PSI - Issue 2_A

Larisa Stepanova et al. / Procedia Structural Integrity 2 (2016) 1789–1796 Stepanova L.V., Roslyakov P.S. / Structural Integrity Procedia 00 (2016) 000–000

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with index m associated to the fracture mode; a m

k coe ffi cients related to the geometric configuration, load and mode;

f m , i j k ( θ ) angular functions depending on stress components and loading mode. Analytical expressions for circumfer ential eigenfunctions are available [Hello, Tahar and Roelandt (2012)]. To each cracked geometry a sequence of coe ffi cients depending on the geometry of the domain, the nature and intensity of the load exists. Analytical defini tions 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 performed by Hello, Tahar and Roelandt (2012), Hello and Tahar (2014) revealed that non-singular terms can have 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 ana lytical 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)]. The present paper is aimed at analytical determination of coe ffi cients in crack tip expansion for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. From practical point of view it is very important to know 1) analytical dependence of coe ffi cients on geometrical parameters of specimens and on applied loads; 2) how many terms of the asymptotic expansions we have to keep in the complete asymptotic expansion of the stress and displacement fields in the neighborhood of the crack tip. The analytical dependence of the coe ffi cients on the geometrical parameters and the system of loads for two finite cracks in an infinite plane medium is given. It is shown that at large distances from the crack tips the e ff ect of the higher order terms of the Williams series expansion becomes more considerable. The knowledge of more terms of the stress asymptotic expansions will allow us to approximate the stress field near the crack tips with high accuracy. Complex variable theory presented in Muskhelishvili (1953) provides a very convenient way to solve many prob lems in plane elasticity. The approach is based on consideration of the Airy stress function Φ ( x 1 , x 2 ) defined as σ 11 ( x 1 , x 2 ) = Φ , 22 , σ 22 ( x 1 , x 2 ) = Φ , 11 , σ 12 ( x 1 , x 2 ) = − Φ , 12 . Following the Koloso ff – Muskhelishvili formalism [Muskhelishvili (1953)] one can obtain the solution of the biharmonic equation in terms of two analytic functions of the complex variable z = x 1 + ix 2 in the form Φ ( x 1 , x 2 ) = Re z ϕ ( z ) + χ ( z ) . Using the Koloso ff - Muskhelishvili relation the in-plane Cartesian stress components σ i j are defined in terms of the complex analytical potentials ϕ � ( z ) and χ � ( z ) as σ 11 ( z ) + σ 22 ( z ) = 4 Re ϕ � ( z ) , σ 22 ( z ) − σ 11 ( z ) + 2 i σ 12 ( z ) = 2 z ϕ �� ( z ) + χ � ( z ) . (2) In the case of biaxial symmetric problems the solution (2) has the form σ 1 11 ( z ) = 2 Re ϕ � 1 ( z ) − 2 x 2 Im ϕ �� 1 ( z ) + ( α − 1) σ ∞ 22 / 2 , σ 1 22 ( z ) = 2 Re ϕ � 1 ( z ) + 2 x 2 Im ϕ �� 1 ( z ) − ( α − 1) σ ∞ 22 / 2 , σ 1 12 ( z ) = − 2 x 2 Re ϕ �� 1 ( z ) . (3) The complex potential ϕ � 1 ( z ) for two collinear cracks of equal lengths in the infinite plate under remote loading is given by the formula [Muskhelishvili (1953)] ϕ � 1 ( z ) = ( σ ∞ 22 / 2)( z 2 − c ) / z 2 − a 2 z 2 − b 2 + ( α − 1) σ ∞ 22 / 4 , c = b 2 E ( π/ 2 , k ) / F ( π/ 2 , k ) , k = 1 − a 2 / b 2 , (4) where F ( π/ 2 , k ) , E ( π/ 2 , k ) are the complete elliptic Legandre of the first and second type. For pure mode II problems the stress state depends on the complex potential ϕ � 2 ( z ) : σ 2 11 ( z ) = 4 Re ϕ � 2 ( z ) − 2 x 2 Im ϕ �� 2 ( z ) , σ 2 22 ( z ) = 2 x 2 Im ϕ �� 2 ( z ) , σ 2 12 ( z ) = − 2 Im ϕ � 2 ( z ) − 2 x 2 Re ϕ �� 2 ( z ) + σ ∞ 12 , (5) where the complex potential ϕ � 2 ( z ) for an infinite plate with two collinear cracks of equal lengths has the form ϕ � 2 ( z ) = − i ( σ ∞ 12 / 2) z 2 − c / z 2 − a 2 z 2 − b 2 + i σ ∞ 12 / 2 . (6) 2. Complex representation of the solution for two collinear cracks of equal lengths in an infinite plane under remote loading