A modified 3D mean strain energy density criterion for predicting shale mixed-mode Ⅰ/Ⅲ fracture toughness

The fracture toughness of rocks is a critical fracturing parameter in geo-energy exploitation playing a significant role in fracture mechanics and hydraulic fracturing.The edge-notched disk bending(ENDB)specimens are employed to measure the entire range of mixed-modeⅠ/Ⅲfracture toughness of Longmaxi shale.To theoretically interpret the fracture mechanisms,this research first introduces the detailed derivations of three established fracture criteria.By distinguishing the volumetric and distortional strain energy densities,an improved three-dimensional mean strain energy density(MSED)criterion is proposed.As the critical volumetric to distortional MSED ratio decreases,the transition from tensiondominated fracture to shear-dominated fracture is observed.Our results indicate that both peak load and applied energy increase significantly with the transition from pure mode I(i.e.,tension)to pure modeⅢ(i.e.,torsion or tearing)since mode-Ⅲcracking happens in a twisted manner and mode-Ⅰcracking occurs in a coplanar manner.The macroscopic fracture signatures are consistent with those of triaxial hydraulic fracturing.The average ratio of pure mode-Ⅲfracture toughness to pure mode-Ⅰfracture toughness is 0.68,indicating that the obtained mode-Ⅲfracture resistance for a tensionbased loading system is apparent rather than true.Compared to the three mainstream fracture criteria,the present fracture criterion exhibits greater competitiveness and can successfully evaluate and predict mixed-modeⅠ/Ⅲfracture toughness of distinct materials and loading methods.

Pulp health and calcific healing of a complicated crown–root fracture with additional root fracture in a maxillary incisor: A case report

BACKGROUND Complicated crown–root fracture (CRF) involves severe injury to the crown, root,and pulp, and may be accompanied by multiple root fractures. The loss of a toothhas lifelong consequences for children and teenagers, but the maintenance of pulphealth and the calcific healing of multiple root fractures are rarely reported in theliterature.CASE SUMMARY This case reports healing of a permanent tooth with complicated crown–root andadditional root fractures, in which pulp health was maintained. A 10-year-old girlfell and fractured the root of her maxillary left central incisor at the cervical level.After the coronal fragment was repositioned, the tooth was splinted until thetooth was no longer mobile, 2 years later. Eight years after treatment, the toothhas remained asymptomatic with vital pulp and localized gingival overgrowth.Cone-beam computed tomography revealed not only calcified healing of the CRFbut also spontaneous healing in an additional undiagnosed root fracture. Thefracture line on the enamel could not be healed by hard tissue and formed agroove in the cervical crown. It was speculated that the groove was related to thelocalized gingival overgrowth.CONCLUSION This case provides a clinical perspective of the treatment of a tooth with acomplicated CRF and an additional root fracture.

Correlation between anxiety, depression, and social stress in young patients with thoracolumbar spine fractures

BACKGROUND Traumatic injuries,such as falling,car accidents,and crushing mostly cause spinal fractures in young and middle-aged people,and>50%of them are thoracolumbar fractures.This kind of fracture is easily combined with serious injuries to peripheral nerves and soft tissues,which causes paralysis of the lower limbs if there is no timely rehabilitation treatment.Young patients with thoracolumbar fractures find it difficult to recover after the operation,and they are prone to depression,low self-esteem,and other negative emotions.AIM To investigate the association between anxiety,depression,and social stress in young patients with thoracolumbar spine fractures and the effect on rehabilitation outcomes.METHODS This study retrospectively analyzed 100 patients admitted to the orthopedic department of Honghui Hospital,Xi’an Jiaotong University who underwent thoracolumbar spine fracture surgery from January 2022 to June 2023.The general data of the patients were assessed with the Hamilton anxiety scale(HAMA),Hamilton depression scale(HAMD),life events scale,and social support rating scale(SSRS)to identify the correlation between anxiety,depression scores,and social stress and social support.The Japanese Orthopedic Association(JOA)was utilized to evaluate the rehabilitation outcomes of the patients and to analyze the effects of anxiety and depression scores on rehabilitation.RESULTS According to the scores of HAMD and HAMA in all patients,the prevalence of depression in patients was 39%(39/100),and the prevalence of anxiety was 49%(49/100).Patients were categorized into non-depression(n=61)and depression(n=39),non-anxiety(n=51),and anxiety(n=49)groups.Statistically significant differences in gender,occupation,Pittsburgh Sleep Quality Index(PSQI)score,and monthly family income were observed between the non-depression and depression groups(P<0.05).A significant difference in occupation and PSQI score was found between the non-anxiety and anxiety groups.Both depression(r=0.207,P=0.038)and anxiety scores(r=0.473,P<0.001)were significantly and positively correlated with negative life events.The difference in negative life event scores as well as SSRS total and item scores was statist-ically significant between patients in the non-depression and depression groups(P<0.05).The difference between the non-anxiety and anxiety groups was statistically significant(P<0.05)in the negative life event scores as well as the total SSRS scores.Additionally,JOA scores were significantly lower in both anxious and depressed patients.CONCLUSION Young patients with thoracolumbar fractures are prone to anxiety and depression.Patients’anxiety and depression are closely associated with social pressure,which reduces the life pressure of young patients with thoracolumbar fractures,enhances social support,and improves the psychology of anxiety and depression.,which affects patients’recovery.

