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.     

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.     

Weight function method for transient thermomechanical fracture analysis of a functionally graded hollow cylinder possessing a circumferential crack

This article introduces a weight function method for fracture analysis of a circumferentially cracked functionally graded hollow cylinder subjected to transient thermomechanical loading. Analytical solutions for transient temperature and stress distributions in the uncracked cylinder are derived by applying finite Hankel transformation. These solutions are utilized to determine stress acting on the faces of the circumferential crack in the local perturbation problem. Thermomechanical material properties are assumed to be power functions of the radial coordinate in the derivations. Coefficients of the weight function are found using reference stress intensity factors computed through the finite element method. Domain form of the J-integral is used in the finite element calculations. Comparisons of the numerical results calculated by the proposed weight function method to those generated by finite element analysis demonstrate the high level of accuracy attained by the application of the developed procedures. Further parametric analyses are presented to illustrate the influences of dimensionless time, crack depth to thickness ratio, power law index, and convection coefficient upon transient mode I thermomechanical stress intensity factors.     

Fracture analysis of hydrogen storage composite cylinders with liner crack accounting for autofrettage effect

An understanding of fracture behavior is crucial to the safe installation and operation of high-pressure composite cylinders for hydrogen storage. This work has developed a comprehensive finite element model to investigate axial surface flaws in cylinder liners using the fracture mechanics and a global–local finite element technique. Since the autofrettage process has a strong influence on cylinder fracture behavior, it is also considered in this analysis. The simulation process is broken down into three steps in order to precisely extract fracture parameters and incorporate the autofrettage effect. In the first step, the global model performs the autofrettage simulation to study the residual stress with consideration of both material hardening and the Bauschinger effect. In the second step, the global model uses residual stress to compute displacement for the local model. Finally, in the third step, the local model extracts the values of stress intensity factor and J-integral. Comparison is conducted on the fracture parameters with various autofrettage levels and crack shapes. The vicinity of the crack front is also studied by the size and shape of the plastic zone, and the validity of stress intensity factor and J-integral dominances is examined.     

An analysis of the numerical path dependence of the J-integral

Path dependent behaviour of the J-integral computed from the results of a finite element analysis may have two sources. The first is the history dependence of the strain energy, which causes J to lose its crack tip strain field characterizing property. The second is rooted in the principle of the finite element displacement method which ensures equilibrium only for each element as a whole, not for points within an element. It is of the utmost importance for a J-based safety assessment that these two sources be clearly distinguished as the latter may be reduced by mesh refinement whereas the former may serve as a criterion for assessing whether J can still be interpreted in its crack tip characterizing sense.The present paper proposes a numerical procedure for this distinction. Two examples confirm the procedure's validity and illustrate the practical necessity of careful evaluation of computed J-values.     

J-integral and local damage fracture analyses for a pump casing containing large weld repairs

Three-dimensional J-integral and two-dimensional Local Approach finite element studies are described for postulated crack-like defects in a large repair weld to the casing of a light water reactor circulation pump. The repair weld residual stress field is simulated and plant operating pressure and thermal transient loads are applied. Crack tip constraint effects are quantified through detailed analysis of the cracked structure and compact tension fracture toughness specimens. Fracture initiation crack sizes are shown to be larger than conceivable fabrication defects that are detectable using modern ultrasonic inspection techniques. The Local Approach study demonstrates the benefits of quantifying crack tip constraint conditions, compared with conventional J-estimation schemes and cracked body J-integral analysis. The method of introducing the crack into the finite element model is shown to have a large effect on calculated crack tip fracture parameters; a slowly developing crack in the residual stress field being more benign.     

IMPORTANCE OF INERTIA TERM IN DYNAMIC CRACK PROBLEMS CONSIDERING LORD–SHULMAN THEORY OF THERMOELASTICITY

A numerical technique is presented for the accurate calculation of stress intensity factors as a function of time for generalized coupled thermoelastic problems. In this task, the effect of the inertia term is investigated, considering different theories of thermoelasticity, and its importance is shown.

A boundary element method using the Laplace transform in time domain is developed for the analysis of fracture mechanics; dynamic coupled thermoelasticity problems with relaxation time are considered in the two-dimensional finite domain. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method. Actual physical quantities in the time domain are obtained using the numerical inversion of the Laplace transform method.

