Static Stress Intensity Factors Research Articles

Mechanics of crack initiation, propagation and crack grain boundary interaction in bicrystal silicon using near-tip localized stress calculation

Crack propagation and arrest are important phenomena in polycrystalline silicon. This phenomenon considerably affects the effectiveness of solar cells made up of polycrystalline silicon. To understand the crack propagation in polycrystalline silicon, it is essential to study the bicrystal first. Earlier, crack propagation has been studied in bicrystal materials primarily from the perspective of material behaviour. However, the mechanics part is largely missing. In the present work, crack propagation of bicrystal silicon during uniaxial tension has been studied at the atomistic scale. The silicon bicrystal was formed by joining two single crystals: crystal 1 and crystal 2 of different orientations. Various states like crack initiation, propagation, arrest, re-initiation and crack grain boundary (GB) interaction have been analysed using Stress Intensity Factor (SIF), calculated from the local crack tip virial stress field. Considering the effect of crack tip velocity, the dynamic SIFs at crack propagation state have been converted to the SIF of an equivalent static crack using the generalized expression for anisotropic material. It has been found that for the propagation state, crack propagation continues as long as the equivalent static SIF is greater than the critical SIF (CSIF). In crystal 1, crack propagation is along the low CSIF direction. Therefore, the crack did not change its path and propagated along a straight line. On the other hand, when the crack enters to crystal 2, it follows a zigzag path. Both of these observations have been justified in terms of near-tip calculated SIFs.

Investigation of dynamic three–point bending fracture properties of SCB sandstone

The stress intensity factor (SIF) is not only an important parameter in the field of fracture mechanics but also a key indicator reflecting the strength of the stress field at the crack tip. Therefore, this paper uses the Split–Hopkinson–Pressure–Bar (SHPB) device to conduct dynamic fracture experiments on semi–circular bend (SCB) sandstones of different fracture modes and then uses an experimental–numerical method to simulate the dynamic loading process to explore the changes law in SIF. Finally, combined with the high–speed digital image correlation (HS–DIC) technology, the fracture characteristics of SCB sandstones are studied. The results show that the rising rate of dissipated energy acting on the specimen fracture and damage increases rapidly when it is close to dynamic fracture, which makes the dynamic SIF also get a substantial increase, and the influence of dissipated energy on dynamic SIF is seen to be very strong. Moreover, prior to the occurrence of dynamic initiation fracture, the damage to the specimen is primarily triggered by microscopic tensile effects at the tip of the preset crack, which results in the value of KI always being greater than KII before the dynamic fracture occurs. Meanwhile, the distribution curve of dynamic SIF exhibits better symmetry or a trend towards symmetry compared to the distribution curve of static SIF. This suggests that the tip of the preset crack distributes with more uniform tensile and shear stresses during dynamic loading. Using the HS–DIC technique, when the specimen occurs dynamic pure mode I initiation fracture, there was a good symmetrical trend in both the vertical strain around the fracture crack and the horizontal displacement field at the specimen. However, when specimens occur dynamic mixed mode I/II initiation fracture, the horizontal displacement fields showed asymmetry, and the specimens all suffered varying degrees of shear damage.

A nonlocal energy‐informed neural network for isotropic elastic solids with cracks under thermomechanical loads

AbstractIn this paper, a nonlocal energy‐informed neural network is proposed to characterize the deformation behaviors of elastic solids with cracks subjected to thermomechanical loads in the framework of ordinary state‐based peridynamics. Based on principle of virtual work, a nonlocal energy approach is developed to recast the solution to peridynamic equilibrium equation as a problem of minimizing the potential energy of the system, which automatically satisfies the traction‐free boundary conditions. Meanwhile, the energy representation of physical system can be treated as the loss function for machine learning methods. Therefore, a nonlocal energy‐informed neural network is constructed to approximate the solution of the system. A distinct advantage of the proposed neural network is that the strain energy is expressed in terms of spatial integration instead of spatial derivatives, which avoids the invalidation of automatic differentiation at the crack surfaces in original physics‐informed neural networks. To demonstrate the convergence and accuracy of the proposed neural network, a series of problems in solid and fracture mechanics are conducted, and compared with results from analytical solutions or classical numerical methods. Additionally, for elastic material containing initial cracks, displacement extrapolation method is encoded into the proposed neural network to evaluate the static stress intensity factor.

