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.

A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

A novel phase-field model for the propagation of mixed-mode hydraulic fractures,characterized by the formation of mixed-mode fractures due to the interactions between fluids and solids,is proposed.In this model,the driving force for the phase field consists of both tensile and shear components,with the fluid contribution primarily manifesting in the tension driving force.The displacement and pressure are solved simultaneously by an implicit method.The numerical solution's iterative format is established by the finite element discretization and Newton-Raphson(NR)iterative methods.The correctness of the model is verified through the uniaxial compression physical experiments on fluid-pressurized rocks,and the limitations of the hydraulic fracture expansion phase-field model,which only considers mode I fractures,are revealed.In addition,the influence of matrix mode II fracture toughness value,natural fracture mode II toughness value,and fracturing fluid injection rate on the hydraulic fracture propagation in porous media with natural fractures is studied.

Mixed-mode fracture behavior in deep shale reservoirs under different loading rates and temperatures

In the last years,shale gas has gradually substituted oil and coal as the main sources of energy in the world.Compared with shallow shale gas reservoirs,deep shale is characterized by low permeability,low porosity,strong heterogeneity,and strong anisotropy.In the process of multi-cluster fracturing of horizontal wells,the whole deformation process and destruction modes are significantly influenced by loading rates.In this investigation,the servo press was used to carry out semi-circular bend(SCB)mixedmode fracture experiments in deep shales(130,160,190℃)with prefabricated fractures under different loading rates(0.02,0.05,0.1,0.2 mm/min).The fracture propagation process was monitored using acoustic emission.The deformation characteristics,displacementeload curve,and acoustic emission parameters of shale under different loading rates were studied during the mixed-mode fracture propagation.Our results showed that during the deformation and fracture of the specimen,the acoustic emission energy and charge significantly increased near the stress peak,showing at this point the most intense acoustic emission activity.With the increase in loading rate,the fracture peak load of the deep shale specimen also increased.However,the maximum displacement decreased to different extents.With the increase in temperature,the effective fracture toughness of the deep shale gradually decreased.Also,the maximum displacement decreased.Under different loading rates,the deformation of the prefabricated cracks showed a nonlinear slow growthelinear growth trend.The slope of the linear growth stage increased with the increase in loading rate.In addition,as the loading rate increased,an increase in tension failure and a decrease in shear failure were observed.Moreover,the control chart showing the relationship between tension and the shear failure under different temperatures and loading rates was determined.

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.

A New Clustering-Based Partitioning Method for VLSI Mixed-Mode Placement

An efficient partitioning algorithm for mixed-mode placement,extended-MFFC-based partitioning,is presented.It combines the bottom-up clustering and the top-down partitioning together.To do this,designers can not only cluster cells considering logic dependency but also partition them aiming at min-cut.Experimental results show that extended-MFFC-based partitioning performs well in mixed-mode placement with big pre-designed blocks.By comparison with the famous partitioning package HMETIS,this partitioning proves its remarkable function in mixed-mode placement.

Adsorption behaviors of avian immunoglobulins and purification of immunoglobulin Y from chicken serum with mixed-mode resins

The importance of immunoglobulin Y(IgY) as a specific antibody equivalent to mammalian immunoglobulin G(IgG) is well recognized. However, production of highly purified IgY is still difficult due to the lack of specific purification methods. In this study, adsorption behaviors of Ig Y on four mixed-mode resins with functional ligands of 4-mercatoethyl-pyridine(MEP), 2-mercapto-1-methyli-midazole(MMI), 5-aminobenzi-midazole(ABI) and tryptophan-5-aminobenzi-midazole(W-ABI) were evaluated. The results showed that high adsorption ratio were found at p H 6.0–7.0 with little adsorption under acidic conditions. The resin with ABI ligand was then used to separate IgY from immunized chicken serum. An efficient process with Ig Y purity of 95% and recovery of 90% was developed after optimization of loading and elution p H and injection volume. The biological activity of the purified Ig Y was fully maintained. These results indicated that mixed-mode chromatography with specially-designed ligands has great potential for the separation of Ig Y from crude feedstock.