Evaluation of varying ductile fracture criterion for 7075 aluminum alloy

As one of the principal failures,ductile fracturing restricts metal forming process.Cockcroft-Latham type fracture criterion is suited for ductile fracture in bulk metal-forming simulation.Finding a way to evaluate the ductile fracture criterion（DFC） and identify the relationship between DFC and deformation conditions for a strain-softening material,7075 aluminum alloy;however,it is a non-trivial issue that still needs to be addressed in a greater depth.An innovative approach is brought forth that the compression tests and numerical simulations provide mutual support to evaluate the ductile damage cumulating process and determine the DFC diagram.One of the results shows that for a fixed temperature,the maximum cumulated damage decreases regularly with increasing strain rate.The most important result shows that DFC of 7075 aluminum alloy at temperatures of 573-723 K and strain rates of 0.01-10 s-1 is not a constant but a change in a range of 0.255-0.453,thus it has been defined with varying ductile fracture criterion（VDFC） and characterized by a function of strain rate and temperature.According to VDFC diagram,the exact fracture moment and position during various forming processes will be predicted conveniently,in addition to which,the deformation domains with lower fracture risk corresponding to higher VDFC can be identified.

Modified MK model combined with ductile fracture criterion and its application in warm hydroforming

A modified MK model combined with ductile fracture criterion（DFC-MK model） is proposed to compute the forming limit diagrams（FLDs） of 5A06-O aluminum alloy sheet at different temperatures.The material constant（C） of ductile fracture criterion and initial thickness imperfection parameter（f0） at various temperatures are determined by using a new computing method based on wide sheet bending test.The FLDs at 20 and 200 °C are calculated through the DFC-MK model.The DFC-MK model,which includes the influence of through-thickness normal stress,is written into the subroutine VUMAT embedded in Abaqus/ Explicit.The cylindrical cup hydroforming tests are carried out to verify the model.The results show that compared with experimental observations,the predicted FLDs based on DFC-MK model are more accurate than the conventional MK model;the errors between the simulations and experiments in warm hydroforming are 8.23% at 20 °C and 9.24% at 200 °C,which verify the effectiveness of the proposed model.

Analytic Expression of Meso-Criterion on Dynamic Fracture

This study presents a meso-criterion of dynamic fracture, on the basis of stress in integral form In such way the difficulty due to the singularity of stress distribution at the crack tip is overcome. A micro-parameter, the atom radius, is introduced into the criterion.Meanwhile a characteristic time concept is taken into account for describing the inertia effect of material. The criterion reveals The criterion reveals the effects of loading rate, defect and sample geometry,material constants including the micro-structure parameter.

Numerical simulation study on shear resistance of anchorage joints considering tensile-shear fracture criterion of 2G-NPR bolt

2G-NPR bolt (the 2nd generation Negative Poisson’s Ratio bolt) is a new type of bolt with high strength, high toughness and no yield platform. It has signifcant efects on improving the shear strength of jointed rock mass and controlling the stability of surrounding rock. To achieve an accurate simulation of bolted joint shear tests, we have studied a numerical simulation method that takes into account the 2G-NPR bolt's tensile–shear fracture criterion. Firstly, the indoor experimental study on the tensile–shear mechanical properties of 2G-NPR bolt is carried out to explore its mechanical properties under diferent tensile–shear angles, and the fracture criterion of 2G-NPR bolt considering the tensile–shear angle is established. Then, a three-dimensional numerical simulation method considering the tensile–shear mechanical constitutive and fracture criterion of 2G-NPR bolt, the elastoplastic mechanical behavior of surrounding rock and the damage and deterioration of grouting body is proposed. The feasibility and accuracy of the method are verifed by comparing with the indoor shear test results of 2G-NPR bolt anchorage joints. Finally, based on the numerical simulation results, the deformation and stress of the bolt, the distribution of the plastic zone of the rock mass, the stress distribution and the damage of the grouting body are analyzed in detail. The research results can provide a good reference value for the practical engineering application and shear mechanical performance analysis of 2G-NPR bolt.

