Non-singular Stress Research Articles

Extension of Tsai-Hill failure concept for mixed-mode I/II fracture investigation of orthotropic materials considering T-stress effects

An essential study in the discourse surrounding composite materials is the investigation of their fracture mechanics due to the intricate nature of composite structures, it is necessary to take into account the presence of cracks and faults throughout the design process. In this research, within the context of linear elastic fracture mechanics, the fracture criterion for orthotropic materials under mixed mode II/I loading is presented by expanding the Tsai-Hill failure criterion. According to the crack, it may be aligned or perpendicular to the fibers; the criterion is stated for both cases. Also, the effects of the non-singular stress component on crack growth have been investigated. The mechanical and fracture data of Russian pine wood was extracted through tensile testing and the finite element (FE) method. The validity of the proposed criterion, in comparison with the experimental findings extracted in this research, has been proved. The precision of the previous proposed fracture criteria has been reviewed in comparison with the proposed criterion. The appropriate behavior of the fracture limit curves compared to the experimental data shows the efficacy of the proposed criterion to predict the initiation and propagation of cracks in orthotropic materials.

An FFT based adaptive grid framework to represent non-singular dislocations

Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics requires very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated in the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed into a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

Influence of distinct testing methods on the mode-I fracture toughness of Longmaxi shale

Abundant shale gas in shale reservoirs can be successfully extracted by the fracking technique. The formation of hydraulic fractures is related to the mode-I fracture toughness that represents the resistance to cracking propagation. Two typical Longmaxi shale samples with horizontal and vertical bedding planes are selected in this work. To adequately measure the mode-I fracture resistance of shale, this research adopts four mode-I test specimens with different geometries (cuboid, cylinder, disk, and semi-disk). The experimental results demonstrate that there is an acceptable linear correlation between mode-I fracture resistance and critical T-stress. Further, the discrepancies in mode-I fracture resistance are due to distinct T-stresses that exist in the mode-I test specimens. Consequently, the mode-I fracture toughness KIc for the shale with horizontal bedding is 1.57 MPa·m0.5, while that for the shale with vertical bedding is 1.36 MPa·m0.5. Considering the influences of singular and non-singular stress terms, the GMTSN (generalized maximum tangential strain) and GMTSEDF (generalized maximum tangential strain energy density factor) fracture criteria can provide theoretical explanations for the variation of mode-I fracture resistance measurements. Based on the AE (acoustic emission) technique, the classical RA-AF method can identify the fracture mechanisms of distinct mode-I test specimens. However, the differences in the high-density zones of the RA-AF distribution are conspicuous for distinct mode-I testing methods.

Numerical modeling of crack propagation from open and closed flaws in rock

The traditional fracture criterion performs well when calculating crack propagation under tensile conditions, but a zigzag crack propagation path was obtained under compression, which is inconsistent with experimental results. This study indicates that the positive and negative oscillations of the mode-II stress intensity factor (KII) under compressive loading during crack propagation is the reason for the zigzag trajectory of crack propagation. Such oscillation of KII can be effectively eliminated when the non-singular stress (T-stress) at the flaw tips is considered, and thus, the path of crack propagation can also be smoothed. However, due to the complicated calculations of the stress intensity factors and T-stress at each step of crack propagation, it is not suitable for simulation in fast, complex environments and multi-flaws. Also, the lack of a shear fracture criterion in traditional fracture methods leads to difficulty in studying shear crack propagation in rock. Therefore, a strength-based localized maximum stress (SLMS) criterion was proposed in this study to model both tensile and shear crack propagations in rock more efficiently, conveniently, and accurately. Then, the crack propagation processes in both plate and Brazilian disc specimens with a single flaw under uniaxial and biaxial compression were modeled to investigate the effects of the flaw inclination angle, friction coefficient, and loading level on the crack propagation path. The influence of the contact friction between the flaw surfaces on the contact status and the crack propagation path during the crack propagation process was also analyzed. Also, the crack propagation in a plate with a single flaw under biaxial tension was modeled, which shows bifurcation crack propagation when the lateral tension is several times larger than the axial tension. All the modeled results indicated that the proposed SLMS criterion can get better results when modeling crack propagations for open or closed flaws under tension or compression conditions.

