Maximum Tangential Stress Criterion Research Articles

Numerical prediction of fretting fatigue crack growth in scaled railway axles considering fretting wear evolution

To predict the fretting fatigue crack growth (FFCG) life of railway axles, a series of interrupted fatigue experiments were conducted on scaled railway axles. Subsequently, the evolutions of fretting wear and fatigue cracks in the press-fitted region were analyzed. Based on the test results, finite element models incorporating fretting wear evolution were established, and the FFCG was investigated using the maximum tangential stress criterion, cyclic resistance curve, and the modified NASGRO equation. The analysis revealed that the fretting wear evolution leads to stress redistribution at the press-fitted region, thereby promoting FFCG. When considering fretting wear evolution, the equivalent stress intensity factor (SIF) range of the crack remains above the threshold value throughout the short crack stage. However, neglecting fretting wear evolution results in the SIF range being below the threshold value for cracks shallower than 0.30 mm. This implies that considering fretting wear evolution enables life prediction throughout the short crack stage. As the crack length increases, the influence of fretting wear evolution on crack growth gradually diminishes. By accounting for fretting wear, a more accurate stress distribution in the press-fitted region can be obtained, leading to a more precise and conservative prediction of crack growth life.

Accuracy verification of fast adaptive galerkin finite volume solver for thermal and mechanical load crack propagation in triangular meshes

This study presents the details of the Galerkin finite volume method for triangular unstructured meshes and its application for the fracture analysis of 2D elasticity problems under simultaneous thermal and mechanical loading. The heat diffusion and elasticity equations are discretized using the Galerkin finite volume method (which, due to omitting matrix manipulation calculations, computations are considerably faster than FEM and XFEM Solvers). The maximum tangential stress criterion is presented after briefly describing the interaction integral formulation used to calculate the stress intensity factors in thermo-mechanical problems. The accuracy of the computed results of the present strategy under thermo-mechanical loads is demonstrated using some benchmark test cases. First, a thermal stress analysis of a plate with an inclined central crack under thermal boundary conditions is performed. Second, the development of a crack under mechanical stress boundary conditions is modeled. Third, crack propagation under both thermal and mechanical boundary conditions is simulated. The present modeling strategy's results are compared with those reported in previous numerical works to verify the accuracy. The stress intensity factors and predicted crack trajectories are utilized to assess the accuracy of computed results and investigate the quality of crack simulation by the proposed numerical method.

Fracture parameters and crack initiation Assessment Employing Mixed-mode I/II Fracture Criterion in Laminate Composites Using Digital Image Correlation Method

The present study aims to have fracture parameters, such as stress intensity factors (SIFs) and J-integral, as well as the crack initiation angle, extracted through the utilization of a novel mixed-mode (I/II) fracture criterion founded on the maximum energy release rate (MERR) and maximum tangential stress (MTS). The digital image correlation (DIC) method is employed in the analysis of crack behavior. Of note, several key findings are highlighted by our research. Results show a slight decrease in the mode I stress intensity factor (KI) with an increase in the crack inclination angle (φ). Interestingly, a bell-shaped curve characterizes the mode II stress intensity factor (KII), with its peak occurring at φ = 45. Comparison of experimental outcomes with finite element (FE) based modeling reveals consistent trends in both KI and KII from both sources. The comparison of experimentally determined crack initiation angles with predictions from the MERR and MTS criteria offers insightful observations. For φ ≤ 30, both criteria closely match experimental outcomes, as the crack initiation corresponds to φ. However, for φ > 30, these criteria underestimate the crack initiation angles, leading to a slower decrease in kink angles compared to observations. Further analysis involves a comparison between the two criteria and experimental results. Here, MERR emerges as being more accurate in prediction than MTS. Lastly, Simpson's rule is employed to calculate the J-integral, contributing to a comprehensive understanding of crack behavior. The reliability of the new mixed-mode fracture criteria and the DIC method in evaluating crack tip conditions and predicting minor kink angles is established by our study. Not only is understanding enhanced by our findings, but practical applications in predicting crack initiation and behavior in complex materials are also offered.

