Computation of Thermal Fracture Parameters for Inclined Cracks in Functionally Graded Materials Using J k -Integral

This article describes the formulation and implementation of the J _k -integral for the analysis of inclined cracks located in functionally graded materials (FGMs) that are subjected to thermal stresses. The generalized definition of the J _k -integral over a vanishingly small curve at the tip of an inclined crack is converted to a domain independent form that consists of area and line integrals defined over finite domains. A numerical procedure based on the finite element method is then developed, which allows the evaluation of the components of the J _k -integral, the modes I and II stress intensity factors and the T-stresses at the crack tips. The developed procedure is validated and the domain independence is demonstrated by providing comparisons to the results obtained by means of the displacement correlation technique (DCT). Detailed parametric analyses are conducted by considering an inclined crack in an FGM layer that is subjected to steady-state thermal stresses. Numerical results show the influences of the thermal conductivity and thermal expansion coefficient variation profiles and the crack inclination angle on the mixed-mode fracture parameters.     

A sub-interface thermal crack problem for bonded dissimilar plates with interfacial thermal resistance

The present work aims to investigate the effect of interfacial thermal resistance on thermal fracture behavior of bonded and composite materials. We consider a sub-interface crack parallel to the interface between two semi-infinite dissimilar plates subjected to remote heat flux thermal loading. A constant thermal resistance is assumed to exist along the interface. The temperature distribution along the crack, the thermal stress intensity factors (TSIFs), and the crack opening/sliding displacements (COD/CSD) are obtained using an integral transform/superposition method. The numerical results for Al₂O₃/Si₃N₄ bimaterial systems show that the magnitude of the mode I TSIF generally decreases with increasing thermal resistance of the interface but increases with increasing thermal resistance for cracks that are very close to the interface. On the other hand, the model II TSIF increases with increasing thermal resistance if the crack is in the Al₂O₃ semi-infinite plate, and decreases if the crack is in the Si₃N₄ semi-infinite plate. The COD/CSD are also significantly influenced by the thermal resistance of the interface.     

Computational Methods for Inclined Cracks in Orthotropic Functionally Graded Materials Under Thermal Stresses

This article sets forth two different computational methods developed to evaluate fracture parameters for inclined cracks lying in orthotropic functionally graded materials, that are under the effect of thermal stresses. The first method is based on the J _k -integral, whereas the second entails the use of the J ₁-integral and the asymptotic displacement fields. The procedures introduced are implemented by means of the finite element method and integrated into a general purpose finite element analysis software. Numerical results are generated for an inclined edge crack in an orthotropic functionally graded layer subjected to steady-state thermal stresses. Comparisons of the mixed-mode stress intensity factors computed by the use of the proposed methods to those calculated by the displacement correlation technique point out that both approaches lead to numerical results of high accuracy. Further results are provided in order to illustrate the influences of inclination angle, material property gradation, and crack length upon the thermal fracture parameters.     

A coupled meshless-finite element method for fracture analysis of cracks

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with the traditional finite element method (FEM) for analyzing linear-elastic cracked structures subject to mode-I and mixed-mode loading conditions. The EFGM was used to model material behavior close to cracks and the FEM in areas away from cracks. In the interface region, the resulting shape function, which comprises both EFGM and FEM shape functions, satisfies the consistency condition thus ensuring convergence of the method. The proposed method was applied to calculate mode-I and mode-II stress–intensity factors (SIFs) in a number of two-dimensional cracked structures. The SIFs predicted by this method compare very well with the existing solutions obtained by all-FEM or all-EFGM analyses. A significant saving of computational effort can be achieved due to coupling in the proposed method when compared with the existing meshless methods. Furthermore, the coupled EFGM–FEM method was applied to model crack propagation under mixed-mode loading condition. Since the method is partly meshless, a structured mesh is not required in the vicinity of the cracks. Only a scattered set of nodal points is required in the domain of interest. A growing crack can be modeled by simply extending the free surfaces, which correspond to a crack. By sidestepping remeshing requirements, crack-propagation analysis can be dramatically simplified. A number of mixed-mode problems were studied to simulate crack propagation. The agreement between the predicted crack trajectories with those obtained from existing numerical simulation and experiments are excellent.     