Uncertainty qualification in evaluating dynamic and static stress intensity factors using SBFEM based on model order reduction

This paper presents a novel framework for Monte Carlo simulation (MCs) of dynamic and static fracture problems. The stress intensity factors can be directly extracted by using polygon scaled boundary finite element method (SBFEM) to analyze dynamic and static fracture problems. Uncertainty qualification analysis focuses on the output uncertainty caused by the uncertainty of system input. MCs is considered as a general tool to solve complex multi-dimensional uncertainty problems because of its simplicity and directness. However, its applicability is hindered due to the huge calculating cost. The use of singular value decomposition (SVD) and radial basis function (RBF) can reduce the computational cost, improve the computational efficiency, and realize the rapid uncertainty qualification analysis based on MCs. Considering the influence of uncertainty of structural material property parameters and structural shape parameters on the calculation results, MCs is used to analyze the statistical characteristics of structural response under random variables. Finally, several examples are given to verify the correctness and effectiveness of the algorithm.

Extended wavelet Galerkin method for mixed-mode cracked FGM plate under static and dynamic loads

The extended wavelet Galerkin method (XWGM) is presented for fracture mechanics analysis of functionally graded material (FGM) cracked plates under static and dynamic loads. The nonhomogeneous stress distribution and stress wave transition can be approximated by the piecewise truncated polynomial (B-spline) functions. The steep stress gradients around the crack tip can be captured by superposing the higher resolution wavelet functions with the nature of multiresolution analysis (MRA) in the wavelet theory. Additionally, a trigonometric function and Heaviside function are employed to enrich the wavelet bases for the fracture modeling. The interaction integral method is utilized to evaluate the static and dynamic mixed-mode stress intensity factors taking the inertia and material nonhomogeneity into account. Applicability of XWGM to the FGM crack problems is discussed through the numerical examples.

Boundary-integral analysis of a 3-d problem for a body with a periodic array of flexible inclusions

The static problem of discontinuous loading of an elastic body with a periodic array of flexible inclusions is considered. The problem is reduced to solving a 2-D boundary integral equation of the second kind of the Newton potential type with respect to an unknown inclusion opening function. With the help of solutions of the equation, the static stress intensity factors mode I are determined and their dependences on the distance between defects, the contrast between the rigidity of the matrix and the inclusion, and the filling of cracks with the injection material are analyzed.

A two-level nesting smoothed extended meshfree method for static and dynamic fracture mechanics analysis of orthotropic materials

This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method.

A new enrichment scheme for the interfacial crack modeling using the XFEM

A new enrichment scheme is proposed for the interfacial crack modeling in the framework of extended finite element method (XFEM), in which the approximation of displacement in the element including the crack tip is reconstructed by only the Heaviside function and the material interface enrichment function. The proposed enrichment scheme can model not only the discontinuity of displacement across the crack surface but also the discontinuity of displacement gradient across the material interface in the crack tip element with the weight functions. In addition, no blending elements are introduced and it is unnecessary to define the ramp function in the new XFEM formulation. The static and dynamic stress intensity factors (SIFs) of interfacial cracks are analyzed by the new enrichment scheme and the results show that the effect of material interface in the crack tip element plays a significant role in the accuracy of fracture mechanics parameters and the proposed method can predict the deformation more accurately and obtain a more stable dynamic SIFs. The proposed new enrichment scheme can be extended to three dimensional crack modeling in future works.

Fracture parameter investigations of functionally graded materials by using ordinary state based peridynamics

Static and dynamic fracture parameter analyses of functionally graded materials (FGMs) are conducted by using the ordinary state based peridynamic theory (OSPD). As a meshfree method, OSPD applies an integral equation to describe the motion of objects, avoiding the geometrical singularity in conventional fracture analysis measures. Domain integral method by introducing material gradient terms is employed in evaluating the static and dynamic stress intensity factors (SIFs). Meanwhile, peridynamic differential operator (PDDO) is also applied for the calculation of physical components derivatives. Different FGM modeling schemes are also examined in the OSPD framework. Cracked FGM structures with mode-I and mixed-mode crack scenarios are under investigation and results are validated by reference solutions. Accuracy and reliability of the proposed method will be examined and discussed.

