Maximum Circumferential Stress Research Articles

Fracture Mechanism of Unstable Rock With Double-Crack Control Discontinuity Subjected to High Earthquake Intensities

Unstable rock in the limestone area has typical characteristics of quasimasonry structure and control discontinuity, where the failure essence is the fracture propagation under loads. The unstable rock control discontinuity in the limestone zone was analyzed, and the geological model for a doublecrack unstable rock mass was obtained according to the theory of geomorphologic evolution. The mechanical model and fracture mechanical model for the complex control discontinuity were constructed. Based on the rock weight, the fissure water pressure and the earthquake load, the formula of the fracture stability coefficient was obtained under the maximum circumferential stress criterion. The fracture stability coefficient expression for the unstable rock is rational according to the verifying case analysis. The fracture stability coefficient corresponding to each intensity decreases with the crack length ratio. Under earthquake intensities Ⅷ and Ⅸ, the unstable rock will collapse; the stability of the unstable rock is liable to the coupling effects of the crack length ratio of the main control discontinuity and the earthquake intensity. The critical crack length ratio decreases with the earthquake intensity. Under earthquake intensity Ⅶ, the theoretical value of the critical crack length ratio is 25.8%, which is slightly less than the actual value of 27.7%, and on the slightly safer side. The fracture stability coefficient expression has good applicability. The research results provide an important theoretical support for the treatment of such unstable rock masses.

Computational Investigation of the Stability of Stenotic Carotid Artery under Pulsatile Blood Flow Using a Fluid-Structure Interaction Approach

Stenosis can disrupt the normal pattern of blood flow and make the artery more susceptible to buckling which may cause arterial tortuosity. Although the stability simulations of the atherosclerotic arteries were conducted based on solid modeling and static internal pressure, the mechanical stability of stenotic artery under pulsatile blood flow remains unclear while pulsatile nature of blood flow makes the artery more critical for stresses and stability. In this study, the effect of stenosis on arterial stability under pulsatile blood flow was investigated. Fluid–structure interaction (FSI) simulations of artery stenosis under pulsatile flow were conducted. 3D idealized geometries of carotid artery stenosis with symmetric and asymmetric plaques along with different percentages of stenosis were created. It was observed that the stenosis percentage, symmetry/asymmetry of the plaque, and the stretch ratio can dramatically affect the buckling pressure. Buckling makes the plaques (especially in asymmetric ones) more likely to rupture due to increasing the stresses on it. The dominant stresses on plaques are the circumferential, axial and radial ones, respectively. Also, the highest shear stresses on the plaques were detected in [Formula: see text] and [Formula: see text] planes for the symmetric and asymmetric stenotic arteries, respectively. In addition, the maximum circumferential stress on the plaques was observed in the outer point of the buckled configuration for symmetric and asymmetric stenosis as well as at the ends of the asymmetric plaque. Furthermore, the artery buckling causes a large vortex flow at the downstream of the plaque. As a result, the conditions for the penetration of lipid particles and the formation of new plaques are provided.

Numerical analysis of surface integrity in parallel turning PART B: Influence of cutting tool chamfer angle and chamfer width

Residual stresses play a vital role in determining the quality of the turned components. Numerical modelling of residual stresses provides a deep insight of the process mechanism and increases the productivity by conserving time, material and manpower. In parallel turning process, two tools are engaged simultaneously to perform the turning process. In this research work, numerical investigation was done to determine the influence of chamfer width and chamfer angle for various cutting speeds and feeds on the surface residual stresses. The numerical analysis was carried out utilising a software program, ABAQUS 6.14. The work and tool material are AISI 4340 steel and carbide. CPE4RT four node plane strain element was used for the analysis. The chip was separated from work material based on Johnson-Cook damage criterion. Surface friction between cutting tool and chip is modelled by employing penalty contact approach. The coefficient of friction was taken as 0.3. With the increase in chamfer angle and chamfer width, the surface residual stresses increase. This is caused due to the increase in cutting temperature and plastic strain. Maximum circumferential and axial compressive residual stress of 415 MPa and 445 MPa is obtained for a chamfer width of 0.2 mm. The cutting speed was 150 m/min and feed 0.1 mm/rev. On the whole, the usage of chamfered cutting tools proved to be beneficial in imparting higher compressive residual stresses in parallel turning process.