On the decay of fracture toughness of sandstone subjected to freeze–thaw cycles

AbstractIn cold regions, progressive cracking, caused by freeze–thaw cycles, plays an important role in deteriorating engineering rock mass. Employing central cracked Brazilian disks (CCBDs) of sandstone, this paper aims at investigating the variations of falling rate among different modes of fracture toughness exposed to freeze–thaw cycles and the applicability of fracture criterion in predictions. Results revealed that different modes of fracture toughness exhibited comparable descending trends after varied freeze–thaw cycles. There was an agreement between the predictions via the generalized maximum tangential stress criterion (GMTS) and the test results. In specified conditions, the GMTS criterion incorporating fracture process zone of 0 freeze–thaw cycle can be applied to estimations for different freeze–thaw cycles. With increasing freeze–thaw cycles the fractal dimensions of mode I and mixed‐mode I–II fracture surfaces displayed a growing trend, indicating that the fracture surfaces became rougher and more complicated.

Loading rate effect on mixed-mode I + II fracture of V-notched Brazilian disc rock specimen under impact loads

Dynamic failure processes of rock masses containing numerous discontinuities including pores, cracks and notches have been the major issues in structural safeties of underground buildings subjected to impact loads, such as earthquake, blast and vehicle vibration. At present, most of the studies are focused on the dynamic fracture behaviors of rocks containing crack-like defects, while the studies on rocks containing notch-type defects are very rare, especially under mixed-mode loading conditions. This paper aims to study the effects of loading rate on mixed-mode I + II fracture characteristics of flattened V-notched Brazilian disc rock specimen under impact loads by means of an split Hopkinson pressure bar (SHPB) system combined with a high-speed camera. Dynamic fracture tests were performed on a total of 125 groups of V-notched rock specimens with different notch angles (0°, 30°, 45°, 60°, and 90°) and loading angles (0°, 30°, 45°, 60°, and 90°) at different air pressures (0.16 MPa, 0.18 MPa, 0.20 MPa, 0.22 MPa, and 0.24 MPa). Based on dimensional analysis, a numerical method was proposed to establish the relationship between mixed-mode notch stress intensity factor (NSIF) at the V-notch tip and external axial load. The dynamic notch fracture toughness values of flattened V-notched Brazilian disc rock specimens were calculated from the maximum values of dynamic force in the SHPB tests according to the proposed numerical method. The effects of loading rate, notch angle, and loading angle on the mixed-mode I + II notch fracture toughness value and crack initiation angle were quantitatively analyzed, which were verified using a modified maximum tangential stress (MTS) criterion by considering the loading rate effect. It is found that the loading rate has a significantly positive effect on the notch fracture toughness value, but has a negative effect on the crack initiation angle due to the inertial effect under dynamic impact loads. Generally, pure mode I fracture occurs for the loading angle of 0° or 90°, pure mode II fracture occurs for the loading angle varying from 35° to 45°, and mixed-mode fracture occurs for other loading angles. Good consistencies of the mixed-mode I + II notch fracture toughness values and crack initiation angles of the V-notched rock specimens are observed between the modified MTS criterion prediction and experimental results. This work may improve the understanding of the dynamic fracture behaviors of rock masses containing notch-type defects during tunnel excavation and underground construction process.

Thermally nonlinear analysis of propagating cracks under generalized thermal shock

In this paper, the thermally nonlinear response of stationary and moving cracks in isotropic media subjected to a hyperbolic thermal shock in the framework of the eXtended Finite Element Method (XFEM) is examined. For this purpose, the thermally nonlinear fully coupled Lord-Shulman (LS) model is considered as governing equations. In addition, two cases of material properties, temperature-dependent (TD) and temperature-independent (TID), are assumed. The geometry, including a predefined crack, is modeled using the XFEM for spatial discretization. For modeling the dynamic crack growth within the XFEM, the Maximum Tangential Stress (MTS) criterion is modified based on the stress field near the tip of a growing crack to calculate the dynamic crack growth angle. Besides, the nonlinear form of the Newmark integration scheme is implemented for the numerical integration in the time domain. Finally, a method based on one of the Barnet-Lothe tensors for isotropic materials is employed to extract the stress intensity factors (SIFs). In the provided examples, the effects of thermal nonlinearity, the temperature-dependency of material properties, and the LS relaxation time on the crack propagation path are explored.

