Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilton principle.Three typical electric boundary conditions are involved in the present model to characterize the fracture behaviors in various physical situations.A staggered algorithm is used to simulate the crack propagation.The polynomial splines over hierarchical T-meshes(PHT-splines)are adopted as the basis function,which owns the C1continuity.Systematic numerical simulations are performed to study the influence of the electric boundary conditions and the applied electric field on the fracture behaviors of piezoelectric materials.The electric boundary conditions may influence crack paths and fracture loads significantly.The present research may be helpful for the reliability evaluation of the piezoelectric structure in the future applications.

Crack propagation simulation in brittle elastic materials by a phase field method

To overcome the difficulties of re-meshing and tracking the crack-tip in other computational methods for crack propagation simulations,the phase field method based on the minimum energy principle is introduced by defining a continuous phase field variable(x)∈[0,1]to characterize discontinuous cracks in brittle materials.This method can well describe the crack initiation and propagation without assuming the shape,size and orientation of the initial crack in advance.In this paper,a phase field method based on Miehe's approach[Miehe et al.,Comp.Meth.App.Mech.Eng.(2010)]is applied to simulate different crack propagation problems in twodimensional(2D),isotropic and linear elastic materials.The numerical implementation of the phase field method is realized within the framework of the finite element method(FEM).The validity,accuracy and efficiency of the present method are verified by comparing the numerical results with other reference results in literature.Several numerical examples are presented to show the effects of the loading type(tension and shear),boundary conditions,and initial crack location and orientation on the crack propagation path and force-displacement curve.Furthermore,for a single edge-cracked bi-material specimen,the influences of the loading type and the crack location on the crack propagation trajectory and force-displacement curve are also investigated and discussed.It is demonstrated that the phase field method is an efficient tool for the numerical simulation of the crack propagation problems in brittle elastic materials,and the corresponding results may have an important relevance for predicting and preventing possible crack propagations in engineering applications.

Quantitative characterization of the brittleness of deep shales by integrating mineral content,elastic parameters,in situ stress conditions and logging analysis

Deep shale reservoirs(3500–4500 m)exhibit significantly different stress states than moderately deep shale reservoirs(2000–3500 m).As a result,the brittleness response mechanisms of deep shales are also different.It is urgent to investigate methods to evaluate the brittleness of deep shales to meet the increasingly urgent needs of deep shale gas development.In this paper,the quotient of Young’s modulus divided by Poisson’s ratio based on triaxial compression tests under in situ stress conditions is taken as SSBV(Static Standard Brittleness Value).A new and pragmatic technique is developed to determine the static brittleness index that considers elastic parameters,the mineral content,and the in situ stress conditions(BIEMS).The coefficient of determination between BIEMS and SSBV reaches 0.555 for experimental data and 0.805 for field data.This coefficient is higher than that of other brittleness indices when compared to SSBV.BIEMS can offer detailed insights into shale brittleness under various conditions,including different mineral compositions,depths,and stress states.This technique can provide a solid data-based foundation for the selection of‘sweet spots’for single-well engineering and the comparison of the brittleness of shale gas production layers in different areas.

Particle Discontinuous Deformation Analysis of Static and Dynamic Crack Propagation in Brittle Material

Crack propagation in brittle material is not only crucial for structural safety evaluation,but also has a wideranging impact on material design,damage assessment,resource extraction,and scientific research.A thorough investigation into the behavior of crack propagation contributes to a better understanding and control of the properties of brittle materials,thereby enhancing the reliability and safety of both materials and structures.As an implicit discrete elementmethod,the Discontinuous Deformation Analysis(DDA)has gained significant attention for its developments and applications in recent years.Among these developments,the particle DDA equipped with the bonded particle model is a powerful tool for predicting the whole process of material from continuity to failure.The primary objective of this research is to develop and utilize the particle DDAtomodel and understand the complex behavior of cracks in brittle materials under both static and dynamic loadings.The particle DDA is applied to several classical crack propagation problems,including the crack branching,compact tensile test,Kalthoff impact experiment,and tensile test of a rectangular plate with a hole.The evolutions of cracks under various stress or geometrical conditions are carefully investigated.The simulated results are compared with the experiments and other numerical results.It is found that the crack propagation patterns,including crack branching and the formation of secondary cracks,can be well reproduced.The results show that the particle DDA is a qualified method for crack propagation problems,providing valuable insights into the fracture mechanism of brittle materials.

