PSI - Issue 2_A

Stepanova Larisa et al. / Procedia Structural Integrity 2 (2016) 1797–1804 Stepanova L.V., Roslyakov P.S., Lomakov P.N. / Structural Integrity Procedia 00 (2016) 000–000 3 of the tested procedures. According to Vesely et al. (2015) the second procedure provides a backward reconstruction of the crack-tip stress field analytically by means of truncate d Williams expansion. The developed procedures are based on the support of Java programming language and considerabl y simplify analyses of the mechanical fields description in a farther distance from the crack-tip. The research (Vesely et al. (2015)) is focused on optimization of selection of FE nodal results for improvement of accuracy of the approximation. In the present study the multiparametric description of the near crack tip fields under mixed mode loading is presented. The description of the stress field near the crack tip is based on the phototelasticity technique and finite element method. Characterization of crack tip stresses has been an area of active research for many decades (see, for instance, Willams (1957), Lychak and Holyns’kyi (2016), Stepanova (2008a), Hua et al. (2015), Berto and Lazzarin (2013) Stepanova (2009), Malikova and Vesely (2015), Sestakova (2013), Sestakova (2014), Stepanova (2008), Stepanova and Fedina (2008), Stepanova and Igonin (2015)). M. Williams in his landmark paper (Willams (1957)) showed that the crack tip stress fields in an isotropic elastic material can be expressed as an infinite series where the leading term exhibits a r − 1 / 2 singularity and the second term is independent of r . Since then, the Williams series expansion appears to be the most favored analytical tool for the description of mechanical fields near crack-tips in planar domains and presents a general framework for the description of the stre ss field in the vicinity of the crack tip in an isotropic linear elastic medium: f m , i j k ( θ ) angular functions depending on stress components and loading mode. Analytical expressions for circumfer ential eigenfunctions are available in (Hello et al. (2012)) f 1 , 11 k ( θ ) = k (2 + k / 2 + ( − 1) k ) cos( k / 2 − 1) θ − ( k / 2 − 1) cos( k / 2 − 3) θ / 2 , f 1 , 22 k ( θ ) = k (2 − k / 2 − ( − 1) k ) cos( k / 2 − 1) θ + ( k / 2 − 1) cos( k / 2 − 3) θ / 2 , f 1 , 12 k ( θ ) = k − ( k / 2 + ( − 1) k ) sin( k / 2 − 1) θ + ( k / 2 − 1) sin( k / 2 − 3) θ / 2 , f 2 , 11 k ( θ ) = − k (2 + k / 2 − ( − 1) k ) sin( k / 2 − 1) θ − ( k / 2 − 1) sin( k / 2 − 3) θ / 2 , f 2 , 22 k ( θ ) = − k (2 − k / 2 + ( − 1) k ) sin( k / 2 − 1) θ + ( k / 2 − 1) sin( k / 2 − 3) θ / 2 , f 2 , 12 k ( θ ) = k − ( k / 2 − ( − 1) k ) cos( k / 2 − 1) θ + ( k / 2 − 1) cos( k / 2 − 3) θ / 2 . (2) To each cracked geometry a sequence of coe ffi cients depending on the geometry and the system of loads exists. Analytical definitions of the coe ffi cients are widely available for the first two terms leading to the finite energy in the crack tip region (the stress intensity factors, the T -stress). However a higher order representation of the stress field around the crack or sharp notch requires determination of the coe ffi cients of higher order terms for the each cracked configuration. Recent investigations (Hello et al. (2012), Hello and Tahar (2014), Stepanova (2008a), Stepanova (2009)) revealed that non-singular terms can hav e a significant e ff ect on the stress field description for di ff erent cracked specimens. The coe ffi cients of crack tip stress expansions can be calculated analytically only for very simple cases. Thus it is important to have analytical expressions for the coe ffi cients of the higher-order terms of Williams expansion for di ff erent cracked specimens and loads (Holyns’kyi (2013), Mirlohi and Aliha (2013), Stepanova and Igonin (2014), Stepanova and Igonin (2015)). 2. Developments in the description of the stress field equati ons near the crack tip σ i j ( r , θ ) = 2 m = 1 ∞ k = −∞ a m k f m , i j k ( θ ) r k / 2 − 1 (1) with index m associated to the fracture mode; a m k coe ffi cients related to the geometric configuration, load and mode ;

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3. Photoelasticity method experiments and types of the test specimens investigated

In this study photoelastic and numerical analysis of a series of test specimens are performed. Using the photoe lasticity method the near crack tip fields in the cracked spec imens under mixed mode loading are obtained. The test specimens and isochromatic fringe patterns obtained experimentally are shown in Figs. 1 – 4. We analysed the plates