Adaptive Finite Element Prediction of Fatigue Life and Crack Path in 2D Structural Components

The existence of a hole near a growing fatigue crack can cause the crack trajectory to deviate. Unless the hole is too close to the crack, the crack is arrested at the edge of the hole and does not progress further. The purpose of this paper was to predict the crack propagation and lifetime of two-dimension geometries for linear elastic materials in mixed-mode loading using a finite element source code program written in Visual Fortran language. The finite element mesh is generated using the advancing front method. The onset criterion of crack propagation was based on the equivalent stress intensity factor which provides the most important parameter that must be accurately estimated for the mixed-mode loading condition. The maximum circumferential stress theory was used as a direction criterion. The modified compact tension (MCTS) was studied to demonstrate the influence of the hole’s presence on the direction of crack growth and fatigue life for different configurations. The Paris’ law model has been employed to evaluate the mixed-mode fatigue life for MCTS in different configurations under the linear elastic fracture mechanics (LEFMs) assumption. The framework involves a progressive crack extension study of stress intensity factors (SIFs), crack growth direction, and fatigue life estimation. The results show that the fatigue growth was attracted to the hole either changes its direction to reach the hole or floats by the hole and grows as the hole is missed. The results of the study agree with several crack propagation experiments in the literature revealing similar crack propagation trajectory observations.

Numerical Simulation and Experimental Study on Residual Stress in the Curved Surface Forming of 12CrNi2 Alloy Steel by Laser Melting Deposition.

The performance and service life of the nuclear emergency diesel engine shaft made of 12CrNi2 alloy steel is very important for the safety of nuclear power. Laser melting deposition (LMD) is a challenging camshaft-forming technology due to its high precision, rapid prototyping, and excellent parts performance. However, LMD is an unsteady process under the local action of laser, especially for curved surface forming, which is more likely to generate large residual stress on components, resulting in cracks and other defects. At present, the stress research on LMD curved surface forming is relatively insufficient. In the present paper, material parameter testing, high-temperature mechanical properties analysis, single-track sample preparation, and heat source checks are conducted. At the same time, the ABAQUS software and the DFLUX heat source subroutine are used to compile the curved double-ellipsoidal moving heat source, and the effects of the temperature-dependent thermophysical parameters and phase change latent heat on the temperature field are considered. A three-dimensional finite element model is established to analyze the thermal stress evolution and residual stress distribution of multi-track multi-layer on a curved surface by LMD, and the effect of the scanning method and interlayer cooling time on the residual stress of the formed components is studied. The results show that with the increase in temperature, the strength of the material reduces, and the fracture morphology of the material gradually transitions from ductile fracture to creep fracture. The material parameters provide a guarantee for the simulation, and the errors of the width and depth of the melt pool are 4% and 9.6%, respectively. The simulation and experiment fit well. After cooling, the maximum equivalent stress is 686 MPa, which appears at the junction of the substrate and the deposited layer. The larger residual stress is mainly concentrated in the lower part of the deposited layer, where the maximum circumferential stress and axial stress are the tensile stress. Compared with the axial parallel lap scanning method, the arc copying lap scanning method has a relatively smaller maximum thermal stress and residual stress after cooling. The residual stress in the deposited layer is increased to some extent with the increase in the interlayer cooling time.

A new robust theoretical prediction model for flange wrinkling in conventional spinning

Flange wrinkling is one of the most common defects affecting the forming quality of workpieces in conventional spinning. To this end, the flange wrinkling mechanism is analyzed and a new robust theoretical prediction model for the flange wrinkling is developed in this work. It is found that the initiation of flange wrinkling is closely related to the excessive circumferential compressive stress in the zone affected by the roller action rather than the whole flange. Then, a theoretical model for the critical circumferential stress producing flange wrinkling is developed based on the generalized variational principle of limit analysis. By comparing the instant maximum circumferential stress in the zone affected by the roller action with the calculated critical stress, the flange wrinkling can be predicted. Compared with the prediction model based on energy method, the main improvements of the new model can be summarized as follows. First of all, the incremental displacement amplitude of deflection mode in the effective compressive region, which dramatically affects the critical stress but overlooked in the model based on energy method, is considered in the new model. Secondly, the sheet blank is regarded as a rigid plastic material in the new model, so the calculation of elastoplastic matrix which is very sensitive to the mechanical states does not need to be considered, thereby avoiding the adverse effect of the extraction deviation of mechanical states on prediction accuracy. Thus, compared with the prediction model based on energy method, the new model has higher prediction accuracy and better robustness under the extraction deviation of mechanical states. In addition to realizing the prediction of flange wrinkling, the new model can also be used to analyze the influence of sheet blank geometric parameters and material parameters on the initiation of flange wrinkling under various forming conditions. In view of this, the new model is applied to reveal the influence of material parameters (material strengthening coefficient K, hardening index n and elastic modulus E) on the initiation of flange wrinkling by combining response surface methodology and central composite design (CCD). The results show that K has the greatest influence on the initiation of flange wrinkling, followed by E and n. Considering the interaction between the parameters, the interaction between K and E also has a greater effect on the initiation of flange wrinkling. The smaller E and larger K are helpful to reduce the risk of flange wrinkling. Changing the value of n has no significant effect on delaying the occurrence of flange wrinkling. The results can deepen the understanding on the prediction and rules of flange wrinkling in conventional spinning.