3-D elastic-plastic finite element analysis of interface cracks under mixed mode load

A three-dimensional finite element computation was performed for a throughedge cracked bimaterial steel specimen under mixed mode loadings in which the crack was lying on an interface between an elastic-plastic material and a perfectly rigid substratum. In order to take account of the average effect of microvoid nucleation and growth in the deformation, the modified Gurson's constitutive equation suggested by Tvergaard and Needleman was used. It was found that due to the interaction between the singularity along the crack front and that along the interface on the specimen surface, the distributions of stresses, plastic deformations, J-integral and void volume fraction in a bimaterial specimen were significantly different from those in a homogeneous specimen. Based on the numerical results on the distributions of void volume fraction and J-integral, the locations of fracture initiation in bimaterial and homogeneous specimens under mixed mode loadings are discussed.     

Stress intensity factors for an axially oriented internal crack embedded in a buried pipe

In this paper, the finite element method has been used to study the effect of soil weight on the stress intensity factors of an axially oriented semi-elliptical crack located on the inner surface of a buried pipe. The Burns and Richard model has been utilized to take into account the interaction between the soil and the pipe. The finite element results revealed that the cracks in a buried pipe are subjected to mixed mode loading. The mode I and mode II stress intensity factors depend on the circumferential location of internal crack. K_I is always significantly larger than K_II and is maximum when the internal crack is along the vertical direction. A comparison between the results of two-dimensional and three-dimensional cracks also signified that the two-dimensional analysis always represents more conservative results. Depending on the crack aspect ratio (a/c), the discrepancy between the results of two and three-dimensional analyses can be significant.     

A transmission electron microscopy study of crack formation and propagation in electrochemically cycled graphite electrode in lithium-ion cells

Subsurface defects and local compositional changes that occurred in graphite anodes subjected to cyclic voltammetry tests (vs. Li/Li⁺, using an electrolyte consisting of 1 M LiClO₄ in a 1:1 volumetric mixture of ethylene carbonate and 1,2-dimethoxy ethane) were investigated using high-resolution transmission electron microscopy (HR-TEM). Cross-sections of anodes prepared by focused ion beam (FIB) milling indicated that graphite layers adjacent to solid electrolyte (SEI)/graphite interface exhibited partial delamination due to the formation of interlayer cracks. The SEI layer formed on the graphite surface consisted of Li₂CO₃ that was identified by {1 1 0} and {0 0 2} crystallographic planes. Lithium compounds, LiC₆, Li₂CO₃ and Li₂O, were observed on the surfaces of separated graphite layers. Deposition of these co-intercalation compounds near the crack tip caused partial closure of propagating graphite cracks during electrochemical cycling, and possibly reduced the crack growth rate. Graphite fibres that were observed to bridge crack faces likely provided an additional mechanism for the retardation of crack propagation.     

Importance of fracture criterion and crack tip material characterization in probabilistic fracture mechanics analysis of an RPV under a pressurized thermal shock

Using a probabilistic fracture mechanics code, the importance of the choice of fracture criterion and the material fracture resistance characterization at the crack tip is elucidated in the failure probability analysis of an reactor pressure vessel. The paper describes the procedure to evaluate the crack extension based on R6, where an increase in fracture resistance by ductile crack extension is considered. Two standard J–resistance curves are prepared for applying the elasto-plastic fracture criterion by R6 tearing.Case studies concerning the effect of the tearing fracture criterion were carried out using a severe pressurized thermal Shock transient. Results are discussed with respect to the fracture criterion, the material J–resistance curve, the algorithm for evaluating the crack tip fracture toughness and the initial crack geometry.The introduction of the elasto-plastic fracture criterion significantly contributes to removal of over-conservatism in applying a linear elastic fracture criterion. It was also confirmed that the algorithm of the re-evaluation method for crack tip fracture toughness has to be correctly applied.     

CONTACT ZONE MODELS FOR AN INTERFACE CRACK IN A THERMOMECHANICALLY LOADED ANISOTROPIC BIMATERIAL

An interface crack between two anisotropic semi-infinite spaces under the action of remote mixed-mode mechanical loading and a temperature flux is considered. Assuming that all fields are independent of the x ₃ -coordinate co-directed with the crack front, the stresses and the temperature flux as well as the jumps of the displacements and the temperature at the interface are presented via a set of holomorphic functions in the whole (x ₁ , x ₂ )-plane with a cut along the crack area. By means of this representation a solution for an open crack model can be given in an analytical form, and further an inhomogeneous combined Dirichlet-Riemann boundary value problem could be formulated for a crack with an artificial contact zone. An exact analytical solution of this problem has been found, and the stress intensity factors are presented for different contact zone lengths with a special success in the case of a small length of the contact zone. Furthermore, it is shown that the obtained solution can be used for the numerical solution of interface crack problems for finite-sized bimaterials. Real contact zone lengths and the associated stress intensity factors are found from the obtained solution; in addition, their dependencies on the intensity of the temperature flux and the mechanical load mixity are demonstrated.     

