PSI - Issue 28

R.M. Zhabbarov et al. / Procedia Structural Integrity 28 (2020) 1768–1773 Author name / Structural Integrity Procedia 00 (2019) 000–000

1769

2

of the multi-parameter description of the crack-tip fields is using the Williams series expansion. The coefficients of the Williams series expansion can be found experimentally (Ramesh and Sasikumar (2020), Jobin et al. (2020), Patil et al. (2017), Stepanova (2020), Ayatollahi (2011), Dolgich and Stepanova (2020), Stepanova and Roslyakov (2016)) and numerically. It should be noted that the second approach is widely used nowadays. Thus, Su and Feng (2005) evaluated leading coefficients of the Williams expansion by using the fractal finite-element method (FFEM). By means of the self-similarity principle, an infinite number of elements is generated at the vicinity of the crack tip to model the crack tip singularity. The Williams expansion series with higher-degree coefficients is used to capture the singular and non-singular stress behaviour around the crack tip and to condense the large amount of nodal displacements at the crack tip to a small set of unknown coefficients. New sets of coefficients up to the sixth degree for mode I and fourth degree for mode II problems are solved. The important fracture parameters such as stress intensity factors and T-stress can be obtained directly from the coefficients without employing any path independent integrals. Convergence study reveals that the present method is simple and very coarse finite element meshes with 12 leading terms in the William expansion can yield very accurate solutions. The effects of the influence of crack length on the higher-degree coefficients of some common plane crack problems are studied in detail. Shanlong et al. (2018) proposed a characteristic analysis method coupled with the finite element method to calculate stress intensity factors for V-notches, in which the notched structure is divided into a singular stress sector and the remained part. The asymptotic expansions are introduced into the singular stress region to transform elastic governing equations into characteristic ordinary differential equations. The established equations are solved by the interpolating matrix method to provide stress singular orders and characteristic angular functions. The asymptotic solution in the singular stress region is then coupled with finite element equations built on the remained structure by interfacial continuous conditions to establish the system equations. Thus, the amplitude coefficients in the stress asymptotic expansions can be determined. The notch stress intensity factors can be evaluated after the stress angular functions and amplitude coefficients being obtained. A symmetric V-notch and an inclined V-notch are respectively investigated to verify the accuracy of calculated notch stress intensity factors by the proposed method through comparing with the reference ones. Due to only conventional finite element method is needed, the proposed method can be freely implanted into commercial finite element software to analyze the singular stress for the V-notched engineering structures. This method can be also easily generalized to analyze the singular physical field for V-notches in composite materials. Li et al. (2019) proposed a new singular Voronoi cell finite element method (S-VCFEM) to study the initiation and propagation of interfacial debonding and matrix cracking in particulate reinforced composite. Composites with large scale inclusions and cracks can be easily simulated with this model. A new assumed stress hybrid variational functional is derived from the element energy function to accommodate the interface traction reciprocity on bonded interface and the interface traction-free on debonded interface and matrix crack surface. S-VCFEM is achieved through enhancements in stress functions in the assumed stress hybrid formulation. In addition to polynomial terms and interaction terms, singular stress terms of Williams expansion are added for stress functions, which are adaptively enriched to accurately capture the crack-tip stress concentrations. After obtaining the stress field, calculation of stress intensity factors is performed, using the least square method based on stress value obtained numerically in the S VCFEM. The efficiency of the method is verified against numerical results obtained by ABAQUS with XFEMmodule for the same problem considering a matrix-crack and a debonded interface.

Nomenclature ij 

stress tensor components around the crack tip

, r 

polar coordinates

m

k a

coefficients of the Williams series expansion

, I II K K , ( ) k m ij f  ( )

stress intensity factors

angular functions related to the geometric configuration, load and mode

m

index associated to the fracture mode





1 1 1 1 2 , ,...., K A a a a 

m

k a

the vector of unknowns



the vector consisting the numerical data