Mode Stress Intensity Factors Research Articles

Applications of the fast multipole fully coupled poroelastic displacement discontinuity method to hydraulic fracturing problems

In this study, a fast multipole method (FMM) is used to decrease the computational time of a fully-coupled poroelastic hydraulic fracture model with a controllable effect on its accuracy. The hydraulic fracture model is based on the fully-coupled poroelastic formulation of the displacement discontinuity method (DDM) which is a special formulation of the boundary element method (BEM). DDM is a powerful and efficient method for problems involving fractures. However, this method involves the multiplication of a dense matrix with a vector in several places. Thus, the DMM algorithm slows down drastically as the number of elements increases, and even more so with the inclusion of necessary details such as poroelasticity, which makes the solution history-dependent. Rather, FMM is a technique to expedite matrix-vector multiplications, within a controllable error range, by approximating the far-range interactions without forming the matrix explicitly. In doing so, FMM offers a leverage in the computational efficiency by modifying the algorithm of the fully-coupled poroelastic displacement discontinuity method (PDMM) in two places. The first modification is in the time-marching scheme, which accounts for the solution of previous time steps to compute the current time step. The second modification is in the generalized minimal residual method (GMRES), where the unknowns are solved for iteratively.Several examples are provided to show the efficiency of the proposed fast multipole fully-coupled poroelastic displacement discontinuity model (FMPDDM) in problems with large degrees of freedom (in time and space). These examples include hydraulic fracturing of a horizontal well and randomly distributed pressurized fractures at different orientations with respect to principal horizontal stresses. The results of FMPDDM are compared to PDDM in terms of computational time and accuracy, which demonstrates that FMPDDM decreases the computation time by up to a factor of 70 for a case with 20000 piece-wise constant elements and a single time step. The difference between the computational times of FMPDDM and PDDM can be higher for a larger number of elements and time steps. Moreover, the associated absolute value of relative error for a problem of this size is observed to be less than 4% for the hydraulic fracture example, and less than 0.5% for the example of randomly distributed pressurized fractures. Furthermore, the solution of tip displacements using both methods are used to compare the computation of stress intensity factors (SIF) in mode I and II, which are needed for fracture propagation. The calculated displacement discontinuities reveal that the SIF calculation using FMPDDM has a negligible error compared to PDDM. Therefore, FMPDDM will not affect the estimation of the fracture propagation direction. Accordingly, the proposed algorithm may be used for fracture propagation studies to substantially reduce the computational time.

Stochastic mixed mode stress intensity factor of center cracks FGM plates using XFEM

Extended finite element method (XFEM) and second-order perturbation technique (SOPT) were combinedly utilized using interaction integral (M-integral) through partition of unity method to find out the mean and variance of mixed mode stress intensity factor (MMSIF). Uncertain system parameters are considered in material properties, crack length, crack orientation, gradient coefficients in the present study. MMSIF in a numerical example with center crack is computed to validate the accuracy of the presented model. Finally, typical numerical results are presented to examine the different modulus ratios, crack angle, crack length, position of crack and tensile, shear and combined loadings with uncertain system properties on the MMSIF.

Application of digital image correlation method for determination of mixed mode stress intensity factors in sharp notches

In the present study, the applicability of the digital image correlation (DIC) method in deriving the notch stress intensity factors (NSIFs) is examined for sharp V-notched components under mixed mode I/II loading. For this purpose, in the first step, the analytical mixed mode stress and displacement fields around the sharp V-notches and the concept of NSIFs are presented. Then, in the second step, the V-notched diagonally loaded square plate (V-DLSP) specimen is chosen for performing the DIC experiments. Three different notch opening angles with four various combinations of mode I and mode II loadings are considered for the V-DLSP specimens tested. Subsequently, in the third step, the DIC method is employed to determine the experimental displacement fields in the V-DLSP samples. Finally, the displacement fields obtained experimentally from the DIC technique are fitted to the analytical displacement field to derive the NSIFs of sharp V-notches by using a numerical method called the over-deterministic method. Comparing the results obtained from DIC method with those obtained from the finite element (FE) analysis shows the effectiveness of DIC method in determining the stress field around sharp V-notches.