Three-dimensional thermal weight function method for the interface crack problems in bimaterial structures under a transient thermal loading

Different from previous two-dimensional thermal weight function (TWF) method, a three-dimensional (3D) TWF method is proposed for solving elliptical interface crack problems in bimaterial structures under a transient thermal loading. The present 3D TWF method based on the Betti's reciprocal theorem is a powerful tool for dealing with the transient thermal loading due to the stress intensity factors (SIFs) of whole transient process obtained through the static finite element computation. Several representative examples demonstrate that the 3D TWF method can be used to predict the SIFs of elliptical interface crack subjected to transient thermal loading with high accuracy. Moreover, numerical results indicate that the computing efficiency can be enhanced when dealing with transient problems, especially for large amount of time instants.     

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 2: Numerical method

The extended displacement discontinuity (EDD) boundary integral equation and boundary-element method are extended and developed to analyze an arbitrarily shaped, planar interface crack in a three-dimensional, transversely isotropic, magnetoelectrothermoelastic bimaterial under combined, thermoelectromagnetomechanical loadings. The fundamental solutions for uniformly distributed EDDs applied over a constant triangular element are obtained through integrating the fundamental solutions for the unit-point EDDs given by Part 1 over the triangular area. To eliminate the oscillatory singularity near the crack front, the Dirac delta function in the integral–differential equations is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended stress intensity factors without oscillatory singularities, the energy release rate, and the local J-integral in terms of intensity factors are all obtained. To validate the solution, the EDD boundary-element method is proposed. As an application, an elliptical interface crack is numerically simulated. The influences of the applied combined loadings and material-mismatch as well as the ellipticity ratio on the multiphysical response are studied.     

A crack in a finite medium under transient non-Fourier heat conduction

This paper investigates the problem of a finite crack in a material layer under the theory of non-Fourier heat conduction. The concept of thermal flow intensity factor is introduced to show the singularity of the thermal flow at the crack tip. Dependence of the crack tip thermal flow field on the thermal flow intensity factor is established in closed-form. Time-varying crack tip thermal flow intensity factors are obtained with sufficient accuracy. In addition to the single crack problem, solution technique and numerical results for the problem of two collinear cracks are given. Effects of crack length and layer thickness on the thermal flow intensity factors are discussed in detail.     

Thermoelastic Models of Continua

Axially symmetric thermal stresses in an elastic pipe weakened by two cylindrical cracks are provided. The surfaces of the cracks are assumed to be thermally insulated. The outer surface of the pipe is heated to maintain a constant temperature T _d , and the inner surface of the pipe is cooled to maintain a constant temperature T _b . As a first step, the boundary conditions related to the temperature field are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the cracks, the temperature difference at the crack surfaces is expanded into a series of functions that diminish to zero outside the cracks. The unknown coefficients in the series are determined by the Schmidt method so as to satisfy the thermal insulation inside the cracks. Next, the boundary conditions related to the stress field are reduced to dual integral equations. To solve the equations, the differences in the displacements at the crack surfaces are again expanded in a series of functions that diminish to zero outside the cracks. The Schmidt method is also used to solve the unknown coefficients in the series so as to satisfy the stress-free conditions inside the cracks. The stress intensity factors are defined and calculated numerically for several configurations of the pipe.     

MULTIPLE SURFACE CRACKING IN GLASS-FIBER-REINFORCED PLASTICS DUE TO A THERMAL SHOCK AT A LOW TEMPERATURE

We consider the transient thermal singular stress problem of multiple surface cracking in glass-fiber-reinforced plastics due to a thermal shock at a low temperature. The layered composite is made of a layer bonded between two layers of different physical properties, and it is suddenly cooled on the surfaces. The surface layers contain parallel arrays of the embedded or edge cracks perpendicular to the boundaries. The thermal and elastic properties of the material are dependent on the temperature. For the case of the crack that ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Finite element calculations are carried out, and the transient thermal stress intensity factors are shown graphically.     

