A Unified Second Order Boundary Element Method for Structures Analysis

Abstract

The classical modelizations used in structures analysis fail for singular situations giving high gradients. All numerical methods lead then to inaccurate or instable results, due to the geometrical and/or loading singularities.

High gradients zones (even very localized) are precisely damaging areas, especially responsible for fatigue collapses. Therefore, it seems particularly interesting to introduce a better modelization in order to eliminate the mentioned difficulties, and insuring by the same way a complete compatibility in complex structures analysis.

From the mathematical point of view, we are lead to construct a second order modelization with free rotations: kinematics of continua are defined by the first gradient of displacements (classical strains) and, further, by second order terms, homogeneous to second gradient components. In addition to classical (symmetrical) stress, we have to take into account skew-symmetric components and couple-stress which are second order terms. The second order quantities are negligible compared to the first order ones in low gradients zones, but they cannot be omitted in singular high gradients zones.

The main advantage of the new modelization is the existence of a common variables system used for all kinds of structures: three dimensional thick structures, shells, beams.

From the numerical point of view, the existence of a common variational principle gives after discretization a common system of algebraic equations, which is automatically reduced in function of geometrical data and in function of a test related to the mesh fineness, according to singular zones.

For the isotropic elastostatic case presented above, the four additional elasticity coefficients are defined in terms of the two classical ones and in function of a mesh size parameter, in order to recover the classical formulation in regular zones.

By using distribution’s theory, the field equations are transformed into two integral equations relating displacements, rotations, stress and couple-stress. Taking limits at collocation points on these relations, we obtain the boundary integral equations to be numerically solved.

The integral equations kernels are the elementary solutions corresponding to unit concentrated forces and couples. These elementary solutions are calculated by FOURIER’s transform and HORMANDER’s method. The numerical treatment is then the same as used for standard B.E. programs of CETIM.