A statistical damage-based constitutive model for shearing of rock joints in brittle drop mode

Some rock joints exhibit significant brittleness,characterized by a sharp decrease in shear stress upon reaching the peak strength.However,existing models often fail to accurately represent this behavior and are encumbered by numerous parameters lacking clear mechanical significance.This study presents a new statistical damage constitutive model rooted in both damage mechanics and statistics,containing only three model parameters.The proposed model encompasses all stages of joint shearing,including the compaction stage,linear stage,plastic yielding stage,drop stage,strain softening stage,and residual strength stage.To derive the analytical expression of the constitutive model,three boundary conditions are introduced.Experimental data from both natural and artificial rock joints is utilized to validate the model,resulting in average absolute relative errors ranging from 3%to 8%.Moreover,a comparative analysis with established models illustrates that the proposed model captures stress drop and post-peak strain softening more effectively,with model parameters possessing clearer mechanical interpretations.Furthermore,parameter analysis is conducted to investigate the impacts of model parameters on the curves and unveil the relationship between these parameters and the mechanical properties of rock joints.Importantly,the proposed model is straightforward in form,and all model parameters can be obtained from direct shear tests,thus facilitating the utilization in numerical simulations.

Effect of δ Phase on Hydrogen Embrittlement of Inconel 718 by Notch Tensile Tests

The effect of δ phase on the hydrogen embrittlement (HE) sensitivity of Inconel 718 was investigated by conducting notch tensile tests. Notch tensile specimens with various precipitation morphologies of δ phase were prepared with different heat treatments, and hydrogen was charged into the tensile specimens before tensile tests via a cathodic charging process. The loss of notch tensile strength (NTS) due to the charged hydrogen was used to evaluate the hydrogen embrittlement sensitivity. The results show that δ phase has deleterious effect on NTSs, and the fracture of hydrogen-charged specimens initiated near the notch surfaces. The loss of NTS caused by precharged hydrogen can be greatly decreased by dissolving δ phase. δ-free Inconel 718 alloy is proposed for the applications in hydrogen environments.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling.An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field.The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines.The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem.A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method.Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments.The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Analysis of the Application Effect of Structured Healthcare Education in Brittle Diabetic Patients

Objective:To explore the application effect of structured healthcare education in patients with brittle diabetes mellitus.Methods:188 brittle diabetic patients admitted to our hospital from May 2021 to December 2023 were selected as the study subjects,and were divided into the control group(n=94)and the observation group(n=94)according to the random number table method.The control group used conventional nursing intervention and the observation group used structured healthcare education.The general information,glycemic indexes,self-efficacy,compliance,and nursing satisfaction of patients in the two groups were observed.Results:There was no statistical significance in the basic information of the two groups of patients(P>0.05);after the intervention,the fasting plasma glucose,2-hour postprandial blood glucose,and HbA1c of the patients in the observation group were lower than those of the control group(P<0.001);after the intervention,the self-efficacy scores of the patients in the two groups increased,and the scores of the observation group were significantly higher than those of the control group(P<0.001);the total adherence rate of the patients in the observation group(90/95.75%)was significantly higher than that of the control group(80/90.10%)(χ^(2)=6.144,P<0.05);and the total satisfaction rate of patients in the observation group(92/97.87%)was significantly higher than that of the control group(78/82.98%)(χ^(2)=12.042,P<0.05).Conclusion:In patients with brittle diabetes mellitus,structured healthcare education can effectively control patients’blood glucose levels,improve patients’self-efficacy and adherence,and enhance patient satisfaction.

Y-Containing Phase in Fe-20Cr-4Al-Y Alloys and Its Effect on Embrittlement

In Fe-20Cr-4Al-Y alloys with (0.13～0.73) wt% Y,Y-containing phase was observed.Its Y content and crystal structure were identical with those Of YFe_9 phase basically,and microhardness was 4～5 times higher than that of α-phase matrix.The phase can strongly prevent grain growth and dislocation annihilation,and trap impurity atoms,as a result,and inhibit the embrittlement caused by treating over 1000℃.