Determination of singular and higher order non-singular stress for angularly heterogeneous material notch

The complete stress distribution around the tip of angularly heterogeneous material V-notch includes singular stress term and higher order non-singular stress term. The latter has important role on the prediction of the fracture, while it is usually neglected due to the evaluation difficulty. The idea of combing the singularity analysis near the notch vertex with the finite element analysis is established to deal with this difficulty. Firstly, a set of variable coefficient differential equations for the singular analysis of angularly heterogeneous material notch are built. The interpolating matrix method is introduced to solve the singular characteristic equations to prepare the singularity order and characteristic angular function. The V-notched structure is then analyzed by the coarse finite element meshes. The displacements from the finite element analysis are substituted into the singularity asymptotic expression to form a set of over-determined equation. The least square method is applied to yield the amplitude coefficients in the series asymptotic expansion. The complete stress field near the angularly heterogeneous material V-notch tip is then constructed according to the obtained amplitude coefficients, singularity orders, characteristic angular functions, together with their first order derivatives. The singular and non-singular stress term can be respectively calculated according to the truncated series terms. The reconstructed stresses have the same precision of the displacement from the view point of the finite element analysis. The role of the non-singular stress on the stress intensity factors of angularly heterogeneous material notch is then investigated.

Mechanics and Energetics of Kink Deformation Studied by Nonlinear Continuum Mechanics Based on Differential Geometry

This study conducts dislocation-based modeling and numerical analysis for kink deformation using the theory of nonlinear continuum mechanics based on differential geometry. The kinematics of the continuum is represented by the reference, intermediate, and current states, which are related to the multiplicative decomposition of the deformation gradient into plasticity and elasticity. The intermediate state is determined by the Cartan first structure equation, whereas the current state is obtained from the stress equilibrium equation. Kink deformation is modeled by a planar array of edge dislocations, and the governing equations are solved numerically using the finite element method. Quantitative verification of the model is conducted by comparing the bending angle at a kink interface and the theoretical prediction given for the tilt grain boundary. We construct two types of ortho-type kink models for several interface lengths to understand the energetics and mechanics of the kink growth process. The present numerical analysis demonstrates the significant stress concentration at the growth front due to the formation of wedge disclination. We also discuss the material strengthening mechanism from elastic strain energy, nonsingular stress fields, and nonlinear elastic interaction between the disclinations.

Steady-State Crack Growth in Nanostructured Quasi-Brittle Materials Governed by Second Gradient Elastodynamics

The elastodynamic stress field near a crack tip propagating at a constant speed in isotropic quasi-brittle material was investigated, taking into account the strain gradient and inertia gradient effects. An asymptotic solution for a steady-state Mode-I crack was developed within the simplified strain gradient elasticity by using a representation of the general solution in terms of Lamé potentials in the moving framework. It was shown that the derived solution predicts the nonsingular stress state and smooth opening profile for the growing cracks that can be related to the presence of the fracture process zone in the micro-/nanostructured quasi-brittle materials. Note that similar asymptotic solutions have been derived previously only for Mode-III cracks (under antiplane shear loading). Thus, the aim of this study is to show the possibility of analytical assessments on the elastodynamic crack tip fields for in-plane loading within gradient theories. By using the derived solution, we also performed analysis of the angular distribution of stresses and tractions for the moderate speed of cracks. It was shown that the usage of the maximum principal stress criterion within second gradient elastodynamics allows us to describe a directional stability of Mode-I crack growth and an increase in the dynamic fracture toughness with the crack propagation speed that were observed in the experiments with quasi-brittle materials. Therefore, the possibility of the effective application of regularized solutions of strain gradient elasticity for the refined analysis of dynamic fracture processes in the quasi-brittle materials with phenomenological assessments on the cohesive zone effects is shown.

Nonsingular Stress Distribution of Edge Dislocations near Zero-Traction Boundary.

