A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems

ABSTRACT

The influence matrix may be of deficient rank in the specified scale when we have solved the 2D elasticity problem by using the boundary element method (BEM). This problem stems from $lnr$ in the 2D Kelvin solution. On the other hand, the single-layer integral operator can not represent the constant term for the degenerate scale in the boundary integral equation method (BIEM). To overcome this problem, we have proposed the enriched fundamental solution containing an adaptive characteristic length to ensure that the argument in the logarithmic function is dimensionless. The adaptive characteristic length, depending on the domain, differs from the constant base by adding a rigid body mode. In the analytical study, the degenerate kernel for the fundamental solution in polar coordinates is revisited. An adaptive characteristic length analytically provides the deficient constant term of the ordinary 2D Kelvin solution. In numerical implementation, adaptive characteristic lengths of the circular boundary, the regular triangular boundary and the elliptical boundary demonstrate the feasibility of the method. By employing the enriched fundamental solution in the BEM/BIEM, the results show the degenerate scale free.

KEYWORDS:

• Boundary element method
• 2D elasticity problem
• degenerate scale
• characteristic length

Nomenclature

a, b	=	The lengthes of semi-major and semi-minor axes of an ellipse
${\alpha }_j\left(\mathit{s}\right)$	=	The boundary density corresponding to the j direction.
$a_0^{\left(j\right)}$,$a_n^{\left(j\right)}$, $b_n^{\left(j\right)}$	=	The unknown coefficients of the fictitious densities for the j direction
B	=	Boundary of the problem
D	=	Domain of the problem
$f\left(x\right)$	=	Dirichlet boundary condition
$G$ and $\lambda$	=	Lame constants
$\kappa$	=	Material constant
L	=	An adaptive characteristic length
$l_T$	=	A side length of the triangle
$p_0^{\left(i\right)}$,$p_n^{\left(i\right)}$, $q_n^{\left(i\right)}$	=	The known coefficients of the boundary data for the i direction
r	=	The distance between source point and field point
$\gamma$	=	A weighting of the rigid body rotation
$R_0$	=	The radius of the circular domain
$\left(R,\theta \right)$	=	Position vector of the source point s in terms of polar coordinates
s	=	Position vector of the source point
$u\left(\mathit{x}\right)$	=	Vector of the displacement of the field point x
$u_i\left(x\right)$	=	Displacement of the i direction at the field point x
$U_{ij}(\mathit{x},\mathit{s})$	=	The fundamental solution or called Kelvin solution
$U_{ij}^c\left(\mathit{x},\mathit{s}\right)$	=	The enriched fundamental solution
$U_{ij}^r\left(\mathit{x},\mathit{s}\right)$	=	Modified fundamental solution
$\nu$	=	Poisson ratio
x	=	Position vector of the field point
$\mathrm{\nabla }$	=	Gradient operator
${\mathrm{\nabla }}^2$	=	Laplace operator
${\delta }_{ij}$	=	Kronecker delta function
$\tau$	=	An arbitrary constant
$\left(\rho ,\varphi \right)$	=	Position vector of the field point x in terms of polar coordinates

Disclosure statement

No potential conflict of interest was reported by the authors.