An extended J-integral for evaluating fluid-driven cracks in hydraulic fracturing

J-integral has served as a powerful tool in characterizing crack tip status. The main feature, i.e. path- independence, makes it one of the foremost fracture parameters. In order to remain the path- independence for fluid-driven cracks, J-integral is revised. In this paper, we present an extended J-in- tegral explicitly for fluid-driven cracks, e.g. hydraulically induced fractures in petroleum reservoirs, for three-dimensional （3D） problems. Particularly, point-wise 3D extended J-integral is proposed to char- acterize the state of a point along crack front. Besides, applications of the extended J-integral to porous media and thermally induced stress conditions are explored. Numerical results show that the extended J- integral is indeed path-independent, and they are in good agreement with those of equivalent domain integral under linear elastic and elastoplastic conditions. In addition, two distance-independent circular integrals in the K-dominance zone are established, which can be used to calculate the stress intensity factor （SIF）.

CRACK TIP FIELD AND J-INTEGRAL WITH STRAIN GRADIENT EFFECT

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani.The theory retains the essential structure of the incremental version of the conventional J_2 deformation theory.No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required.The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus.The strain gradient measures are included into the tangent modulus as internal parameters.Therefore the boundary value problem is the same as that in the conventional theory.Two typical crack problems are studied:(a)the crack tip field under the small scale yielding condition induced by a linear elastic mode-I K-field and(b)the complete field for a compact tension specimen.The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory.The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it.Consequently,the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.

J-Integral of Interfacial Crack Between Metal-Base Ceramic Coating and Steel

The definition of J-integral of interfacial crack was introduced. The three-point bending tests were can＇ied out to obtain the critical loading values when the interfacial crack initiation occurred between coatings and substrates. The finite element analysis （FEA） was adopted to analyze the stress distribution in the specimens and compute the J-integral of the interracial crack between LX88A coating and Chinese Q345 steel. The results showed that the average value of critical J-integral is 0.70 N/m, which can be taken as the fracture parameter to evaluate the interracial fracture behavior for the three-point bending specimens of LX88A coating/Q345 steel system.

PATH-INDEPENDENT J-INTEGRAL AND ITS DUAL FORM IN ELASTIC-PLASTIC SOLIDS WITH FINITE DISPLACEMENTS

In this paper, based on energy variational principles of elastic-plastic solids, the path-independent J-integral and its dual form in elastic-plastic solids with finite displacements are presented. Whose testification is given there after.

Theory and Calculation of the J-Integral for Coupled Chemo-Mechanical Fracture Mechanics

In this paper,by introducing a chemical field,the J-integral formulation is presented for the chemo-mechanical coupled medium based on the laws of thermodynamics.A finite element implementation of the J-integral was performed to study the mode I chemo-mechanical coupled fracture problem.For derivation of the coupled J-integral,the equivalent domain integral(EDI)method was applied to obtain the mode I J-integral,with expression of the area integrals based on constitutive relationships of a linear elastic small deformation for chemo-mechanical coupling,instead of the finite deformation problem.A finite element procedure is developed to compute the mode I J-integral,and numerical simulation of the y-direction stress field is studied by a subroutine UEL(User defined element)developed in ABAQUS software.Accuracy of the numerical results obtained using the mode I J-integral was verified by comparing them to a well-established model based on linear elastic fracture mechanics(LEFM).Furthermore,a numerical example was presented to illustrate path-independence of the formulated J-integral for a chemo-mechanical coupled specimen under different boundary conditions,showing a high accuracy and reliability of the present method.The variation laws of J-integral and the y-direction stress field with external chemical,mechanical loading and time are revealed.The J-integral value increases with larger external concentration loading in the same integral domain.The extent of diffusion is much greater with larger concentration,which leads to a stronger coupling effect due to the chemical field.This work provides new insights into the fracture mechanics for the chemo-mechanical coupled medium.

A NEW CYCLIC J-INTEGRAL FOR LOW-CYCLE FATIGUE CRACK GROWTH

The constitutive equation under the low-cycle fatigue （LCF） was discussed, and a two-dimensional （2-D） model for simulating fatigue crack extension was put forward in order to propose a new cyclic J-integral. The definition, primary characteristics, physical interpretations and numerical evaluation of the new parameter were investigated in detail. Moreover, the new cyclic J-integral for LCF behaviors was validated by the compact tension （CT） specimens. Results show that the calculated values of the new parameter can correlate well with LCF crack growth rate, during constant-amplitude loading. In addition, the phenomenon of fatigue retardation was explained through the viewpoint of energy based on the concept of the new parameter.