Modelling of reactor pressure vessel subjected to pressurized thermal shock using 3D-XFEM

We describe the model developed for integrity assessment of a reactor pressure vessel (RPV) subjected to pressurized thermal shock (PTS). The assessment is based on a multi-step simulation scheme, which includes the thermo-hydraulic, thermo-mechanical and fracture mechanics analyses. The thermo-mechanical model uses the three-dimensional finite element method (FEM) to calculate the stress fields of the reactor pressure vessel (RPV) subjected to the thermal load and pressure during large break loss of coolant accident (LBLOCA). In this contribution, the multistep simulation scheme is extended to the analysis of two-phase steam-water flow occurring in the LBLOCA situation and the fracture analysis of postulated cracks under this accident event. It is demonstrated that the principle of the combined one-way coupled system code, CFD and structural mechanics codes can be used for the fracture analysis in case of a transient accident, which involves two-phase flow. The prediction of the temperature field is achieved by using two-phase computational fluid dynamics (CFD) simulation. To perform the stress analysis, an appropriate finite element discretization of the RPV wall is used, which considered the cladding and the ferritic low allow steel. The calculation of the stress intensity factor (SIF) in mode I for hypothetical cracks located in critical positions of the RPV is based on the linear elastic fracture mechanics (LEFM).The fracture mechanics model is based on the node-based submodeling technique in combination with XFEM in Abaqus. This is used to refine the mesh required by the fracture analysis in the region of interest, especially those regions under the cooling plume formed during the LBLOCA. The submodel may contain three types of cracks: axial, circumferential and inclined. The performed integrity assessment compares the stress intensity factor calculated in the deepest point of a surface crack with the material’s fracture toughness. Two approaches to extract the stress intensity factor were applied in the present paper, namely, the classical FEM and eXtended FEM or XFEM. The classical FEM and XFEM solutions are compared. Both techniques give similar results. However, XFEM is more convenient as it is mesh independent.

Evaluation of the mixed mode (I/II) fracture toughness of cement emulsified asphalt mortar (CRTS-II) using mixture design of experiments

Cement emulsified asphalt mortar (CEAM) is a key component of ballastless rail track systems which is tasked with damping the train movement energy and transferring loads to the concrete roadbed. This mortar consists of asphalt emulsion, sand, cement, and water and enjoys better toughness and flexibility properties than cement mortar thanks to the combined effect of cement and asphalt emulsion. One of the primary causes of failure in this mortar is cracking at low temperatures. This study investigated the fracture toughness of CEAM (CRTS- II) with different volume percentages of mortar constituents, namely cement, sand, water, and asphalt emulsion, at low temperature in the linear elastic fracture mechanics (LEFM) framework. The critical stress intensity factors of the mortar for pure tensile (mode I), pure shear (mode II), and mixed tensile-shear (mixed mode I/II) fractures with Me values of respectively 1.0, 0, and 0. 67 were obtained. Considering the necessity of examining the effect of mix design ratios and their interactions and the need to limit these ratios to ensure acceptable rheological and mechanical properties, the experiments were designed using the D-Optimal mixture design method at 5% significance level. By assuming a quadratic model between the critical stress intensity factors of each fracture mode and the volume percentages of mortar constituents, a total of 20 randomized mix designs were obtained. For each mix design, the semi-circular bending (SCB) test was performed with at least 3 replications. The regression models based on the experimental results were developed. The findings demonstrated the dependence of mode I, mode II, and mixed mode I/II critical stress intensity factors on the mix design ratios. The lowest and highest fracture toughness estimates were related to the mixed tensile-shear and pure shear modes, respectively. Finally, the mixture design method was used to delimit the optimum mix design region for maximizing the fracture toughness of all three modes.

Revisit of antiplane shear problems for an interface crack: Does the stress intensity factor for the interface Mode III crack depend on the bimaterial modulus?