A DUCTILE FRACTURE CRITERION BASED ON THREEDIMENSIONAL VOID MODEL

A fracture criterion derived from a microscopic point of view is proposed and has proved to be effective in the analysis of uniaxial tension. On the one hand, a method of predicting a ductile fracture is proposed using a three-dimensional void model and the assumption of velocity discontinuity. The relationship between the void volume fraction and the critical strain to fracture, calculated with the help of the new model, shows the same tendency as that obtained from the modified Thomason model. On the other hand, the mechanical and metallographic analyses of the uniaxial tension experiment are performed using four kinds of carbon steel. The relationship between the void volume fraction and the critical strain to fracture, calculated from the new model, agrees better with the result obtained from the experiment, rather than that calculated by the modified Thomason model, which confirms the validity of the ductile fracture criterion based on the three-dimensional void model.

Thermo-hydro-mechanical-chemical(THMC)coupling fracture criterion of brittle rock

Based on analysis of thermo-hydro-mechanical-chemical(THMC)coupling mechanism for brittle rock,THMC coupling indicator in terms of rock porosity was introduced to represent the influencing degree of THMC coupling field on stress field in order to establish THMC coupling fracture criterion.A novel real-time measurement method of permeability(related to porosity)was proposed to determine the THMC coupling indicator,and self-designed THMC coupling tests and scanning electron microscope tests were conducted on pre-cracked red sandstone specimens to study the macroscopic and microscopic fracture mechanism.Research results show that the higher the hydraulic pressure is,the smaller the crack initiation load is and the easier the Mode I fracture occurs.Test results are in good agreement with prediction results(crack initiation load and angle,and fracture mode),which can verify the effectiveness of the newly established THMC coupling fracture criterion.This new fracture criterion can be also further extended to predict THMC coupling fracture of multi-crack problem.

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s^（-iε）, so that K = ■ s^（-iε）, s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin（ASME J. Appl. Mech. 24：361–364, 1957） did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor（SIF）-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.

A New Ductile Fracture Criterion and Its Application to the Prediction of Forming Limit in Deep Drawing

With considering the influence of equivalent plastic strain on void-damage and taking Lemaitre damage equivalent stress as plastic potential, based on continuous damage mechanics theory, a new criterion for ductile fracture is derived. The two key material constants in the criterion are determined by the combination of tension tests with FE (finite element) simulation. On the basis of the values of stress and strain calculated from commercial finite element software, the forming limit in cylindrical deep drawing of annealed aluminum alloy LY12(M) is predicted by means of the new ductile fracture criterion. Experiments verify that the predicted results are in agreement with the experimental ones. Hence, it is reliable to predict the forming limit in deep drawing by means of the new ductile fracture criterion.

Fracture strength evaluation of titanium alloys using modified average stress criterion

Structural integrity procedures were used to demonstrate the fitness for the purpose of engineering components transmitting loads. The prediction of the fracture strength of titanium alloys containing sharp notches through the damage model depends on the un-notched strength and the critical length of the damage zone ahead of the notch. In general, the critical length of the damage zone depends on the material, specimen, and size of the sharp notch. Modifications were made in one of the stress fracture criteria known as the average stress criterion for accurate prediction of notched tensile strength of titanium alloy specimen containing sharp notches. To examine the adequacy of these modifications, fracture data of center-cracked titanium alloys with various thicknesses are considered. The notched （fracture） strength estimates are found to be close to the test results. The modified average stress criterion is very simple to predict the notched tensile strength.

Fracture criterion for predicting surface cracking of Ti40 alloy in hot forming processes

Hot compression tests were conducted on Ti40 burn resistant titanium alloy in the temperature range of 900-1 100 ℃ and strain rate range of 0.01-10 s-1 to investigate its fracture behavior and critical fracture conditions in hot forming. It was observed that the failure of Ti40 alloy is attributed to longitudinal surface cracking due to severe oxidation of element V and the secondary tensile stresses. The critical fracture strain increases with increasing temperature and decreasing strain rate. From these observations and parallel FEM simulations,it was concluded that the critical fracture strain is a function of a single argument Zener-Hollomon parameter,and there is a linear relationship between them. An Oyane criterion successfully predicted the location of crack initiation. The critical fracture values also exhibit a liner relationship with lnZ. Based on these results,a new fracture criterion of Ti40 alloy based on Zener-Hollomon parameter was established.

