Evaluation of the stress field around a notch tip using contour integrals

The mode I and mode II asymptotic stresses around a notch tip are in general governed by different orders of singularity. Direct computation of the mixed-mode near-tip stress field therefore appears to be difficult. In this paper, we propose a pair of contour integrals J_kR. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. As an aside, by defining a pair of generalized stress intensity factors (SIFs) (K_I)_β and (K_II)_β, the relationship between J_kR and the SIFs is derived and expressed as functions of the notch angle β. Once the J_kR-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. The feasibility of our formulation is demonstrated in two numerical examples, where various instances with different notch angles are considered. No particular singular elements are used in this study.     

Calculation of stress intensity factors for curved cracks in anisotropic FGMs

Abstract

A combined analytical and numerical method is proposed for computation of mixed-mode stress intensity factors (SIFs) for arbitrary curved cracks in anisotropic functionally graded materials (FGMs). By developing a pair of closed-form expressions that relate the SIFs and the J_k-integrals, it is anticipated that the SIFs can be properly extracted should the J_k-integrals be accurately evaluated. To this end, a novel method for calculating the J_k-integrals is presented and has proved reasonably accurate in numerical computations. Since neither a priori information nor extra auxiliary solutions corresponding to the singular behavior is required, this proposed scheme appears to be applicable to problems containing arbitrary shapes of curvature in generally anisotropic FGMs.     

Computation of mixed-mode stress intensity factors for curved cracks in anisotropic elastic solids

A numerical procedure, based on the concept of the J_k-integrals, is presented for computation of the mixed-mode stress intensity factors for curved cracks. Since no auxiliary solutions are required in practice, this approach can be applicable for problems containing cracks of arbitrary curved shape in generally anisotropic elastic solids. Nevertheless, while these integrals are demonstrated to be path-independent in a modified sense, it is noted that the near-tip region is always inevitably included. Attention is therefore addressed to modeling of the singular behavior in the vicinity of the crack tip in such manner that the calculation is based on a combination of full-field finite element analysis and near-tip asymptotic results. With this, no particular singular elements are required in the calculation.     

Fracture toughness of unidirectional fiber-metal laminates: crack orientation effect

Fiber-metal laminates (FMLs) are structural composites developed for aeronautical applications. The application of FMLs to structures demands a deep knowledge of a wide set of properties, including fracture toughness. The objective of this work was to evaluate the effect of crack orientation on the fracture toughness (critical J-integral and CTOD δ₅) of unidirectional FMLs. Small C(T) and SE(B) specimens with notches parallel and perpendicular to the fibers direction were tested. A study of the relation and equivalence between J_C and δ_5C, which heavily depend on the yield strength and on the stress state, was performed motivated by apparently contradictory experimental results. These results can be explained by the direction-dependent yielding properties of unidirectional FMLs. The best overall equivalence between J_C and δ_5C was obtained considering plane stress state and using the effective yield strength, both for unidirectional FMLs notched parallel and perpendicular to the fibers direction.     

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

This paper presents a new analytical expression relating the J₂‐integral and stress intensity factors (SIF) in an in‐plane traction‐free crack between two orthotropic elastic solids using the complex function method. The singular oscillatory near tip field of a bimaterial interfacial crack is usually characterized by a pair of SIFs. In linear elastic interfacial fracture mechanics, the majority of numerical and experimental methods rely on the analytical equations relating J_k‐integrals and SIFs. Although an analytical equation relating J₁‐integral or strain energy release rate and SIFs is available, a similar relation for J₂‐integral in debonded anisotropic solids is non‐existent. Using this new analytical expression, in conjunction with the values of J_k, the SIFs can be computed without the need for an auxiliary relation. An example with known analytical solutions for SIFs is presented to show the variation of the J₂‐integral near the crack tip of a bimaterial orthotropic plate. Different bimaterial combinations are considered, and the effect of material mismatch on J_k is demonstrated.     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. K _I and K _II are represented by path integrals of the first stress invariant and its partial derivative. K _III is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for K _I. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

Effect of crack depth on the shift of the ductile-brittle transition curve of steels

