( ) ( ) ( ) [ ] ∫ − σ = σ + −µ µ + ∂ ∂ −µ x a xz * zz * x * dt Gh( x) G u x 0 1 1 1 2 1 1 1 1 1 1 1 1 , ( ) 0 1 2 2 1 1 2 2 1 1 1 1 1 1 1 1 = ⋅ − µ −µ − ∂ ∂ − µ µ − σ h( x) ) [u ] ( (u ) x G z * x * zz * , (1) where the symbols and ( describe the jumps and sums of a function on passing through the surfaces of inclusion, i.e. , * [ ] *) A]* + − + − = + = − [ A A , ( A) A A * z h( x ) A A =± ± = , u are the components of a displacements vector in inclusion; σ - are the constituents of a vector of stresses inside a defect. Eqn. 1 can describe all kinds of elastic inclusion deformations. If G or 1 1 x z ,u 0 1 1 zz xz , σ 1 = →∞ 1G the dependences for a cavity or an absolutely rigid inclusion respectively can be obtained from Eqn. 1. The system of dependences forms the mathematical model of elastic inclusion deformation. STRESSED STATE DETERMINATION IN A BODY WITH INCLUSION At first we present [2] the elastic problem for a body with thin inclusion as a composition of two problems: the problem a) for the homogeneous body under the given applied external loading inducing the vector of stresses ( ) o zz o xz o z , σ σ σ r , and the problem b) for the body with the cavity subjected to unknown stresses { =± −∞< <∞ y z h( x), } 1 z o z z ~σ =−σ +σ r r r on inclusion surfaces. We represent the displacement vector as the sum ,u ) u~(u~ ,u~ ) u (u ,u ) x z z u (u o z o x o x r r r + = . Using the supposition about thinness of inclusion we can replace [1] a task b) by the singular problem c) for the body with the mathematical cut { −∞< < y }∞ ) ≤ x a, with the stresses ~ ( ~ , ~ zz xz z σ σ σ r applied to it surfaces. The solution of problem a) is known: ) p ( [u ] p( )h( x)/G * o zz * o z 2 1 σ = = −µ ; . (2) Using Fourier integral transformation the solution of the problem c) can be obtained [3] in kind of such dependences concerning the stress and displacement jumps on the inclusions surfaces. ; ' ∫ ∫ − − − σ π −µ − µ + π −µ − σ = a a xz * a a z * zz * dt t x [ ~ ] ) ( dt t x [u~ ] ) ( G ( ~ ) 2 1 1 2 1 ( ) ∫ ∫ − − − σ π −µ − µ − − ′ π −µ − µ ′ = a a xz * a a z * x * dt t x [ ~ ] ) G( dt t x [u~ ] ) ( u~ 4 1 3 4 2 1 1 2 . (3) Here and further symbol prime means the derivate on x. Substituting Eqn. 3 to Eqn. 1 we obtain the system of singular integro-differential equations for unknown vectors z * [ ~ ] σ r and * [u~ ] r in such form ∫ ∫ ∫ − − − =− σ + − σ + − ′ a a a a xz * a a xz * z * dt B [ ~ ] dt M ; t x [ ~ ] dt B t x [u~ ] B 1 13 12 11 , x a, M , z h( x) [u~ ] dt B t x [ ~ ] dt B t x [u~ ] B a a z * a a xz * z * ≤ = =− − − σ + − ′ ∫ ∫ − − 0 2 23 22 21 (4)
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