Effects of arbitrary holes/voids on crack growth using local mesh refinement adaptive XIGA

This paper is devoted to numerical investigation of holes/voids effects on crack growth in solids using a locally refined (LR) B-splines extended isogeometric analysis (XIGA). We particularly focus our attention on crack-hole interaction analysis. Special enrichment functions captured discontinuity and singularity are used, by which the computational mesh is thus independent of both cracks and holes, eliminating re-meshing when modeling crack growth. Geometries of arbitrary shaped cracks and holes are accurately described with the NURBS curves, and the coordinates of control points and weights of the NURBS curves are produced from the iges file of Rhino. An efficient stress recovery based error estimator in combination with the structured mesh refinement strategy is adopted to conduct local mesh refinement. The interaction integral method is applied to evaluate the stress intensity factors (SIFs), and the direction of crack growth is determined with the maximum circumferential stress criterion. Analysis of crack-hole interaction is examined through several numerical examples, and computed results are validated with respect to reference solutions.

Hoop Stress Modification, Stress Hysteresis and Degradation of a REBCO Coil Due to the Screening Current Under External Magnetic Field Cycling

Degradation of a REBCO coil under external magnetic fields is one of the major technical problems in the field of HTS magnet technology. A possible cause of such degradation is an inhomogeneous hoop stress distribution, or hoop stress modification (both increase and decrease), induced by the screening current. In this work, we investigate such a hoop stress modification by a small coil experiment with a strain measurement and a numerical simulation. An experimental result shows a very high stress increase factor of >4.1, defined by the maximum circumferential stress over BzJR stress, and the simulated result is in qualitative agreements. The strain (stress) shows a hysteresis effect corresponding to the screening current behavior. A large hoop stress modification causes not only a hoop stress increase, but also buckling of the conductor, which induces delamination and micro-clacks of the superconducting layer. We also show the stress modification can be reduced by bonding turns with epoxy.

Optimal design of wave generator profile for harmonic gear drive using support function

The wave generator profile for harmonic drive was designed using a closed convex curve generated by the support function. An optimization model of the wave generator profile was established, and the objective function was defined to minimize the maximum circumferential stress of the flex spline assembled with the wave generator. Then the curvature difference of the flex spline neutral line and pure kinematic error were investigated considering different wave generator profiles. Finally, the finite element method was applied to establish the assembly model. The stress and deformation of the flex spline were analyzed, as well as the stress of the flexible bearing. Results show that the maximum circumferential stress of the flex spline with the proposed wave generator profile was significantly reduced compared to the harmonic gear with a traditional elliptic wave generator. The stress in the major axis region of the flex spline is distributed more evenly after assembly with the wave generator. And the maximum Von-Mise stress of the flexible bearing was obviously reduced, as well as the contact force for outer ring and rollers in the major axis region of the wave generator.

Non-tip failure of smooth fissure in 3D printed specimens

Non-tip cracking for kinked fissures has been investigated in our previous study. This paper reveals that non-tip cracking also occurs for smooth fissures without kinks. 3D printed photosensitive resin specimens with a smooth S-shape fissure are tested under uniaxial compression. The influence of geometry and orientation of the S-shape fissure on the specimen deformation and strength behaviour, especially the non-tip-cracking failure is investigated. Results show that two cracking patterns, tip-cracking and non-tip cracking, occurs in a similar trend with that observed for kinked fissures. The non-tip cracking is sensitive to the inclination angle β and the ratio between the amplitude and ratio of the amplitude to the half length of the S-shape fissure, i.e., the effective curvature A/a, as it only occurs when β equals to 30°, 45°, 60° and the A/a is equal to or greater than 2/7.5. A simplified piecewise linear crack model is proposed to predict the occurrence of the non-tip cracking failure. The S-shape fissure is approximated by three straight segments and the three failure models A, B and C corresponding to their equivalent straight segments are identified based on the maximum circumferential stress criterion. The cracking pattern of the S-shape fissure is governed by model A and C failure and the predicted cracking patterns match well with the respective experimental ones. The results from the present research complement the traditional fracture mechanics by extending the understanding about non-tip cracking failure.