An overview of the results for Mode III interface crack problems available in the literature, as well as those obtained by the authors, is presented in the paper. The analytical and semi-analytical solutions for the main fracture characteristics (i.e. stress intensity factors and energy release rates) are discussed with respect to their dependency on bimaterial constants. Detailed formulation for the antiplane shear problem for an interface crack interacting with internal cracks is presented using the methods of complex potentials and singular integral equations. Some particular solutions (e.g. for the problem of a singularity interacting with an interface and for the Mode III interface crack) are obtained, and these can serve as benchmark solutions or as parts of other similar problems. The approximate closed-form solution is derived for a special case of the interaction between the interface crack and internal micro-cracks. The resulting form of the solution for the stress intensity factors is rather simple and its structure is easily analyzed. A theoretical analysis of the explicit form of expressions for the stress intensity factors shows the conditions under which the solution for the interface crack depends on the bimaterial mismatch, and those under which it does not depend. This formula also allows to define the condition, when a small internal crack does not influence the stress intensity factor of the interface crack (i.e. invisible for the interface crack).

Novel analytical approach for evaluating the mechanical properties of friction stir spot joints through constitutive modeling

Proper assessment of mechanical properties for friction stir spot welds is controversial due to complex stress conditions within the weld nugget. Because of that, mechanical testing results are expressed in terms of load displacement curves; therefore, interpretation of the plastic behavior by stress-strain curve is theoretically impossible. In the present work, stress intensity factors and complex structural stresses of spot welds during tensile shear test were estimated through analytical simulation based on fracture mechanics methods for elastic and plastic regions. Subsequently, the stress-strain curve and the corresponding work hardening rate were estimated and interpreted with acceptable precision. The experiments were performed using a high manganese twinning induced plasticity steel which is one of the commonly used steel grades in automotive industry. The best combination of tensile-shear strength and extension was found for the joints welded at rotation speed of 1600 rpm where stress intensity factor of mode I contribution was the highest. The occurrence of twinning induced plasticity in a higher extent was the main reason for an increase in extension and work hardening of this sample.

Fatigue crack behaviors under asynchronous biaxial loading

Crack behavior of aluminum alloy 2A12-T4 is investigated experimentally and numerically using cruciform specimens subjected to in-plane biaxial loadings applied in horizontal (x axis) and vertical (y axis) directions. Different cruciform configuration plans, with variant sizes of slots in loading arms and different transitional fillet radius between adjacent arms, are analyzed by means of FEM to confirm a final specimen geometry with relative uniform stress distribution in its central area. The tests for fatigue crack growth behavior under cyclic biaxial sinusoidal loadings with phase differences of 0° or −60° between two orthogonal fatigue loads are conducted by using a biaxial fatigue test system. The biaxial load ratio, Pmaxx/Pmaxy, varies from −0.5 to 1.0. Initial central cracks are inclining in angles of 0°, 45° and 60° to x-axis. Crack growth path and lives are obtained. It is observed that the crack growth behavior under biaxial loading differs from that with only uniaxial loading. A widely accepted fatigue crack growth model, which combines Paris equation, maximum tangential stress (MTS) criterion and an equivalent stress intensity factor (SIF) proposed by Tanaka, is used for mixed mode-I/II crack subjected biaxial loading. The combined model is evaluated by the SIFs analysis based on analytical and numerical solution. The mixed mode SIFs are calculated based on finite element analysis with mesh updating. The simulated crack growth path and growth lives correlate well with the experiments in most cases. Both the experiments and the simulations demonstrate that the tensile load parallel to crack path behaves as the retarder to the crack growth, and the compressive load parallel to crack path behaves as the promoter to the crack growth. It is found that equivalent SIF used in the model overestimates the effect of mode II SIF on the fatigue crack growth rate by Paris equation. It is questionable to use the combined model for the cases with asynchronous biaxial loadings or those leading to crack surface contact.

