Vicinity Of Crack Tip Research Articles

On the theoretical and phase field modeling of the stress state associated with ferroelastic twin nucleation and propagation near crack tip

Ferroelastic domain switching is considered as that a twin undergoes a lattice orientation described by a deviatoric eigenstrain in this work. A continuum analysis, based on Eshelby equivalent inclusion theory, is derived to study the nucleation and evolution of twin near a pre-existing crack tip (Mode-I) in a mono-phase tetragonal crystal, in an attempt to achieve a better understanding of the contribution that twinning makes to the toughening effect. The analysis develops progressively from the homogenous isotropic, inhomogeneous isotropic to the general anisotropic conditions; and several analytical solutions are evidenced through phase field simulations. Two important critical stresses, defined as the nucleation stress and the fracture-induced twinning (FIT) stress, are found to determine the twin evolution; specifically, the nucleation stress is related to the twinning in a defect-free solid, and the FIT stress is related to the twinning in the vicinity of crack tip. Moreover, considering the inhomogeneity effect, the change of the stress intensity factor (SIF) due to a ferroelastic twin is derived, which provides evidence that the modulus toughening effect originating from the elastic misfit between the parent and twin can be ignored at a relative low applied SIF. Theoretical analysis and phase field simulations both clarify that the shielding/anti-shielding effect to the crack tip is dependent on the nucleation location and the twin size during evolution. The best toughening formula of a ferroelastic ceramic can be obtained by decreasing the FIT stress and increasing the change of SIF simultaneously through adjusting certain material parameters.

On the Computational Derivation of Bond-Based Peridynamic Stress Tensor

The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

Meshfree thermomechanical crack growth simulations with new numerical integration scheme

This paper presents for the first time an improved meshfree particle method without the need for background cells in terms of numerical integration for thermal-mechanical crack growth analysis. In this work, both adiabatic and isothermal crack surfaces are considered. Asymptotic solutions-based enriched functions are incorporated into the approximation scheme to mathematically capture jump across crack surfaces of both the temperature and displacements, as well as the singularities of heat fluxes and stresses in the vicinity of crack tip. Once the stress intensity factors (SIFs) are evaluated, the direction of crack growth can be determined. The meshfree analysis is based on radial point interpolation method (RPIM), in which Cartesian transformation method (CTM) is adopted for numerical integration, instead of the conventional Gaussian quadrature. The utilization of Gaussian scheme requires background cells in shape of quadrilaterals or triangles. In contrast, the CTM is advantageous in the manner that the background cells are no longer required, leading to truly meshfree formulation. The novel contribution of the current work is the extension of CTM scheme in which the integration domain is updated according to crack growth. The crack surfaces are viewed as part of the boundaries of the problem domain. Therefore, it is guaranteed that no discontinuities exist within the domain. The accuracy of the proposed approach is demonstrated by comparison of computed results with reference ones from analytical solution, and other existing numerical methods.

Efficient Toughening of Short-Fiber Composites Using Weak Magnetic Fields.

Short fibers may serve as toughening agents of composite materials because of the high energy dissipated during fracture, associated with numerous fiber pullouts. An ongoing challenge is to improve their toughness even further, by directing and concentrating fibers near highly stressed structural regions. Weak magnetic fields are utilized to increase the fracture toughness of an epoxy matrix reinforced by short magnetized glass fibers by directing and concentrating fibers near highly stressed structural regions. The orientation and local concentration of the fibers are controlled by the vector components of the magnetic field, and by the gradient in field intensity, respectively. Optimized fracture toughness was achieved by using two pairs of permanent magnets, combining enhanced concentration of fibers in the crack-tip vicinity with alignment of the fibers along the load direction. This optimized value was well above the reference fracture-toughness measured for composites with the same filler content in the absence of a magnetic field, as well as above the value achieved by exploiting unidirectional alignment, without fiber translation, using a solenoid. The method suggested in this study—localized reinforcement using magnetic translation of fillers through the formation of magnetic gradients—enables efficient and controllable improvement in the composite’s overall resistance to fracture, without the involvement of additional phases or material.

Nonsingular Gradient Solutions in Crack Mechanics and the Concept of Stress Concentration

We consider the fracture mechanics problem for the finite and semi-infinite cracks in the gradient elasticity. Local stress fields that define the fracture the strength of materials are found as solutions of the inhomogeneous Helmholtz equations in which the inhomogeneity is determined by classical stresses. To construct solutions, the radial factors method and the Papkovich-Neuber representation are used. It is shown that, in problems of crack mechanics. We show that the local stresses in the vicinity of crack tips are non-singular, have the form characteristic of stress concentration, and depend only on the level of acting stresses and the scale parameter, which is found as a result of mechanical testing of material samples.

