Torsional rigidity of an elliptic bar with multiple elliptic inclusions using a null-field integral approach

Abstract

Following the success of using the null-field integral approach to determine the torsional rigidity of a circular bar with circular inhomogeneities (Chen and Lee in Comput Mech 44(2):221-232, 2009), an extension work to an elliptic bar containing elliptic inhomogeneities is done in this paper. For fully utilizing the elliptic geometry, the fundamental solutions are expanded into the degenerate form by using the elliptic coordinates. The boundary densities are also expanded by using the Fourier series. It is found that a Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integral and cancel out to each other. Null-field points can be exactly collocated on the real boundary free of facing the principal values using the bump contour approach. After matching the boundary condition, a linear algebraic system is constructed to determine the unknown coefficients. An example of an elliptic bar with two inhomogeneities under the torsion is given to demonstrate the validity of the present approach after comparing with available results.

Bilateral bounds for the shear and torsion factors: comments on elementary derivations

“…An upper bound can be computed, following Prager (cf. Renton [11] and Chen et al [5]), by considering that a congruent (but not necessarily equilibrated) displacement field gives an upper bound of the total potential energy of the equilibrated solution of the static problem. However, for a "generic" cross-section, it is very hard to guess the appropriate form of the warping in order to obtain a realistic small estimate of the required upper bound.…”

“…In the last months, a number of papers addressed the problem of shear and torsion factors of an elastic beam, χ V and χ T , respectively, with special regard to the coherence of their definitions (Dong et al [1], Vasil'ev [2]) and the evaluation of some bounds with or without recourse to an explicit solution of the skew shear-bending and torsion problem (Favata et al [3], Chen [4], Chen et al [5]) (reference will be made to χ i , even if the reduction coefficient k i = 1/χ i often appears in literature).…”

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

“…Chen and Y.Z. Chen in [8][9][10][11][12][13][14][15][16] seems to be more focused on the degenerate scale problems. Our efforts are mainly paid for establishing effective algorithms by following [23,27].…”

“…In Ang and Kang [1], complex boundary elements are studied. For circular and elliptic domains, the NFM is developed by Chen's group, see [8][9][10][11][12][13]. Recently, the explicit collocation equations and semi-analytic solutions are derived in Li et al [25], and the conservative schemes are developed in Lee et al [19].…”

“…The particular solutions (PS) and fundamental solutions (FS) in polar coordinates can be found in many textbooks, but with much less coverage in elliptic coordinates [12,28,29]. Since the elliptic domains with elliptic holes may be found in some engineering problems, the PS and the FS expansions in elliptic coordinates are essential for numerical computations.…”