The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. The thermal dynamic stress intensity factor for mode I is evaluated using the J-integral method. The accuracy of the method is investigated through comparison of the results with the data available in literature.

The J integral, which represents the dynamic energy release rate for propagating cracks, contains a boundary integral and a domain integral. The boundary integral contains strain energy, tractions, and strains whereas the domain integral contains inertia and strains. The J-integral method allows these two terms to be calculated separately. In this way, the importance of each term may be investigated by considering different theories of dynamic thermoelasticity.     

Establishing a methodology to predict the crack initiation load in through-wall cracked components using tensile specimen test data

Among several other definitions, the crack initiation fracture toughness (J_i) based on critical stretch zone width (SZWc), called J_SZWc, is being considered as a geometry independent material property. The problem in SZWc experimental evaluation is in identifying the size of stretch zone on a blunted crack front as this requires a high degree of precision and expertise in measuring the SZW. The present study addresses finite element determination of SZWc value using tensile test data. The role of stress tri-axiality in standard fracture specimens and the smooth round tensile specimen is also studied. Based on the ASTM-E1820 standard, the present study also showed that fracture specimen thickness greater than the specified size is to be used for numerical prediction of valid SZWc value. The numerically predicted SZWc that leads to J_SZWc matches well with experimental values. Using J_SZWc, the crack initiation load is also determined in circumferentially through-wall cracked (TWC) elbows and compares well with experimental results. Thus the paper establishes the methodology to predict crack initiation in cracked piping components using numerically obtained valid J_SZWc from material’s tensile test data.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part II: J-integral estimation

Evaluation of the J-integral plays a central part in evaluation of the critical crack length for unstable fracture for piping systems. Simplified evaluation methods for the J-integral for a circumferential through-wall crack in pipes subjected to axial and bending loading or their combination is reviewed in this paper. Use of the LBB.ENG2 method and a similar approach based on the η-factor concept were found to result in significant underestimation of the J-integral for small and medium crack angles. On the other hand, the reference stress method based on the solutions for stress intensity factor and limit load recommended in the companion paper (Part I) provides solutions which agree well with the available non-linear finite-element solutions and can be utilized as a powerful tool for J-integral evaluation for arbitrary materials, not restricted to simple power-law hardening.     

Influences of some materials on J-estimation methods for pipes with circumferential surface cracks

The approximate calculation methods (SC.ENG1 and SC.ENG2) for the J-integral for pipes with circumferential surface cracks are discussed and three-dimensional elastic–plastic finite element models for circumferentially surface-cracked pipe are conducted to evaluate the accuracy of these methods for different pipe materials used in China. The numerical studies verify that the SC.ENG2 method provides more accurate estimates of J than SC.ENG1. Based on three-dimensional elastic–plastic fracture analysis, the distribution of the local J-integral along the front of a circumferential constant-depth internal surface crack is investigated and the influences of different pipe materials with different yield plateaux on J-integral values are discussed. The validity of SC.ENG1 and SC.ENG2 J-integral estimation methods for pipe steel materials with different yield plateaux used in China are examined in detail and the SC.ENG2 method is found to provide reasonable estimates of J for materials with yield plateaux.     

Characterizing the crack initiation load in circumferentially through-wall cracked elbows under bending load

Ductile fracture assessments of circumferentially through-wall cracked elbows, based on the elastic–plastic J-integral concept, are discussed with particular interest in its ability and accuracy to predict experimental results corresponding to the initiation of stable crack growth. 3D non-linear finite element analysis is backed up with experimental results to determine the crack initiation load. Non-linear finite element analyses were performed considering both material and geometrical non-linearity using the advanced fracture analysis code WARP3D. Numerical analyses have been carried out to understand the role of crack tip constraint in standard specimens and the elbow component. An attempt has been made to obtain a unique multiaxiality quotient (q) for evaluation of the level of constraint. The work provides benchmark data to assist in the engineering treatment of cracked piping elbows.     