Prediction of the effect of shot peening residual stress on fretting fatigue behaviour

Fretting occurs at two contact bodies, which sustain oscillation loading. The fretting phenomenon decreases the fatigue lifetime dramatically. Shot peening is a commonly used surface treatment to enhance the resistance to damage because the residual stress is introduced to the surface of the material. Some studies showed that residual stress was helpful to improve fatigue lifetime. Under fretting fatigue condition, there are two main stages: crack initiation stage and the crack propagation stage. In this paper, the authors apply the Critical Plane (CP) method combined with the Theory of Critical Distance (TCD) to study the fretting fatigue crack initiation behaviour under the influence of shot peening residual stress, including the crack initiation position, orientation and the crack initiation lifetime. Meanwhile, the Linear Elastic Fracture Mechanics (LEFM) theory and the Extension Maximum Tangential Stress (EMTS) criterion are adopted to study crack propagation behaviour considering shot peening residual stress effect. The results show that the methods applied in this paper can make good predictions of fretting fatigue behaviour, including both initiation and propagation stages. Meanwhile, the influence of residual stress in these two stages are studied as well.

Investigation of Constraint Effect and Fracture Mode for Mixed Mode Inclination Surface Crack in Infinite Plate Under Compression

Abstract The stress field, constraint effect, and fracture mode transition at crack tip of mixed mode I-II-III inclination surface crack under compression have been investigated. The effects of geometrical configurations (relative crack depth and aspect ratio), friction coefficient, and biaxial scale factor on stress intensity factor (KII and KIII) and in-plane constraint parameter T-stress are quantitatively studied, the stress field at different crack inclination angles under tension and compression are compared, the failure mode at special locations along crack front of inclination surface crack is analyzed according to the generalized maximum tangential stress criterion (GMTS). The relative crack depth has slight effect on stress intensity factor and T-stress, and aspect ratio has a significant effect on stress intensity factor and T-stress. The friction coefficient decreases the magnitude of stress intensity factor and increases the magnitude of T-stress, the greater the crack inclination angle is, the more pronounced the effect is when crack inclination angle greater than 30 deg. The stress distribution around crack tip under tension and compression is completely different. At free surface, the crack will failure in-plane shear mode II sliding crack, and at the deepest part of crack, the crack will start as out-plane shear mode III tearing crack under compression.

Analysis of Fracture Characteristics of Ore Rock Based on GMTS Criterion

In this paper, the Tongkaiyu ore-rock is taken as the research object, aiming to analyze the fracture characteristics of the mixed type I, II and I–II faults in ore-rock by using the appropriate fracture criterion, and analyze the fracture development mode. Considering the non-singular term (T stress) in the formula of stress field sequence at crack tip, the generalized maximum tangential stress (GMTS) criterion is established, which is consistent with the boundary conditions of the central straight Brazilian disk (CSTBD) specimen. Through CSTBD splitting test and three-dimensional particle flow code (PFC3D) numerical simulation, the Angle between the central direct joint and the loading direction was changed, and the fracture toughness and fracture development law of type I, type II and type I–II mixed fracture were obtained. The theoretical and simulation results show that, compared with the maximum tangential stress (MTS) criterion, the GMTS criterion has an advantage in judging the fracture characteristics of mixed types I, II and I–II in ore rock.

A Theoretical and Experimental Investigation on the Fracture Mechanism of Center-Symmetric Closed Crack in Compacted Clay under Compression–Shear Loading

In this study, a modified maximum tangential stress criterion by considering T-stress and uniaxial compression tests have been utilized to theoretically and experimentally reveal the fracture initiation mechanism of a center-symmetric closed crack in compacted clay. The results show that wing cracks occur in the linear elastic phase of the stress-strain curve. In the plastic phase of the stress-strain curve, the wing cracks extend gradually and the shear cracks occur. The crack initiation stress and peak stress of compacted clay first decrease with the rise in pre-crack inclination angle (β = 0°–40°), and then increase with the rise in pre-crack inclination angle (β = 50°–90°). When the pre-crack inclination angle is relatively small or large (β ≤ 10° or β ≥ 70°), the crack type is mainly tension cracks. Secondary shear cracks occur when the pre-crack inclination angle is 10°–80°. When the dimensionless crack length is larger than 0.35, the crack types include wing-type tension cracks and secondary shear cracks. The experimental results were compared with the theoretical values. It was found that the critical size rc of compacted clay under compression-shear loading was 0.75 mm, smaller than the value calculated by the empirical formula (12 mm). The MTS criterion considering T-stress can be used to predict the compression-shear fracture behavior of compacted clay.