Polyurethane foam behavior in mixed mode bending

Mixed mode four-point bending testing is performed on polyurethane foams. This paper presents only the results on the stress intensity factors (SIFs) obtained experimentally for a density of 325 kg/m3 , although tests were done for three foam densities. An asymmetric four-point bending setup was used for determining the critical SIFs in Mode I and Mode II, and discussions on the influence of the initial crack length on the SIF values are done. As initial crack length is increased the theoretical predictions give a better comparison to experimentally obtained results. Mode II testing is also performed on a polyurethane foam as a particular case of mixed-mode four-point testing. Crack initiation and propagation in Mode II depends on the geometrical parameters of the testing configuration. Numerical XFEM simulations are done in order to clarify the situations in which the crack doesn't propagate, and failure is produced in the region of supports. A combined experimental-XFEM analysis is recommended to fully understand the particularities of the behavior of cellular materials.

The effects of some parameters on perforation tip initiation pressures in hydraulic fracturing

Perforations play an important role in hydraulic fracturing as channels of fracturing fluid penetrating into reservoirs. The perforation geometry and the geo-stresses of reservoirs could affect the initiation pressures (i.e. the critical hydraulic pressure as perforation tips start to initiate), and the fracture initiation direction which could affect the fracturing fluid flowing into reservoirs. In this paper, two collinear perforations emanating from a wellbore were considered, and by using conformal mapping techniques, the stress intensity factors (SIFs) of mode I and mode II at the perforation tips were obtained. According to the maximum tangential stress criterion, i.e. MTS criterion, the analytical solutions of the initiation pressures and the initiation angles at perforation tips were obtained. The theoretical results were compared with the previous test results by using cubic concrete models and the previous in-situ measuring results. The prediction results of initiation pressures generally agreed with the test results and the in-situ results. Finally, the effects of perforation lengths, perforation angles with the major principal stress and the principal stress difference on hydraulic fracturing were analyzed. The results showed that the optimal perforation angles should lie in the range from 0° to 30°, and the optimal perforation length should 2 or 3 times the wellbore diameter.

Three-dimensional mixed-mode stress intensity factors for deflected internal surface cracks in thin and midsize-thick-walled spherical pressure vessels

In this paper, three-dimensional mixed-mode stress intensity factor solutions are obtained for deflected inner surface cracks contained in thin- and midsize-thick-walled spherical pressure vessels. In an effort to build a library of stress intensity factor solutions for different realistic cases, main parameters affecting the problem, which are crack deflection angle, normalized crack depth and crack shape aspect ratio, are changed systematically to cover all combinations. The results show that as the deflection angle increases, mode-I stress intensity factors decrease and mode-II and mode-III stress intensity factors increase for a given case. It is also observed that as the normalized crack depth increases, stress intensity factors increase along the crack front. The results show that magnitudes of the normalized mixed mode stress intensity factors for midsize-thick-walled vessels are higher than those of thin-walled spherical vessels. Crack deflection angle is shown to have the highest influencing effect among all parameters. It is also observed that especially for higher crack deflection angles, which have higher mode-mixity, crack surfaces may come into contact partially or fully.

Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites based on multi-scale method

Boron nitride nanotubes (BNNTs) possess superior mechanical, thermal and electrical pro- perties and are also suitable for biocomposites. These properties make them a favorable reinforcement for nanocomposites. Since experimental studies on nanocomposites are time- consuming, costly, and require accurate implementation, finite element analysis is used for nanocomposite modeling. In this work, a representative volume element (RVE) of epo- xy/BNNT nanocomposites based on multi-scale modeling is considered. The bonds of BNNT are modeled by 3D beam elements. Also non-linear spring elements are employed to simu- late the van der Waals bonds between the nanotube and matrix based on the Lennard- -Jones potential. Young’s and shear modulus of BNNTs are in ranges of 1.039-1.041TPa and 0.44-0.52TPa, respectively. Three fracture modes (opening, shearing, and tearing) have been simulated and stress intensity factors have been determined for a pure matrix and nanocom- posite by J integral. Numerical results indicate that by incorporation of BNNT in the epoxy matrix, stress intensity factors of three modes decrease. Also, by increasing the chirality of BNNT, crack resistance of shearing and tearing modes are enhanced, and stress inten- sity factor of opening mode reduced. BNNTs bridge the crack surface and prevent crack propagation.

