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.