Mixed Mode Loading Research Articles

Effective stress intensity factor range for fatigue cracks propagating in mixed mode I-II loading

The actual service axles are often subjected to mixed-mode loading, and predicting the mixed-mode I-II crack propagation behaviour using the mode I effective stress intensity factor (ΔKI) differs from the real service conditions. To effectively predict the fatigue crack propagation behaviour of actual service structures, the I-II stress intensity factor range (ΔKP-R) considering two closure effects was adopted to describe the fatigue crack propagation under mixed mode loading. A test database was established based on monitoring data of mode I and mixed-mode I-II (30/45/60°) crack propagation tests under different stress ratios. Combining domain knowledge and symbolic regression (SR) methods, an angle factor was proposed for constructing correlation functions between ΔKI and ΔKP-R. The results showed that the loading angle (α) only affects the initial projection of the load parallel and perpendicular to the fatigue crack growth (FCG) direction. Compared with the geometric correction factor, the correlation function acquired by the angle factor constructed by the SR method has higher accuracy, and the balance parameters (SCORE) obtained by the former are significantly higher than those obtained by the latter under the same function complexity. The SR verification results demonstrated that constructing mode I and I-II correlation functions with angle factors has a good predictive effect.

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

Three-dimensional fracture modes decoupling by means of invariant integrals in wood under mixed mode loading

This paper addresses the determination of the energy release rate in an orthotropic elastic material such as wood with a three-dimensional view of the problem. The mixed mode loading associated with the anisotropy of the material required mode decoupling by isolating the energy release rates in mode I, mode II and in mode III. These energy release rates are calculated using the invariant integral M of Linear Elastic Fracture Mechanics (LEFM). This decoupling has necessitated the introduction of kinetically and statically admissible three-dimensional virtual displacement and stress fields in the vicinity of the crack front. The fracture mode decoupling strategy is presented and implemented through finite element modelling of a Mixed Mode Crack Growth (MMCG) specimen. Different thicknesses are investigated to highlight the 3D effects. The modelling highlighted the effect of Mode II at the edges of free surfaces for mixed loadings including mode III (mode III, mode (II+III) and mode (I+II+III)), while mode III was the predominant at the middle of the specimen. In cases of mixed mode (I+II), the energy release rate GI was highest at the middle of the specimen and lowest at the edges of the free surfaces, regardless of the degree of mixing for a given thickness. The correlation between the results obtained using the proposed approach and the non-dependence of the integration domain, as well as with the usual approaches employed in the literature, is demonstrated.

Dyke emplacement under mixed loading conditions: Insights from the Dharwar Craton, India

Dykes are intrusive igneous bodies that play crucial role in the supply and ascent of magma to the Earth’s crust. Magma can intrude along pre-existing anisotropies such as fractures or foliations present within the host rock or it may create its own path by fracturing the host rock. In the latter scenario, when fractures are formed by the pressure exerted by the invading magma, a dyke’s outcrop shape and geometry are diagnostic of the conditions under which it evolved. Here, we report mafic dykes emplaced within the younger granites of Dharwar Craton, peninsular India. Outcrop attributes of these dykes are characteristic of emplacement under conditions of mixed mode loading. We discuss different discrete modes of fracture formation and their possible combinations to understand the generation and eventual emplacement of dykes under mixed mode loading. This leads to the development of a comprehensive sequence of progressive dyke evolution under mixed mode I-III loading and thereby distinguishing incremental orders of dyke horn formation. We further apply this knowledge along with collected field evidence on dyke body geometries to propose an evolutionary model of dyke formation and emplacement within the Chitradurga granite under varying regional stress fields of the Chitradurga Schist Belt, Western Dharwar Craton, India. We infer, that the dykes initiated as extensional fractures within an earlier NE-SW directed compressive stress field and were subsequently sheared sinistrally by the effect of the adjacent Chitradurga Shear Zone on account of a later E-W to ESE-WNW directed compression.