MOVING DEFECT WITH IRREVERSIBLE DEFORMATION ZONE AND CRITERION OF NONLINEAR DYNAMIC FRACTURE OF SOLIDS

This letter reports a fully dynamic analysis of a moving defect with a plastic zone.The fields of stress and displacement of this defect are determined,so that the opening displacement at the tip of the defect is obtained.Based on it the author proposes a criterion of nonlinear dynamic fracture of solids.

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.

The Fatigue Fracture Criterion Based on the Latent Energy Approach

The extensive literature on the fatigue problem, published for more than one hundred years, is reviewed by the known scientists [1,2]. As it follows from these investigations, the fundamental amount of failures in engineering practice connected with the fatigue fractures of materials and structure elements. The fatigue problem is complicated one and it is not solved yet. So the theoretical and experimental investigations of this problem will be continued. In our paper the energy approach to formulate the fatigue strength criterion is proposed. The criterion is based on the conception of the latent energy [3-7]. This conception was not applied previously to the fatigue problem. The latent energy is consumed to generate the irreversible deformation and to damage and fracture of metallic materials. So the fatigue fracture criterion can be formulated using the results of latent energy measurements in the macro experiments. This is most impotent advantage of the proposed approach. The logistic function is used to describe the dependence of latent energy from the value of irreversible deformation. It is assumed that the cyclic strength of metals is defined by the latent energy, stored in specimen, when it is reached the critical value in accordance with the logistic curve in a saturation zone. This proposal is used to formulate the fatigue strength criterion. The functions and parameters of received criterion are concretized and comparisons with experimental results for axial cyclic tension for sheet aluminum alloy specimens are given.

Incubation time fracture criterion for FEM simulations

The paper is discussing problems connected with embedment of the incubation time criterion for brittle fracture into finite element computational schemes. Incubation time fracture criterion is reviewed; practical questions of its numerical implementation are extensively discussed. Several examples of how the incubation time fracture criterion can be used as fracture condition in finite element computations are given. The examples include simulations of dynamic crack propagation and arrest, impact crater formation （i.e. fracture in initially intact media）, spall fracture in plates, propagation of cracks in pipelines. Applicability of the approach to model initiation, development and arrest of dynamic fracture is claimed.

Transferability of Charpy Absorbed Energy to Fracture Toughness Based on Weibull Stress Criterion

The relationship between Charpy absorbed energy and the fracture toughness by means of the (crack tip opening displacement (CTOD)) method was analyzed based on the Weibull stress criterion. The Charpy absorbed energy and the fracture toughness were measured for the SN490B steel under the ductile-brittle transition temperature region. For the instrumented Charpy impact test, the curves between the loading point displacement and the load against time were recorded. The critical Weibull stress was taken as a fracture controlled parameter, and it could not be affected by the specimen configuration and the loading pattern based on the local approach. The parameters controlled brittle fracture are obtained from the Charpy absorbed energy results, then the fracture toughness for the compact tension (CT) specimen is predicted. It is found that the results predicted are in good agreement with the experimental. The fracture toughness could be evaluated by the Charpy absorbed energy, because the local approach gives a good description for the brittle fracture even though the Charpy impact specimen or the CT specimen is used for the given material.

Evaluation of Energy Saving Operational Modes for Industrial Fracture Connected Processes on the Basis of Incubation Time Fracture Criterion

A problem for a central crack in a plate subjected to plane strain conditions is investigated. Mode I crack loading is created by a dynamic pressure pulse applied at a large distance from the crack. It was found that for a certain combination of amplitude and duration of the pulse applied, the energy transmitted to the sample has a strongly marked minimum, meaning that with the pulse amplitude or duration moving away from the optimal values, minimum energy required for initiation of crack growth increases rapidly. The results obtained indicate a possibility to optimise energy consumption of different industrial processes connected with fracture. Much could be gained in, for example, drilling or rock pounding where energy input accounts for the largest part of the process cost. Presumably further investigation of the effect observed can make it possible to predict optimal energy saving parameters, i.e. frequency and amplitude of impacts, for industrial devices, e.g. bores, grinding machines, and hence significantly reduce the process cost. The pre- diction can be given based on the parameters of the media fractured （material parameters, prevalent crack length and orientation, etc.）.

A New Fracture Criterion for a Crack under Combined Mode Loading

A new fracture criterion was proposed. The physical explanation of the criterion is that crack will propagate when the minimum strain energy density in iso hoop stress curve reach a critical strength of the material considered. The resulting curve of critical fracture of mixed mode cracks shows that the present fracture is efficient and more accurate than the previous criteria.