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This paper presents the calculation of stress intensity factor (K) solutions for surface cracks in the thread ground of bolts subjected to axial loading directly applied by the nut. The stress-strain computations have been done by means of the finite element method with quarter-point singular isoparametric elements along the crack front. The stress intensity factor is calculated through the stiffness derivative method, by using a virtual crack extension technique to compute the energy release rate. Two modifications are made to improve the accuracy of the results: the displacement not only of the main node, but also of the quarter-point nodes located in the normal plane and the adjacent nodes in the crack line, avoiding both the change of the singularity and the crack curving. The results show that direct loading on the thread flank by a nut increases the stress intensity factor. This effect decreases with the crack length. For the deepest circular cracks, however, nut loading relaxes the K-value, mainly at the crack surface.
All numerical models simulating dynamic crack growth must incorporate a scheme for extending the crack. For example, in FEM, a node is released in the crack plane or in explicit MPM cracks the crack path is extended by a small amount [15,16]. Even in cohesive zone modelling, a crack grows when the crack opening displacement at the crack root reaches the cohesive law's critical value (δc) and traction drops to zero. In computational mechanics code that correctly conserves total energy, all these virtual crack extensions can cause an increase in kinetic energy that can quickly deteriorate numerical results. But, this conversion to kinetic energy does not reflect crack extension in real materials where that energy is instead absorbed by some surface processes representing the material's fracture toughness. One solution to dealing with artefacts in dynamic crack propagation simulations is to add damping to mimic energy absorption in real materials, but it is challenging to add realistic damping. In previous orthogonal cutting simulations, it was noted that a new form of damping, denoted as PIC damping , worked very well for crack propagation simulations. In brief, this damping focuses damping effects in regions with high velocity gradients and, therefore, selectively dampens regions around a propagating crack tip. Simulations with PIC damping enabled are extremely stable for all cutting conditions, while simulations without PIC damping were only stable for a few conditions. When they both work, they give nearly identical cutting forces except for far less noise when using PIC damping. All simulations here used the PIC damping method .
Based on the finite element software ABAQUS and graded elementmethod, we developed a dummy node fracture element, wrote the usersubroutines UMAT and UEL, and solved the energy release rate component offunctionally graded material (FGM) plates with cracks. An interface elementtailored for the virtual crack closure technique (VCCT) was applied. Fixedcracks and moving cracks under dynamic loads were simulated. The results werecompared to other VCCT-based analyses. With the implementation of a crackspeed function within the element, it can be easily expanded to the cases ofvarying crack velocities, without convergence difficulty for all cases.Neither singular element nor collapsed element was required. Therefore, dueto its simplicity, the VCCT interface element is a potential tool forengineers to conduct dynamic fracture analysis in conjunction with commercialfinite element analysis codes.
In recent years, there are growing concerns on how crackedfunctional material body responds to collision under impulse loading. Toaccurately evaluate the fracture mechanics under dynamic loading, researchersproposed dynamic fracture parameters, such as dynamic stress intensity factor(DSIF) and strain energy release rate (SERR). The dynamic fracture parameterof simple geometric model, ideal material model, or special load model can bedetermined by the analytical method. However, this method is not applicableto complex structure or boundary conditions, and its experimentalmeasurements are very expensive and time-consuming. Nevertheless, this typeof problems can be well resolved by numerical calculations.
At present, the finite element method (FEM) is widely used forfracture analysis in FGMs. For instance, a pair of FEM-based elastodynamiccontour integrals was developed to calculate the elastodynamic asymptoticmixed-mode stress field for plane elastic materials containing a stationarynotch tip . Graded finite elements can be used in fracture analysis inFGMs where the elastic moduli are smooth functions of spatial coordinates,which are integrated into the element stiffness matrix. The stress intensityfactors for mode I and mixed-mode two-dimensional problems can becomparatively evaluated through three FGMs-tailored approaches:path-independent J-integral, modified crack closure integral, anddisplacement correlation . The feasibility of FEM in cracked or uncrackedFGM plates was studied. The J contour integral of ABAQUS was used tocalculate stress intensity factors for an edge cracked FGM plate .Matthews used the finite element analysis (FEA) for large displacementJ-integral test to analyze mode I interlaminar fracture in compositematerials . The dynamic crack tip fields were determined, and the crackpropagation of anisotropic materials was also characterized . Theseprevious works are important; however, they only focus on the dynamic cracksof isotropic and orthotropic materials, but not on the direction of crackpropagation.
The methods used to resolve the fracture parameters includeJ-integral, M-integral, extrapolation, and virtual crack closure technique(VCCT). Among all fracture parameters, SERR is used increasingly inconjunction with linear elastic fracture mechanics (LEFM) and can be computedby VCCT together with FEA. VCCT requires a preexisting crack with a sharpneat tip within a material for crack initiation as well as conditions ofsmall-scale yield to hold. With material nonlinearity at the crack tip (smallprocess zone) ignored, LEFM-based approaches were proven effective inpredicting crack initiation and subsequent growth [21, 22].
A cohesive theory assumes the presence of a process zone in frontof the crack tip whose fracture properties consist of upper and lowersurfaces controlled by the cohesive traction-displacement discontinuityrelationship and allows non-self-similar crack propagation . An automatedfracture procedure implemented in the large-scale, nonlinear, and explicit,finite element code DYNA3D can be used to simulate dynamic crack propagationin arbitrary directions . Manolis et al. used boundary element method(BEM) to analyze the dynamic fracture of a smoothly inhomogeneous anddefective plane . Solving crack growth problems, the recent approach onsmoothed finite element methods is really a good candidate [40, 41]. The DSIFaround the antiplane crack in an infinite strip FGM under impact loading wasinvestigated . FG cracked plates under different loads and boundaryconditions were numerically simulated using NURBS-based XIGA . XIGA hasbeen applied to stationary and propagating cracks in 2D , plasticcollapse load analysis of cracked plane structures , and crackedplate/shell structures .
In this study, based on the commercial FEA software ABAQUS andgraded element method, we developed a dummy node fracture element, wrote theuser subroutines UMAT and UEL, and solved the energy release rate componentof cracked FGM plates.
 K. N. Shivakumar, P. W. Tan, and J. C. Newman Jr., "Avirtual crack-closure technique for calculating stress intensity factors forcracked three dimensional bodies," International Journal of Fracture,vol. 36, no. 3, pp. R43-R50, 1988.
 D. Xie, A. M. Waas, K. W. Shahwan, J. A. Schroeder, and R. G.Boeman, "Computation of energy release rates for kinking cracks based onvirtual crack closure technique," Computer Modeling in Engineering &Sciences, vol. 6, no. 6, pp. 515-524, 2004.
 S. A. Fawaz, "Application of the virtual crack closuretechnique to calculate stress intensity factors for through cracks with anelliptical crack front," Engineering Fracture Mechanics, vol. 59, no. 3,pp. 327-342, 1998.
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 Q. Qian and D. Xie, "Analysis of mixed-mode dynamiccrack propagation by interface element based on virtual crack closuretechnique," Engineering Fracture Mechanics, vol. 74, no. 5, pp. 807-814,2007.
 G. Bhardwaj, I. V Singh, B. K. Mishra, and T Q. Bui,"Numerical simulation of functionally graded cracked plates using NURBSbased XIGA under different loads and boundary conditions," CompositeStructures, vol. 126, pp. 347-359, 2015. 2b1af7f3a8