Alloying with Ge and Hollowing Reduces Lithiation-Induced Stresses in Si Nanopillar Anodes

High energy-density batteries are crucial to energy storage solutions. In Lithium Ion Batteries (LIB), Si nanopillars are promising anodes due to their highest theoretical specific capacity. However, volume expansion and fracture during cycling inhibit its widespread adaptation. Ge, which is isomorphic with Si, shows better fracture resistance and higher cycle life but has higher molecular weight and cost. Alloying Si with Ge offers a trade-off in optimizing stresses, weight and cost. Here, we computationally evaluate the effect of alloying Si with Ge in reducing stresses generated during lithiation. Hollowing, which creates additional free surface for expansion is also considered. First, we model the stress evolution in nanopillars of Si, Ge, Si-Ge core-shell and Si0.5Ge0.5 alloy. Alloying Si with Ge uniformly, reduces the maximum circumferential stress by around 17%, however, the Si core-Ge shell structure shows stress reduction only when Lithiation is confined only to the Ge. Stresses in Si/Ge alloyed nanotubes considering lithiation from the outer boundary as well as from both boundaries are considered. We find a non-monotonous change in Lithiation stress with varying radius ratio (Rin/Rout) and Rin/Rout=0.4 leads to the least maximum Hoop stress. The stress reduction in Si-nanotubes in such configuration is found to be 16%.

Crack growth adaptive XIGA simulation in isotropic and orthotropic materials

Reliable prediction and thorough understanding of crack growth in engineering materials and structures are of challenging problems, but scientific and technical community has been continuously pursuing an efficient numerical approach to crack propagation simulation. We present in this paper a significant extension and development of the adaptive extended isogeometric analysis (XIGA) based on LR B-splines (locally refined B-splines), which facilitates the local refinement, for accurately modeling crack growth in isotropic and orthotropic media. The discontinuity and singularity of displacements and stresses induced by cracks are locally captured by special enrichment functions, for both isotropic and orthotropic media. This allows the representation of crack to be independent of the computational mesh and avoids the requirement for re-meshing in crack growth simulation. The numerical integration accuracy is improved with the use of the almost-polar integration and sub-triangle technique for the crack-tip and cut elements, respectively. The posteriori error estimator based on Zienkiewicz–Zhu error estimation is employed to define refinement domains, and thereby, drives the adaptivity. On the other hand, the maximum circumferential stress criterion is adopted to determine the direction of crack growth. Several crack growth numerical examples for both isotropic and orthotropic media are studied in different scenarios to show the performance and accuracy of the developed approach. Numerical results reveal that the present approach integrated with the adaptive local refinement scheme offers higher accuracy and convergence rate than the standard XIGA. Furthermore, the adaptive XIGA in a tandem with Nitche’s method is applied to simulate the crack growth of complicated structures. We additionally explore the effects of some numerical aspects or factors on crack path. Smaller crack growth increment could be used to improve the accuracy of crack path within a finer mesh in the adaptive XIGA.

Predicting hydraulic fracture propagation based on maximum energy release rate theory with consideration of T-stress