Relationship Between Impact Toughness and Microstructure for the As-Rolled and Simulated HAZ of Low-Carbon Steel Containing Ti-Ca Oxide Particles

In this study, the microstructure and mechanical properties of Ti-Ca deoxidized low-carbon steel compared with Al-Ca deoxidized steel were investigated. Hot rolling experiment and high heat input welding heat-affected zone (HAZ) simulation were carried out. In situ transformation behavior and crack propagation during impact in simulated HAZ were analyzed. The results indicated that interlocking acicular ferrite (AF) was the dominant microstructure after Ti-Ca deoxidation treatment, and TiOx-CaO-Al2O3-MnS-type complex inclusion with maximum proportion was effective for AF nucleation. However, Al2O3-CaO-(Ca, Mn)S-type inclusion that was dominant in Al-Ca steel exhibited significantly lower ability for AF nucleation. The high ratio of high-angle grain boundaries/low-angle grain boundaries in AF microstructure hindered the initiation and propagation of crack. The stress concentration in the vicinity of crack tip can be released because of superior deformability of AF structure as compared to bainite packets. By introducing Ti-Ca oxide particles, the mechanical properties of hot-rolled steel plate and HAZ, particularly impact toughness, were significantly enhanced because of the formation of fine AF.

Dynamic fracture of soda-lime glass plates studied using two modified Digital Gradient Sensing techniques

Two modified full-field Digital Gradient Sensing (DGS) methods of higher measurement sensitivity, transmission-reflection DGS (tr-DGS) and double-transmission DGS (t2-DGS), are extended in this work to examine their feasibility to study fast fracture events in soda-lime glass by measuring stress gradients. Full-field optical measurement of deformations and stresses in such high-stiffness and low-toughness brittle material is very challenging. Extremely high crack speeds (~1400 m/s), low failure strain (εf < 0.1%), and highly localized sub-micron scale deformations are among the challenges to overcome in this material system to succeed in this regard. The higher measurement sensitivity of tr-DGS and t2-DGS make them particularly suitable for studying transparent glasses and ceramics relative to previous versions of DGS method. These two new methods are demonstrated for studying dynamically growing cracks in soda-lime glass plates subjected to dynamic impact loading in a modified Hopkinson pressure bar device and ultrahigh-speed photography (1 million fps). Previously proposed reflection DGS (r-DGS) is also implemented to comparatively show the benefits of higher measurement sensitivity offered by tr-DGS and t2-DGS methods. The measured stress gradients are numerically integrated to estimate σxx+σyy stress fields around the crack tip vicinity. The optical measurements are also used to obtain fracture parameter histories of crack speed, stress intensity factors and energy release rates under mode-I conditions. The energy release rates increase rapidly when the crack speed approaches ~1400 m/s. The measured fracture parameters are also assessed using a complementary elasto-dynamic finite element model up to crack initiation. The agreement between measurements and simulations supports the feasibility of these two optical methods for studying dynamic fracture mechanics of this challenging material.

Experimental and numerical investigation of effect of size, position and geometry of some cutouts on fatigue life and crack growth path on AISI1045 steel plate

The main factor in determining path and rate in fatigue crack growth (FCG) problems is the stress field around the crack tip that is defined as stress intensity factor (SIF). Cutouts cause a stress concentration that depends on their size and geometry. This stress concentration can affect the stress field near the crack tip vicinity and deviates the FCG path. In this paper in order to study the effect of cutout parameters on FCG, first, a taguchi statistical analysis is used. Then, controlling and changing the FCG path and rate in specimens with different cutouts are studied. The results show that the size and position of cutouts are the most effective parameters on the crack growth path. According to the cutout’s features, an increase or decrease in fatigue life can be expected. An effective range for the distance between a cutout and a crack tip was determined. To verify the results of the extended finite element method (XFEM), the mode I SIF was obtained experimentally and compared with XFEM results. It was concluded that numerical results are in such a good agreement with experimental results.

Path-independent J-integral for cracks in decagonal quasicrystals

The path-independent J-integral is derived for fracture mechanics analysis of decagonal quasicrystals (QCs). The gradient theory of quasicrystals is developed here to consider large strain gradients at the crack tip vicinity. The constitutive equations contain phonon and phason stresses, and the higher-order stress tensor. The higher-order elastic material parameters are proportional to the internal length material parameter and the conventional elastic coefficients. The FEM equations are derived to solve general boundary value problems for the strain gradient theory of the QCs.

