A new boundary element approach of modeling singular stress fields of plane V-notch problems

In this paper, a new boundary element (BE) approach is proposed to determine the singular stress field in plane V-notch structures. The method is based on an asymptotic expansion of the stresses in a small region around a notch tip and application of the conventional BE in the remaining region of the structure. The evaluation of stress singularities at a notch tip is transformed into an eigenvalue problem of ordinary differential equations that is solved by the interpolating matrix method in order to obtain singularity orders (degrees) and associated eigen-functions of the V-notch. The combination of the eigen-analysis for the small region and the conventional BE analysis for the remaining part of the structure results in both the singular stress field near the notch tip and the notch stress intensity factors (SIFs). Examples are given for V-notch plates made of isotropic materials. Comparisons and parametric studies on stresses and notch SIFs are carried out for various V-notch plates. The studies show that the new approach is accurate and effective in simulating singular stress fields in V-notch/crack structures

An experimental (digital photoelastic experiments) and numeric study of the stress field in the vicinity of two interacting cracks: stress intensity factors, T-stresses and coefficients of higher order terms

The multi-parameter description of the stress field in the neighbourhood of the crack tips of two interacting crack based on the photoelastic study and finite element analyses is given. Digital photoelasticity method is used for obtaining the isochromatic and isoclinic patterns in the plate with two collinear horizontal and two inclined cracks in anisotropic linear elastic material. Stress intensity factors, T-stresses and coefficients of the higher order terms in the multi-parameter Williams series expansion are experimentally determined. The finite element analysis for the cracked plates with the same configurations has been performed. Stress intensity factors, T-stress and coefficients of nine-term asymptotic expansion for the stress field are numerically obtained and are compared with the experimental results. The comparison shows good agreement of experimental and numerical estimations of these fracture mechanics parameters. Very good agreement is shown to exist between the digital photoelasticity method and finite element results confirming the effectiveness of the photoelasticity technique in obtaining the coefficients of higher order terms of the Williams series expansion from the experimental stress field around the crack tip.The reported study was funded by RFBR, project 19-01-00631

The New Algorithm for the Determination of the Williams Asymptotic Expansion Coefficients for Notched Semidiscs Using the Photoelasticity Method and Finite Element Method

The study proposes the algorithm for the determination of the coefficients of the Williams series expansion in notched semidisks with different angles of the notch. The algorithm is based on the experimental procedure of the photoelasticity method and the finite element analysis. The large series of experiments for semidiscs was performed. Digital photoelasticity method is used to analyse experimentally the complete Williams series expansion of the stress and displacement fields in the vicinity of the crack tip in isotropic linear elastic plates under Mixed Mode loading. The distribution of the isochromatic fridge patterns is employed for obtaining the stress field near the crack tip by the use of the complete Williams asymptotic expansion for various classes of the experimental specimens (plates with two collinear cracks under tensile loading and under mixed mode loading conditions). The higher order terms of the Williams series expansion are taken into account and the coefficients of the higher order terms are experimentally obtained. The stress field equation of Williams up to fifty terms in each in mode I and mode II has been considered. The comparison of the experimental results and the calculations performed with finite element analysis has shown the importance and significant advantages of photoelastic observations for the multi-parameter description of the stress field in the neighborhood of the crack tip.Financial support from the Russian Foundation of Basic Research (project No. 19-01-00631) is gratefully acknowledged

Fatigue crack propagation under corrosion of high-strength steel

Corrosion is nearly unavoidable phenomenon in most of the metal materials. In this paper, its effect on the sharp edge crack propagation under tensile loading is investigated. A rectangular plate with a perpendicular crack and an elliptical corrosion pit nearby is modelled via finite elements and fracture behavior of the crack is analyzed. The multi-parameter fracture mechanics concept is applied, i.e. the higher-order terms of the Williams expansion are calculated by means of the over-deterministic method and utilized for tangential stress approximation in the vicinity of the crack tip. Thus, the generalized MTS criterion could be used for estimation of the crack deflection angle. The calculations were performed for a selected corrosion pit size and location considering various crack lengths. The results are discussed and a crack length with the highest probability to deflect from its original perpendicular propagation direction is found