The interaction between hydraulic and natural fractures is a key factor in determining hydraulic fracture complexity. Based on the theory of linear elastic fracture mechanics and the fracture toughness discontinuity model, the maximum energy release rate (MERR) theory-based fracture propagation model with consideration of T-stress is discussed in this paper. A calculation model of fracture propagation orientation is established via the maximum circumferential stress criterion based on the influence of non-singular terms in the Williams expansion, and the influence of transverse stress (T stress) on the fracture propagation orientation is analyzed. Within considering the influence of the far-field stress, the limitation of the influence of the traditional MERR theory on the fracture propagation is modified, and a more reasonable criterion is established to reflect the influence of the stress difference on fracture propagation. In addition, the influences of the approaching angle, stress difference, net pressure, fracture tip plastic zone radius, and dip angle on fracture propagation is analyzed. Results show that when the T-stress is compressive stress or the T-stress is tensile stress and Bδ < 0.375, the propagation orientation remains 0° for pure Mode I fractures. As for mixed Mode I-II fractures, the deflection angle increases with an increase in T stress, and that the existence of negative T stress can restrain the fracture deflection. Under the conditions of low approaching angle, low horizontal stress difference, high net pressure, and high dip angle, hydraulic fractures are more prone to deflection along the weak plane. The modified theoretical model is verified by laboratory fracturing experiments. This study reveals the interaction between hydraulic fractures and weak planes, and can provide guidance for predicting complex fracture geometry in field constructions.

Alloying with Ge and Hollowing Reduces Lithiation-Induced Stresses in Si Nanopillar Anodes

High energy-density batteries are crucial to energy storage solutions. In lithium-on batteries (LIBs), Si nanopillars are promising anodes due to their highest theoretical specific capacity. However, volume expansion and fracture during cycling inhibit its widespread adaptation. Ge, which is isomorphic with Si, shows better fracture resistance and higher cycle life but has higher molecular weight and cost. Alloying Si with Ge offers a trade-off in optimizing stresses, weight and cost. Here, we computationally evaluate the effect of alloying Si with Ge in reducing stresses generated during lithiation. Hollowing, which creates additional free surface for expansion is also considered. First, we model the stress evolution in nanopillars of Si, Ge, Si–Ge core-shell and Si0.5Ge0.5 alloy. Alloying Si with Ge uniformly, reduces the maximum circumferential stress by around 17%, however, the Si core-Ge shell structure shows stress reduction only when lithiation is confined only to the Ge. Stresses in Si/Ge alloyed nanotubes considering lithiation from the outer boundary as well as from both boundaries are considered. We find a non-monotonous change in lithiation stress with varying radius ratio (Rin/Rout) and Rin/Rout = 0.4 leads to the least maximum Hoop stress. The stress reduction in Si-nanotubes in such configuration is found to be 16%.

Numerical investigation of creep crack growth in plastically graded materials using C(t) and XFEM

In this work, a numerical methodology is proposed to predict the creep response and creep crack growth for functionally graded material/plastically graded material in the extended finite element method framework. The stress relaxation and redistribution as a result of creep are incorporated into the analysis of graded material. The Ct-integral (which includes all creep stages from small-scale creep to extensive creep) based relation analogous to Paris Law is employed to calculate the creep crack growth rate. The numerical difficulties faced at the time of simulation, such as strain-energy rate density derivative and maintaining plastic/creep irreversibility during crack growth are well addressed in this paper. The strain-energy rate density derivative is evaluated by the surface approximation. The plastic/creep irreversibility is maintained by the appropriate data transfer and null step analysis scheme. The crack growth direction is decided by the maximum circumferential stress criterion. The proposed methodology is first validated with the experimental results for homogeneous material and later on, it is implemented to calculate the crack growth for various cases of gradation. The capabilities of the present scheme are demonstrated by performing the mixed-mode crack growth simulations at the specimen and component level.

Complete stress–strain constitutive model considering crack model of brittle rock

The microcrack propagation-coalescence mechanism and stress–strain constitutive model of brittle rock is of great significance for prediction of mechanical properties. To establish a more applicable constitutive model of brittle rock, the crack model based on fracture mechanics was introduced. First, based on the crack slip model and maximum circumferential stress theory, a theoretical equation for crack propagation length was established. The microcrack propagation-coalescence mechanism of brittle rock material under loading was studied. On this basis, a macro-coalescence zone model was proposed. Through the relationship between the deformation characteristics of stress–strain curves at different crack growth stages and the behavior of microcracks, a complete stress–strain constitutive model of brittle rock based on the crack model was theoretically deduced. The constitutive model was verified by the results of a coupling hydro-mechanical test of granite carried out after different heat treatment temperatures. The advanced intelligent algorithm, genetic algorithm, was adopted to simulate experimental data. The research results showed that: (1) The derived stress–strain constitutive model of brittle rock is not only high in accuracy of stress–strain simulation, but also can simulate both the axial and circumferential stress–strain curves. Moreover, the dilatancy and crack initiation stresses can be obtained directly. (2) The parameters of the stress–strain constitutive model not only have clear physical meaning, but the mechanisms of rock mechanical properties, such as dilatancy, acoustic emission, and stress fluctuation, can be explained.

