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.

Closure Effect ofⅠ+ⅡMixed-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.

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.

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.

ANISOTROPIC PLASTIC STRESS FIELD AT A MIXED-MODE CRACK TIP UNDER PLANE AND ANTI-PLANE STRAIN

On condition that the perfectly plastic stress component at a crack tip was a function of plane polar angle only, and made use of equilibrium equations, anisotropic plastic stress-strain-rate relations, compatibility equations and Hill anisotropic plastic yield condition, the generally analytical expressions of the anisotropic plastic stress field at a mixed-mode crack tip under plane and anti-plane strain were obtained. Applying these expressions to the mixed-mode cracks, the analytical expressions of anisotropic plastic stress fields at the tips of mixed-mode I-III, II-III and I-II-III cracks were derived.

Mixed-mode fast-slow oscillations in the frequency switching Duffing system with a 1:n frequency ratio

Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work.The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable.As a result,a novel route to composite fast-slow dynamics is obtained.We find that,when presented with variable frequency ratios in a 1:n fashion,the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations,and hence the mixedmode fast-slow oscillations.Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not,these interesting oscillations are divided into two classes,both of which are investigated numerically.Our study shows that,when the frequency ratio n is increased from n=3,newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line,which is divided into smaller parts,showing sliding and downward crossing dynamical characteristics.This is the root cause of the clusters,showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations.Thus,a novel route to composite fast-slow dynamics by frequency switching is explained.Besides,the effects of the forcing on the mixed-mode fast-slow oscillations are explored.The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters.The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.

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 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.

Flexible Strain Sensors with Ultra‑High Sensitivity and Wide Range Enabled by Crack‑Modulated Electrical Pathways

This study presents a breakthrough in flexible strain sensor technology with the development of an ultrahigh sensitivity and wide-range sensor,addressing the critical challenge of reconciling sensitivity with measurement range.Inspired by the structure of bamboo slips,we introduce a novel approach that utilises liquid metal to modulate the electrical pathways within a cracked platinum fabric electrode.The resulting sensor demonstrates a gauge factor greater than 108 and a strain measurement capability exceeding 100%.The integration of patterned liquid metal enables customisable tuning of the sensor’s response,while the porous fabric structure ensures superior comfort and air permeability for the wearer.Our design not only optimises the sensor’s performance but also enhances the electrical stability that is essential for practical applications.Through systematic investigation,we reveal the intrinsic mechanisms governing the sensor’s response,offering valuable insights for the design of wearable strain sensors.The sensor’s exceptional performance across a spectrum of applications,from micro-strain to large-strain detection,highlights its potential for a wide range of real-world uses,demonstrating a significant advancement in the field of flexible electronics.

Elastic-viscoplastic field at mixed-mode interface crack-tip under compression and shear

For a compression-shear mixed mode interface crack, it is difficult to solve the stress and strain fields considering the material viscosity, the crack-tip singularity, the frictional effect, and the mixed loading level. In this paper, a mechanical model of the dynamic propagation interface crack for the compression-shear mixed mode is proposed using an elastic-viscoplastic constitutive model. The governing equations of propagation crack interface at the crack-tip are given. The numerical analysis is performed for the interface crack of the compression-shear mixed mode by introducing a displacement function and some boundary conditions. The distributed regularities of stress field of the interface crack-tip are discussed with several special parameters. The final results show that the viscosity effect and the frictional contact effect on the crack surface and the mixed-load parameter are important factors in studying the mixed mode interface crack- tip fields. These fields are controlled by the viscosity coefficient, the Mach number, and the singularity exponent.

