Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

CLOSED-FORM WEIGHT FUNCTION FOR EDGE CRACK PROBLEMS

An accurate crack surface opening displacement formula for edge cracks in finite bodies has been derived,and was subsequently used to develop wide-range closed-form weight functions for edge cracks. The analytical weight function is general and can be easily used to determine accurate stress intensity factors and crack surface opening displacements for arbitrarily loaded edge cracks with high efficiency.Examples have been given for problems of edge crack(s)with a number of typical load conditions for illustration.

Strip-coalesced interior zone model for two unequal collinear cracks weakening piezoelectric media

In this paper, a mathematical strip-saturation model is proposed for a poled transversely isotropic piezoelectric plate weakened by two impermeable unequal-collinear hairline straight cracks. Remotely applied in-plane unidirectional electromechanical loads open the cracks in mode-I such that the saturation zone developed at the interior tips of cracks gets coalesced. The developed saturation zones are arrested by distributing over their rims in-plane normal cohesive electrical displacement. The problem is solved using the Stroh formalism and the complex variable technique. The expressions are derived for the stress intensity factors （SIFs）, the lengths of the saturation zones developed, the crack opening displacement （COD）, and the energy release rate. An illustrative numerical case study is presented for the poled PZT-5H ceramic to investigate the effect of prescribed electromechanical loads on parameters affecting crack arrest. Also, the effect of different lengths of cracks on the SIFs and the local energy release rate （LERR） has been studied. The results obtained are graphically presented and analyzed.

SLOW PROPAGATION OF FATIGUE CRACK NEAR THRESHOLD

The effect of pearlite interlamellar spacing(d)from 0.31 to 0.77 μm of T8 steel on fatigue crack propagation near threshold has been investigated at different stress ratios(R).The closure stress intensity factor(K_(cl)),the threshold value(ΔK_(th))and crack tip opening dis- placement range(ΔCOD)were measured.The results show that pearlite interlamellar spac- ing in this range has no effect on K_(cl),but has effect on the threshold value in low stress ratio. The threshold value is higher for materials with larger interlamellar spacing.No matter what the stress ratio is,during the propagation near threshold,the propagation rate will be the same,provieded the crack tip opening displacement range is the same. Metallurgical examinations were also carried out to show that the crack propagation path is sensitive to mierostructure.The crack always propagates along either boundary of pearlite colony or ferrite lamellar in pearlite.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

The concept of eigen crack opening displacement （COD） can be defined as the COD of a crack in infinite plate under the tractions acting on the crack surface. By introducing this concept, the eigen COD formulation of boundary integral equation is proposed in this paper, together with the solution procedures for multiple crack problems in plane elasticity. With the proposed approach, the multiple crack problems can be solved with the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix as that in the numerical Green’s function （NGF） approach but without the trouble to determine the complementary solutions since the standard boundary element discretization on the crack surface is no longer required with the proposed approach. Some numerical examples computing the stress intensity factors are presented and compared with those in literature to show the accuracy and the effectiveness of the proposed approach.

THE ANALYSES OF THE THREE-DIMENSIONAL STRESS STRUCTURE NEAR THE CRACK TIP OF MODE I CT SPECIMENS IN ELASTICPLASTIC STATE(Ⅰ)—THE ANALYSES OF CONSTRAINT PARAMETERS AND FRACTURE PARAMETERS

In the present paper, three dimensional analyses of some general constraint parameters and fracture parameters near the crack tip of Mode I CT specimens in two different thicknesses are carried out by employing ADINA program. The results reveal that the constraints along the thickness direction are obviously separated into two parts: the keeping similar high constraint field (Z1) and rapid reducing constraints one(Z2). The two fields are experimentally confiremed to correspond to the smooth region and the shear lip on the fracture face respectively. So the three dimensional stress structure of Mode I specimens can be derived through discussing the two fields respectively. The distribution of the Crack Tip Opening Displacement (CTOD) along the thickness direction and the three dimensional distribution of the void growth ratio (Vg) near the crack tip are also obtained. The two fracture parameters are in similar trends along the thickness direction, and both of them can reflect the effect of thickness and that of the loading level to a certain degree.