SIF-based fracture criterion of rock-concrete interface and its application to the prediction of cracking paths in gravity dam

Experimental tests were conducted on the composite rock-concrete specimens with granite, sandstone, and C30/C50 concrete to study the interfacial fracture process under three-point bending and four-point shear conditions. A unified interfacial crack initiation criterion expressed by the stress intensity factors (SIFs) was fitted by the SIFs corresponding to the interfacial crack initiation based on the experimental data. By combining with the fictitious crack model, the crack initiation criterion can be transformed into the crack propagation criterion by considering the contributions of the cohesive forces in the fracture process zone. By assessing the relationships of the interfacial propagation criterion and the maximum circumferential stress criteria of the rock and concrete, the potential interfacial crack propagation paths can be predicted. Furthermore, numerical analyses were carried out by introducing these criteria to simulate the complete fracture process of the rock-concrete interface, where the predicted P-CMOD curves and crack trajectories showed good agreements with the experimental data. Finally, by taking a classic gravity dam in practical engineering as an example, the effects of the water levels, initial crack length and crack propagation length on the fracture behaviour of concrete, rock and their interface were investigated. The whole fracture processes for various fracture parameters of the rocks, concretes and their interfaces were simulated numerically. The results indicated that the employed method was effective for the safety assessment of the gravity dam and the application of these propagation criteria is convenient because only the initial fracture toughnesses of the rocks, concretes and their interfaces are required.

Comprehensive Comparisons of Two-and Three-Dimensional Numerical Estimation of Stress Intensity Factors and Crack Propagation in Linear Elastic Analysis

This paper presents a comprehensive comparison of two finite element software in two and three dimensions. The first is a two-dimensional crack growth simulation program developed as source code by using Visual Fortran Language whereas the second is ANSYS mechanical program (3D structural analysis for fracture mechanic analysis). The procedure consists of computing stress intensity factors (SIFs), the crack growth path and the stresses and strain distributions via an incremental analysis of the crack extension, considering two and three-dimensional analysis. Comparisons between these two software were performed for different case studies for stress intensity factors as well as crack growth trajectory. This source code includes the mesh generator based on the advancing front method as well as all the pre and post process for the crack growth simulation under linear elastic fracture mechanics theory with a user-friendly interface. The maximum circumferential stress criterion was used for prediction of the crack growth in isotropic materials under mixed-mode loading. Furthermore, the equivalent domain integral method has been used for calculating the stress intensity factors values during crack growth. Verification of the predicted crack path and stress intensity factors values were validated with relevant experimental data and numerical results obtained by other researchers with a good agreement.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

Abstract The numerical manifold method (NMM) can be viewed as an inherent continuous-discontinuous numerical method, which is based on two cover systems including mathematical and physical covers. Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants. Therefore, higher-order NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy. In addition, it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics. Thus, some other enriched local approximations are introduced to model the stress singularity at the crack tip. Generally, higher-order NMM, especially first-order NMM wherein local approximations are first-order polynomials, has the linear dependence problems as other partition of unit (PUM) based numerical methods does. To overcome this problem, an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method (FEM), which has no linear dependence issue. Meanwhile, the stresses at the nodes of mathematical mesh (the nodal stresses in FEM) are continuous and the degrees of freedom defined on the physical patches are physically meaningful. Next, the extended NMM is employed to solve multiple crack propagation problems. It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation. Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM. The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions. Thus the effectiveness and correctness of the developed NMM have been validated.

An energy-based criterion of crack branching and its application on the multidimensional space method

In this paper, a criterion of crack branching based on the energy release rate is proposed. Theory of multidimensional space method for crack branching is established in the framework of the extended finite element method (XFEM), in which the formula of the boundary shift energy is deduced. The boundary cracking energy is obtained because the boundary shift model is abstracted into the boundary cracking model. The energy release rates of the crack branching in four quadrants are determined based on boundary cracking energy. Further, the energy release rate of the crack branching is introduced into multidimensional space method to determine the crack branching. If the crack tip branching, the criteria of crack branching are applied to determine the direction of the branching cracks, otherwise the criteria of maximum circumferential stress are applied to determine the direction of the cracks. Moreover, a classical numerical example is investigated to verify the correctness of the criterion of crack branching in the framework of the multidimensional space method.