Effects of crystalline orientation, twin boundary and stacking fault on the crack-tip behavior of a mode I crack in nanocrystalline titanium

In this work, molecular dynamics simulation and linear elastic fracture mechanics were employed to analyze the crack-tip behavior of a Mode I crack in nanocrystalline titanium. The effects of crystalline orientation, twin boundary and stacking fault on crack propagation were taken into account. Simulation results demonstrate that the crack-tip behavior and thus the crack propagation mode strongly depend on crystalline orientations and plane defects. Cracks lying on the hexagonal close-packed basal plane, the prismatic plane, or along a stacking fault plane defect propagate in a brittle manner, without involving proper dislocation emissions and twin nucleations in the crack-tip vicinity. In contrast, cracks show a ductile propagation behavior when aligned along the pyramidal plane, the {101¯2} plane, or the {101¯2}〈1¯011〉 twin boundary. For these cases, local crack-tip plasticity and crack-tip reconstructions are found to play significant roles. The impact of strain rate on the crack-tip behavior of a basal crack was also investigated in detail. Five strain rates varied between 109 s−1 and 1010 s−1 were considered. With increased strain rates, a brittle-to-ductile transition was clearly observed for crack propagation. The desired transition can be attributed to the high stress and energy concentrations near the crack-tip under elevated strain rates, leading to the successive emission and propagation of partial dislocations. To verify and validate the simulation results, a theoretical analysis on the competition between brittle and ductile crack propagations was also implemented. The theoretical predictions based on the linear elastic fracture mechanics were found to reasonably agree with the simulation results. The observations and conclusions deduced from the combined modeling and theoretical study are helpful to the better understanding of fracture mechanics in nanocrystalline titanium.

Influence of Applied Load on Oxidation in the Vicinity of Crack Tips of Stainless Steel under High Temperature Water

Influence of Applied Load on Oxidation in the Vicinity of Crack Tips of Stainless Steel under High Temperature Water

INFLUENCE OF THE HIGHER ORDER TERMS IN WILLIAMS’ SERIES EXPANSION OF THE STRESS FIELD ON THE STRESS-STRAIN STATE IN THE VICINITY OF THE CRACK TIP. PART II

The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams asymptotic series expansion. While the general structure is well known, complete expressions are rarely available for specific crack problems. The paper is devoted to the multi-parameter description of the stress field in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field are obtained. The procedure used in the paper relates Williams series coefficients and the complex potentials of the plane elasticity. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept. Complete closed-form expressions can be used to derive, test and improve numerical and experimental approaches involving higher order terms in crack-tip expansions.

Elastic field for a blunt crack in a decagonal quasicrystalline material

We show that the full-field solution for the plane strain deformations of a decagonal quasicrystalline material with traction-free parabolic boundary can be used to directly obtain the elastic field describing phonon and phason stresses as well as displacements in the vicinity of a blunt crack tip in a decagonal quasicrystalline material. The aforementioned full-field solution is derived by applying complex variable methods to the boundary value problem in the physical plane without any recourse to conformal mapping and a corresponding image plane.

Computational Analysis of Linear Elastic Crack Growth in Functionally Graded Bodies Using Non-Uniform Steps Integrated in the MLPG

This paper describes the estimation of mixed-mode crack growth path in functionally graded materials under mechanical and thermal loads by the meshless method. Simple and complex geometries with single and double cracks are provided to verify the merits of a recently proposed modified meshless local Petrov–Galerkin method and to show the applicability of this method in crack growth simulation. Polygon test function domains are constructed locally at every crack growth step in order to numerically calculate contour and domain integrals. The classical form of thin plate spline radial basis function without enrichment with a regular distribution of points in the vicinity of crack tips are used. In this efficient method, the total number of domain points are kept fixed during crack growth steps forever. For this end, a simple method of new crack tip determination at every growth step is proposed. The MLPG results are compared with reference solutions including experimental, finite element, boundary element and other meshless methods and the accuracy of this method is investigated in different numerical test problems.

Распределение внутренней энергии в окрестности вершины трещины нормального отрыва

Распределение внутренней энергии в окрестности вершины трещины нормального отрыва

Analytical investigations of interface crack growth between two dissimilar rock layers under compression and tension

Since most geological contacts in Earth's crust represent potential sites of interface cracks, these cracks are of particular importance in geological engineering. Fracture propagation of interface cracks growth between dissimilar rock layers depends on the contrasting elastic properties across the bedding interface. Most of the topics associated with interfaces in rock mechanics and geological engineering have focused on the reciprocal influence between ordinary cracks and layer surfaces without taking into account the presence of interfacial flaws between rock layers. In this research, the case of an inclined internal interface crack between two dissimilar rock layers under compression and tension was studied in detail. The effect of crack inclination angle on the stress intensity factor and on the stress distribution functions in the vicinity of an interface crack tip was investigated. Our results indicated that interface crack growth can occur in stiffer or more compliant layer depending on the elastic stiffness contrast (differences in Young's Modulus between two layers of rock) and crack inclination angle. The critical value of crack inclination angle, after which interface crack growth can occur in one of two dissimilar rock layers, was determined by taking into account the elastic stiffness contrast and Poisson's ratio of rock formations. Fracture envelopes for the heterogeneous rock medium are located above those of the homogeneous rock medium and by increasing the elastic stiffness contrast of dissimilar rock layers, the area under fracture envelope increases and the interface cracks needs more energy to satisfy fracture condition.

