Application of quadratic isoparametric finite elements in linear fracture mechanics

Abstract

Quadratic isoparametri c elements are shown to embody 1//r singularity f or c alculating s tress i ntensity f actors of elastic f racture mechanics. The singularity is obtained by placing the mid-side node on any side at the quarter p oint. Figure 1 shows the 2-dimensional, 8-noded quadrilateral (a) and 6-noded triangle (b), isoparametric e lements with the mid-side nodes near the crack tip at the quarter nodes. Figure 2 shows the S-dimensional elements w ith the mid-side node near the crack edge at the quarter p oints. The local s trains in these e lements vary as 1//r throughout the element. In the 3-D case, the strains along the crack edge are non-singular. A very important feature of these elements is that they satisfy the necessary requirements for convergence [I] in their singular form as well as in their non-singular form. They, therefore, pass the patch test [2], possess rigid body motion (R.B.M.), constant strain modes, interelement compatibility, and continuity of displacements. In contrast, other special crack tip elements [3,4], do not possess rigid body motion modes and do not pass the patch test, thus making their use in the problems cited below questionable. The existence of rigid body motion and constant strain modes in the proposed isoparametric elements allows the calculation of stress intensity factors for thermal gradients in 2- and 3-D problems and in problems where symmetry about the crack cannot be invoked (R.B.M. exists). In addition, since these elements are part of the element library of most general purpose programs, their use in linear fracture mechanics is very tractable. The element formulation in its non-singular form is well documented ([I], pp. 103-154). The element in the singular form is formulated exactly in the same manner except for a restriction on the location of the nodal points. In summary, the element is formulated by mapping its geometry from the cartesian space into a unit curvilinear space using special quadratic functions [i]. The same functions, in the curvilinear space, are used to interpolate the displacements within the element, hence the name isoparametric. In order to achieve the required singularity, the Jacobian of transformation [J], from the cartesian to the curvilinear space, is made singular by placing the mid-side nodes near the crack tip at the quarter points. The singularity occurs only at the crack tip point. It can be easily shown, for example, that for the rectangular form of the case in Figure la, the strain in the local x-direction along the line i-2, is given by