Mixed-mode fracture assessment of largely deformable hyperelastic materials highlighting crack removal phenomenon

Due to the special mechanical characteristics, rubbers are nowadays utilized in various industries. In this research, the effect of an existing crack on the behavior of Nitrile Butadiene Rubber (NBR) filled with Carbon Black (CB), which has shown high deformability, is investigated experimentally, numerically and analytically under mixed-mode (I/II) loading. It is observed experimentally that in the cracked largely deformable rubbers, after the initial crack blunting, the crack is nearly removed and the specimen is deformed in such a way that almost no sign of stress concentration factor exists in the sample. Thus, the rupture of these rubbers is divided into two separate phases; named here as the crack removal phase and the final rupture phase, and for each separate phase, a distinct criterion is developed to assess the fracture of cracked rubbers. The good agreement between the theoretical predictions based on each criterion and the corresponding experimental data confirms the very good accuracy of the presented criteria. The independence of critical J-integral value (Jc) from the mode mixity is also investigated.

Fatigue delamination damage analysis in composite materials through a rule of mixtures approach

The present study investigates delamination damage initiation and propagation within a homogenization theory of mixtures, using the concept of virtual layers and virtual interfaces. It eliminates spatial discretization of layers, introducing a resultant damage variable to capture structure’s bulk response under both monotonic and cyclic loads. Fatigue-induced deterioration is classified into sub-critical, critical, and over-critical stages based on interfacial stresses. Calibration is conducted employing the widely-available Wöhler curves for each loading mode independently. An advance-in-time strategy is included in the model to enhance the simulation speed. The reliability of the approach is assessed for crack initiation and propagation separately through standard test coupons, showing good correlation with experimental data in mode I, mode II, and mixed-mode loading conditions. Depending on the calibration procedure adopted, the model is applicable to a wide range of stress ratios. In addition, it could be integrated into any standard finite element framework using the desired number of elements through the thickness regardless of the physical amount of layers. This allows easy modification of stacking sequences or the number of layers within the constitutive law without mesh structure changes, facilitating simulation of large-scale composite laminates with minimal accuracy loss and reduced computational costs.

A strain-based criterion for predicting size-dependent fracture resistance of quasi-brittle materials under mixed mode loading

In the present study, a strain-based approach called the modified maximum tangential strain criterion has been developed for evaluating size effect on the mixed-mode fracture resistance of quasi-brittle materials. This approach relies on the generalized maximum tangential strain criterion, which considers the singular (K) and the first non-singular (T) terms of Williams series expansion. Furthermore, as an important and size-dependent parameter in the proposed criterion, a new formulation is presented to calculate the critical distance (rc). The predictions of the new criterion are compared not only with the available experimental data, but also with the results estimated by another size effect criterion named the modified maximum tangential stress criterion. It is shown that the proposed strain-based criterion is highly capable of estimating the size-dependent mixed-mode fracture behavior of quasi-brittle materials without requiring to compute the other higher order terms.

Numerical Fatigue Crack Growth on Compact Tension Specimens under Mode I and Mixed-Mode (I+II) Loading.

This study focused on standard Compact Tension (CT) specimens and two loading modes during the numerical analyses carried out, namely: pure mode I and mixed-mode loading (Modes I+II). Numerical stress intensity factors, KI, were calculated using Abaqus® 2022 and compared with those given analytically under pure mode I loading, showing very good agreement. Additionally, KI, KII, and KIII results obtained from Abaqus® were presented for mixed-mode loading, analyzing crack growth and variation through the thickness of the CT specimen. Moreover, fatigue crack growth simulations under mode I loading were conducted on standard CT specimens using the Extended Finite Element Method (XFEM) and the Paris Law parameters of an AISI 316L stainless steel. It was shown that XFEM effectively determines crack propagation direction and growth, provided that an appropriate mesh is implemented.

