BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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18 Ion Crăciun and 0 0 0 2 2 ⎧⎪ Ω = 2 4+ 4 α ∇0, ⎨ 0 0 ⎪⎩ Θ = ( β + γ−ε) 2 −4α 2 . Writing the amplitudes of displacement and rotations appearing in equations (30) in terms of reduced Galerkin vectors Φ= Φθ (,) rzeθ , Ψ = Ψr(,) rzer + Ψz(, rz) ez, we have 0 0 ⎧uθ = 4Φθ −2 α 3( ∂zΨr −∂rΨz), ⎪ 0 0 0 2 ⎨ϕr = 2 α∂ zΦθ + 2 3Ψr − [( β+ γ−ε) 2−4 α ] ∂rΨ , ⎪ ⎪⎩ ϕ =− 2 α( r +∂ ) Φ + Ψ − [( β+ γ−ε) −4α −1 2 z r θ 2 3 z 2 ] ∂ zΨ , (35) where −1 ⎧ div Ψ = ∇· Ψ = r ∂ r( rΨr) +∂zΨz = Ψ, ⎪ ⎨rot Ψ = ∇× Ψ = ( ∂zΨr −∂rΨz) eθ , ⎪ ⎩grad Ψ= ∇ Ψ= ∂ rΨer +∂zΨez. Equations for the stress functions Φ θ , Ψ r and Ψ z are obtained from Eqs. (13) by taking into account both the cylindrical coordinates and the form of fields in the case of the second axially-symmetric problem. We have 0 ⎧Ω Φθ + X θ = 0, ⎪ 0 0 ⎨ 3 Ω Ψ r + Yr = 0, (36) ⎪ 3Ω Ψ z + Yz = 0. ⎩ Therefore, if the set of functions { Φ θ , Ψr , Ψz} is a solution in B of the system (36), then (35) is a solution in B of the second axially-symmetric problem (33). The strain fields in B are given by the matrices ⎛ 0 γrθ 0 ⎞ ⎛κrr 0 κ rz ⎞ ⎜ ⎟ ⎜ ⎟ γ = γθr 0 γθz , κ = 0 κθθ 0 , ⎜ 0 γzθ 0 ⎟ ⎜ ⎝ ⎠ ⎝κzr 0 κ ⎟ zz ⎠ whose elements are given by the geometric relations in B γ = γ =∂ u − φ γ = φ −r u −1 rθ r θ z , θr z θ , γ =∂ u + φ , γ =−φ , zθ z θ r θz r κ =∂ φ , κ =∂ φ , rz r z zr z r −1 ( κrr , κθθ , κzz ) = ( ∂rφr , r φr , ∂zφz ). The stress fields in B are defined by the matrices (37) (38)
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 19 ⎛ 0 σrθ 0 ⎞ ⎛μrr 0 μrz ⎞ ⎜ ⎟ ⎜ ⎟ σ = σθr 0 σθz , μ = 0 μθθ 0 . ⎜ 0 σ 0 ⎟ ⎜ ⎝ zθ ⎠ ⎝μzr 0 μ ⎟ zz ⎠ (39) Knowing the amplitudes of the rotations in (32), we determine the stresses the constitutive relations in B σij φ , φ and the displacement u θ given r z and the couple-stresses ⎧( σrθ, σθr) = ( μ+ α)( γr θ, γθr) + ( μ−α)( γθr, γr θ), ⎪( σθz, σzθ ) = ( μ+ α)( γθz, γz θ) + ( μ−α)( γz θr, γθz), ⎨ ⎪( μrz , μzr ) = ( γ+ ε)( κrz , κzr ) + ( γ−ε)( κzr , κrz ), ⎪ ⎩( μrr , μθθ ) = (2 γrr + βκ, γθθ + βκ), μzz = γzz + βκ, where κ = κ + κ + κ = ∇·. φ rr θθ zz 6. Action of Body and Couple-body Loadings Let us consider the special case of body loadings given by μ ij by means of (40) X= (0,0, X ), Y = (0,0, Y ). (41) z The loading X z is connected with the triple ( ur , φθ , uz ) which appears in the first axially-symmetric problem characterized by equations (32), so that only the body moment Y z enters in the second axially-symmetric problem. This situation may imply that only the component Ψ z does not vanish identically in relation (36). Consequently, we obtain the following representation for the triple ( uθ , φr, φ z) in B : z where the stress function 0 ⎧ uθ = 2 α 3 ∂rΨz, ⎪ 0 2 2 ⎨φr = − [( β+ γ−ε) 2 −4 α ] ∂rzΨz, . ⎪ 2 2 ⎩φz = { 2 3− [( β+ γ−ε) 2−4 α ] ∂z z} Ψz, Ψ z satisfies (42) 3 Ψ z Yz 0. Ω + = (43) Introducing another version of the stress function of that kind is connected with body loadings of the form X= ( X ,0,0), Y = ( Y ,0,0). (44) r r
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