Investigation of the effect of load distribution along the face width and load sharing between the pairs in contact on the fracture parameters of the spur gear tooth with root crack

Abstract The magnitude and location of stress intensity factor (SIF) in mode I (KI), mode II (KII), mode III (KIII) relies mainly on the geometry and location of the crack tip, variation in magnitude of load due to multiple pair contact along the active profile and the distribution of the load along the face width of a spur gear drive. Mostly, in literature the calculation of SIF in gear were carried out either by using a two dimensional (2D) Finite element model (FEM) with point load for unit width or a three dimensional (3D) FEM with uniform / parabolic load distribution along the face width without considering load sharing effect between the multiple pairs in contact. In the present study, an attempt has been made to explore the significance of the load distribution along the face width and the load sharing between the pairs when the contact moves from the highest point of tooth contact (HPTC) to the lowest point of tooth contact (LPTC) on the SIF calculation. A comparative study has been made by using a validated 3D single tooth spur gear FEM with four different load cases such as case 1: Uniform load distribution along face width without considering load sharing between the pairs, case 2: Uniform load distribution along face width by including the effect of load sharing between the pairs, case 3: Actual load distribution along the face width without considering load sharing between the pairs, case 4: Actual load distribution along the face width by including the effect of load sharing between the pairs. The stiffness based approach was used to calculate the load distribution ratio (LDR) of a contact point along the contact line and the load sharing ratio (LSR) between the pairs in contact. The detailed analysis showed that case 1, case 2 and case 3 load cases overestimate the SIFs comparing to the case 4. The first three load cases overestimate the KI by 39%, 2.5%, and 45% and KII by 390%, 36%, and 129% and KIII by 282%, 47%, and 132% respectively when compared to load case 4. The results of this study provide valuable guidelines for the crack propagation as well as life prediction calculations for spur gear drives.

Термопружна поведінка біматеріалу з міжфазною щілиною, заповненою нестисливою рідиною

Thermo-stressed state of the bimaterial with the interfacial crack filled with an incompressible fluid is investigated. The bimaterial with zero Dundurs parameter is considered. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for a temperature jump between the cracks faces and an opening crack. The pressure of the fluid is determined from the equation of the incompressible fluid. Using the method of successive approximations a numerical solution of the problem is constructed. The opening crack and temperature jump are determined by the finite sum for Chebyshev polynomials of the second kind. The coefficients for Chebyshev polynomials are determined from the system of linear algebraic equations. Having made numerical calculations, the dependences of the pressure of the fluid and the opening crack from the mechanical loads and density and direction of the heat flow have been constructed. The influence of fluid pressure on the stress intensity factor mode I are analyzed.

A Numerical Study of Creep Crack Growth in an Aero-engine Turbine Disc using XFEM

In this paper, a turbine disc of an aero-engine having a macro crack under creep loading is studied to predict the creep crack growth. Extended finite element method (XFEM) is used as it does not require conformal meshing and remeshing due to change in the topology during crack growth. The elasto-plastic behavior of the material is modelled by Ramberg-Osgood model and von-Mises yield criterion. The creep behavior is modelled by coupling of spatial and temporal dimensions to capture the effect of stress relaxation and redistribution due to creep strain. The creep crack growth rate is computed by the C(t)-integral which includes small-scale creep, transition creep and extensive creep. The crack growth direction is estimated by the maximum principal stress criterion using mode-I and mode-II stress intensity factors (SIFs). The interaction integral approach is implemented for the evaluation of stress intensity factors of different modes. The history fields of plasticity and creep are transferred properly from the old configuration to new configuration after the crack growth which is the main challenge in this analysis. This proposed numerical scheme is then utilized to obtain the creep crack growth in the component made of elasto-plastic-creeping material. The creep crack growth variation with time is estimated by the XFEM to evaluate the life of an aero-engine turbine disc under creep conditions.