Preparation and evaluation of mixed-mode resins with tryptophan analogues as functional ligands for human serum albumin separation

Five tryptophan analogues with a hydrophobic indole ring and an amino group on each molecule were used as functional ligands of mixed-mode resins for human serum albumin(HSA) purification. Their adsorption performance was evaluated and the effects of p H and salt addition on HSA adsorption were studied. The resins prepared showed typical p H-dependent adsorption and the highest adsorption capacity and affinity were found at pH 5.0for all the resins tested. The saturated adsorption capacity was 138.02 mg·g^(-1)with the tryptaminefunctionalized resin, which significantly decreased at p H below 4.0 due to electrostatic repulsion between ligands and HSA. Moreover, the addition of Na Cl or(NH_4)_2SO_4in media reduced HSA adsorption capacity, although the two salts showed different affecting profiles. The tryptamine-functionalized resin showed the best salt-tolerant performance, and its high adsorption capacity was maintained under high salt concentrations. In addition, the five resins prepared showed good adsorption selectivity for recombinant HSA from Pichia pastoris broth. Molecular docking results between tryptamine and HSA indicated that tryptamine was favorable to bind on Site II(indole-binding site) of HSA.

Development of X-FEM methodology and study on mixed-mode crack propagation

The extended finite element method（X-FEM） is a novel numerical methodology with a great potential for using in multi-scale computation and multi-phase coupling problems. The algorithm is discussed and a program is developed based on X-FEM for simulating mixed-mode crack propagation. The maximum circumferential stress criterion and interaction integral are deduced. Some numerical results are compared with the experimental data to prove the capability and efficiency of the algorithm and the program. Numerical analyses of sub-interfacial crack growth in bi-materials give a clear description of the effiect on fracture made by interface and loading condition.

Stabilization of Networked Control Systems Using a Mixed-Mode Based Switched Delay System Method

The phenomenon of mixed-mode is one of the most important characteristics of switched delay systems. If a networked control system(NCS) with network induced delays and packet dropouts(NIDs & PDs) is recast as a switched delay system, it is imperative to consider the effects of mixed-modes in the stability analysis for an NCS. In this paper, with the help of the interpolatory quadrature formula and the average dwell time method, stabilization of NCSs using a mixed-mode based switched delay system method is investigated based on a novel constructed Lyapunov-Krasovskii functional. With the Finsler's lemma, new exponential stabilizability conditions with less conservativeness are given for the NCS. Finally, an illustrative example is provided to verify the effectiveness of the developed results.

FATIGUE GROWTH MODELING OF MIXED-MODE CRACK IN PLANE ELASTIC MEDIA

This paper presents an extension of a displacement discontinuity method with cracktip elements （a boundary element method） proposed by the author for fatigue crack growth analysis in plane elastic media under mixed-mode conditions. The boundary element method consists of the non-singular displacement discontinuity elements presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or right crack-tip element is placed locally at the corresponding left or right crack tip on top of the non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the maximum circumferential stress criterion. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the numerical approach. Crack growth is modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characteristics of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the fatigue growth process of cracks emanating from a circular hole in a plane elastic plate is simulated using the numerical simulation approach.

MESHLESS METHOD FOR 2D MIXED-MODE CRACK PROPAGATION BASED ON VORONOI CELL

A meshless method integrated with linear elastic fracture mechanics(LEFM)is presented for 2D mixed-mode crack propagation analysis.The domain is divided automatically into sub-domains based on Voronoi cells,which are used for quadrature for the potential energy. The continuous crack propagation is simulated with an incremental crack-extension method which assumes a piecewise linear discretization of the unknown crack path.For each increment of the crack extension,the meshless method is applied to carry out a stress analysis of the cracked structure.The J-integral,which can be decomposed into mode Ⅰ and mode Ⅱ for mixed-mode crack,is used for the evaluation of the stress intensity factors(SIFs).The crack-propagation direction,predicted on an incremental basis, is computed by a criterion defined in terms of the SIFs. The flowchart of the proposed procedure is presented and two numerical problems are analyzed with this method.The meshless results agree well with the experimental ones,which validates the accuracy and efficiency of the method.

A 90° mixed-mode twisted nematic liquid-crystal-on-silicon with an insulating protrusion structure

A 90°mixed-mode twisted nematic liquid-crystal-on-silicon(90°-MTN LCoS) with protrusion located between the adjacent pixels is proposed to reduce the effect of fringing field. The influence of the protrusion with different widths from0.5 μm to 0.9 μm and different heights from 0.3 μm to 0.7 μm is investigated. The results demonstrate that the invalid pixel region width can be reduced by 31.5% via using the protrusion with the suitable width and height compared with no protrusion case, which provides a higher display quality, such as the higher reflectance and contrast ratio.