Novel Methodologies for Preventing Crack Propagation in Steel Gas Pipelines Considering the Temperature Effect

Using the software ANSYS-19.2/Explicit Dynamics,this study performedfinite-element modeling of the large-diameter steel pipeline cross-section for the Beineu-Bozoy-Shymkent gas pipeline with a non-through straight crack,strengthened by steel wire wrapping.The effects of the thread tensile force of the steel winding in the form of single rings at the crack edges and the wires with different winding diameters and pitches were also studied.The results showed that the strengthening was preferably executed at a minimum value of the thread tensile force,which was 6.4%more effective than that at its maximum value.The analysis of the influence of the winding dia-meters showed that the equivalent stresses increased by 32%from the beginning of the crack growth until the wire broke.The increment in winding diameter decelerated the disclosure of the edge crack and reduced its length by 8.2%.The analysis of the influence of the winding pitch showed that decreasing the distance between the winding turns also led to a 33.6%reduction in the length of the straight crack and a 7.9%reduction in the maximum stres-ses on the strengthened pipeline cross-section.The analysis of the temperature effect on the pipeline material,within a range from-40℃ to+50℃,resulted in a crack length change of up to 5.8%.As the temperature dropped,the crack length decreased.Within such a temperature range,the maximum stresses were observed along the cen-tral area of the crack,which were equal to 413 MPa at+50℃ and 440 MPa at-40℃.The results also showed that the presence of the steel winding in the pipeline significantly reduced the length of crack propagation up to 8.4 times,depending on the temperature effect and design parameters of prestressing.This work integrated the existing methods for crack localization along steel gas pipelines.

Crack Propagation Path in Two-Directionally Graded Composites Subjected to Mixed-Mode Ⅰ+Ⅱ Loading

Crack propagation path in two-directionally graded composites was investigated by the finite element method.A graded extended finite element method(XFEM)was employed to calculate displacement and stress fields in cracked graded structures.And a post-processing subroutine of interaction energy integral was implemented to extract the mixed-mode stress intensity factors(SIFs).The maximum hoop stress(MHS)criterion was adopted to predict crack growth direction based on the assumption of local homogenization of asymptotic crack-tip fields in graded materials.Effects of material nonhomogeneous parameters on crack propagation paths were also discussed in detail.It is shown that the present method can provide relatively accurate predictions of crack paths in two-directionally graded composites.Crack propagates in the decreasing direction of effective Young′s modulus.The shape and steepness of property gradient perpendicular to the crack surface have great influences on crack paths.Through redesigning material property reasonably,crack growth in graded material can be changed to improve mechanical behaviours of cracked structures.

Effect of surface crack on the bearing capacity of strip footing placed on rock mass

In hilly regions,the existence of surface cracks in rock mass induces a potential threat to structural stability.Thus,the present research aims to explore the impact of surface cracks on the loadbearing capacity of strip footing placed on the rock mass.By taking into account the various boundary constraints across the surface of crack edges,the study investigates the presence of two categories of surface cracks,namely(1)separated crack,and(2)fine crack.The lower bound limit analysis is employed in conjunction with the finite element method(LBFELA)to conduct the numerical analysis.In order to evaluate rock mass yielding,the power conic programming(PCP)method is utilized to implement the generalized Hoek-Brown(GHB)failure criterion.The stability of the strip footing is analyzed by determining the bearing capacity factor(Nσγ),which is presented in the form of design charts by varying the strength parameters of rock,including the Geological Strength Index(GSI),Hoek-Brown material parameter(mi),Disturbance factor(D),and Normalised Uniaxial Compressive Strength(σci/γB),whereγis the unit weight of rock mass,and B is the width of strip footing.The study also investigates the impact of cracks on strip footings,considering different positions of the crack(LC)and depths of the crack(DC).The results demonstrate that the influence of the fine crack is only noticeable until the LC/B ratio reaches 6.However,for the separated crack,its impact remains significant even when the LC/B ratio exceeds 16.The appearance of fine crack at the edge of the footing results in a decrease in the magnitude Nσγof up to 45%,indicating a substantial reduction in the stability of the footing.The failure patterns are presented and discussed in detail for various cases in this study to examine the effect of surface cracks on the strip footing and to address the extent of the plastic collapse.

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.

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.

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.

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.