In this work, an improvement in the stiffness derivative method based on a shape design sensitivity analysis is proposed, so that the error inherent in the finite difference procedure is avoided. For a global estimation of G from a given finite element solution, this approach is shown to be equivalent to the well-known J-integral when the latter is numerically implemented through its equivalent domain integral. However, it is verified that its direct application to 2D mixed mode problems of linear elastic fracture mechanics through the field decomposition technique yields estimates for GI and GII which are in general more accurate for the proposed method. The importance of the velocity field is also remarked and some suggestions for its choice are given.
A number of numerical techniques exist which permit to obtain the SIFs or equivalently the SERR in linear elastic fracture mechanics (LEFM) from a FE solution. Among them, the so-called indirect methods are considered as the most accurate and efficient [1], [2], [3] and [4]. These methods are based on an energetic approach, and for them, the SERR G is the characterizing parameter. By contrast, direct methods yield an estimate of the SIF K through a local approach, without calculating the corresponding value of G. Therefore they need a refined mesh around the crack tip and the use of special crack tip elements. Generally speaking, all these methods are post-processing techniques which are applied after performing a numerical analysis, such a FE analysis (FEA).
Some of the indirect methods most commonly used in the literature are certain contour integrals, such as the J integral [5], the stiffness derivative method, proposed independently by Parks [2] and Hellen [3], the EDI method [4], [6], [7] and [8] and other methods based on a virtual crack closure approach [9] and [10]. One of the main disadvantages of indirect or energetic methods when compared to local methods in LEFM is their difficulty to give independent estimations of the SIF associated with each crack opening mode, KI, KII and KIII in mixed mode problems. In fact, this is due to their global character, which does not lend itself to the separate evaluation of the SERR associated with each mode GI, GII and GIII. Several authors have proposed different techniques to uncouple the values of G for each mode, making use of contour integrals, usually restricted to 2D problems. These methods are basically related to either the field decomposition technique [11], [12] and [13], the use of Jk integrals (k=1,2) [14] combined with their relationships with KI,KII [15], [16] and [17], or other techniques such as the M1 integral method [18]. Some authors have combined the application of these mode separation techniques with the EDI method [19] and [20], which has proved specially useful for 3D analyses.
On the other hand, certain energetic methods usually utilized for the estimation of G or J (particularly all those related to the virtual crack extension concept) can be interpreted as a shape design sensitivity analysis (SDSA). Typical SDSA methods are all based on a domain whose contour varies by means of a change of one or more design variables. These SDSA methods can be applied to a LEFM problem in a straightforward manner, simply by assuming that there is a single design variable in 2D problems: the crack length. Its extension produces a change in the contour (i.e. a shape change), and consequently a change in some quantities of interest, such as the total potential energy of the system Π or the strain energy U. The aim of a SDSA is precisely to obtain the variation of the structural response when a change in a design variable is induced.
The approach presented in this paper makes use of the so-called discrete analytical method of sensitivity analysis. This method allows the analytical evaluation of the stiffness derivative, avoiding the finite difference approximation introduced when the stiffness derivative method is used in its classical way [2], [3] and [21]. It is also shown that this improved way of applying the stiffness derivative method is exactly equivalent to the EDI method, when both are implemented numerically through the FE method (FEM). Moreover, the proposed method is applied to mixed mode 2D problems using a field decomposition technique, yielding in general better estimations of GI, GII than the values obtained through the direct application of the field decomposition technique to the J contour integral [11], [12] and [13] or the EDI method [19] and [20]. This improvement can be crucial when dealing with error estimation, since an error estimator needs a confident evaluation of the quantity of interest, such as the one provided by the proposed method.
The remainder of the paper is organized as follows. First, a brief review of the stiffness derivative method and its interpretation as a discrete analytical method of sensitivity analysis will be given. Then, this approach is related to the EDI method, with a detailed proof of their equivalence given in Appendix A. Following that, the proposed methodology is extended to 2D mixed mode problems. This extension is numerically verified by means of three mixed mode examples. Finally, the results are discussed and some remarks regarding the choice of the so-called velocity field, which is of paramount importance in every SDSA, are given.
In this work an improvement in the FE implementation of the stiffness derivative method based on a SDSA has been proposed. The discrete analytical approach of SDSA avoids the classical finite difference approximation and the corresponding truncation error. It has been shown that this procedure is numerically equivalent to the evaluation of J through a domain integral (EDI method) when applied to LEFM problems, provided the same velocity field is used. Moreover, it has been verified through numerical examples that when both approaches are applied to mixed mode problems using a domain decomposition technique, the proposed method yields better results and gives a smoother convergence when an h-adaptive procedure is used. This improvement can be of foremost importance when dealing with error estimation. Numerically the proposed method has the advantage of being consistent with a single decomposition of the FE displacement field. Several velocity fields have been utilized to solve the numerical examples and some remarks about their features have been included in order to achieve good numerical estimations.