Fatigue crack growth rate under mixed-mode loading conditions (I+III) of a carbide-free bainitic steel designed for rail applications

Carbide-free bainitic steel was designed for railway infrastructure application, focusing on high-speed and heavy-loaded freight tracks. Considering the complex state of stresses occurring on the rails running surface, mode III plays a significant role in the initiation and propagation of fatigue cracks of rails during service. Thus, the Fatigue Crack Growth Rate (FCGR) under mixed-mode loading conditions (I+III) was evaluated. It was revealed, that fatigue lifetime increases with loading angle modes. In the area of fatigue fracture, transgranular cracking mechanisms dominated. For the stable fatigue crack growth, a trend was observed related to the decrease in the fraction of intergranular fracture with the increasing loading angle modes (α). Secondary cracks indicated privileged cracking directions related to the crystallographic structure of bainite. The influence of the mechanical stability of retained austenite during mixed-mode FCGR requires further in-depth research. These studies contribute to understanding the factors influencing the reliability of railway tracks in terms of designing new materials and modeling the rate of crack growth to precise assessment of the life cycle of rails.

A new strain-based approach to investigate the size and geometry effects on fracture resistance of rocks

In this paper, a new strain-based criterion is suggested for assessing the effects of size and geometry of specimen on the fracture resistance of rocks under mixed-mode (I/II) loading. The new approach named the modified maximum tangential strain (MMTSN) criterion is based on the classical maximum tangential strain (MTSN) criterion, in which the first non-singular term (T) of Williams series expansion is considered in addition to the singular terms (K). Furthermore, to provide more coherence, the critical distance (rc) from the crack tip is defined according to a new strain-based failure model. Unlike similar strain-based fracture models available in the literature, the critical distance rc in the MMTSN criterion is assumed to be size-dependent and a semi-empirical formulation is utilized for describing this size-dependency. To assess the ability of MMTSN for considering the size and geometry effects, the experimental data existing in the literature for a number of cracked Brazilian disk (CBD) and semi-circular bend (SCB) specimens manufactured from Guiting limestone are taken into account. It is demonstrated that the MMTSN criterion can predict the experimental data very well by taking into consideration the size and geometry effects without needing to calculate the other higher order terms.

Mesoscale mechanism of damage in fracture process zone of CFRP laminates simulated with triaxial stress state-dependent constitutive equation of matrix resin

The three-dimensional failure process experimentally observed by synchrotron radiation X-ray computed tomography (SR X-CT) regarding the influence of the interfiber distance is discussed on the basis of the results of numerical experiments. Triaxial stress states in the fracture process zone of carbon fiber reinforced polymers were analyzed on the mesoscale under mode I and mixed-mode (mode I + II) loading. Yield and damage models depending on stress triaxiality were used to accurately simulate three-dimensional stress states in the damage zone around the crack tip. Owing to the heterogeneity of composites, deviatoric stress is prominent in the thin resin region where the interfiber distance is small under mode I loading. On the other hand, matrix resin is triaxially stressed in the middle point between carbon fibers in the thick resin region where the interfiber distance is large. Under mode II loading, the shapes of fiber/matrix debonding depended on the interfiber distance. Areas with stress concentration were found owing to a large debonding area in the thick resin region resulting in a matrix cracking-prone stress state. These findings explain the damage and failure processes well observed by SR X-CT and provide a fundamental understanding of the damage mechanism at a mesoscale.

Low-Temperature Mixed Mode (I/II) Fracture Characterization of Polymerized Sulfur Modified AC Mixtures

In this study, asphalt concrete (AC) mixtures were modified with polymerized sulfur, using PG58-22 bitumen, and crushed siliceous aggregate. Modifications involved replacing the base binder with 20%, 30%, and 50% polymerized sulfur, compared to a control mix with no replacement. The mixtures were subjected to Single Edge Notched-Beam (SE(B)) fracture tests under mixed mode (I/II) conditions with notch offset value of 48 mm, with temperatures ranging from 0 °C to -20 °C. These tests, focusing on the mixtures' response to mixed mode loading, provided load-displacement curves, enabling the determination of fracture energy. Results indicated an increase in fracture energy for 20% and 30% sulfur-modified mixtures. However, a trend towards increased embrittlement was also observed, as fractures occurred at lower displacements. Significantly, higher sulfur content correlated with similar or decreased mixed-mode (I/II) fracture energy, suggesting an improved resistance to low-temperature cracking for lower replacement percentages.

