Special Crack Tip Elements Research Articles

A Three-Dimensional Finite Element Analysis of the Double-Torsion Test

A special three-dimensional crack-tip element has been developed to investigate a simple and widely applicable fracture toughness test method. Previous experimental work with the double-torsion method has shown that the use of a relatively thin sectioned specimen may be permitted. The section concerned is considerably thinner than that used in conventional techniques, while the technique also simplifies the determination of the fracture toughness parameter. K IC values, which are independent of the crack length, have been obtained for glasses, ceramics, polymers, and a variety of metals and alloys. The numerical solution presented is supportive of many experimental observations made during testing. Excellent correlation between the finite element and experimental results has been obtained. The maximum stress intensity factor is shown to be almost independent of crack length over a considerable range.

The collapsed cubic isoparametric element as a ingular element for crack probblems

AbstractFor the 12−node quadrilateral isoparametric elements, it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by the simple technique of collapsing the quadrilateral elements into triangular elements around the crack tip and placing the two side nodes of each side of the triangles at 1/9 and 4/9 of the length of the side from the tip. This is analogous to placing the mid−side nodes at quarter points in the vicinity of the crack tip for the quadratic isoparametric elements. The advantages of this method are that the displacement compatibility is satisfied throughout the region and that there is no need of special crack tip elements. The stress intensity factors can be accurately obtained by using general purpose programs having isoparametric elements such as NASTRAN.

On the use of special crack tip elements in cracked elastic sheets

This paper presents a finite element method for determining the stress intensity factor in a cracked elastic sheet. Special cracked elements are placed around each crack tip; in each special element the stresses and displacements are derived from the exact stress function while the continuity of the displacements at the nodes is satisfied in a least square sense. A general procedure for evaluating the stiffness matrix of a cracked isotropic, or orthotropic, element is presented, and the numerical results obtained are compared with exact analytical results.

On the use of isoparametric finite elements in linear fracture mechanics

AbstractQuadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three‐dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three‐dimensional problems.

Crack tip finite elements are unnecessary

AbstractIt is shown that a singularity occurs in isoparametric finite elements if the mid‐side nodes are moved sufficiently from their normal position. By choosing the mid‐side node positions on standard isoparametric elements so that the singularity occurs exactly at the corner of an element it is possible to obtain quite accurate solutions to the problem of determining the stress intensity at the tip of a crack. The solutions compare favourably with those obtained using some types of special crack tip elements, but are not as accurate as those given by a crack tip element based on the hybrid principle. However, the hybrid elements are more difficult to use.

Application of quadratic isoparametric finite elements in linear fracture mechanics

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

A stiffness derivative finite element technique for determination of crack tip stress intensity factors

A finite element technique for determination of elastic crack tip stress intensity factors is presented. The method, based on the energy release rate, requires no special crack tip elements. Further, the solution for only a single crack length is required, and the crack is “advanced” by moving nodal points rather than by removing nodal tractions at the crack tip and performing a second analysis. The promising straightforward extension of the method to general three-dimensional crack configurations is presented and contrasted with the practical impossibility of conventional energy methods.