MODELING OF THERMAL CRACKING IN ELASTIC AND ELASTOPLASTIC TWO-PHASE SOLIDS

A review is given about fracture mechanical investigations concerning the thermal crack initiation and propagation in one of the segments or in the material interface of two-arid three-dimensional self-stressed two-phase compounds. The resulting boundary value problems of the stationary thermoelasticity and thermoplasticity for the cracked two- and three-dimensional bimaterial structures considered are solved using the finite element method. Furthermore, by applying an appropriate crack growth criterion based on the numerical calculation of the total energy release rate G of a quasistatic mixed-mode crack extension the further development of thermal crack paths starting at the intersection line of the material interface with the external stressfree surface of the two- and three-dimensional elastic bimaterials could be predicted. In the case of the disklike two-phase compounds, the theoretically predicted crack paths show a very good agreement with results gained by associated cooling experiments. Several specimen geometries consisting of different material combinations and subjected to uniform and nonuniform temperature distributions have been studied using the relevant methods of fracture mechanics. Thereby thermal cracks propagating in one segment of an elastic bimaterial only obey the condition G_II = 0, whereas for interface cracks a mixed-mode propagation is always existent where the G_II values play an important role. Moreover, by applying the proposed crack growth criterion the possible crack kinking direction ? of an interface crack tip out of the interface could be predicted by taking into consideration the finite thickness of an interlayer (interphase). In addition, an analysis of the stress and strain fields in the vicinity of thermal interface cracks in the discontinuity area of two- and three-dimensional elastoplastic two-phase compounds has been performed by using the FE-method. Thereby a heat source Q was assumed in one of the two materials in the neighborhood of an interface crack tip. The corresponding stress states in the bimaterial structuresand especiallyin the vicinity of the interface crack tip have been calculated by applying the incremental I₂-plasticity and using a bilinear hardening material law and based on a sequentially coupled solution of the heat transfer and the thermal stress boundary value problems. Finally, the failure assessment has been performed on the basis of the local J-integral which, for three-dimensional interface cracks, was recently generalized by two of the authors.     

The applications of 2D elasto-plastic stochastic finite element method in the field of fracture mechanics

In this paper the stochastic finite element method (SFEM) is applied to the field of fracture mechanics and cracked structures are analysed by using an elasto-plastic methodology. The main random variables considered are Young's modulus, Poisson's ratio and the crack size. The formulae of the mean and variance value of the J-integral for elasto-plastic deformation are discussed and an effective method for the probabilistic safety assessment of cracked structures is given. Some elasto-plastic problems are solved by this method, and numerical examples demonstrate that this method is appropriate for engineering problems and that the related computer program developed in this paper has sufficient precision.     

Derivation of ‘γ’ parameter from limit load expression of cracked component to evaluate J–R curve

Experimental evaluation of the J-integral requires the ‘η_pl’ function, proposed by Rice et al. [Progress in flaw growth and fracture toughness testing (1973) 231], to multiply the area under the load vs. plastic load-line-displacement curve. However, the J-integral, thus evaluated, requires modification if crack growth occurs. A ‘γ’ term was proposed by Hutchinson and Paris [Elastic–plastic fracture (1979) 37] and later generalised by Ernst et al. [Fracture mechanics (1979) 581] and Ernst and Paris [Techniques of analysis of load–displacement records by J-integral methods (1980)] to correct the J-integral to account for crack growth. The η_pl and γ functions are available for very few geometries under specific loading conditions. A limit load-based general expression of η_pl was given by Roos et al. [Int J Pres Ves Piping 23 (1986) 81], but no such expression is available for γ functions. The advantage of having limit load-based general expressions for η_pl and γ functions is that the limit load for a particular geometry subjected to a specific loading condition is easily available in the open literature. In the present paper, a limit load-based general expression for the γ function is derived. The general expression is then validated by deriving the known γ functions of various geometries subjected to various loading conditions, which are available in the open literature. The general expressions are then used to derive new η_pl and γ functions for same pipe and elbow geometries with various crack configurations under different loading conditions, for which no solutions are available in the open literature. Finally, experiments have been carried out on 200 mm nominal bore (NB) elbows with throughwall circumferential cracks under in-plane bending moment. The proposed new expressions of η_pl and γ functions for this geometry are used to obtain the J–R curve from the experimental load vs. load-line-displacement and load vs. crack growth data.     