Study on J-integral with SENB specimens in a condenser material

A series of tests was performed with three-point single-edge-notched-bend (SENB) specimens in a condenser material (Titanium alloy). Results show that the J-integral values of welded joint and HAZ are obviously smaller than those of the base metal. It signifies that the welding process can result in a reduced toughness of Titanium alloy and the effect of crack orientation on toughness value is not negligible for engineering applications. Besides, the J-integral values of L-T direction specimens are much higher than those of L-S ones. The J-integral values of rolled ring are:JC-R>JC-L>JL-R.

STUDY ON DYNAMIC J-INTEGRAL OF MECHANICAL HETEROGENEOUS WELDED JOINT

Welded joint is a mechanical heterogeneous body, and mechanical heterogeneity has great effect on dynamic fracture behaviour of welded joints. In the present investigation, dynamic response curve and dynamic J-integral of practical undermatched welded joint and whole base and whole weld three-point-bend (TPB) models containing longitudinal crack are com- puted. Dynamic J-integral is evaluated using virtual crack extension (VCE) method and the computation is performed using MARC finite element code. Because of the effect of inertia, dynamic load response curve of computed model waves periodically. Dynamic J-integral evaluated by VCE method is path independent. The effect of inertia has little influence on dynamic J-integral curve. The value of dynamic J-integral of undermatched welded joint is lower than that of whole base metal and higher than that of whole weld metal. The results establish the foundation of safety evaluation for dynamic loaded welded structures.

ON J-INTEGRALS IN THE PLANE FRACTURE OF COMPOSITE MATERIALS

By using a complex function method in this paper, the complex form of J-integral of mixed mode crack tip for unidirectional plate of linear-elastic orthotropic composites is obtained first by substituting crack tip stresses and displacements into general formula of J-integral. And then, the path-independence of this J-integral is proved. Finally, the computing formula of this J-integral is derived. As special examples, the complex forms, path-independence and computing formulae of J-integrals of mode I and mode II crack tips for unidirectional plate of linear-elastic orthotropic composites are given.

A J-INTEGRAL ANALYSIS OF CRACKS IN OVERMATCHED WELDED JOINTS

The effects of material inhomogeneity and weld width of the overmatched welded joints with a longitudinal crack or flaw upon the J-interal,a characterizing parameter for elastoplastic fracture,have been studied.The overmatched weld is idealized to be a parallelly cracked and sandwiched hard layer,the yieht stress of which is higher than that of the base metal.Numeri- cal solutions were obtained by using elastoplastic finite element method under the assumption of plane stress.The resuits show that the values of J-integral are remarkably affected by the material inhomogeneity ant the weld width especially when the nominal stress or nominal strain is comparatively high,which,therfore,should be taken into account when evaluating the crack driving force for an overmatched joint.

THE FULLY PLASTIC J-INTEGRAL OF CORNER CRACKS AT A HOLE IN A PLATE

The line-spring model of surface cracks is applied to the fully plastic analysis of corner cracks at a hole in a plate. The generalized fully plastic constitutive relations and the fully plastic J-integral, as well as its coefficients in polynomial expressions are given. The model obtained is incorporated into a finite element program. The corner cracks at a hole in a plate subjected pure tensions are calculated by the present model. The fully plastic J-integral is then estimated. The results obtained show that the line-spring model is effective for the analysis of corner cracks. The influence of the crack depth and the hardening exponent on the fully plastic J-integral is also discussed.

RELATIONSHIP BETWEEN J-INTEGRAL AND FRACTURE SURFACE AVERAGE PROFILE

To investigate the causes that led to the formation of cracks in materials, a novel method that only considered the fracture surfaces for determining the fracture toughness parameters of J-integral for plain strain was proposed The principle of the fracture-surface topography analysis （FRASTA） was used. In FRASTA, the fracture surfaces were scanned by laser microscope and the elevation data was recorded for analysis. The relationship between J-integral and fracture surface average profile for plain strain was deduced. It was also verified that the J-integral determined by the novel method and by the compliance method matches each other well.

STUDY ON THE RELATIONSHIP BETWEEN J-INTEGRAL AND COD

J-integral arid crack opening displacement 8 （COD） were important parameters for characterization of fractures in engineering materials. The relationship between J-integral arid COD has been investigated for a long time and was generally represented as J = m σy, δ, where σy is the yield strength arid m is a function of specimen geometry arid material properties. To determine the value of m, extensive studies and experiments have been performed. The method that used the fracture-surface topography analysis （ FRASTA ） for determining J-integral from fracture surfaces of materials was introduced. On the basis of the relationship between COD arid fracture surface average profile, the relationship between J-integral arid COD was deduced arid compared with the generally used equation. The method was experimentally confirmed to be able to provide a new way to determine the relationship between J-integral and COD.