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 Straightforward Direct Traction Boundary Integral Method for Two-Dimensional Crack Problems Simulation of Linear Elastic Materials

This paper presents a direct traction boundary integral equation method(DTBIEM)for two-dimensional crack problems of materials.The traction boundary integral equation was collocated on both the external boundary and either side of the crack surfaces.The displacements and tractions were used as unknowns on the external boundary,while the relative crack opening displacement(RCOD)was chosen as unknowns on either side of crack surfaces to keep the single-domain merit.Only one side of the crack surfaces was concerned and needed to be discretized,thus the proposed method resulted in a smaller system of algebraic equations compared with the dual boundary element method(DBEM).A new set of crack-tip shape functions was constructed to represent the strain field singularity exactly,and the SIFs were evaluated by the extrapolation of the RCOD.Numerical examples for both straight and curved cracks are given to validate the accuracy and efficiency of the presented method.

Study on fracture characteristic of welded high-density polyethylene pipe

Crack opening displacement(COD) was applied to characterize the fracture initiation of the tough high density polyethylene. Normal single side notched three point bend specimens and silica rubber replica techniques were used to study the characteristic COD of high density polyethylene pipe and its butt fusion joints including the weld fusion zone and heat affected zone at different temperature from -78 ℃ to 20 ℃ . Testing results show that the characteristic COD appears to depend on the structural features that are determined by welding process and the testing temperature. As the temperature is lowered, the characteristic COD of all zones studied decreases. Because the welding process significantly changes some structural feature of the material, characteristic COD of the weld fusion zone is the smallest one among those of the three zones. The results can be used for the engineering design and failure analysis of HDPE pipe.

Engineering critical assessment on welding flaws of the X65 UOE girth weld based on CRACKWISE

Focused on the girth weld of a φ762 x 28.6 mm （X65） offshore UOE pipe, the fracture toughness properties at different weld joint positions were tested. Meanwhile,the largest size of different cracks under service condition was calculated by using CRACKWISE. The service life of the UOE pipe with postulated crack-like flaws was calculated by considering various fatigue factors, such as vibrations caused by ocean current and fluctuations of inner pressure. The assessment process and results can be used to direct the repair of weld flaws laid in pipes or to assess the reliability of in-serving pipes.

Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization;2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix;3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

APPLICATION OF SINTAP TO THE SAFETY ASSESSMENT OF X65 PIPELINE STEEL

The recently developed European flaw assessment procedure, structural integrity assessment procedure（SiNTAP） is applied to assessment for welded joints of the API 5L X65 pipeline steel with an assumed embedded flaw and surface flaw at the weld toe. As one of the basic input data, fracture toughness crack tip opening displacement （CTOD） tests are conducted at 0℃ and performed according to the requirements of the standard of BS7448. For the heat affected zone （HAZ） specimens, the microstructure observation is performed to insure that the tip of the crack is located in the coarse grain zone. The result explains the dispersity of the test values. In structural integrity assessment procedure-fracture assessment diagram（SINTAP FAD） method, the failure curves of welded joints at level 1 and 3 are derived from the tensile test results. The results of the assessment show that all assessment points are located within the failure lines of analysis level 1 and 3. So the welded joint of the pipeline is safe. This study laid the foundation of application of SINTAP to pipeline structure assessment.

PERIODICAL INTERFACIAL CRACKS IN ANISOTROPIC ELASTOPLASTIC MEDIA

By using Fourier transformation the boundary problem of periodical interfacial cracks in anisotropic elastoplastic bimaterial was transformed into a set of dual integral equations and then it was further reduced by means of definite integral transformation into a group of singular equations. Closed form of its solution was obtained and three corresponding problems of isotropic bimaterial, of a single anisotropic material and of a bimaterial of isotropy- anisotropy were treated as the specific cases. The plastic zone length of the crack tip and crack openning displacement ( COD) decline as the smaller yield limit of the two bonded materials rises, and they were also determined by crack length and the space between two neighboring cracks . In addition , COD also relates it with moduli of the materials .