Three-dimensional dynamic fracture analysis using scaled boundary finite element method: A time-domain method

A time-domain method for modeling three-dimensional transient dynamic fracture problems is developed based on the scaled boundary finite element method (SBFEM). For this purpose, a cracked polyhedron modeled by the SBFEM is constructed to simulate the through-thickness crack in the domain. The mass and stiffness matrices of polyhedrons are derived and assembled to form the three-dimensional elastodynamic equations in the time domain. High-order elements are used to improve computational accuracy. The dynamic response is evaluated by the Newmark method, and the stress field is expressed semi-analytically. Based on the theory of linear ealstodynamic fracture mechanics, the static stress intensity factors are extended to the dynamic stress intensity factors (DSIFs) by considering the dynamic effect. The DSIFs are directly extracted from the analytical solution in the radial direction of the cracked polyhedron. Numerical examples are modeled to validate the presented method. Good agreement is observed between the computed results and the published results in the literature. The effects of the time step, mesh density, and material damping coefficient on the computational accuracy are also investigated. It is found that moderately sized third-order elements can lead to very good solutions, and an increase of both the orders or number of elements does not significantly improve the accuracy of the simulation. The distribution of the DSIFs along the crack front of the 3D models is investigated and it is found that the DSIFs vary strongly along the crack front.

Determination of fracture toughness for steel 22k from the results of tests of different types specimens

Nowadays, in various industries, in particular in nuclear energy, to determine the fracture toughness, along with standard tests of compact specimens, which are quite expensive and complex, methods are developed to determine these characteristics by impact tests of Charpy specimens using different correlations between Charpy impact fracture energy (CVN) and critical stress intensity factor (J-integral). The paper analyzes correlation and analytical methods, the authors of which consider them universal for a certain class of steels. Correlation methods are divided into one-stage and two-stage. One-stage methods allow to obtain the value of the critical stress intensity factor by the known fracture energy. Two-stage methods in the first stage offer the calculation of the dynamic critical stress intensity factor, in the second the temperature shift and obtaining a static critical stress intensity factor. Analytical methods according to the іmpact fracture diagram of the specimen allow to construct a J-R curve and calculate the value of the J-integral. A series of fracture tests of CT specimens made of heat-resistant steel 22K was carried out, the reference temperature T0 was determined according to the single-temperature method of the ASTM-1921 standard and the Master curve was constructed. A series of standard Charpy specimens impact tests in the temperature range -50…+100°С was performed using an instrumented drop-weight impact testing machine equipped with a high-speed registration system. According to the results of Charpy specimens impact tests, the fracture toughness were determined using different methods. It is established that both analytical and correlation methods cannot be universal and can be used to determine the fracture toughness of 22K steel. Therefore, a new exponential correlation was proposed between the fracture energy of the Charpy specimens and the critical stress intensity factor for heat-resistant steel 22K.

Study on the static critical stress intensity factors of sandstone in a water environment based on semicircular bending specimens

Fracturing behavior of rock is significantly affected by water. To study the influence of water on the critical values of stress intensity factors (the SIFc, including KIf and KIIf) corresponding to the onset of fracture of rocks, edge-cracked semicircular bend (SCB) testing was performed on sandstone specimens (collected at a site in Linyi City, Shandong Province, China) with different water contents. First, water absorption was tested to study the evolution of the water content in the specimens. Then, the SIFc of SCB specimens after different soaking durations in water (yielding conditions ranging from dry to saturation) was tested under three-point loading. The results showed a linear decrease in SIFc with increasing water content for all the specimens with different notch angles; however, with increasing notch angle, the KIf exponentially decreased, while the KIIf increased. To study the effect of sustained load on SIFs of rock in water, a new experiment was designed on the specimen subjected to loading while soaking for a certain duration, then the SIFc was tested. Comprehensive predictive empirical relationships are established between SIFc and water content/soaking duration for the sandstone. A novel observation is for two specimens with the same water soaking duration, the SIFc of a specimen with loading during soaking is lower than that when it was tested after soaking, which is attributed to the fact that water is more likely to penetrate into the rock, because the bending load opens pores and cracks then leads to an increase in water content under such conditions. In other words, the water content of sandstone specimens used in this study is the key factor affecting its SIFc.