A STUDY OF J-INTEGRAL OF THE ORTHOTROPIC COMPOSITE MATERIAL

The relation between J-integral near model I crack tip in the orthotropic plateand displacement derivative is derived in this paper. Meanwhile,Ihe relation betweenstress intensity factor K_1 and displacement is also given in this paper.With stickingfilm moire interferometry method, the three-point bending beam is tested,thus thevalues of J-integral and K_1 can be obtained.from the displacement field,and then thetruth of relation formula between J-integral and K_1 in the orthotrtopic compositematerials is experimentally verified.

Review:J-integral resistance curve testing and evaluation

In this paper a critical review is presented of the history and current state of the art of J-integral resistance curve testing and experimental evaluation methods in conjunction with a discussion of the development of the plane strain fracture toughness test standard ASTM E1820 developed by American Society for Testing and Materials (ASTM). Early research efforts on this topic are reviewed first. These include the J-integral concept, experimental estimates of the J-integral for stationary cracks, load line displacement (LLD) and crack mouth opening displacement (CMOD) based η factor equations, different formulations of J-integral incremental equations for growing cracks, crack growth corrected J-R curve determination, and experimental test methods. Recent developments in J-R curve testing and evaluation are then described, with emphasis on accurate J-integral incremental equations, a normalization method, a modified basic method, a CMOD direct method with use of incremental equations, relationships of plastic geometry factors, constraint-dependent J-R curve testing and correction approaches. An overview of the present fracture toughness test standard ASTM E1820-08a is then presented. The review shows that after more than 40 years of investigation and development, the J-integral resistance curve test methods in ASTM E1820 have become simpler, more cost-effective and more accurate.

Study on the J-Integral and Failure Assessment Line of Electron Beam Welded Thick TC4-DT Alloy Joints

The effects of strength mis-matching on J-integral and failure assessment line (FAL) of electron beam welded thick TC4-DT titanium alloy joints have been investigated.Elastic and elastic-plastic finite elements (FE) calculations on centre crack tensile (CCT) specimens were made,and the condition of 20% strength mis-matching was considered.The results indicate that the limit load of the joint can be substituted by that of the base metal.The error of substitution is less than 5%.The failure assessment lines for both over-matching and under-matching joints are increasingly close to that of the base metal,with the ratio of weld width to specimen width (H/W) decreasing.Moreover,the failure assessment lines of the joints can be devised using the stress-strain curves of the base metal,which will be convenient for the practical application of engineering

A CATASTROPHE ANALYSIS ON THE STABILITY OF THE CRACK GROWTH IN THREE-POINT BENDING SPECIMENS

This paper presents an attempt at the application of catastrophe theory to the stability analysis of J-controlled crack growth in three-point bending specimens. By introducing the solutions of J-integral in the completely yielding state for the ideal plastic material, the critical condition of losing stability for the crack propagation in the specimen is obtained, based on the cusp catastrophe theory. The process of the crack growth from geometrical sense is described.

CONSERVATION LAW AND APPLICATION OF J- INTEGRAL IN MULTI-MATERIALS

The conservation law of J-integral in two-media with a crack paralleling to the interface of the two media was firstly proved by analytical and numerical finite element method. Then a schedule model was established that an interface crack is inserted in four media. According to the J-integral conservation law on multi-media, the energy release ratio of Ⅰ-type crack was considered to be conservation when the middle medium layers are very thin. And the conservation law was also convinced by numerical method. By means of the dimension analysis on the model, the asymptotic results and formula calculating the energy release ratio and complex stress intensity factor are presented.

Mechanical characterization of sisal reinforced cement mortar

This work aims at evaluating the mechanical behavior of sisal fiber reinforced cement mortar. The composite material was produced from a mixture of sand, cement, and water. Sisal fibers were added to the mixture in different lengths. Mechanical characterization of both the composite and the plain mortar was carried out using three point bend, compression, and impact tests. Specimens containing notches of different root radii were loaded in three point bending in an effort to determine the effect of the fibers on the fracture toughness of the material. The results obtained indicate that, while fiber reinforcement leads to a decrease in compressive strength, J-integral calculations at maximum load for the different notch root radii have indicated, particularly for the case of long fibers, a significant superiority of the reinforced material in comparison with the plain cement mortar, in consistence with the impact test data.

The energy release rate for hygrothermal coupling elastic materials

In the fracture problems of hydrophilic elastic materials under coupling effects of heat conduction, moisture diffusion and mechanical deformation, the conventional J-integral is no longer path independent. The value of J is unequal to the energy release rate in hygrothermal coupling cases. In the present paper, we derived a general form of the energy release rate for hygrothermal fracture problems of the hydrophilic elastic materials on the basis of energy balance equation in cracked areas. By introducing the constitutive relations and the essential equations of irreversible thermodynamics, a specific expression of the energy release rate was obtained, and the expression can be reformmulated as path independent integrals, which is equivalent to the energy release rate of the fracture body. The path independence of the integrals is then verified numerically.