Fracture assessment of U-notched diagonally loaded square plates additively manufactured from ABS with different raster orientations

This paper investigates the fracture behavior of 3D-printed notched specimens under various combinations of mode I and mode II loadings, including pure mode I and pure mode II loadings. The study employs fused deposition modeling (FDM) to fabricate U-notched diagonally loaded square plate (UNDLSP) specimens using Acrylonitrile Butadiene Styrene (ABS) filaments. To experimentally measure the fracture load and fracture initiation angles, 120 UNDLSP specimens are 3D printed with different notch tip radii, notch orientation angles, and [0, 90], [45, −45] raster orientations. Tensile characteristics and translaminar fracture toughness of ABS plates with [0, 90] and [45, −45] raster orientations are obtained using dog-bone and cracked DLSP samples, respectively. Considering the layered nature and anisotropy of the material, the virtual isotropic material concept (VIMC) is utilized along with two well-known stress-based brittle fracture models: the maximum tangential stress (MTS) and the mean stress (MS) criteria. The test results are predicted using the fracture limit curves and fracture initiation angle curves developed for these fracture models in terms of the notch stress intensity factors (NSIFs). Furthermore, a general equation is established to estimate the fracture initiation angle of U-notched ABS specimens under mixed mode I/II loading, irrespective of geometry, raster orientation, and notch tip radius, based on the curves obtained from the MTS and MS criteria. The fracture surfaces of the UNDLSP specimens are examined using scanning electron microscopy (SEM) to study the micro-mechanisms involved in the fracture process. The findings indicate that both the VIMC-UMTS and VIMC-UMS criteria accurately predict the experimentally achieved fracture loads. Additionally, it is observed that the fracture initiation angle curves for both criteria are nearly identical and accurate. The significance of this research is not only providing numerous new experimental results on mixed mode I/II fracture of notched 3D-printed ABS plates, but also providing simple, fast, accurate, and robust criteria for predicting the critical loads of such plates, without the need for conducting layer-by-layer failure analysis using progressive damage models.

Crack deflection in shale-liked layered rocks under three-point bend loading

The bedding angle has a significant influence on the fracturing of anisotropic geomaterials, such as shale rock, and crack deflection usually takes place during crack propagation. To better understand and estimate the tensile crack deflection in shale, 35 large notched deep beam (NDB) specimens with different bedding angles—0° (Short-Transverse), 15°, 30°, 45°, 60°, 75°, and 90° (Arrester)—were tested under three-point bend (3PB) loading. For mode I dominated loadings, significant expected anisotropies in both fracture toughness and crack deflecting behavior were observed. The nominal mode I fracture toughness showed a progressive decrease from the arrester to the short-transverse orientation, with values from 1.371 MPa·m1/2 to 0.731 MPa·m1/2. When the bedding angle was less than or equal to 60°, the crack deflection induced the difference between the nominal and actual fracture toughness. By introducing the weak plane model to describe the fracture toughness of shale, an anisotropic K-type maximum tangential stress criterion (KMTS criterion) was presented. Using the KMTS criterion, the deflection angle and fracture load for an arbitrary bedding angle could be obtained conveniently with fracture toughness values in both arrester and short-transverse orientations. The comparison between theoretical predictions and experimental data demonstrated that the KMTS criterion could provide a dependable prediction, particularly for the slight deflection when the crack propagates within the matrix.

Thermal effects on prediction accuracy of dense granite mechanical behaviors using modified maximum tangential stress criterion

Thermal effects on prediction accuracy of dense granite mechanical behaviors using modified maximum tangential stress criterion

Fretting fatigue crack propagation under out-of-phase loading conditions using extended maximum tangential stress criterion

In this paper, a numerical crack growth analysis is conducted by using the Maximum Tangential Stress (MTS) criterion and its extension, which considers the contact stress between the crack surfaces. Moreover, fretting fatigue is a multiaxial non-proportional loading problem, and most components are subjected to asynchronous loads. Therefore, this paper mostly focuses on the influence of loading phase difference (Φ) on the crack propagation behavior. Meanwhile, for the fretting fatigue problem, as there is a relationship between the crack initiation and propagation, the effect of initiation characteristics, including crack initiation position and direction is studied under different loading phase differences using Linear Elastic Fracture Mechanics (LEFM) theory. It is observed that the predicted crack path and lifetime by using the extended MTS criterion and Paris’ law is in good agreement with the experimental results in the literature. It is found that the phase difference has a strong effect on the crack propagation behavior. Furthermore, we found that the crack initiation position has a greater influence on crack propagation behavior than the crack initiation direction.