Approximation of the crack-tip field in fatigue cracks in bridge steel specimens: DIC analysis of different constraint levels

A study on the accuracy of the approximation of the displacement field around of crack tip in a sample made from bridge steel (S355) is main objective of contribution. Linear elastic fracture mechanics (LEFM) theory in framework of multi-parameter formulation, i.e. postulated by Williams is used to determine of coefficients of Williams power series terms. Over deterministic method was used to calculate the terms based on the least squares regression technique, applied on data from numerical simulation and experiment on S355 steel grades. Comparison between the stress fields (by principal stress s1 and von Mises stress sHMH) obtain from experimental measurement DIC, Hybrid method and obtain from reconstruction by using various number of Williams power terms are quantified in order to get key information around the crack tip region on bridge steel specimens

Extended Boundary Element Method approach for Direct and Accurate Evaluation of Stress Intensity Factors

This thesis introduces an alternative method to evaluate Stress Intensity Factors (SIFs) in computational fracture mechanics directly, using the Extended Dual Boundary Element Method (XBEM) for 2D problems. A novel auxiliary equation introduced which enforces displacement continuity at the crack tip to yield a square system. Additionally, the enrichment method has been extended to 3D, so that the J-integral with XBEM and a direct technique are used to evaluate SIFs. This includes a complete description of the formulation of enrichment functions, a substitution of the enriched form of displacement into boundary integral equations, treatment of singular integrals, assembly of system matrices and the introduction of auxiliary equations to solve the system directly. The enrichment approach utilizes the Williams expansions to enrich crack surface elements for accurate evaluation of stress intensity factors. Similar to other enrichment methods, the new approach can yield accurate results on coarse discretisations, and the enrichment increases the 2D problem size by only two degrees of freedom per crack tip. In the case of 3D, the number of the new degrees of freedom depends on the desired number of crack front points where SIFs need to be evaluated. The auxiliary equations required to yield a square system are derived by enforcing continuity of displacement at the crack front. The enrichment approach provides the values of singular coefficients KI, KII and KIII directly in the solution vector; without any need for postprocessing such as the J-integral. Numerical examples are used to compare the accuracy of these directly computed SIFs to the J-integral processing of both conventional and XBEM approximations

Asymptotic Fields for Cracks Terminating at Bi-Material Interface with Arbitrary Angles

The bi-material crack problem is an interesting and important topic in the field of fracture mechanics. The existing mainstream solutions, either analytical or computational, are commonly focused on some specific cases, e.g., a crack lying on exactly the bonded border of dissimilar materials, or a crack impinging upon a bi-material interface at a right angle. However, little attention is paid to the general cases, i.e., cracks approaching or attacking the material divided border arbitrarily, which is more likely to happen in the engineering products. With any possibility of the crack\u27s incidence angle, the asymmetric nature of the geometry and the materials property induces more difficulties in the mathematical formulation of the crack-tip stress field. The conventional analytical methods may not be a convenient way for the derivation, especially of the fracture parameters. For this end, in this study, the Williams\u27 expansion method is exploited to investigate the two-dimensional/three-dimensional fracture problem in which the crack terminates at a biomaterial interface with an arbitrary angle of incidence. The characteristic equation is obtained and solved to investigate the distribution of dominant roots. Mathematically, a matrix-based system is developed, which can be easily used to formulate the general asymptotic solution of the singular stress and displacement fields surrounding the crack-tip. The theory of singularities is introduced to represent the mixed-mode nature of the solution for the arbitrarily-oriented crack. This concept is further employed for the cases with complex singularities. After that, the relationship of the asymptotic field and the linear elastic fracture parameters is established directly through a linear system. In addition, taking advantage of the enriched element approach, the derived formulation in this study is programmed and implemented in a finite element analysis. This provides an efficient and effective method for simulating and solving different types of crack problems, especially with complicated geometries, loading patterns and material combinations. Then different mixed-mode fracture criteria for predicting the direction of crack growth are introduced. With the method discussed in this study, the maximum circumferential stress criterion is considered to be the most appropriate one, but needs to be slightly modified for multiple material problems. Finally, some examples of numerical solution of the asymptotic fields are demonstrated using the computed stress intensity factors and the developed matrix system for the general crack cases with an arbitrary impinging angle with respect to an interface. The numerical results for specific cases are compared with the existing references

