A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

An assessment of far field effects on the photoelastic determination of mixed mode stress intensity factors

A limiting approach inquiry into far field effects on local field equations for mixed mode surface flaws is investigated by computer analysis. Curves showing the diminishing effect of the far field stress towards the cracks tip are plotted using fringe radii ratios vs fringe number. The zone used in the experimental determination of K₁ and K₂ is superimposed on these curves to investigate the extent to which K₁ and K₂ determined by the two parameter method are influenced by the far field stress.     

The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

A remark on the direct numerical determination of stress intensity factors at crack tips

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.     

Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods

A method for evaluating mode I, mode II and mixed-mode stress intensity factors from in-plane displacement fields using the method of nonlinear least-squares is proposed in this paper. Along with stress intensity factors, crack tip location and rigid body displacement components are determined simultaneously from both displacement components obtained using full-field optical methods or numerical methods. The effectiveness is validated by applying the proposed method to mixed-mode displacement fields obtained through digital image correlation, displacement fields obtained by analysis using elasto-plastic finite element method, and displacement fields around a fatigue crack obtained by electronic speckle pattern interferometry. Results show that the proposed method can extract stress intensity factors from the displacement fields both accurately and easily. Furthermore, they can be determined even if the material at a crack tip exhibits small-scale yielding. It is expected that the proposed method is applicable to various fracture problems during experimental and numerical evaluation of structural components.     

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

T-stress near the tips of a cruciform crack with unequal arms

The T-stress near the tips of a crack of cross shape embedded in an isotropic elastic solid is analyzed. The integral transform technique is employed to convert the associated boundary value problem to a system of singular integral equations. According to the stress difference method, T-stresses can be expressed as a sum of an integral involving crack opening displacement (COD) and applied loading at infinity. Obtained results indicate that, in addition to applied loading, T-stresses at the horizontal (vertical) crack tips depend on the COD of the vertical (horizontal) crack surface. COD plays a leading role in determining T-stresses, in particular for a cruciform crack of two crack-arm lengths of the same order. Moreover, T-stresses for a single-crack limiting case can be recovered from the present results as the length of one arm approaches zero. For a biaxial tension of the same magnitude, T-stresses are present for a cruciform crack, but absent for a single crack. Finally, for several cases of interest, T-stresses for a cruciform crack are evaluated and compared with those for a single crack, and the influence of the ratio of two crack-arm lengths b/a and the COD on the T-stress of a cruciform crack is examined.     

The analysis of thermoelastic isopachic data from crack tip stress fields

A computer program—FACTUS (fracture analysis of crack tips using SPATE)—has been developed for the efficient analysis of thermoelastic data obtained from around a crack tip. The program is based on earlier work for the determination of stress intensity factors (SIFs), and also includes a novel solution procedure for the derivation of the non-singular stress term σ _{0 x} . The program has been used in the analysis of a series of large plate specimens with central or edge slots/cracks. The derived SIFs are compared with independent values. Issues, e.g. crack closure and the extent and effect of the plastic zone, are discussed.     

The in-plane and out-of-plane stress constraint factors and K-T-Tz description of stress fields near the border of a quarter-elliptical corner crack

The elastic T-stress and stress intensity factor K for quarter-elliptical corner cracks have been investigated in elastic plates by detailed three-dimensional finite-element calculations. The distributions of normalized K and T-stress have been obtained along the crack front with aspect ratios (a/c) of 0.2, 0.3, 0.4, 0.5, 0.6, 0.8 and 1.0, and far-field tension and the effect of Poisson's ratio have also been considered. The normalized K increases and the normalized T-stress decreases with the increase of Poisson's ratio v. For v= 0.3, the normalized K gradually increases in the range of crack-face angle φ≥ 22.5° and decreases in the range of φ≤ 22.5° with the increase of a/c. The normalized T-stress increases in the beginning and then decreases with increasing φ except for a/c= 0.2 and a/c= 0.3. By fitting the numerical results with the least squares method, empirical formulae have been given for the convenience of engineering applications. Combining with the corresponding out-of-plane constraint factor T_z, the three-parameter K-T-T_z approach has been provided, which can accurately describe the stress field around the crack front.     