Verification of strain energy splits of phase field fracture model using Westergaard’s problem under mixed-mode loading

One challenge in verifying strain energy splits in phase field fracture models is the lack of reference problems with exact solutions in mixed-mode loading that could serve as benchmark problems. In this work, different strain energy splits used in the variational phase field model have been applied to the classical Westergaard’s problem. In contrast to other benchmark problems used in the literature, the proposed problem not only includes the singular term as a boundary condition but also implicitly considers all terms. To the authors’ knowledge, this is the first time in the literature that this reference problem has been used to rigorously verify the accuracy of different energy splits of the phase field model. Several critical factors such as the pre-crack definition, mesh size, mixed mode ratio and the length scale parameter have been analysed. A good correlation between the numerical estimations and the analytical predictions was found for some pre-crack definition approaches under mode I. However, we demonstrated that the compatible regularisation length scale and mesh size are significantly lower under mode II loading (i.e., when the crack kinks) than under mode I.

Closure Effect of I + II Mixed-mode Crack for EA4T Axle Steel

The crack-closure effect is a crucial factor that affects the crack growth rate and should be considered in simulation analysis and testing. A mixed-mode I + II loading fatigue crack growth test was performed using EA4T axle steel specimens. The variation of the plastic-induced crack closure (PICC) effect and the roughness-induced crack closure (RICC) effect during crack deflection in the mixed-mode is examined in this study. The results show that the load perpendicular to the crack propagation direction hinders the slip effect caused by the load parallel to the crack propagation direction under mixed-mode loading, and the crack deflection is an intuitive manifestation of the interaction between the PICC and RICC. The proportion of the RA value change on the crack side caused by contact friction was reduced by the interaction between PICC and RICC. The roughness of the crack surface before and after the crack deflection is different, and the spatial torsion crack surface is formed during the crack propagation process. With the increase of the crack length, the roughness of the fracture surface increases. During the crack deflection process, the PICC value fluctuates around 0.2, and the RICC value is increased to 0.15.

Delamination bridging response of Z‐pins under mixed mode loading: The influence of carbon and Kevlar Z‐pins

AbstractThis paper presents the effect of varying the type of Z‐pin on the delamination bridging performance of unidirectional composite laminates. Carbon and Kevlar Z‐pins were inserted in thick composite laminates using the pre‐hole insertion Z‐pinning process and their bridging performance characterized across the different mode‐mixity range. The bridging response indicated that Kevlar Z‐pin experienced load reduction driven by interfacial friction (Stage III), which was absent in the Carbon Z‐pin. When Mode II dominated the bridging crack, Kevlar Z‐pin showed partial pullout and fiber shear cracking, while the Carbon Z‐pin exhibited clean transverse fracture. Although the bridging load obtained by the Carbon Z‐pin was twice that of Kevlar Z‐pin, the Kevlar Z‐pin provided a higher interlaminar fracture energy. The superior bridging performance of Kevlar Z‐pin is correlated with the enhanced ductility. Moreover, the resin‐rich area was involved in deformation and provided the extra axial load of Z‐pin.Highlights The comparison of mixed‐mode bridging mechanism between Kevlar Z‐pin and Carbon Z‐pin. Reveal the variation of multi‐directional load and energy consumption with mixed‐mode angle in thick laminates. Detailed scanning electron microscope observation to understand failure characteristics of ductile Z‐pin. Evidence that the extra axial load of Z‐pin provided by resin‐rich area.

Effect of Post-Cured through Thickness Reinforcement on Disbonding Behavior in Skin-Stringer Configuration.

An experimental investigation of interlaminar toughness for post-cured through-thickness reinforcement (PTTR) skin-stringer sub-element is presented. The improvement in the crack resistance capability of skin-stringer samples was shown through experimental testing and finite element analysis (FEA) modeling. The performance of PTTR was evaluated on a pristine and initial-disbond of the skin-stringer specimen. A macro-scale pin-spring modeling approach was employed in FEA using a non-linear spring to capture the pin failure under the mixed-mode load. The experimental results showed a 15.5% and 20.9% increase in strength for the pristine-PTTR and initial-disbond PTTR specimens, respectively. The modeling approach accurately represents the overall structural response of PTTR laminate, including stiffness, adhesive strength, crack extension scenarios and progressive pin failure modes. This modeling approach can be beneficial for designing damage-tolerant structures by exploring various PTTR arrangements for achieving improved structural responses.