Nuclear reactor pressure vessel (RPV) steels degrade due to neutron irradiation during normal operation. As a result, the ductile-brittle transition curve of the steel shifts to higher temperature which decreases operation margins in both the temperature and pressure. The loss of these margins however can be offset somewhat by appealing to arguments based on constraint of potential/postulated shallow cracks. In this paper, it is demonstrated that the fracture toughness values for shallow flaws are higher than those determined from standard deep cracked test specimens based on constraint consideration. The J-A₂ three-term solution is used to characterize the crack-tip stress field where J represents the level of loading and A₂ quantifies the level of constraint. Based on the RKR cleavage model, procedures to quantify the temperature shift between specimens with different constraint levels are developed. The experimental data by Sherry et al. [Sherry AH, Lidbury DPG, Beardsmore DW. Validation of constraint based structural integrity assessment methods. Final report, Report No. AEAT/RJCB/RD01329400/R003, AEA Technology, UK, 2001] for the A533B RPV steel are used to demonstrate the procedure and it is shown that the ductile-brittle transition curve shifts to lower temperature from high constraint to low constraint specimens.     

Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials

A new numerical method to calculate the stress intensity factors (SIFs) of a three-dimensional interface crack between dissimilar anisotropic materials was developed. In this study, the M-integral method was employed for mode separation of the SIFs. The moving least-square method was utilized to calculate the M-integral. Using the M-integral with the moving least-square method, SIFs can be automatically calculated with only the nodal displacements from the finite element method (FEM). Here, SIFs analyses of some typical three-dimensional problems are demonstrated. Excellent agreement was achieved between the numerical results obtained by the present method and the corresponding results proposed by other researchers. In addition, the SIFs of a single-edge crack, a through crack, and a semi-circular crack between two anisotropic solids in three-dimensional structures were analyzed.     

Soft chemistry synthesis of MxV2O5 + ε·nH2O (MCo, Ni, Cu, Zn) nanowires

Recently a simple method was used for preparing nanowires M_xV₂O_5 + ε·nH₂O (MMg, Mn). It consists of mixing a boiling solution of vanadium oxide with the corresponding metal nitrate. In the present paper, this method was explored for preparing other nanowires layered M_xV₂O_5 + ε·nH₂O compounds (MTi, Cr, Fe, Co, Ni, Cu, and Zn). The results obtained show that If the precipitates are formed immediately after mixing the two solutions, such in case of titanium, chromium and iron, the structure and particles shape of the products are different from the layered compound M_xV₂O_5 + ε·nH₂O. However, if the precipitates start to appear only after several minutes of stirring, such in case of cobalt, nickel, copper and zinc, the phases obtained are similar to layered nanowires M_xV₂O_5 + ε·nH₂O. The amount of the metal inserted “M” was found to be lower than 25 at.% in all as-prepared samples. Nickel containing sample shows the highest inserted amount.     

Evaluation of J1 and J2 integrals for curved cracks using an enriched boundary element method

This paper introduces an enriched Boundary Element Method in which functions are introduced that are known to model singularities or discontinuities from a priori knowledge of the solution space. Additional fundamental solutions are introduced to solve for the additional unknowns created by enrichment and a numerical integration routine is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. The solution of a curved crack in an infinite domain by Muskhelishvili is used to assess the accuracy of the method. Using an appropriate technique to evaluate J₁ and J₂ integrals, it is found that very good agreement with the exact solution is seen with improvements in accuracy over similar FEM implementations.     

Analytical-variational and generalized weighted conservative integral method on solution of interfacial cracks with their applications to bi-metal glued joints

In this paper, first of all, the eigen expansions of displacement and stress fields satisfying all of basic equations, boundary conditions along cracked portion of interface and conditions of continuity along uncracked portion of interface are derived in a bi-metal glued joint with interfacial crack. Next, the expressions of generalized weighted conservative integrals (WCIs) and the relationships between generalized SIFs and WCIs are established. Furthermore, the coefficients in above expansions are determined by generalized variational method, then the values of conservative integrals can be obtained. The generalized SIFs found directly from leading terms of above expansions agree with that found indirectly from conservative integrals satisfactorily. The convergency of results with increase of number of terms in above expansions is very good and the magnitudes of conservative integrals along different paths keep constant accurately.     

