Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique
Introduction
In recent years, due to the growing demands for predicting the behavior of cracked structures during impact and crash events, the importance of fracture mechanics has increased considerably in engineering applications. To evaluate structural integrity under dynamic loading, it is fundamental to know the transient fracture parameters such as dynamic stress intensity factors and/or dynamic strain energy release rates. Some analytical solutions are available but are limited to idealized materials, simple geometrical configurations and specific loading histories [1], [2], [3], [4], [5], [6]. These fracture parameters may also be obtained by experimental measurements [7], [8], [9], [10], [11], [12]. However, the experiment procedure is very time consuming and expensive and is not applicable to most engineering materials.
Due to rapid and extensive developments in both computational hardware and software, numerical methods are becoming economical and feasible for performing dynamic fracture analyses on practical structures with novel materials, having complex geometry and under varied loading history. Atluri and Nakagaki [13] even suggest that numerical methods are mandatory to deal with practical situations of fractures. Many numerical methods such as finite difference method (FDM) [14], [15], boundary element method (BEM) [16], [17], [18], [19], [20], [21] and meshless Petrov-Galerkin method (MPGM) [22], [23] have been attempted.
With their own finite element method (FEM) codes, Aberson et al. [24] and Aturi and Nishioka [25], [26] developed singular elements to compute dynamic stress intensity factors (DSIF) for both propagating and stationary cracks. Sun and Jih [27] used the virtual crack closure technique (VCCT) to compute dynamic strain energy release rates. Their work is invaluable in providing a basic understanding for dynamic fracture mechanics. However, these codes have not been commercialized and therefore, are not available to engineers who need to perform dynamic fracture analysis in an industrial setting. On the other hand, the universal finite element analysis (FEA) software which is widely and successfully used in industry may not provide fracture parameters. Extra post-processing is commonly needed to calculate fracture parameters based on the FEA output primary variables such as nodal displacements and forces.
To fill the gap between the commercial FEA and fracture mechanics analysis, this paper describes an interface element developed to accommodate fracture analysis as a user modification to FEA commercial software. The interface element uses VCCT as described by the authors [28], [29], [30], [31]. The VCCT was initially proposed by Rybicki and Kanninen [32]. It is one of the most efficient and robust tools to compute strain energy release rates within one step analysis without using singular elements. The proposed interface element has five nodes with a very stiff spring between the node pair at the crack tip to calculate the internal forces. Three dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. All the components required to calculate the strain energy release rates (GI and GII) based on VCCT are available within the interface element and therefore, the strain energy release rates can be calculated and output simultaneously as the FEA software performs conventional analysis. It has been successfully implemented within ABAQUS® through a user element UEL for crack kinking [28] and crack growth in 2D [29] and 3D [30], [31] under static loading.
This paper further validates the interface element for stationary cracks subjected to the dynamic loading. Five examples of 2D panels with different crack locations and orientations under impulsive loading (Heaviside function) were examined. The dynamic stress intensity factors converted from dynamic strain energy release rates output from the present interface element are compared with those obtained by others with FDM, BEM or FEM. Further, Kalthoff tests on the notched bend specimen under impact loading were fully simulated. The losts of contact (LOC), the most significant observation in the test, was captured by applying contact condition between the load and support rollers and the specimen. The dynamic stress intensity factors calculated by the present interface element with ABAQUS® agree well both quantitatively and qualitatively with those measured by Kalthoff using the optical method of caustics. Therefore, this interface element technique is a powerful tool to perform failure analysis during crash and collision for engineering structures at industrial level via commercial FEA software.
Section snippets
Interface element based on VCCT
The VCCT was first proposed for 2D crack configurations by Rybicki and Kanninen [32] and was extended to three dimensions (3D-VCCT) by Shivakumar et al. [33]. Recently, Krueger [34] presented a summary of historical development and a discussion with respect to different applications.
Fig. 1 shows the definition and node numbering of the VCCT interface element for 2D fracture problems. When the element is applied, it is placed in such a way that the nodes 1 and 2 are located at the crack tip,
Validation examples
To demonstrate the capability and reliability of the VCCT interface element for solving dynamic crack problems in linear fracture mechanics, 2D crack problems with existing numerical or experimental results are solved and results are compared. In order to compare with the dynamic stress intensity factors ( and ) available in the literature, they are related to the strain energy release rate (, ) for isotropic materials as follows [35].
For plane stress
Conclusions
This paper describes an interface element for 2D-VCCT for use in dynamic fracture analysis. The element has been implemented into commercial FEA software ABAQUS® via the user element subroutine UEL. The element is evaluated with five examples of impulsive loading and two examples of impact loading. The results from the present interface element agree well with the available numerical and experimental results obtained by the others. With this interface element, fracture mechanics can be directly
Acknowledgements
Financial support for this work was provided by NASA, the State of South Carolina, and Clemson University through the EPSCoR grant “Development and Enhancement of Research Capability for Aircraft Structures and Materials.”
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