Pharmacological intervention for chronic phase of spinal cord injury

Spinal cord injury is an intractable traumatic injury. The most common hurdles faced during spinal cord injury are failure of axonal regrowth and reconnection to target sites. These also tend to be the most challenging issues in spinal cord injury. As spinal cord injury progresses to the chronic phase, lost motor and sensory functions are not recovered. Several reasons may be attributed to the failure of recovery from chronic spinal cord injury. These include factors that inhibit axonal growth such as activated astrocytes, chondroitin sulfate proteoglycan, myelin-associated proteins, inflammatory microglia, and fibroblasts that accumulate at lesion sites. Skeletal muscle atrophy due to denervation is another chronic and detrimental spinal cord injury–specific condition. Although several intervention strategies based on multiple outlooks have been attempted for treating spinal cord injury, few approaches have been successful. To treat chronic spinal cord injury, neural cells or tissue substitutes may need to be supplied in the cavity area to enable possible axonal growth. Additionally, stimulating axonal growth activity by extrinsic factors is extremely important and essential for maintaining the remaining host neurons and transplanted neurons. This review focuses on pharmacotherapeutic approaches using small compounds and proteins to enable axonal growth in chronic spinal cord injury. This review presents some of these candidates that have shown promising outcomes in basic research(in vivo animal studies) and clinical trials: AA-NgR(310)ecto-Fc(AXER-204), fasudil, phosphatase and tensin homolog protein antagonist peptide 4, chondroitinase ABC, intracellular sigma peptide,(-)-epigallocatechin gallate, matrine, acteoside, pyrvate kinase M2, diosgenin, granulocyte-colony stimulating factor, and fampridine-sustained release. Although the current situation suggests that drug-based therapies to recover function in chronic spinal cord injury are limited, potential candidates have been identified through basic research, and these candidates may be subjects of clinical studies in the future. Moreover, cocktail therapy comprising drugs with varied underlying mechanisms may be effective in treating the refractory status of chronic spinal cord injury.

Low‑Temperature Oxidation Induced Phase Evolution with Gradient Magnetic Heterointerfaces for Superior Electromagnetic Wave Absorption

Gradient magnetic heterointerfaces have injected infinite vitality in optimizing impedance matching,adjusting dielectric/magnetic resonance and promoting electromagnetic(EM)wave absorption,but still exist a significant challenging in regulating local phase evolution.Herein,accordion-shaped Co/Co_(3)O_(4)@N-doped carbon nanosheets(Co/Co_(3)O_(4)@NC)with gradient magnetic heterointerfaces have been fabricated via the cooperative high-temperature carbonization and lowtemperature oxidation process.The results indicate that the surface epitaxial growth of crystal Co_(3)O_(4) domains on local Co nanoparticles realizes the adjustment of magnetic-heteroatomic components,which are beneficial for optimizing impedance matching and interfacial polarization.Moreover,gradient magnetic heterointerfaces simultaneously realize magnetic coupling,and long-range magnetic diffraction.Specifically,the synthesized Co/Co_(3)O_(4)@NC absorbents display the strong electromagnetic wave attenuation capability of−53.5 dB at a thickness of 3.0 mm with an effective absorption bandwidth of 5.36 GHz,both are superior to those of single magnetic domains embedded in carbon matrix.This design concept provides us an inspiration in optimizing interfacial polarization,regulating magnetic coupling and promoting electromagnetic wave absorption.

A new mixed-mode phase-field model for crack propagation of brittle rock

Study on crack propagation process of brittle rock is of most significance for cracking-arrest design and cracking-network optimization in rock engineering.Phase-field model(PFM)has advantages of simplicity and high convergence over the common numerical methods(e.g.finite element method,discrete element method,and particle manifold method)in dealing with three-dimensional and multicrack problems.However,current PFMs are mainly used to simulate mode-I(tensile)crack propagation but difficult to effectively simulate mode-II(shear)crack propagation.In this paper,a new mixed-mode PFM is established to simulate both mode-I and mode-II crack propagation of brittle rock by distinguishing the volumetric elastic strain energy and deviatoric elastic strain energy in the total elastic strain energy and considering the effect of compressive stress on mode-II crack propagation.Numerical solution method of the new mixed-mode PFM is proposed based on the staggered solution method with self-programmed subroutines UMAT and HETVAL of ABAQUS software.Three examples calculated using different PFMs as well as test results are presented for comparison.The results show that compared with the conventional PFM(which only simulates the tensile wing crack but not mode-II crack propagation)and the modified mixed-mode PFM(which has difficulty in simulating the shear anti-wing crack),the new mixed-mode PFM can successfully simulate the whole trajectories of mixed-mode crack propagation(including the tensile wing crack,shear secondary crack,and shear anti-wing crack)and mode-II crack propagation,which are close to the test results.It can be further extended to simulate multicrack propagation of anisotropic rock under multi-field coupling loads.