Among many types of defects present in crystalline materials, dislocations are the most influential in determining the deformation process and various physical properties of the materials. However, the mathematical description of the elastic field generated around dislocations is challenging because of various theoretical difficulties, such as physically irrelevant singularities near the dislocation-core and nontrivial modulation in the spatial distribution near the material interface. As a theoretical solution to this problem, in the present study, we develop an explicit formulation for the nonsingular stress field generated by an edge dislocation near the zero-traction surface of an elastic medium. The obtained stress field is free from nonphysical divergence near the dislocation-core, as compared to classical solutions. Because of the nonsingular property, our results allow the accurate estimation of the effect of the zero-traction surface on the near-surface stress distribution, as well as its dependence on the orientation of the Burgers vector. Finally, the degree of surface-induced modulation in the stress field is evaluated using the concept of the -norm for function spaces and the comparison with the stress field in an infinitely large system without any surface.

On the prediction of the stress field in adhesive joints using a combined analytical-numerical method

In this paper, the stress field in adhesively bonded joints is predicted by employing an analytical approach together with the finite element method (FEM). To accomplish this, the adhesive joint is considered as a bi-material notch. Then, a unified analytical formulation for the in-plane stress and displacement field of bi-material notches is presented in the form of asymptotic series. Subsequently, a numerical procedure based on an over-determined system of equations is used to obtain the coefficients of the series solution for various types of adhesive joints including single lap joint (SLJ), single fillet lap joint (SFLJ), butt joint (BJ), scarf joint (SJ), and single lap joint with an inside taper and adhesive fillet (SLJ-ITAF). By comparing the results obtained from the analytical stress field with the numerical results, the accuracy of the calculated coefficients is evaluated. It is shown that by using the proposed approach one can obtain the stress field within the adhesive layer by using the stress or displacement fields of the adherend. The obtained results also show that neglecting the higher-order terms can result in significant errors while considering the first three non-singular stress terms besides the singular term reveals very good results.

Crack front instability in mixed-mode I+III: The influence of non-singular stresses

In a previous paper (Leblond et al., 2011), a theoretical instability threshold was derived for the currently observed phenomenon of crack front fragmentation under mixed-mode I+III loading conditions. Instability modes were shown to emerge when the mode mixity ratio KIII0/KI0 exceeds some critical value [KIII0/KI0] that only depends on Poisson’s ratio. Unfortunately, the predicted threshold was found to be much larger than that observed in general. Numerical simulations of crack front fragmentation (Chen et al., 2015), based on a phase-field model, subsequently evidenced an important role of the specimen size on the non-coplanar instability. Here, we explore theoretically the influence of existence of some finite characteristic length(s), arising from the loading, by accounting for the presence of non-singular stresses Tij0 in the unperturbed configuration of the crack. By re-examining the linear stability analysis of Leblond et al. (2011) with these more general assumptions, we show that a negative non-singular stress Txx0 in the direction of propagation does not affect crack front fragmentation, while a negative non-singular Tzz0 in the direction of the crack front strongly hampers it. On the contrary, positive non-singular stresses Txx0 and Tzz0, or non-zero non-singular (antiplane shear) stresses Txz0, promote the fragmentation process, sometimes through the formation of facets that drift along the front as propagation proceeds. Large values of all three of these non-singular stresses may result in a significant lowering of the threshold value [KIII0/KI0] of the mode mixity ratio for instability, even possibly down to zero; which stresses out the potential existence of the fragmentation instability even in pure mode I. Yet, our results cannot explain the often observed formation of non-coplanar facets at extremely low mode mixity ratios; indeed the wavelength of the instability modes predicted are comparable to the finite characteristic length(s) introduced, of at least centimetric order of magnitude, while the formation of facets has been observed experimentally at scales as low as some tens of microns.