Effect of Yield Strength Distribution Welded Joint on Crack Propagation Path and Crack Mechanical Tip Field.

Due to the particularity of welding processes, the mechanical properties of welded joint materials, especially the yield strength, are unevenly distributed, and there are also a large number of micro cracks, which seriously affects the safety performance of welded joints. In this study, to analyze the effect of the uneven distribution of yield strength on the crack propagation path of welded joints, other mechanical properties and residual stresses of welded joints are ignored. In the ABAQUS 6.14 finite element software, the user-defined field (USDFLD) subroutine is used to define the unevenly distributed yield strength, and extended finite element (XFEM) is used to simulate crack propagation. In addition, the static crack finite element model of the welded joint model is established according to the crack propagation path, which is given the static crack model constant stress intensity factor load, and the influence of an uneven yield strength distribution on mechanical field is analyzed. The results show that the crack length of welded joints as well as the plastic deformation range of the crack tip in high stress areas can be reduced with the increase of yield strength along the crack propagation direction. Moreover, the crack deflects to the low yield strength side. This study provides an analytical reference for the crack path prediction of welded joints.

Hybrid meshless displacement discontinuity method (MDDM) in fracture mechanics: Static and dynamic

This paper investigated a hybrid Meshless Displacement Discontinuity Method (MDDM) for a cracked plate subjected to static and dynamic loadings. The purpose of MDDM is to model displacement discontinuity on a cracked surface by the displacement discontinuity method in an infinite plate. This was achieved by considering a meshless approach, the equilibrium equations, and the boundary conditions for a domain with an irregular nodes distribution. Also, by imposing the principle of superposition, accurate and convergent solutions can be obtained. In this paper, the static and dynamic stress intensity factors, and the crack growth for different initial crack length and crack slant angles are investigated. The Laplace transform method is applied to deal with dynamic problems and the time-dependent values are obtained by the Durbin inversion technique. Validations of the presented technique are demonstrated by four numerical examples of plates with a central embedded crack.

Extended meshless moving Kriging method for crack propagation analyzing in orthotropic media

Orthotropic composite material is the particular type of anisotropic materials and their products have been extensively used in a wide range of engineering applications. Study on mechanical behaviors of such materials under working conditions is very essential. In this study, an extended meshfree moving Kriging interpolation method (namely as X- MK) is presented for crack analyzing in 2D orthotropic materials models. The Gaussian function is used for constructing the moving Kriging shape functions. Typical advantages of the MK shape function are the high-order continuity and the satisfaction of the Kronecker’s delta property. To calculate the stress intensity factors (SIFs), interaction integral method is used with orthotropic auxiliary fields. Several numerical tests including static SIFs calculating and crack propagation predicting are performed to verify the accuracy of the present approach. The obtained results are compared with available refered results and they have shown a very good performance of the present method.

A polygonal XFEM with new numerical integration for linear elastic fracture mechanics

The extended finite element method (XFEM) has become a powerful and effective technique for modeling fracture problems without remeshing. In this paper, we introduce a novel and effective computational approach that is based on polygonal XFEM (named as PolyXFEM) for the analysis of two-dimensional (2D) linear elastic fracture mechanics problems. The PolyXFEM is equipped with a new numerical integration technique that uses the concept of Cartesian transformation method (CTM) over polygonal domains. The underlying idea of the CTM is to transform a domain integral into a boundary integral and a one-dimensional integral. This is computationally more efficient compared to the two-level mapping integration commonly used on polygons. Furthermore, the PolyXFEM is effective in modeling crack problems due to the local enrichment based on the partition of unity method. Our numerical results show that improvements in accuracy are reached thanks to the higher-order of the polygonal shape functions of the PolyXFEM. In addition, efficiency in computational time is achieved because of the usage of the CTM integration scheme, which appears to be quite suitable for polygonal domains. To demonstrate the accuracy and performance of the developed PolyXFEM approach, several numerical examples for 2D linear elastic fracture problems are considered, in which static stress intensity factors and crack propagation are investigated and compared with reference solutions. The convergence and mesh independence of the PolyXFEM for crack analysis is also analyzed.