Anomalous yield and intermediate temperature brittleness behaviors of directionally solidified nickel-based superalloy

A nickel-based superalloy with good corrosion resistance was fabricated by directional solidification, and its microstructure and tensile properties at elevated temperatures were investigated. Microstructure observations reveal that the γ＇ precipitates are arrayed in the y matrix regularly with some MC, Ni5Hf and M3B2 particles distributed along the grain boundary. The tensile tests exhibit that the tensile properties depend on temperature significantly and demonstrate obvious anomalous yield and intermediate-temperature brittleness （ITB） behavior. Below 650℃, the yield strength decreases slightly but the ultimate tensile strength almost has no change. When the temperature is between 650 ℃ and 750 ℃, the yield and ultimate tensile strengths rise rapidly, and after then they both decrease gradually with temperature increasing further. The elongation has its minimum value at about 700 ℃. The TEM examination exhibits that sharing of the γ＇ by dislocation is almost the main deformation mechanism at low temperatures, but the γ＇ by-pass dominates the deformation at high temperatures. The transition temperature from shearing to by-pass should be around 800 ℃. The anomalous yield and intermediate-temperature brittleness behaviors should be attributed to the high content of γ＇. In addition, the carbides and eutectic structure also contribute some to the ITB behaviors of the alloy.

Thermal-hydro-mechanical coupling stress intensity factor of brittle rock

A new calculation formula of THM coupling stress intensity factor was derived by the boundary collocation method, in which an additional constant stress function was successfully introduced for the cracked specimen with hydraulic pressure applied on its crack surface. Based on the newly derived formula, THM coupling fracture modes （including tensile, shear and mixed fracture mode） can be predicted by a new fracture criterion of stress intensity factor ratio, where the maximum axial load was measured by self-designed THM coupling fracture test. SEM analyses of THM coupling fractured surface indicate that the higher the temperature and hydraulic pressure are and the lower the confining pressure is, the more easily the intergranular （tension） fracture occurs. The transgranular （shear） fracture occurs in the opposite case while the mixed-mode fracture occurs in the middle case. The tested THM coupling fracture mechanisms are in good agreement with the predicted THM coupling fracture modes, which can verify correction of the newly-derived THM coupling stress intensity factor formula.

Evaluation of Urban Village Renovation System in Xi’an City of China Based on Brittleness Analysis

Renovation system of urban villages in Xi'an City was evaluated. Influence factors of urban village renovation were analyzed on the basis of brittleness theory, and an evaluation index system established through multi-level inconsistency decomposing. By incorporating the catastrophe theory with fuzzy mathematical theory, the mathematic model was created, and catastrophe membership function was obtained as well as evaluation results. Policies for the renovation of urban villages and new direction of the renovation were interpreted. The application case proved that catastrophe progression method was objective and effective and it could provide new concepts for the evaluation and adjustment of urban village renovation. Moreover, application of brittleness theory in the research on urban village renovation is of great instruction and reference value for the present urban construction.

APPLICATION OF DIAMOND TWIST DRILL FOR DRILLING HARD-BRITTLE MATERIALS

A new conception, which combines the advantages of both twist drill and diamond grit, is proposed to develop a new tool for drilling hard brittle materials. The manufacturing process of the drill is introduced, and drilling experiments are carried out by using of the drill developed. As a result, not only it can drill holes with a high efficiency, but also a good quality of hole inlet and outlet can be obtained for such materials as glasses, marble, granite, Al 2O 3, etc. The prospect in practical application of the technique developed is also discussed.

Brittleness index and seismic rock physics model for anisotropic tight-oil sandstone reservoirs

Brittleness analysis becomes important when looking for sweet spots in tightoil sandstone reservoirs. Hence, appropriate indices are required as accurate brittleness evaluation criteria. We construct a seismic rock physics model for tight-oil sandstone reservoirs with vertical fractures. Because of the complexities in lithology and pore structure and the anisotropic characteristics of tight-oil sandstone reservoirs, the proposed model is based on the solid components, pore connectivity, pore type, and fractures to better describe the sandstone reservoir microstructure. Using the model, we analyze the brittleness sensitivity of the elastic parameters in an anisotropic medium and establish a new brittleness index. We show the applicability of the proposed brittleness index for tight-oil sandstone reservoirs by considering the brittleness sensitivity, the rock physics response characteristics, and cross-plots. Compared with conventional brittleness indexes, the new brittleness index has high brittleness sensitivity and it is the highest in oil-bearing brittle zones with relatively high porosity. The results also suggest that the new brittleness index is much more sensitive to elastic properties variations, and thus can presumably better predict the brittleness characteristics of sweet spots in tight-oil sandstone reservoirs.