Theoretical and Experimental Study considering the Influence of T-Stress on the Fracture Behavior of Compression-Shear Crack

The influence of the nonsingular stress term (T-stress), existing in the Williams expansion of the crack tip stress field, on the fracture behavior of the compressive-shear crack is comprehensively considered in this paper. According to the stress boundary conditions on the crack surfaces, the theoretical solution of the stress field around the closed crack tip is established using the complex potential theory, which includes not only the singular term containing the stress intensity factor KII but also three nonsingular terms (T-stress: Tx, Ty, and Txy) without KII. Then, the differences between tangential stresses and shear stresses at the crack tip obtained in the cases with and without T-stress are compared, respectively. Outcome indicates that there are tangential tensile stress and compressive stress zones at the crack tip when considering T-stress, while only the former exists when ignoring T-stress. In the case of considering T-stress, an improved crack initiation criterion is proposed, where the maximum tangential tensile stress and maximum shear stress are utilized simultaneously to predict the crack initiation and failure mode under the multiaxial compression stress state. Besides, the triaxial compression tests were carried out on the prismatic rock-like material samples with a central crack, and the effectiveness and validity of the proposed criterion are verified by comparing the experimental results and theoretical predictions. It reveals that the proposed criterion can reliably predict the crack initiation angles and failure modes with higher accuracy than traditional fracture criterion.

Crack front instability in mixed-mode I+III: the influence of non-singular stresses

Crack front instability in mixed-mode I+III: the influence of non-singular stresses

A criterion for a hydraulic fracture crossing a frictional interface considering T-stress

Hydraulic fracturing has been widely used to stimulate oil and gas production from low permeability reservoirs. However, the geologic discontinuities, such as natural fractures, joints, and bedding planes, can significantly affect the propagation direction of the hydraulic fracture. Considering the effect of the non-singular stress term of Williams’ series expression for stress field near the fracture tip (i.e., T-stress), an analytical criterion for a hydraulic fracture crossing a frictional interface (such as the natural fracture and bedding plane under compression-shearing) was proposed. The present criterion was validated by comparing it to the published experiments, and the effect of T-stress on the propagation of the hydraulic fracture crossing a frictional interface was analyzed by comparing the present criterion to the published criteria. The results show that the accuracy of the present criterion depends on the difference in geomechanical properties of the rock on either side of the interface. The effect of T-stress on the hydraulic fracture propagation can be divided into the facilitation and the inhibition, which depend on the intersection angle between the hydraulic fracture and the frictional interface. No matter what the intersection angle between the hydraulic fracture and the frictional interface is, the effect of T-stress can inhibit the change of propagation direction when the hydraulic fracture crosses the interface.

Numerical simulation for T-stress for complex multiple branching and intersecting cracks based on continuous-discontinuous cellular automaton

In this work, a discontinuous cellular automaton combined non-singular stress term of T-stress is developed for multiple cracks problems. In order to reflect the influence both singular term of SIFs and nonsingular constant term of T-stress, a new enriched shape function based on crack tip near-field asymptotic functions is proposed, which can simultaneously describe singular and non-singular stress around crack tip. Aimed at non-convergence and internal discontinuity for blending cells, a modified formulation for mathematic description of cellular automaton approximation is employed, in which a mixed uniformly varying function is employed. Besides, mathematic descriptions for strong discontinuities of intersecting and branching cracks are developed in this paper, both opening crack surfaces and close crack surfaces are given in detail respectively. Finally, a comprehensive cellular automaton model considering T-stress is constructed, which includes cell, neighborhood, updating rules and so on. Some numerical examples are given to illustrate that the present method is efficient and accurate, and it can be widely used for some practical engineering.

Modified Generalized Maximum Tangential Stress Criterion for Simulation of Crack Propagation and Its Application in Discontinuous Deformation Analysis

This work modified the generalized maximum tangential stress (MTS) criterion by taking parallel and perpendicular nonsingular stresses, i.e., Tx and Ty, into account. When applied to the traditional discontinuous deformation analysis (DDA) procedure, the modified generalized MTS criterion was influential in simulating open crack propagation and modeling closed crack propagation. The generalized J integrals (J1 and J2 integrals) were used to calculate the stress intensity factors (SIFs) and the T-stresses, the basic parameters of the modified generalized MTS criterion. The open crack models, namely the center cracked circular disc (CCCD) model and the edge cracked semicircular bend (SCB) model, were simulated by the DDA method with the modified generalized MTS criterion. Moreover, the closed crack models dealing with the specimens with a center crack subjected to compression stress were simulated. The simulation results of the open and closed crack models were all in good agreement with the experimental data.