Impact of surface preparation method on the fracture toughness of adhesive joints subjected to diverse loading modes

While research has explored surface treatment effects on fracture energies in pure modes (I or II), there is a lack of research under mixed-mode loading. The aim of this paper is to investigate the influence of surface treatment approaches on the fracture behaviour and fracture energy of adhesive joints at different loading modes. To achieve this, double cantilever beam (DCB) adhesive joints were manufactured considering two different surface treatment methods namely grit blasting and mechanical bristle blasting. The joints were subjected to five different mode mixities. Fracture energy for each tested condition was calculated based on a compliance-based approach. The fracture energies in mode I reached an increase of 55 % when grit blasting surface treatment was applied, while for mode II the fracture energy reached a value three times higher for mechanical bristle blasting surface treatment, both energies from the mixed mode tests. The failure mechanisms were also evaluated for joints with different surface treatments as a function of loading mode. In all tested conditions, a cohesive failure mode was observed; however, the results underscore the substantial influence of the surface treatment method on the fracture energy of adhesive joints. This is attributed to the distinct failure mechanisms observed under varying surface treatment conditions.

Monitoring Crack Tip Position in Adhesively Bonded Joints Under Mixed I+II Mode Loading

The use of adhesive joints has gained favour in recent years, due to their capacity for weight reduction, improved stress distribution, and enhanced fatigue resistance. However, the widespread use of adhesive joints in structural applications is hindered, because reliable and mature inspection and defect detection techniques are not yet available. Adhesive joints in structural applications are subject to fatigue loading, during which debonds can initiate and propagate, ultimately leading to potential failure. In the majority of cases, joints operate under mixed-mode loading conditions. In this work, steel Cracked Lap Shear specimens were tested to investigate the crack growth behaviour of adhesive joints under mixed I+II mode loading. To address the challenge of monitoring these joints, a range of experimental techniques was employed. These techniques included Visual testing, Digital Image Correlation, Optical Backscatter Reflectometry, and localization by Acoustic Emission. The different methods were evaluated and compared to determine their effectiveness in locating the crack tip within the joints during fatigue propagation.

A simple, fast, and robust criterion for predicting fracture load and crack growth path in notched elastoplastic plates under mixed-mode loading

This study assesses the combined in-plane tensile/shear failure of aluminum plates containing notches. With this aim, a numerical method, i.e., the so-called equivalent material concept (EMC) in association with the extended finite element method (XFEM), is introduced to simulate the fracture phenomenon. The predicted failure load, initiation angle, and growth path are compared to experimental results reported for aluminum alloys in the literature to evaluate the functionality of the proposed method. Also, the accuracy of the EMC-XFEM model was compared to the conventional Ductile Damage method. The EMC-XFEM combined model is found to be fast and robust with appropriate accuracy in fracture predictions of ductile samples in comparison with the elastoplastic damage model and experimental data.

Effect of Mode II in the mixed-mode on the fatigue crack growth behaviour for SAPH440 material

In this study, the effect of mixed-mode loading on fatigue crack growth under varying applied loads for SAPH440 material was experimentally investigated. Previous research has either examined the effect of mixed-mode loading at a fixed angle or its effect on slanted cracks. This paper uses a CTS specimen and a loading device to apply varying angles of load during Mode I loading conditions. Initially, a numerical analysis was performed to validate existing equations for the stress intensity factor, which quantifies the crack driving force in the experiments. In the crack initiation region, load shedding techniques and load increasing techniques were used to confirm fatigue cracking considering mixed-mode effects. The change in the threshold value according to the loading angle was also evaluated. In the crack propagation region, a multi-level loading technique was employed to assess the mixed-mode effects under varying loading angles. This process verified the mixed-mode effect under conditions where the applied load angle varies. The fatigue crack fracture surfaces were examined using digital camera capture and SEM. It was found that when mixed-mode loading occurs during Mode I loading, the fatigue crack propagation rate varies depending on the mixed-mode loading angle.