Effects of Support Friction on Mixed-Mode I/II Fracture Behavior of Compacted Clay Using Notched Deep Beam Specimens under Symmetric Fixed Support

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, and support friction coefficient, and the normalized fracture parameters (YI, YII, and T*) of NDB specimens were calibrated. The numerical results showed that the values of YI, YII, and T* decreased at different degrees after considering the support friction. Notably, the support friction coefficient could significantly change the loading pattern at the crack tip. To verify this phenomenon, 12 compacted clay NDB specimens were prepared, and a mixed-mode I/II fracture test was performed under fixed support conditions; the phenomenon of asymmetric crack propagation was studied. The test data were processed using the numerical calibration results of YI, YII, and T* with and without consideration of friction. Afterward, the test data were compared and analyzed by combining the generalized maximum tangential stress (GMTS) and the maximum tangential stress (MTS) criteria. The analysis indicated that the real fracture characteristics of compacted clay NDB specimens could not be reflected when conducting mixed-mode I/II fracture tests under symmetric fixed support conditions if the test results were analyzed by YI, YII, and T* without considering support friction, as in previous studies.

Mixed mode crack growth behaviour considering plasticity-induced and roughness-induced closure

A theoretical analytical solution of the Christopher-James-Patterson (CJP) model in mixed mode is presented in this paper, where the influence of roughness-induced crack closure (RICC) on fatigue crack growth is considered and a new effective stress intensity factor range (ΔKP-R) is proposed as the driving force for crack propagation. The solution showed that both the maximum tangential stress criterion (MTS) and the minimum strain energy density criterion (S) can reasonably well predict the crack deflection angle. Experimental testing on EA4T axle steel samples showed that both the 8 % compliance offset criterion in the conventional ASTM procedure and the parabolic-line method were suitable for the determination of the crack-opening force. The solution includes plasticity-induced crack closure (PICC) and RICC effects, therefore the crack propagation behaviour can be described in a more effective way. Compared with the traditional mixed model crack growth models with higher fitting correlation coefficient, the fitting effect of the CJP model has been improved by 12 % in this solution, and the residual stress in the pre-crack stage increased the crack-closure level. Due to the influence of residual stress, the rate of roughness value (RA) showed a fluctuation of 4.5 %, which indirectly affected the fatigue crack growth rate (FCGR) of the test.

A general interaction integral for dynamically propagating cracks in anisotropic materials

In this paper, some relevant tools for numerical analysis of dynamically propagating cracks in anisotropic materials are discussed within the framework of the eXtended Finite Element Method (XFEM). A general interaction integral that is domain-independent is proposed to determine the Stress Intensity Factors (SIFs) for both stationary and dynamically propagating cracks in anisotropic materials. Besides, a set of functions for enriching the nodes of the tip element, which are applicable for modeling both quasi-stationary and dynamically propagating cracks in anisotropic materials, are also proposed. In addition, a modified form of the Maximum Tangential Stress (MTS) criterion is manipulated for estimating the direction of the propagating crack which takes into account the asymptotic stress field near a propagating crack tip in anisotropic materials under mixed mode plane loading. Finally, the accuracy of the suggested method is then confirmed using some numerical examples that involve both stationary and dynamically propagating cracks. Additionally, a thorough investigation is done into the impacts of material anisotropy and material orientation on the path of dynamically propagating cracks in orthotropic structures.