Analysis of biaxial loading effect on fracture toughness of reactor pressure vessel steels

The local stress–strain state (SSS) near the crack tip and its connection with the crack tip opening displacement and J-integral under biaxial loading have been studied by finite element methods in elastic–plastic finite strain statement. Numerical investigations have been performed for various crack lengths and two types of biaxial loading (tension and bending) under conditions of small- and large-scale yielding. To predict the biaxial loading effect on cleavage fracture toughness, the procedure has been elaborated, this being based on the revealed regularities for SSS near the crack tip under biaxial loading and brittle fracture criterion proposed earlier. Prediction of the biaxial loading effect on cleavage fracture toughness has been performed as applied to reactor pressure vessel steel. The calculated results have been compared with available experimental data. Alternative approaches for prediction of the biaxial loading effect on fracture toughness have been discussed.     

R6 and R5 procedures: The way forward

This paper first briefly summarises the existing methods in the low-temperature fracture assessment procedure, R6, and the high-temperature procedure, R5, for treating the effects of secondary stresses on structural integrity. Recently, there have been a number of developments, which identify the way forward for these procedures. A modified J-integral definition has been derived, which is path independent for cases of proportional and non-proportional loading and is ideal for evaluating the crack driving force for defects in secondary and residual stress fields. Results of finite element analysis are presented that show that the use of the modified J-integral can lead to a lower crack driving force for secondary stresses than current simplified R6 methods. More detailed calculations have assessed the effects on fracture of a slowly growing crack and constraint effects associated with secondary stresses. Preliminary results are presented, showing the long-term potential of more advanced methods in providing significant benefits in structural integrity assessments. For high-temperature applications, the paper presents methods for calculating the relaxation of secondary stresses due to both creep strain and creep crack growth, extending current methods in R5 that only allow for relaxation due to creep strain. Related work addressing the combined effects of plasticity and creep on relaxation of the crack tip fields is also presented and the results are illustrated for a typical geometry and loading.     

Validation of surface crack stress intensity factors of a tubular K-joint

Tubular K-joints are encountered widely in offshore structures, and the prediction of damaged joints depends very much on the accuracy of stress intensity factor solutions (SIFs). No parametric equations and very few results have been proposed and published in the literature for estimating the SIFs of any K-joints subjected to complex loading conditions. In this paper, a mesh generation method proposed previously for the Y-joint and T-joint has been extended to the K-joint. This method is realized by dividing the K-joint into several sub-zones with each zone consisting of different types of elements and mesh densities. This method has a distinct advantage of controlling the mesh quality, and most importantly the aspect ratio of the elements along the crack front. When the mesh of all the sub-zones has been generated automatically and completely, they are merged to form the complete model. The two most commonly used methods, namely the J-integral and displacement extrapolation, are used to evaluate the SIF values along the crack front of a typical K-joint. To validate the accuracy of these computed SIFs, a full-scale K-joint specimen was tested to failure under fatigue loading conditions. The standard alternating current potential drop (ACPD) technique was used to monitor the rate of crack propagation of the surface crack located at the hot spot stress region. Using the given material parameters C and m, the experimental SIFs were deduced, and they are found to be in good agreement with the computed SIFs obtained from the generated models. Hence, the proposed finite element models are both efficient and reliable.     

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.     

Evaluation of J-integral for surface cracked plates under biaxial loading using extended reference stress method

In this paper, reference stress solutions for plates with semi-elliptical surface cracks were firstly reviewed, and the applicability of the solutions was examined through the comparison with finite element analysis results under uniaxial loading. Next, an extended reference stress method was newly developed to evaluate J-integral for cracked plates under biaxial loading. The predictive accuracy of the method was validated through the comparison with finite element analysis results under biaxial loading. As a result, it was ascertained that the proposed method together with Lei's reference stress solution predicts J-integral with acceptable accuracy.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part I: stress intensity factor and limit load solutions

The J-integral and the crack opening area are the main parameters required for a leak-before-break evaluation of a piping system. Stress intensity factor and limit load solutions have been widely used for evaluating these parameters in a simplified way. Solutions for the stress intensity factor and limit load for a pipe with a circumferential through-wall crack subjected to axial and bending loads are reviewed and compared in this study. Based on the comparisons, recommendations are then made on expressions for calculating these parameters.