Applications on Three-Dimensional Mixed Mode Fatigue Crack Propagation Using Fracture and Crack Propagation Analysis System (FCPAS)

In this study, results from two applications on three-dimensional fatigue crack propagation using the in-house developed “Fracture and Crack Propagation Analysis System (FCPAS)” are presented. FCPAS uses finite element models to calculate three-dimensional mixed mode stress intensity factors and other sub-modules to predict shapes of the incrementally growing crack fronts and the resulting fatigue crack propagation lives. The evolving crack surfaces are generated by successively adding the incremental growth surfaces and re-meshing and re-solving the finite element model using the standalone program FRAC3D, a finite element-based standalone program employing three-dimensional enriched finite elements. The results show that the fracture surfaces evolve into a mode-I type three-dimensional crack as expected. Very good agreements are obtained between the results of the current studies and those of numerical applications in terms of both evolving crack surfaces and crack propagation lives. Therefore, it is concluded that the applied method is capable of accurately predicting linear elastic three-dimensional mixed mode fatigue crack propagation problems.

Mixed mode stress intensity factors of slanted edge cracked plate with hole subjected to various in-plane loadings using XFEM

Mixed mode stress intensity factors of slanted edge cracked plate with hole subjected to various in-plane loadings using XFEM

Mixed mode stress intensity factors of slanted edge cracked plate with hole subjected to various in-plane loadings using XFEM

This manuscript presents the fracture behaviour of a slanted edge cracked plate with a circular hole subjected to various in-plane loadings, such as tensile, shear and combined (tensile and shear) ...

Three-Dimensional Mixed Mode Stress Intensity Factors for Inclined Elliptical Surface Cracks in Plates under Uniform Tensile Load

In this study, main results from parametric analyses of three-dimensional inclined elliptical surface cracks using FRAC3D are presented in terms of normalized mixed mode stress intensity factors (SIFs) along crack fronts. FRAC3D is the solver of in-house developed “Fracture and Crack Propagation Analysis System (FCPAS)” and is a finite element-based standalone program employing three-dimensional enriched finite elements. In the analyses, values of the main parameters affecting the problem, which are normalized crack depth with respect to plate thickness (a/t=0.2, 0.4, 0.6, 0.8), crack aspect ratio (a/c=0.25, 0.50, 1.0, 2.0, 4.0) and crack inclination angle (β=0°, 15°, 30°, 45°, 60°, 75°), are changed systematically in their realistic ranges to obtain a wide library of solutions representing the general problem and to develop empirical SIF equations that are valid for all values of the main parameters in their pre-determined ranges. This resulted in a total of 120 three-dimensional fracture analyses. The results show that as the inclination angle of the surface crack increases, mode-I SIF decreases along the crack front, while mode-II and mode-III SIFs increase up to β=45°, which contains the plane with maximum in-plane shear stress. For higher values of β, mode-II and mode-III SIFs start decreasing as expected. As the normalized crack depth increases, the normalized SIFs also increase. When the depth point and the free-surface point are compared with each other, it is seen that the crack aspect ratio also has big effect on the relative values of the normalized SIFs.

Crack initiation and propagation on CT specimens of two polymers (ABS and PMMA), under cyclic constant displacement loading

Cyclic constant displacement tests on CT (compact tension) specimens, according to the ASTM E647-00 standard, have been carried out in order to investigate the crack initiation and propagation for two polymeric materials: the ABS (Acrylonitrile Butadiene Styrene) and PMMA (Polymethyl methacrylate). Experimental cyclic tests have been performed on a self-designed and constructed machine working under constant displacement at frequency of 0.25 Hz and with maximum displacement of 390 μm. The displacement control is implemented using the Arduino software, a personal computer together with an inductive proximity sensor, which allows determining the displacement with ±4 μm of precision. The stress intensity factor in mode I (KI), is obtained for different crack lengths “a”, different amplitudes of displacement, for the two polymeric materials. In addition, the energy release rate G for plane stress and plane strain are obtained by using the stress intensity factors in mode I, previously calculated. Finally, the fracture surfaces are analyzed by scanning electron microscope in order to examine the general trends for crack initiation and propagation under the loading conditions being described.