Improved hybrid solid element with a traction‐free cylindrical surface

Abstract

SUMMARYNew 8-node solid elements with two parallel faces and one traction-free cylindrical surface are derived using the assumed stress hybrid model. Six new expressions of stress components are developed by using four stress functions and cylindrical co-ordinates, so that the normal stress 0, on the plane perpendicular to the two parallel faces varies as a parabola, and the assumed stress field satisfies the equilibrium equations as well as the traction-free conditions over the cylindrical boundary. The assumed stress field also satisfies the compatibility conditions when the new element is degenerated into the 2-dimensional case. Examples have clearly demonstrated that these present special elements are far superior in predicting the stress concentration factors, the distributions of circumferential stresses and the normal stress 6 , .

Stress analysis around holes or notches by special finite elements

“…In the following, two different strategies for selecting the higher-order interpolation are examined. The first consists of introducing additional nodes, and in particular midside nodes, as suggested in References [7,8] for three-dimensional elements and in References [11][12][13][14][15][16][17][18][19] for plane elements. The second consists of introducing additional drilling degrees of freedom in the existing corner nodes, as proposed in Reference [11].…”

“…In fact, it has been found that the stress distributions obtained by enforcing the traction-free condition a posteriori may be inaccurate [6]. On the contrary, special mixed finite elements based on assumed stresses which obey a priori the condition of vanishing for tractions on the cavity rim have been successfully applied to various problems, such as for example biand three-dimensional elasticity [6][7][8][9][10][11][12], fracture mechanics [13][14][15][16][17] and coupled deformationdiffusion [18,19]. Of course, such improvements are achieved only if a proper balance between the approximations assumed for static and kinematic variables is preserved as it always occurs for mixed models.…”

“…Modelling the stress as an independent variable [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], could be a viable alternative for representing traction-free boundaries, as the traction boundary constraints can be handled easier. In particular, the representation of tractionfree boundaries can be accomplished a posteriori through the variational principle [5,6], or a priori by choosing appropriate stress approximations [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The latter approach appears to be superior.…”

Studies of Stress Concentration by Using Special Hybrid Stress Elements

“…where the matrices G and H are determined by equation (11). In the following numerical examples the sti ness matrices k of these kinds of element are calculated by using the simple equation (13).…”

Straight free-edge effects in laminated composites

“…In general, the convergence rate for the ÿnite element method is dominated by the nature of the solution in the vicinity of steep stress gradient. The regular high-accuracy element obtained by using polynomials of high order as interpolating functions cannot improve the rate of convergence [19][20][21]. So, more computational di culty is encountered for the small aspect ratios for the free-edge problems.…”