Characteristic Tensor for Evaluation of Singular Stress Field Under Mixed-Mode Loadings

A characteristic tensor is defined using stress tensor averaged in a small circular domain at the crack tip and multiplied by the root of domain radius.It possesses the original stress tensor characteristics and has a simple relationship with conventional fracture-mechanics parameters.Therefore,it can be used to estimate stress intensity factors(SIFs)for cracks of arbitrary shape subjected to multiaxial stress loads.A characteristic tensor can also be used to estimate SIFs for kinked cracks.This study examines the relation between a characteristic tensor and SIFs to demonstrate the correlation between the characteristic tensor and fracture-mechanics parameters.Consequently,a single straight crack and a kinked crack of finite length existing in a twodimensional,infinite isotropic elastic body in a plane stress state,were considered to investigate the properties of the characteristic tensor under mixed-mode loadings.To demonstrate the practical utility of the characteristic tensor,the stress distribution obtained through finite element analysis(FEA)was used to estimate mixed-mode SIFs,and the values of estimated SIFs were compared with those obtained using an analytical solution.Results demonstrate that SIFs estimated under mixed-mode loadings exhibit a good agreement with the analytical values.This indicates that the proposed characteristictensor-based approach is effective in extracting features of singular stress fields at crack tips,and can be employed to estimate values of fracture-mechanics parameters,such as SIFs.Owing to its simplicity,the proposed approach can be easily incorporated in commercial FE codes for practical applications to simulate the crack-growth problem under both static and dynamic loading scenarios.The excellent applicability of the characteristic tensor greatly contributes to efficiency of the design process in industries.

Autonomous Development Strategies for College English Teachers in a Mixed-mode Learning Community

At present,the improvement of the quality of higher education and the improvement of the professional ability and comprehensive quality of college students all depend on the education level and discipline professional ability of teachers in colleges and universities.In the new era of educational reform and development,university teachers also need to continuously learn and progress in order to adapt to changes in the educational environment and the update of the knowledge system.Nowadays,the construction and development of the mixed-mode learning community environment is becoming one of the effective ways for college teachers to improve their learning.From the perspective of the significance of autonomous development of college English teachers in the context of mixed-mode learning community,the current situation and other issues,this article expounds the strategies that promote the autonomous development of college English teachers in the context of mixed-mode learning community to improve the professional quality of college English teachers.

A Mixed-Mode Biquad Employing OTAs and Grounded Capacitors

This paper introduces a mixed-mode biquad employing OTAs （operational trans-conductance amplifiers） and grounded capacitors. The circuit can perform mixed-mode operation by selecting the input and output terminals. Additionally, the circuit enables low-pass, band-pass, high-pass, band-stop and all-pass transfer functions suitably choosing the input terminals. The circuit parameters ω0 and Q can be tuned orthogonally through adjusting the trans-conductance gains of the OTAs. The biquad enjoys very low sensitivities with respect to the circuit active and passive components. The achievement examples are given together with simulation results by PSPICE.

A DVCC-Based Mixed-Mode Biquadratic Circuit

This paper introduces a mixed-mode biquadratic circuit employing DVCCs （differential voltage current conveyors） and grounded passive components. The biquadratic circuit can perform mixed-mode operation selecting the input and output terminals. And the circuit enables LP （low-pass）, BP （band-pass）, HP （high-pass）, BS （band-stop） and AP （all-pass） transfer functions by suitably choosing the input terminals. The circuit parameters o30 and Q can be tuned orthogonally through adjusting the passive components. The biquadratic circuit enjoys very low sensitivities with respect to the circuit components. The achievement example is given together with simulation results by PSPICE.

The Mixed-Mode Fatigue Crack Propagation Model of Piezoelectric Materials Under Electric Fatigue Loading by the J_(k)-Integral

The electric fatigue load has a significant effect on the crack propagation behavior and failure life of piezoelectric materials and devices. In this paper, an electrical mixed-mode fatigue crack propagation model for piezoelectric materials is proposed based on the piezoelectric J_(k)-integral theory. The crack initiation, propagation, and life prediction criteria of piezoelectric materials under electric fatigue loading are given by this model, and the finite element simulation model is established to study the electrical mixed-mode crack propagation behavior of piezoelectric structures. Meanwhile, the electrical mixed-mode fatigue crack propagation model is applied to the fatigue crack propagation behavior of a piezoelectric typical defective structure, the crack–hole interference model. The mixed-mode crack propagation, fatigue life, and the interference behavior between the crack and hole at various hole locations of the crack–hole interference model are well recognized by this model. The crack propagation behavior under different electrical load intensities is also considered. The results show that the hole in front of the crack tip inhibits crack propagation to a certain extent, and the strength of electrical load affects the fatigue life of piezoelectric materials and structures. Therefore, the proposed electrical mixed-mode fatigue crack propagation model provides a reference for predicting the mixed-mode fatigue crack propagation behavior and fatigue life of piezoelectric structures under electric fatigue loading.