The initiation criterion for open-crack considering fissured water under biaxial compression

Fissured rock is under compression-shear load for a long time in the geotechnical engineering, while the blunt crack tip and fissured water also make the compression-shear fracture mechanism of the open-crack more complicated. So, the profound study on the distribution of the stress at the open-crack tip is made and the open-crack initiation criterion under hydrostatic and confining pressure is also established. First of all, according to Muskhelishvili's tensile stress hypothesis, the stress intensity factor KI under compression is studied and the stress field at the open-crack tip is modified, which involves the curvature radius ρ, high-order and non-singular terms, then the revised maximum tangential stress (MTS) criterion is proposed. Second, the effect of the curvature radius ρ and half-length a on KI and the initiation angle θ are discussed respectively. It is found that KI increases with increasing ρ and decreasing a. It is also found that when the loading angle β = 0 ∼ 45°, the initiation angle θ increases with decreasing ρ and increasing a. In the range of β = 45 ∼ 90°, the initiation angle θ is insensitive to ρ, a and tends to converge. The initiation stress goes down firstly in the range of β = 0 ∼ 50° and then gradually goes up with increasing β. Third, the model for the open-crack filled with fissured water under biaxial compression is discussed. The trend of the wing-crack initiation angle θ is similar to that under uniaxial compression with increasing β and a, but the upper limit of θ decreases with increasing the hydrostatic pressure pw. Finally, the revised criterion is preliminarily verified with the indoor test.

Theoretical and experimental investigations of crack propagation behavior and delamination in GLARE specimens affected by non-proportional fatigue loadings

In actual operation conditions, the structural parts of automotive, aerospace and other significant industrial applications undergo the multi axial fatigue loading with non-proportionality states. Fiber metal laminates (known as FML) are the best options to manufacture these structural parts. The current study examines the effects of a specific condition of multi-axial fatigue loadings (repeated tension with static shear stresses) on the behaviors of crack growth and delamination in GLARE specimens (abbreviated of glass laminates aluminum reinforced epoxy). Analytical model with idea of maximum tangential stress criterion modification has been adopted to represent the non-proportionality state of fatigue loading. Paris equation, equivalent stress intensity factor range, fiber-bridging effects and delamination shape were included in this model. The current idea was compared with a previous idea that depends on the principal stresses in representation of non-proportional multi-axial fatigue loadings. Specimens have coding GLARE_2A 2/1 were fabricated by VARTM technique (vacuum assisted resin transfer molding technique). Special treatments have been done on the aluminum surfaces before specimen fabrication to increasing delamination resistance. Novel fatigue apparatus was manufactured to guarantee apply the under study conditions for multi axial fatigue loading and to introduce experimental investigation of crack and delamination growths in GLARE specimens. The obtained results of the current idea were in a considerable agreement with that of the previous idea of the principal stresses. The maximum deviations between analytically extracted results were less than 1% for crack growth, 5.4% for bridging stress, 4.8% for crack opening and 6.3% for delamination growth. Experimentally, the measured results have been compared with analytical results for the purpose of validation. Acceptable matching between results with maximum difference of 7.65% was the important outcome of this study.

Non-local and local criteria based on the extended finite element method (XFEM) for fracture simulation of anisotropic 3D-printed polymeric components

PurposeThe purpose of this study is to develop an efficient numerical procedure for simulating the effect of printing orientation, as one of the primary sources of anisotropy in 3D-printed components, on their fracture properties.Design/methodology/approachThe extended finite element method and the cohesive zone model (XFEM-CZM) are used to develop subroutines for fracture simulation. The ability of two prevalent models, i.e. the continuous-varying fracture properties (CVF) model and the weak plane model (WPM), and a combination of both models (WPM-CVF) are evaluated to capture fracture behavior of the additively manufactured samples. These models are based on the non-local and local forms of the anisotropic maximum tangential stress criterion. The numerical models are assessed by comparing their results with experimental outcomes of 16 different configurations of polycarbonate samples printed using the material extrusion technique.FindingsThe results demonstrate that the CVF exaggerates the level of anisotropy, and the WPM cannot detect the mild anisotropy of 3D-printed parts, while the WPM-CVF produces the best results. Additionally, the non-local scheme outperforms the local approach in terms of finite element analysis performance, such as mesh dependency, robustness, etc.Originality/valueThis paper provides a method for modeling anisotropic fracture in 3D-printed objects. A new damage model based on a combination of two prevalent models is offered. Moreover, the developed subroutines for implementing the non-local anisotropic fracture criterion enable a reliable crack propagation simulation in media with varying degrees of complication, such as anisotropy.