PSI - Issue 13

Ivan Shatskyi et al. / Procedia Structural Integrity 13 (2018) 1482–1487 Author name / Structural Integrity Procedia 00 (2018) 000–000

1483

2

1. Introduction For contemporary material science the problem of the extension of working life of responsible structures is relevant. An efficient means of renovation of damaged products is injectable technology of healing crack-like defects (Marukha et al. (2014)). Filling the crack with another material may significantly lessen the load of the area near its peaks. Yet, the filler of the crack, by lessening the load of its neighbourhood, receives itself part of the external load. That is why, taking into consideration the concentration of the stress in the reinforcement is an obligatory element of forecasting the firmness of renewable. The equilibrium of bodies with cracks filled with compliant material is often viewed through Winkler layer model, and the problem is reduced to solving integrodifferential equations in relation to displacement jumps on the cuts. A lot of plane and spatial problems have been solved in these articulations by Marukha et al. (2014), Kurshin and Suzdal'nitskii (1973), Sotkilava and Cherepanov (1974), Panasyuk et al. (1986).The model of the crack partially healed by soft material was suggested in the paper by Shats’kyi (2015). In terms of problems of deformation of thin-walled structures there are papers by Khachikyan (1970), Grilitskii and Sulim (1975), Dragan and Opanasovich (1979), Grilitskii et al. (1979), Popov (1982), Aleksandrov and Mkhitaryan (1983), Mura (1988), Sulym (2007), where thin inclusions with arbitrary rigidity were examined and papers by Osadchuk (1985), Shatskii (1989), Shats’kyi and Perepichka (2004), Shats’kyi and Makoviichuk (2005), Zehnder and Viz (2005), where the ultimate equilibrium of plate and shells in multiparameter loading were investigated. In this article, we consider the problem of elastic and critical equilibrium of the plate, weakened by a rectilinear slit, filled with low-modulus medium, on conditions of simultaneous impact of tension, shear, bending and twisting. The aim of the research is to analyse in detail the critical equilibrium of the composition, considering the strength of all components. 2. Model and method 2.1. Statement of the problem Let us examine an infinite plate 2 ( , , ) [ , ] x y z h h    R , which contains a through crack-like defect, namely: a slit 2 l long and 2 ( ) b x wide. The slit is fully filled with injection material, which presumably is a lot more compliant than the material of the plate: E 0 << E . This composition is subject to constant membrane forces of tension n , shear s , bending m and twisting p moments, which are uniformly distributed on the infiniteness. We investigate the influence of the low-modulus filler on the elastic and limiting equilibrium of the plate with a crack. We will conduct the analysis in the framework of classical theory of thin plate. To model the filler layer, we will use Winkler model, according to which stresses in the inclusion are proportional to displacements jumps on the slit edges. The boundary value problem under such conditions will be as follows: compatibility and equilibrium equations: 0    , 0 w   ,   2 , \ x y L  R ; (1) boundary conditions on the cut:

  x u b x

  x b x 

  b x   y u

  b x    y

  , x l l   ;

0 1 2

0 3 2

0 2 2

0 4 2

0 y  ,

,

,

,

(2)

,

N B 

y M B  

xy N B 

xy M B  

y

 

 

conditions on infiniteness: 0

x N  , xy N s  , y N n  , (3) Here  , w are function of stresses and plate flexure,  is two-dimensional Laplace operator, N x , N y , N xy are membrane forces, M x , M y , M xy are bending and twisting moments, [ u x ], [ u y ] are discontinuities of displacements in the medsurface, [ ϑ x ], [ ϑ y ] are ruptures of rotation angles of the normal on the cut; 0 1 0 2 B E h  , 0 2 0 2 B G h  , 0 3 3 0 2 / 3 B E h  , 0 3 4 0 2 / 3 B G h  , 0 0 0 / (2(1 )) G E    , 0 x M  , y M m  , xy M p  ,   , x y  .