the attention of other researchers who also discuss alternative procedures to achieve this goal, mainly to combine boundary and finite elements-the latter naturally satisfies the equilibrium. Here, the works by Khun [2], Dumont [3], Elleithy et al [4] and He et al [5], to name but a few, can be mentioned.…”

Some Aspects of the BEM Research in China

“…He found that the conventional boundary integral equations for 2D potential and elasticity problems are not equivalent to the boundary value problem of corresponding partial differential equation. He and his co-authors at Peking University and Zhejiang University, presented the sufficient and necessary boundary integral equation equivalent to the partial differential equation for the boundary value problems [34,35] .…”

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

“…For a degenerate-scale problem containing an elliptical boundary, two different degenerate scales are given in equation (15) and equation (16). The exact solution of displacement field is given by equation (4) and equation (5), subjected to the boundary conditions of equation (6), where the angle of rotation ( r  ), the Lamé constants G and  are given as 30°, 1.0 and 0.25, respectively.…”

“…According to the above analytical result, it is found that only the coefficients of the constant term could not be determined due to the zero denominator when the size of the domain is a degenerate scale of equation (15) and equation (16). Range deficiency by a constant term for the solution space appears.…”

Boundary elements with equilibrium satisfaction: a consistent formulation for plate bending analysis considering Reissner’s theory

“…This subject has already called the attention of other researchers who also discuss alternative procedures to achieve this goal, mainly to combine boundary and finite elements-the latter naturally satisfies the equilibrium. Here, the works by Khun [2], Dumont [3], Elleithy et al [4] and He et al [5], to name but a few, can be mentioned.…”

Some Aspects of the BEM Research in China

“…He found that the conventional boundary integral equations for 2D potential and elasticity problems are not equivalent to the boundary value problem of corresponding partial differential equation. He and his co-authors at Peking University and Zhejiang University, presented the sufficient and necessary boundary integral equation equivalent to the partial differential equation for the boundary value problems [34,35] .…”