Establishing methodology to predict fracture behaviour of piping components by numerically predicting specimen fracture data using tensile specimen test

Using large deformation FEM analyses in SA333Gr.6 carbon steel material, the present study demonstrated the assessment of SZWc value that leads to J_SZWc and finally compares with the respective experimental results. It also includes numerical prediction of specimen J-R curve using Gurson-Tvergaard-Needleman parameters obtain from tensile specimen tests. Using numerically predicted results, the crack initiation and instability stages in circumferentially through-wall cracked elbows is finally predicted and compares with experimental results. The present study gives evidence that the non-linear FEM analysis supported with proper tensile test data can be helpful in assessing the safety of bend pipes with through-wall crack.     

The relationship between Ct and contour integrals under small-scale transient creep

In this paper, the relationship between the C_t and contour integrals for small-scale transient creep conditions is established, via elastic-creep finite element analysis. It is found that the C_t integral is similar to the contour integral value evaluated roughly at the creep zone size defined as the boundaries where time-dependent effective creep strains equal the instantaneous effective elastic strains and measured normal to the crack plane at the crack tip.     

The energy dissipation rate approach to tearing instability

Stable and unstable tearing in metals is currently analysed by J integral theory, or by the G_R curve approach. This paper explains an alternative analysis route based on energy dissipation rate, D. It is shown that the implication of increasing toughness with crack growth in G_R and J_R curves is misleading. Even in small scale yielding (SSY), it is possible to have stable tearing under increasing G or J whilst at the same time D is constant (or even reducing) with crack growth. New terms: C for crack driving force, D^* for geometry normalised D, D_ssy for D in SSY, and crack stability index are explained. A D based fracture analysis diagram is introduced. Comparisons are made between energy dissipation rate, J integral, and G_R curve instability prediction methods. It is shown that, in most instances, these different approaches are compatible; but that the use of J_R curves derived in fully yielded test pieces to predict failure in SSY has the potential to lead to an unconservative instability prediction. The practical advantage of the energy dissipation rate approach is that it can be applied to all product thicknesses at any extent of crack growth. The major advantage compared to the G_R approach is that toughness measurements can be made on much smaller specimens.     

Optical properties of Cu(In,Ga)Se2 and Cu2ZnSn(S,Se)4

The optical properties of CuInSe₂, CuGaSe₂, Cu₂ZnSnS₄, and Cu₂ZnSnSe₄ are investigated using three different first-principles methods, namely the generalized gradient approximation by Perdew, Burke, and Ernzerhof (PBE), the hybrid Hartree-Fock-like functional by Heyd, Scuseria, and Ernzerhof (HSE), and a Green's function approach (GW). The density-of-states, the complex dielectric function ε(ω) = ε₁(ω) + iε₂(ω), and the optical absorption coefficient α(ω) are determined, providing fundamental understanding of these materials. We find that even though the PBE method generates fairly accurate effective crystal potentials, the HSE and GW methods improve considerably the band-gap energies E_g and also the localization of the semicore states, thereby describing the optical properties much better. Furthermore, we also present optimized convergence parameters for the self-consistent HSE calculation in order to reduce the computational time of this orbital-dependent method.     

Creep-fatigue crack propagation in lead-free solder under cyclic loading with various waveforms

Crack propagation tests of lead-free solder were conducted using center-notched plate specimens under cyclic tension-compression of three load waveforms: pp waveform having fast loading and unloading, cp-h waveform having a hold time under tension, and cc-h waveform having a hold time under tension and compression. In the case of fatigue loading, i.e. pp waveform, the path of crack propagation was macroscopically straight and perpendicular to the maximum principal stress direction, showing tensile-mode crack propagation. The introduction of the creep components by hold time in cc-h and cp-h waveforms promoted shear-mode crack propagation. For fatigue loading of pp wave, the crack propagation rate was expressed as a power function of the fatigue J integral and the relation was identical for load-controlled and displacement-controlled conditions. The creep component due to the hold time greatly accelerates the crack propagation rate when compared at the same values of the fatigue J integral or the total J integral (the sum of fatigue J and creep J integrals). The creep crack propagation rate was expressed as a power function of the creep J integral for each case of cp-h and cc-h waveforms. The crack propagation rate for cp-h waveform is higher than that for cc-h waveform. The predominant feature of fracture surfaces was striations for pp waveform and grain boundary fracture for cp-h waveform. Grain fragmentation was abundantly observed on the fracture surface made under cc-h waveform.