Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.

Calculations on compact disc cracking

The Griffith equation for brittle cracking has three problems. First, it applies to an infinite sheet whereas a laboratory test sample is typically near 100 × 100 mm. Second, it describes a central crack instead of the more dangerous and easily observable edge crack. Third, the theory assumes a uniform stress field, instead of tensile force application used in the laboratory. The purpose of this paper is to avoid these difficulties by employing Gregory's solution in calculating the crack behaviour of PMMA (Poly Methyl Meth Acrylate) discs, pin loaded in tension. Our calculations showed that axial disc loading gave nominal strengths comparable with Griffith theory, but the force went to zero as the crack fully crossed the disc, fitting experimental results. Off-axis loading was more interesting because the predicted strength was lower than in axial testing, but also gave unexpected behaviour at short crack lengths, where nominal strength did not rise indefinitely but dropped as crack length went below D/10, quite different from Griffith, where strength rose continuously as cracks were shortened. Such off-axis loading leads to a size effect in which larger discs are weaker, reminiscent of the fine fibre strengthening phenomenon reported in Griffith's early paper (Griffith 1921 Phil. Trans. R. Soc. Lond. A 221 , 163–198. ( doi:10.1098/rsta.1921.0006 )). This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction'.

A Method for Establishing the Central Crack Stress Intensity Factor Database for Probabilistic Risk Assessment Based on the Universal Weight Function

Probabilistic failure risk assessment is becoming important in the field of airworthiness. In the fracture mechanics module of probabilistic failure risk assessment, it is important to efficiently and accurately calculate the stress intensity factors (SIFs). At present, the weight function method (WFM), especially the universal weight functions (UWFs) proposed by Glinka and Shen, has been adopted to calculate SIFs with high accuracy and computational efficiency. However, the concrete coefficients in the universal weight functions remain unknown, and the rules of the geometry parameters and these coefficients have not yet been summarized, which hinders their subsequent use. In this article, the specific type of embedded crack-central crack is under discussion, and the derivation of the UWF is introduced. The response surface method (RSM), as a means of database establishment, is used to construct the relations between the geometric parameters including the length and thickness of a three-dimensional finite plate and coefficients in the UWF. The errors of the SIF calculation between the UWF and finite element results are less than 2 MPa m within a certain range. For the evaluation of the boundary effect on central cracks, the difference between finite and infinite plates is discussed. In addition, considering the complexity of the general off-centre crack, an approximation method has been proposed to transform the off-centre crack to the central crack. The results show that the method can be applied with high precision in specific situations and stresses the necessity of follow-up research on general off-centre cracks.

Study of mechanical behavior by fatigue of a cracked plate repaired by different composite patches

This research is based on the study of the fatigue behavior of an aluminum alloy plate with a central crack. The plate is subjected to a tensile loading on its lower and upper parts. Several parameters were highlighted, such as the loading effect with a load ratio R = 0. The effects of the load ratio on both the repaired and not repaired plates, by two composite patches, which are boron/epoxy and graphite/epoxy, were investigated, as well as the effect of the plate material on plate life, comparing different materials

Study the effect of perforation type for plate with central crack on the stress intensity factor using the XFEM

n this study, loading was carried out for several types of perforated plates, such as circular, rhombic and rectangular holes, where the holes were arranged in two types, namely straight arrangement and alternating arrangement. The stress intensity factor and shape factor were calculated for each case, taking into account the diameter of the holes. So, it is found the SIF increases significantly when the plate is perforated, and the same applies to the shape factor, also increases. In the case of circular holes, the increases in the average value of (SIF) reached to (80.88 %) when the plate was perforated with alternated arranged of circular holes, while the straight arrangement of circular holes the increases of average values of SIF reach to (67.55 %). Either in the case of rhombus holes: the SIF values are increases to (51.07 %) when the plate was perforated with the alternated arrangement, while in the straight arrangement of holes the (SIF) increase to (35.43 %). It was observed through this study, the increases of stress intensity factor and the shape factor with different crack lengths were more stable in the plate that perforated with an alternated arrangement of holes than the straight arrangement. The higher values of stress intensity factor obtained when the plates were perforated with circular holes, due to the circular shape has more stiffness, so the Absorption of force will be small Compared with the rhombus and rectangular shape that will be less stiffness which the absorption of strength is greater.

The J-area integral applied in peridynamics

The J-integral is in its original formulation expressed as a contour integral. The contour formulation was, however, found cumbersome early on to apply in the finite element analysis, for which method the more directly applicable J-area integral formulation was later developed. In a previous study, we expressed the J-contour integral as a function of displacements only, to make the integral directly applicable in peridynamics (Stenström and Eriksson in Int J Fract 216:173–183, 2019). In this article we extend the work to include the J-area integral by deriving it as a function of displacements only, to obtain the alternative method of calculating the J-integral in peridynamics as well. The properties of the area formulation are then compared with those of the contour formulation, using an exact analytical solution for an infinite plate with a central crack in Mode I loading. The results show that the J-area integral is less sensitive to local disturbances compared to the contour counterpart. However, peridynamic implementation is straightforward and of similar scope for both formulations. In addition, discretization, effects of boundaries, both crack surfaces and other boundaries, and integration contour corners in peridynamics are considered.

Finite element method with account of singularity for mixed mode cracks

The paper deals with obtaining an analytical solution for stiffness matrix coefficients at a crack tip area for mixed mode cracks in plane strain conditions. The numerical study is focused on an infinite plate with a straight-through central crack under mixed loading. Analytical solutions are obtained as kinematic boundary conditions for plane strain. We analyzed distribution features of the stress-strain state fields and stress intensity coefficients at the top of the crack area, determined using the finite element method taking into account the singularity. The analytical formulas are obtained which set the kinematic conditions for a general and special case of loading a plate with a defect in the elastic setting for the case of plane deformation. The comparative analysis of the numerical results is presented for two cases of forming the design diagram of the top of the crack: the traditional method of creating a mathematical cut and the finite element method taking into account the singularity. The advantage of using the finite element method considering the singularity is found. We used an example of a plate with a through straight rectilinear central crack with the equal biaxial tension to show that setting the boundary conditions at the top of the crack taking into account the singularity allows one to significantly reduce dimensions of a calculation scheme of the finite element method and keep the calculation accuracy. It is concluded that such a formulation can be applied in an elastic-plastic formulation. The comparison between the classical finite element solution and finite element with singularity is presented. The convenience of the finite element method with singular boundary conditions is demonstrated.

The study of laser deep penetration processing on prefabricated cracks of high carbon steel

This paper describes the testing of laser deep penetration processing on controlled fragmentation of high carbon steel (50SiMnVB). The objective of this research is to make use of defects in deep penetration zones and optimize the quality of prefabricated cracks. The research shows that a deeper central crack appear in the deep penetration zone when the heat input is 60 J/mm, and the appeared crack is affected by martensite in microstructure, high microhardness and a high magnitude of tensile stress. Furthermore, the fracture mode of the deep penetration zones is mixed-mode (ductile and brittle fracture).