PSI - Issue 39

O.N. Belova et al. / Procedia Structural Integrity 39 (2022) 770–785 Author name / Structural Integrity Procedia 00 (2021) 000–000

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1. Introduction Analytical determination of the near crack tip fields in an isotropic linear elastic material is an integral task of the fracture mechanics (Hou et al. (2021), Qiu et al. (2021), Stepanova and Dolgikh (2021), Mirzaei et al. (2020), Ramesh and Sasikumar (2020), Jobin et al. (2020), Vivekanandan and Ramesh (2019), Dolgikh and Stepanova (2020), Patil et al. (2017), Tabanyukhova (2020), Stepanova (2020), Ayatollahi et al (2011), Stepanova and Dolgikh (2018)). The near crack-tip fields in the immediate vicinity of a crack tip can be found analytically by theoretical method of solid mechanics (Hello et al. (2012), Hello (2018), Stepanova and Roslyakov (2016)), numerically by, for instance, finite element method (Stepanova (2020b)) and experimentally by means of optical measurements using interference-optical methods such as caustic, digital holography, moire, digital photoelasticity, speckle interferometry or digital image correlation method (Vivekanandan and Ramesh (2019)). The pivotal goal of the present research is to determine the stress field in the vicinity of the crack tip in the plate with two interacting cracks in an isotropic linear elastic material experimentally by the digital photoelasticity method and computationally by finite element analysis based on the multi-parameter Williams power series expansion keeping the higher-order terms. The usage of the multi-parametric representation of the stress, strain and displacement fields is not just for academic curiosity but an urgent need in many cases of engineering interest ((Jobin et al. (2020), Yang et al. (2021), Vivekanandan and Ramesh (2019), Dolgikh and Stepanova (2020), Patil et al. (2017), Tabanyukhova (2020), Stepanova (2020), Ayatollahi et al. (2011), Stepanova et al. (2017)). The effects and importance of higher-order terms in the Williams series expansion of the crack-tip fields were thoroughly analysed for different cracked specimens by different authors (Jobin et al (2020), Patil et al (2017), Malikova and Vesely (2014)). Currently the multi-point over-deterministic technique for evaluating the multi parameter stress field is used (Yang et al (2021)). The over-deterministic approach can use the experimental measurements of the stress or displacement fields, for instance, by the interference-optical methods of measurements (Dolgikh and Stepanova (2020)), or by the finite element analysis (Li and Zheng (2021)). Nonetheless, many questions both in digital image processing procedures and in the technique of the multi-point over-deterministic method are still unanswered and it is necessary to fill these lacunes. Thus, the aim of the paper is to obtain the stress fields near the crack tip based on the stress data obtained experimentally via optical measurements and to compare the stress field approximations with the stress field derived from finite element analysis. Experimental data such as isochromatic phase maps and isoclinic phase maps obtained from the photoelasticity observation are taken as inputs. It should be noted that digital photoelasticity is an experimental technique used by many engineering applications to evaluate the stress fields in bodies under mechanical loads (Ramesh (2000), Ramesh et al. (1997)). The photoelasticity method is currently undergoing a Renaissance (Ramesh (2021), Su et al. (2021), Liu et al. (2020)). After being developed and then largely abandoned in 2000-2010, the photoelasticity method is now in active use and can be considered as a viable technique for determining stress fields in a structural component. Interest in using the digital photoelasticity is now being fueled by possibility of digital processing of the entire set of experimental information. Owing to the increase in computing resources the digital photoelasticity technique is now one of the powerful tools for investigating the stress field in solids. The advancement of computers coupled with developments in digital image processing has had a great influence in developments of modern photoelasticity (Su et al (2021), Ramesh and Sasikumar (2020)). Hence, one can make a conclusion that the digital optics revolution in recent years enhanced and empowered innovation in all areas of modern life (Ramesh (2020), Ramesh (2021), Ramesh (2015)) and, particularly, photoelasticity is a well-developed method for reliable measurement of stress and strain distributions in engineering practice. In this study for a better understanding of the multi – parameter approximation of the stress and displacement fields in the vicinity of the tip of cracks and notches the experimental technique of photoelasticity has been utilized for calculating the coefficients of higher order terms of the multi-parameter stress field. In the present study a new type of the cracked specimen is presented. This type of the cracked specimen is useful for study of the mixed mode loading. Then, the experimental photoelasticity results were compared with the corresponding values obtained from finite element analysis and a good correlation was observed. The Williams model is based on linear elastic fracture mechanics and uses the mathematical form of an infinite series to describe the crack tip stress field distribution. According to the model the crack tip stress field can be expressed as a function of the number of terms in the series at the region surrounding the crack tip: ( ) 2 / 2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f σ θ θ = =∞ − = =−∞ = ∑ ∑ (1)