Asymptotic Stress Field for the Blunt and Sharp Notches in Bi-Material Media Under Mode III Loading

This study introduces a novel approach to analyze the stress and displacement fields around blunt notches in bi-material media, focusing on mode III loading conditions. The eigenfunction expansion method is used to derive a simplified yet accurate solution, satisfying the boundary conditions for bi-material blunt V-notches. The robustness of the proposed asymptotic solution is validated through several finite element analyses, encompassing a range of notched geometries such as blunt V-notches, VO-notches, and circular holes. Notably, it is demonstrated that when the notch tip radius approaches zero, the solution coincides with the existing sharp V-notch mode

Special problems of fracture mechanics of singular stress concentrators in composite materials

Předkládaná disertace se zabývá obecnými singulárními koncentrátory napětí a to zejména ostrým vrubem neboli V-vrubem, ostrým bi-materiálovým vrubem a ostrou materiálovou inkluzí. V první části práce je stručně nastíněna Kolosovova-Muschelišviliho teorie komplexních potenciálů rovinné pružnosti aplikovaná na problémy lomové mechaniky. Dále je diskutována lineární elastická lomová mechanika trhlin, V-vrubů, bi-materiálových vrubů a bi-materiálových spojů. V rešerši jsou dále zahrnuta kritéria směru iniciace trhliny i její stability a to kritérium maximálního tečného napětí, faktor hustoty deformační energie a sdružené napěťově-energetické kritérium. Následují text uvádí omezení jednoparametrové lomové mechaniky a výhody její multiparametrové formy. Další část představuje metody pro určení nezbytných parametrů pro popsání pole napětí a posuvů v blízkosti obecného singulárního koncentrátoru napětí. Tyto parametry zahrnují vlastní číslo a zobecněný faktor intenzity napětí. Vlastní číslo je určeno jako řešení problému vlastních hodnot zatímco metody pro určení zobecněného faktoru intenzity napětí tvoří Psi-integrál a metoda přeurčitosti. Obě zmiňované metody jsou aplikovány na zde studované obecné singulární koncentrátory napětí a vzájemně porovnány. Kritéria pro vznik trhliny v obecném singulárním koncentrátoru napětí jsou navržena. V rámci numerických příkladů jsou předpovězeny směry iniciace trhliny a podmínky stability pro konkrétní problémy. Kritické síly pro V-vrub jsou předpovězeny pomocí výše zmíněných kritérií a srovnány s experimentálními daty v literatuře. V následující části jsou ukázány metody analýzy multi-materiálového problému. V závěru práce jsou shrnuty způsoby iniciace a šíření trhliny v blízkosti ostré materiálové inkluze.The presented dissertation deals with general singular stress concentrators (GSSC) namely with a sharp notch also known as a V-notch, a sharp bi-material notch and a sharp material inclusion. The review section briefly outlines the Kolosov-Muskhelishvili complex potential theory of the plane elasticity applied on fracture mechanics problems. Next, the linear elastic fracture mechanics of cracks, V-notches, bi-material notches and bi-material junctions is discussed. The review also includes the crack initiation direction and the stability criteria of the maximum tangential stress, the strain energy density factor and the coupled stress-energy criterion. In the following text, limits of the single parameter and advantages of the multi-parameter fracture mechanics are presented. The next section introduces methods to determine the necessary parameters to describe the stress and displacement field near the GSSCs. The parameters include the eigenvalue and the generalized stress intensity factor (GSIF). The eigenvalue is determined as an eigenvalue problem, while the methods to calculate the GSIF are the Psi-integral and the overdeterministic method. Both the methods are applied on the studied GSSCs and mutually compared. Finally the criteria for crack initiation in the GSSCs are proposed in the multi-parameter form. The crack initiation direction and the stability conditions are predicted for particular problems in numerical examples. The failure forces for a V-notch are predicted by above mentioned criteria and compared with experimental data found in literature. In following section methods to analyze multi-material problem are shown. The final section summarizes with means of the crack initiation and propagation near the sharp material inclusion.