Healing of an Internal Plane Crack in a Transversely Isotropic Body
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HEALING OF AN INTERNAL PLANE CRACK IN A TRANSVERSELY ISOTROPIC BODY V. P. Sylovanyuk,1,2 N. A. Ivantyshyn,1 and T. I. Rybak3 We propose a mathematical model of healing of a crack located in the plane of isotropy of a transversely isotropic body subjected to the action of arbitrary external loads performed by gluing its faces. We solve the system of integral equations for the displacements of crack faces. For several cases of uniaxial tension of the body with filled crack, we obtain the analytic relations and plot the dependences of the stress intensity factor (SIF) on the parameters of the filler material and the elastic moduli of the body. It is shown that the SIF of the unfilled crack, unlike the case of filled crack, does not depend on the elastic constants of the material. Keywords: strength, crack, injection technologies, fracture, healing of the cracks.
Introduction The technology of injection hardening of cracked elements of building structures after long-term operation is frequently used in the engineering practice [1–3]. Some computational models and methods aimed at the evaluation of the serviceability of restored structural elements and their residual lifetime were proposed in [2–5]. These models were based on the assumption of homogeneity of the materials and the isotropy of their elastic properties. The models intended for the investigation of creep of the materials were developed in [6, 7]. This enables one to predict the long-term strength of the restored structural elements. In applying the technology of injection hardening to the reinforced structural elements, it is necessary to take into account the anisotropy of elastic properties or the inhomogeneity of materials. In what follows, on the basis of a model of the transversely isotropic body weakened by a crack, we propose a method for the evaluation of the efficiency of the hardening of this body by injections into the crack (healing). Basic Relations of the Linear Elasticity Theory for Transversely Isotropic Bodies [8] Without bulk forces, the solution of the problems of linear elasticity theory for transversely isotropic bodies is reduced to the determination of the components of the vector of displacements u j ( j = 1, 2, 3) from the bal-
ance equations
A11 1 2 3
∂ 2 u1 A11 − A12 ∂ 2 u1 ∂ 2 u1 ∂ ⎡ A11 + A12 ∂u 2 ∂u ⎤ + + A + + (A13 + A44 ) 3 ⎥ = 0 , 44 ⎢ 2 2 2 2 2 ∂x 2 ∂x 3 ⎦ ∂x1 ∂x 2 ∂x 3 ∂x1 ⎣
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected].
Pulyui Ternopil National Technical University, Ternopil, Ukraine.
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 54, No. 6, pp. 68–74, November–December, 2018. Original article submitted August 7, 2018. 1068-820X/19/5406–0827
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Springer Science+Business Media, LLC
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A11 − A12 ∂ 2 u 2 ∂ 2 u2 ∂ 2 u2 ∂ ⎡ A11 + A12 ∂u1 ∂u ⎤ + A11 + A44 + + (A13 + A44 ) 3 ⎥ = 0 , ⎢ 2 2 ∂x1 ∂x 2 2 ∂x1 ∂x 3 ⎦ ∂x 2 ⎣ ∂x 3 ⎛ ∂2 ∂2 A44 ⎜ 2 + 2 ⎝ ∂x1 ∂x 2
T. I
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