General procedure for evaluation of crack closure integral in problems of fracture mechanics

Abstract

Fracturebased design procedures ensuring safety in the presence of cracks, which cannot be detected with hundred percent reliability by NonDestructive Testing (NDT) methods, have become essential for aerospace and many other hightechnology applications. These design procedures, based on Linear Elastic Fracture Mechanics (LEFM), are considered adequate in most practical situations that involve only smallscale yielding around crack tips. Effective utilization of these procedures requires accurate estimation of fracture parameters such as stress intensity factors (SIF) or strain energy release rates (G), and fatigue crack growth rate in cracked bodies. Finite Element Method (FEM) has become the most popular and powerful tool for estimating these fracture parameters. Finiteelement stress analysis methods basically provide displacements and stress or strain distributions in a structure. In the presence of cracks, it becomes essential to use special postprocessing techniques to derive SIF or strainenergyreleaserate information from these distributions. The most popular methods are displacement or stress/force extrapolation for the estimation of SIF, and (i) virtual crack extension, (ii) Modified Crack Closure Integral (MCCI), (iii) Jintegral, and (iv) the recently developed Equivalent Domain Integral (EDI) for estimation of strain energy release rates (G). General loading patterns on conventional metallic structures and recent advances in structural constructions such as laminated composites and adhesively bonded joints have focused attention on the need for indepth study of mixedmode fracture problems. The MCCI method mentioned above has excellent potential for computationally efficient postprocessing of FEM output for the analysis of such problems. A systematic development of the MCCI technique is the main subject of investigation in this thesis. The Crack Closure Integral (CCI) is based on the concept proposed by Irwin that the strain energy release rate during virtual crack extension is equal to the energy required to close the crack back to its original size. Application of this concept to FE analysis is known as the Modified Crack Closure Integral (MCCI) technique. The MCCI method initiated by Rybicki and Kanninen for twodimensional crack problems modelled with 4noded quadrilateral elements has been used by several researchers for many practical problems. The MCCI expressions, which are elementdependent, were further modified for Linear Strain Triangular (LST) elements by Buchholz. However, a formal procedure for deriving appropriate elementdependent MCCI expressions has not been given till now. In the present thesis, a general procedure is presented for deriving MCCI expressions for both two and threedimensional crack problems and its utility is illustrated through several examples. First, a general procedure for estimating strain energy release rates using MCCI is proposed for twodimensional problems. This is demonstrated for both regular and singular elements. In mixedmode problems, the MCCI expressions used for regular elements are valid. For singular elements, the derivation of expressions involves certain modifications. The necessary changes in the procedure are indicated, and a few numerical examples of mixedmode problems are presented to illustrate the procedure. This procedure is later extended to threedimensional crack configurations. Here, the structure is modelled with 8noded brick elements, and the crack closure integral is estimated over an area in the plane of virtual crack extension. The evaluation of this integral is initially carried out over the entire face of each element along the crack front. This provides the distribution of the average value of strain energy release rate G for each element along the crack front. By this process, one obtains an average value of G for each element used along the crack front. To obtain accurate Gdistributions along the crack front, it is necessary to estimate G at a number of points along the crack front. This is achieved during the postprocessing stage by dividing each element into a number of subdivisions and evaluating the energy required to close each of these subdivisions in each element. This subarea integration, an effective postprocessing technique, has been introduced and used in the present thesis. Special cracktip elements such as 5noded triangular and 6noded quadrilateral elements for twodimensional problems, and 12noded brick elements for threedimensional problems, are developed. Numerical studies are presented to demonstrate the computational economy of using these elements at the crack tip/front along with the MCCI method. Application of the special cracktip elements developed along with the general MCCI procedure is demonstrated for the analysis of a pinloaded lug with diametrically opposite cracks originating from the hole. Besides the estimation of SIF, the analysis has to deal with a nonlinear contact problem at the pinhole interface. The present software is used to predict the fatigue crack propagation (FCP) life estimation on the same configuration using LEFM and wellknown crackgrowth models. The analytical life estimates are compared with available experimental results. With the developments presented in this thesis, the application of the MCCI procedure is expected to become much wider, particularly with regard to its application to mixedmode crack problems, and to achieve economical and accurate solutions for many structural problems where LEFM is applied.