The Orientation of Strain-Induced Crystallites in Uniaxially-Strained, Thin and Wide Bands Made from Natural Rubber

Vulcanized natural rubber (unfilled and filled with 20 phr carbon black) is strained. We suppress the macroscopic formation of fiber symmetry by choosing strip-shaped samples ("pure-shear geometry") and investigate the orientation of the resulting crystallites by two-dimensional wide-angle X-ray diffraction (WAXD), additionally rotating the sample tape about the straining direction. Indications of a directed reinforcing effect of the strain-induced crystallization (SIC) in the thin strip are found. In the filled material fewer crystallites are oriented and the orientation distribution of the oriented crystallites is less perfect. The results confirm, that it is important for the evaluation of crystallinity under deformation to check, whether fiber symmetry can be assumed. This has consequences in particular on the quantitative interpretation of space-resolved scanning experiments in the vicinity of crack tips. Furthermore it raises the question, whether there is an asymmetric reinforcing effect of the SIC in the vicinity of crack tips inside natural rubber.

Fracture modeling with the adaptive XIGA based on locally refined B-splines

This paper aims at investigating fracture behavior of single and multiple cracks in two-dimensional solids by an adaptive extended isogeometric analysis (XIGA) based on locally refined (LR) B-splines. The adaptive XIGA is capable of modeling cracks without considering the location of crack faces due to the local enrichment technique based on partition-of-unity concept. The XIGA approximation is locally enriched by Heaviside function and crack tip enrichment functions to capture the discontinuity across crack faces and singularity in the vicinity of crack tips. The LR B-splines, which are generalized by B-splines and NURBS, not only inherent desirable properties of the B-splines and NURBS but also can be locally refined, ideally suitable for adaptive analysis. Structured mesh refinement strategy is applied to perform local refinement for LR B-splines based on a posteriori error estimator. According to the recovery technique proposed by Zienkiewicz and Zhu, the smoothed strain field is obtained to construct the posteriori error estimation based local refinement. The stress intensity factors (SIFs) are evaluated using the contour interaction integral technique. Several benchmark numerical examples are illustrated in comparison to analytical or reference solutions to verify the accuracy and efficiency of the developed approach. The proposed adaptive XIGA method is also applied to a curved crack, multiple cracks and complicated structure with a crack, which sufficiently presents the applicability of the proposed method in crack modeling. In addition, the convergence rate of the adaptive local refinement strategy is numerically studied and compared with that of the global refinement approach. The convergence rate of the adaptive local refinement is shown to be faster than that of the global refinement.

Cyclic deformation characteristics of the metastable β-type Ti–40Nb alloy

The present study investigates the cyclic deformation behavior of a new metastable β-type Ti–40Nb alloy (wt%) in two different microstructural conditions. Severe cold rolling followed by recrystallization annealing were performed on cast and homogenized Ti–40Nb rods to obtain a single β-phase microstructure at room temperature. Additional aging at 573 K was carried out to generate isothermal ω-phase precipitates in the β-matrix. Fatigue tests at cycles up to 2 × 106 and a stress ratio of R = −1 were realized with samples with electrochemically polished surfaces. EBSD analysis was carried out after the fatigue tests. Furthermore, detailed fractographic investigations as well as TEM analysis were executed. Results revealed significant differences in the cyclic deformation behavior and a higher fatigue strength for the aged condition. Reasons for superior fatigue properties of the aged condition are a pronounced precipitation hardening effect of the ω-phase as well as a complete suppression of stress-induced martensite formation and deformation twinning due to a barrier function of the ω-precipitates. In the vicinity of the fatigue crack tip, where the localized plastic zone dominates the damage evolution, precipitation-depleted channels can be observed. Within these ω-depleted channels dislocations have an increased mobility, allowing highly localized plastic deformation. Signs of other deformation features with increasing distance to the fatigue crack are not observed. EBSD analysis of the recrystallized samples with initial single β-phase showed on the contrary changes in the microstructure caused by the cyclic loading and the high instability of the β-phase. A dominant development of stress-induced α’’-martensite towards a full martensitic microstructure dependent on the applied stress amplitudes and cycles as well as additional isolated {332} <113> twinning were detected.