Thermal Fracture Analysis Of Nonhomogeneous Plate with Interfaces Under Uniform Heat Flow

The mixed-mode thermomechanical fracture problem in a nonhomogeneous material plate with two interfaces is studied in this research. Uniform heat flow conditions are considered. The interaction energy integral method for the thermal fracture problem is developed to calculate the thermal stress intensity factors (TSIFs) in nonhomogeneous materials. This method is proved to be domain independent for nonhomogeneous materials even when the integral domain is cut by one interface or many interfaces. Combining the interaction energy integral method with the extended Finite Element Method (XFEM), the temperature fields, the displacement fields, the thermal stress fields, and the TSIFs are calculated. In this article, both the edge crack and the internal crack are considered. Some examples are presented to study the influence of the material properties on the TSIFs. It can be found that the mismatch of the elastic modulus and thermal expansion coefficient can affect the TSIFs dramatically; however, the thermal conductivity interface will not arouse a kinking behavior of the TSIFs. It can be concluded that the existence of an interface (especially for elastic modulus and thermal expansion coefficient) affects the TSIFs greatly.     

On the Use of Path-Independent Integrals in Calculating Mixed-Mode Stress Intensity Factors for Elastic and Thermoelastic Cases

It is well known that the energy release rates associated with translation, rotation and self-similar expansion of cavities or cracks in solids are expressed by path-independent integrals J, L and M, respectively. It is shown that for a crack under a uniform tension and for an insulated crack disturbing a uniform heat flow, the energy release rates can be calculated by first considering an elliptical cavity and then performing a limiting process. This limiting process, with certain special properties of the M-integral and the additional relationship provided by the L-integral makes it possible to find the mixed-mode stress intensity factors.     

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 1: Theoretical solution

The extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.     

The singular behavior of the temperature gradient in the vicinity of a macrocrack tip

The power of singularity of the temperature gradient at a macrocrack tip is analyzed in this work. For a crack in an infinite medium. Williams' method of eigenfunction expansions is extended to heat conduction problems with a crack and comparison with the complex function approach is made. The intensity factor of temperature gradient (IFTG) is introduced to quantify the thermal energy cumulated in the neighborhood of a macrocrack tip. As an entirety, the power of singularity of the temperature gradient is analyzed for a crack in both isotropic and orthotropic media, and an interfacial crack between dissimilar materials. It is shown that the power of singularity of the temperature gradient is not affected by the discontinuous jumps of the thermal properties across the material interface, while that for a crack in an orthotropic medium depends on the ratio of thermal conductivities in the principal directions of material orthotropy.     

Effects of location,thermal stress and residual stress on corner cracks in nozzles with cladding

The stress intensity factors (K_I) for corner cracks in a boiling water reactor feedwater nozzle with stainless steel cladding are obtained for loading by internal pressure and a fluid quench in the nozzle. Conditions both with and without residual stress in the component are considered. The residual stress is simulated by means of a reference temperature change. The stress distribution for the uncracked structure is obtained from a three-dimensional finite element model.A three-dimensional influence function (IF) method, in conjunction with the boundary-integral equation method for structural analysis, is employed to compute K_I values from the uncracked stress distribution. For each type of loading K_I values are given for cracks at 15 nozzle locations and for six crack depths. Reasonable agreement is noted between calculated and previously published pressure-induced K_I values. Comparisons are made to determine the effect on K_I of crack location, thermal stress and residual stress, as compared with pressure stress. For the thermal transient it is shown that K_I for small crack depths is maximised early in the transient, while K_I for large cracks is maximised later under steady state conditions. Computations should, therefore, be made for several transient time points and the maximum K_I for a given crack depth should be used for design analysis. It is concluded that the effects on K_I of location, thermal stresses and residual stresses are significant and generally too complex to evaluate without advanced numerical procedures. The utilised combination of finite element analysis of the uncracked structure and three-dimensional influence function analysis of the cracked structure is demonstrated and endorsed.     

The thermoelastic problem of a stretchable rigid line inhomogeneity at soft bimaterial interface

Abstract

The thermoelastic problem of a stretchable rigid line inhomogeneity at a bimaterial interface is considered. The closed-form solution is presented and the explicit expressions of the stress at the tip of the rigid line inhomogeneity are derived. The effects of thermal expansion of rigid line on the stress intensity factors are investigated. It is found that the singularity of stress maintains the same structure, and the stresses possess an apparent oscillatory character, as in the case of a rigid line without the effects of thermal expansion. However, there is a significant difference in the coefficient of the singular behavior of the stresses at the tip of rigid line, which is due to the temperature changes.