Burial depth interval of the shale brittle–ductile transition zone and its implications in shale gas exploration and production

Brittleness and ductility of shale are closely related to shale gas exploration and production. How to predict brittleness and ductility of shale is one of the key issues in the study of shale gas preservation and hydraulic fracturing treatments. The magnitude of shale brittleness was often determined by brittle mineral content(for example, quartz and feldspars) in shale gas exploration.However, the shale brittleness is also controlled by burial depth. Shale brittle/ductile properties such as brittle, semibrittle and ductile can mutually transform with burial depth variation. We established a work flow of determining the burial depth interval of brittle–ductile transition zone for a given shale. Two boundaries were employed to divide the burial depth interval of shale brittle/ductile properties. One is the bottom boundary of the brittle zone(BZ), and the other is the top boundary of the ductile zone(DZ). The brittle–ductile transition zone(BDTZ) is between them.The bottom boundary of BZ was determined by the overconsolidation ratio(OCR) threshold value combined with pre-consolidation stress which the shale experienced over geological time. The top boundary of DZ was determined based on the critical confining pressure of brittle–ductile transition. The OCR threshold value and the critical confining pressure were obtained from uniaxial strain andtriaxial compression tests. The BZ, DZ and BDTZ of the Lower Silurian Longmaxi shale in some representative shale gas exploration wells in eastern Sichuan and western Hubei areas were determined according to the above work flow. The results show that the BZ varies with the maximum burial depth and the DZ varies with the density of the overlying rocks except for the critical confining pressure.Moreover, the BDTZ determined by the above work flow is probably the best burial depth interval for marine shale gas exploration and production in Southern China. Shale located in the BDTZ is semi-brittle and is not prone to be severely naturally fractured but likely to respond well to hydraulic fracturing. The depth interval of BDTZ determined by our work flow could be a valuable parameter of shale gas estimation in geology and engineering.

Construction of a novel brittleness index equation and analysis of anisotropic brittleness characteristics for unconventional shale formations

The brittleness prediction of shale formations is of interest to researchers nowadays.Conventional methods of brittleness prediction are usually based on isotropic models while shale is anisotropic.In order to obtain a better prediction of shale brittleness,our study firstly proposed a novel brittleness index equation based on the Voigt–Reuss–Hill average,which combines two classical isotropic methods.The proposed method introduces upper and lower brittleness bounds,which take the uncertainty of brittleness prediction into consideration.In addition,this method can give us acceptable predictions by using limited input values.Secondly,an anisotropic rock physics model was constructed.Two parameters were introduced into our model,which can be used to simulate the lamination of clay minerals and the dip angle of formation.In addition,rock physics templates have been built to analyze the sensitivity of brittleness parameters.Finally,the effects of kerogen,pore structure,clay lamination and shale formation dip have been investigated in terms of anisotropy.The prediction shows that the vertical/horizontal Young’s modulus is always below one while the vertical/horizontal Poisson’s ratio(PR)can be either greater or less than 1.Our study finds different degrees of shale lamination may be the explanation for the random distribution of Vani(the ratio of vertical PR to horizontal PR).

Discrete element modeling on the crack evolution behavior of brittle sandstone containing three fissures under uniaxial compression

Based on experimental restilts of brittle, intact sandstone under uniaxial compression, the micro-parameters were firstly confirmed by adopting particle flow code （PFC2D）. Then, the validation of the simulated models were cross checked with the experimental results of brittle sandstone containing three parallel fissures under uniaxial compression. The simulated results agreed very well with the experimental results, including the peak strength, peak axial strain, and ultimate failure mode. Using the same micro- parameters, the numerical models containing a new geometry of three fissures are constructed to investigate the fissure angle on the fracture mechanical behavior of brittle sandstone under uniaxial compression. The strength and deformation parameters of brittle sandstone containing new three fissures are dependent to the fissure angle. With the increase of the fis- sure angle, the elastic modulus, the crack damage threshold, and the peak strength of brittle sandstone containing three fissures firstly increase and secondly decrease. But the peak axial strain is nonlinearly related to the fissure angle. In the entire process of deformation, the crack initiation and propagation behavior of brittle sandstone containing three fissures under uniaxial compression are investigated with respect to the fissure angle. Six different crack coalescence modes are identified for brittle sandstone containing three fissures under uniaxial compression. The influence of the fissure angle on the length of crack propagation and crack coalescence stress is evaluated. These investigated conclusions are very important for ensuring the stability and safety of rock engineering with intermittent structures.