Experimental Study on Mixed-Mode(Ⅰ–Ⅱ)Fracture Toughness of Freshwater Ice

In recent years,the issue of aircraft icing has gained widespread recognition.The breaking and detachment of dynamic ice can pose a threat to flight safety.However,the shedding and fracture mechanisms of dynamic ice are unclear and cannot meet the engineering needs of ice-shedding hazard assessment.Therefore,studying the fracture toughness of ice bodies has extremely important practical significance.To address this issue,this article uses a centrally cracked Brazilian disk(CCBD)specimen to measure the pure modeⅠtoughness and pure modeⅡfracture toughness of freshwater ice at different loading rates.The mixed-mode(Ⅰ–Ⅱ)fracture characteristics of ice are discussed,and the experimental results are compared and analyzed with the theoretical values of the generalized maximum tangential stress(GMTS)criterion considering the influence of T-stress.The results indicated that as the loading rate increases,the pure modeⅠtoughness and pure modeⅡfracture toughness of freshwater ice decrease,and the fracture toughness of freshwater ice is more sensitive to the loading rate.Ⅰn terms of fracture criteria,the theoretical value of the ratio of pure modeⅡfracture toughness to pure modeⅠfracture toughness based on the GMTS criterion is in good agreement with the experimental value,while the theoretical value based on the maximum tangential stress(MTS)criterion deviates significantly from the experimental value,indicating that the GMTS criterion considering the influence of T-stress can better predict the experimental results.

Comparison of mixed-mode forced-convection solar dryer with and without stainless wire mesh in solar collector

A mixed-mode forced-convection solar dryer(MMFCSD)is a device that utilizes both direct and indirect solar energy.The solar col-lector,which stores thermal energy for indirect solar uses,is an essential component of the dryer.Unfortunately,the thermal effi-ciency of this device is generally low.In this study,a technique was employed to improve the heat transfer of the solar collector in a MMFCSD.The technique involved adjusting the air flow pattern into a swirling flow to disturb the thermal boundary layer on the absorber plate under forced convection by using stainless wire mesh.The experiment was conducted under actual conditions and bananas were used as the drying sample.The experimental results of the thermal efficiency of the solar collector(ƞ_(solar))and the drying efficiency(ƞ_(drying))are presented.The results indicated that the air outlet temperature andƞsolar of the solar collector with stainless wire mesh were higher than the case without stainless wire mesh,reaching a maximum temperature of 46.22°C and 37.97°C,and average ƞ_(solar) of 0.26±0.02 and 0.14±0.01,respectively.The MMFCSD with stainless wire mesh had a higher ƞ_(drying) than the case without stain-less wire mesh,with values of 0.048±0.004 and 0.039±0.003,respectively,resulting in an~23.07%increase.This was attributed to the air swirling flow through the stainless wire mesh and the heat accumulation in the drying chamber,which led to an increase in the drying chamber temperature from 54.03°C to 63.60°C,an increase in the effective moisture diffusivity from 7.28×10^(-7) to 1.19×10^(-6) m^(2)/s and a decrease in the drying time of 5 h 30 min.However,further research is needed to investigate the quality of the dried sam-ples and their economic value.

Simulation of 3D fracture propagation under I-II-III mixed-mode loading

Fracture propagation under mixed-mode loading conditions prevails in many natural geological processes and deep engineering projects,while the corresponding numerical simulation is very challenging in rock mechanics,especially in 3D cases.In most previous studies,the complexity of 3D fracture geometry was over-simplified,and model III loading was often not considered.In this study,we propose to use an efficient stress-based Sch€ollmann criterion combined with Displacement Discontinuity Method(DDM)to model 3D fracture propagation under arbitrary I+II+III mixed-mode loading conditions.A novel curve-smoothing algorithm is developed to smoothen the fracture front during propagation,which significantly enhances the model's ability in dealing with complex 3D fracture geometry.In particular,we adopt two different solution schemes,namely staggered and monolithic,to simulate mode I fracture propagation in the case of hydraulic fracturing.The accuracy,efficiency and convergency of the two solution schemes are compared in detail.Our research findings suggest that the degree of coupling between fracture aperture and fluid pressure in hydraulic fracturing lies somewhere between one-way and two-way,which favors the staggered solution scheme.To further test our new model,we provide three additional numerical examples associated with 3D fracture propagation under various mixed-mode loading conditions.Our model shows excellent performance in efficiently locating the new fracture front and reliably capturing the complex 3D fracture geometry.This study provides a generic algorithm to model high-fidelity 3D fracture propagation without simplifying fracture geometry or loading conditions,making it widely applicable to fracture-propagation-related problems.