On the photoelastic determination of complex stress intensity factors

An improvement of the one-parameter extrapolation method of photoelastic determination of complex (mixed-mode) stress intensity factors at straight or curvilinear crack tips in a plane isotropic elastic medium due to Smith et al. [12, 13] can be achieved by measuring the absolute value of such a factor on the isochromatic fringes along properly selected polar directions and not at the maxima of the isochromatic fringes. In this way, the unknown value of the constant term of the stress field near the crack tip is taken into account. It is seen that it is always possible to find at least one appropriate polar direction to measure the absolute value of the stress intensity factor. In the case of Mode I stress intensity factors, these polar angles are = ± 120° and not = ± 90° as generally considered previously. Some numerical results are also presented in this special case and show the efficiency of the present method.     

Determination of the near-tip stress fields for the scattering of P-waves by a crack

Summary A conservation law implied by the field equations of linear elastodynamics is derived, and a procedure based on this conservation law is given for the direct determination of the near-tip stress fields arising from the scattering of normally incident P-waves by a crack in a homogeneous, isotropic elastic medium.     

Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique

In this study the fracture mechanics parameters, including the strain energy release rate, the stress intensity factors and phase angles, along the curvilinear front of a three-dimensional bimaterial interface crack in electronic packages are considered by using finite element method with the virtual crack closure technique (VCCT). In the numerical procedure normalized complex stress intensity factors and the corresponding phase angles (Rice, J Appl Mech 55:98–103, 1988) are calculated from the crack closure integrals for an opening interface crack tip. Alternative procedures are also described for the cases of crack under inner pressure and crack faces under large-scale contact. Validation for the procedure is performed by comparing numerical results to analytical solutions for the problems of interface crack subjected to either remote tension or mixed loading. The numerical approach is then applied to study interface crack problems in electronic packages. Solutions for semi-circular surface crack and quarter-circular corner crack on the interface of epoxy molding compound and silicon die under uniform temperature excursion are presented. In addition, embedded corner delaminations on the interface of silicon die and underfill in flip-chip package under thermomechanical load are investigated. Based on the distribution of the fracture mechanics parameters along the interface crack front, qualitative predictions on the propensity of interface crack propagation under thermomechanical loads are given.     

Some simple formulas to predict the variation of stress intensity factors for mode I crack induced by near crack-tip inclusion

Several simple formulas have been developed to predict the variations of stress intensity factors (SIFs) for mode I crack induced by the inclusion within crack-tip field. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated from the remote applied SIF K₀. As validated by numerical examples, the developed formulas have satisfactory accuracy for wide range of the modulus ratios between inclusion and matrix as long as the inclusion is located in the K₀-controlled field.     

Some simple formulas to predict the variation of stress intensity factors for mode I crack induced by near crack-tip inclusion

Several simple formulas have been developed to predict the variations of stress intensity factors (SIFs) for mode I crack induced by the inclusion within crack-tip field. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated from the remote applied SIF K₀. As validated by numerical examples, the developed formulas have satisfactory accuracy for wide range of the modulus ratios between inclusion and matrix as long as the inclusion is located in the K₀-controlled field.     

An analysis of factors influencing precision of thermal property values during freezing

An unfrozen water fraction prediction equation is proposed that compensates for both unfreezable water (M_b) and the equivalent molecular weight of solids (MW_s) during the food freezing process. The parameters (M_b and MW_s), when obtained from published unfrozen water content data, demonstrate that the proposed equation describes the data very well. When unfrozen water content data is not available, MW_s can be obtained from the initial freezing point (T_f) of the food product, and M_b can be obtained using calorimetric methods. The precision of MW_s is shown to be very dependent on the precision of T_f for a frozen food with medium (75%) to high (95%) moisture content. The precision of the input variables required in the proposed equations (temperature, M_b, MW_s, solids content) along with other physical parameters of the unfrozen food product (density, specific heat, and thermal conductivity) were investigated in an effort to determine the predictability of the frozen food density, apparent specific heat, enthalpy, thermal conductivity and unfrozen water content.