Dynamic stress intensity factors of interacting branched cracks

In the present work the boundary element method (BEM) is applied to analysis of statically and dynamically loaded infinite plates with multiple stationary branched cracks. The material of the plates is linear-elastic, homogenous and isotropic. In the applied BEM approach the displacement and traction boundary integral equations are used simultaneously for nodes on crack surfaces. Contrary to the finite element method (FEM) in the BEM numerical solutions are obtained by discretization of external boundaries and crack surfaces. The dynamic problem is solved by using the Laplace transform method and the solution in the time domain is computed by the Durbin numerical inversion method. Numerical examples of multiple branched cracks in infinite plates subjected to static and dynamic loadings are presented. An influence of orientation, distances between cracks and the number of cracks on static and dynamic stress intensity factors (SIF) is studied.

Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements

In this paper, the recent development of extended consecutive-interpolation 4-node quadrilateral element (XCQ4), which goes beyond certain limitations of conventional methods, is further employed to study dynamic and static thermoelastic fracture problems. In particular, static stress intensity factors (SIFs) and transient dynamic responses of two-dimensional (2-D) stationary cracks in elastic solids under thermal and thermal-mechanical loadings are investigated. In addition, simulation of quasi-static crack propagation under thermal-mechanical loading condition is also performed. To precisely capture the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack, the unknown field variables of mechanical displacements and temperature are approximated in terms of the XCQ4 framework by different enrichment strategies, which include either the use of the conventional branch functions or the new ramp function associated with Heaviside function. Only adiabatic condition is considered on the crack surfaces. For quasi-static crack growth analysis, the maximum hoop stress criterion, which is also valid for thermo-elastic fracture mechanics, is adopted. For transient crack analyses, the inertial effect is taken into account for the interaction integral in the consideration of DSIFs. Through eight numerical examples for thermoelastic fracture problems, the validity of the XCQ4 is confirmed by a comparison of the obtained results to those derived from different approaches in previous studies. In addition, the accuracy, performance, and superior advantages of the XCQ4 over standard extended finite element approach are also demonstrated.

Effect of wire speed on subsurface cracks in wire sawing process of single crystal silicon carbide

The effect of wire speed on dynamic stress intensity factors (DSIFs) at subsurface crack tips in the wire sawing process of single crystal silicon carbide (SiC) is investigated. The analysis of wire speed is converted to dynamic excitation loads due to moving diamond abrasives on wire saw. The excitation loads are obtained from stress field around the abrasives considering the morphology of abrasives and scratching depth. The governing equations are established for the hexagonal crystal structures of SiC, and solutions under the dynamic excitation load are calculated using Laplace transformation and Fourier transformation. Results show that the scratching angle and length of cracks have a great influence on the DSIFs. However, the effect of wire speed on the ratio of maximum DSIFs and maximum static stress intensity factors (SSIFs) is relatively small when the wire speed is less than 100m/s. This indicates that the wire sawing process can be simplified as a quasi-static process when analyzing subsurface cracks problem.

Integral equation analysis for cracked strip of orthotropic functionally graded material

In this paper, the integral equations are formulated and solved numerically for a cracked orthotropic strip with Functionally Graded Materials (FGMs) subjected to both static and dynamic loads. The Green's function of displacement discontinuity in an orthotropic FGM strip is derived by using the Fourier transform and the Laplace transform techniques with the shear modulus and the mass density varying exponentially, and constant Poisson's ratio for functionally graded materials. It has been shown that the transformed Green's solutions have the same order of singularities as that in the static case. The normalized time-dependent stress intensity factors are obtained with the Durbin’s inversion method for the Laplace transform. The static stress intensity factors are compared with the results given by finite element analysis and excellent agreements are achieved.