Construction of Nonsingular Stress Fields for Non-Euclidean Model in Planar Deformations

A material with a microstructure is considered. A material is described on the basis of a non-Euclidean model of a continuous medium. In equilibrium, the total stress field is represented as the sum of elastic and self-balanced stresses, the parameterization of which is given through the scalar curvature of the Ricci tensor. It is proposed to use the spectral biharmonic equation to calculate the scalar curvature. Using the example of a plane strain state of a material, it is shown that the amplitude coefficients of elastic and self-balanced fields can be chosen so that singularities of the same type compensate each other in the full stress field

Influence of a static torque on plasticity and asperity-induced closure effects during fatigue crack growth in bainitic steel cylinders loaded in push-pull

Mode I fatigue crack growth with a superimposed static torque was investigated in bainitic steel. The torque had little effect on crack growth kinetics and path. However, the non-singular stress parallel to the crack front, and bending stresses due to eccentric crack growth triggered crack twisting, even in mode I. A progressive increase in crack sliding displacement was due to a rise in elastic torsional compliance, but also to plastic ratchetting at the crack tip, and to a progressive shearing of crack face asperities, which reduced the enhancement of roughness-induced closure by the torque, however shown to enhance plasticity-induced closure.

Analysis of multi-crack propagation by using the extended boundary element method

A new way is proposed to simulate the crack propagation paths of the multi-cracked structure. Firstly, the complete displacement and stress fields of the multi-cracked structure are calculated by the extended boundary element method (XBEM) coupled with characteristic analysis of the crack tip. Secondly, based on the maximum circumferential stress criterion by considering the contribution of the non-singular stress terms, the crack initiation angles are obtained. Thirdly, the cracks propagate forward along the crack initiation angles to form a new multi-cracked structure. During this process, boundary element adaptive mesh technique is established and the XBEM is used to analyze the newly formed multi-cracked structure repeatedly. Finally, the crack propagation paths of the multi-cracked structure are obtained. Numerical examples show that the results obtained by the XBEM are in good agreement with the experimental results.

Investigation of the effect of the blast waves on the opposite propagating crack

The stress waves have a significant effect on crack propagation during blasting. In this paper, a microsecond controlled blasting case involved with the interactions between blast waves and an opposite propagating crack is studied using dynamic photoelasticity method. The results show that the stress waves influence the local stress field of dynamic crack tip , and alter the crack propagation behavior. Moreover, during crack-wave interactions, both the singular stresses and the nonsingular stresses around the crack tip changed apparently. With the incidence of dilatational wave , the non-singular stresses σ ox and σ oy around the tip of opposite propagating crack increase remarkably, inducing an increase in the crack extension resistance, and decreasing both the crack velocity and the dynamic stress intensity factor K I d . Whereas, when the shear wave impinges onto the opposite propagating crack, the crack tends to increase the crack velocity. In addition, the fractal dimension of the crack velocity decreases during the incidence of dilatational wave, and increases when the shear wave impinges upon the opposite propagating crack.

The effect of T-stress on mixed mode I/II fracture of composite materials: Reinforcement isotropic solid model in combination with maximum shear stress theory

Due to the existence of cracks and defects in composite materials during the manufacturing process, damage and crack growth in these materials must be anticipated before application. In the present research, an efficient theoretical mixed mode I/II fracture criterion is proposed for fracture investigation of orthotropic materials considering the effects of non-singular stress term in William’s series expansion. This criterion is developed combining the maximum shear stress (MSS) theory with reinforcement isotropic solid concept (RIS) as a superior material model. RIS is based on the fact that cracks in orthotropic materials generally propagate along the fibers in the isotropic matrix media regardless of initial crack-fiber orientation. According to this fact, RIS offers to employ an isotropic stress field around the crack tip with defined stress reduction coefficients as fiber effects. This criterion is applicable for two cases wherein crack is placed both along and across the fibers. A comparison between available experimental data for two wood species and the results of the proposed criterion shows that this criterion can accurately predict both crack initiation and crack propagation direction and is in full compliance with the fracture nature of composite materials.