CLOSED CONVEX SHELLS Meunargia T. Mixed forms of stress-strain relations are given in the form. λ + 2µ θ + 1

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1 Seminar of I. Vekua Institute of Applied Matematics REPORTS, Vol. 43, 207 CLOSED CONVEX SHELLS Meunargia T. Abstract. If Ω is a closed convex sell, ten S : x 3 = 0 is an ovaloid. It is proved tat in tis case te equation of equilibrium may ave only te unique regular solution and ence, te corresponding omogenous equation as no non-zero solution on S. Keywords and prases: Closed sells, stress and strain tensors. AMS subject classification (200): 74B20. Mixed forms of stress-strain relations are given in te form σ i j = θg i j + 2µe i j (i = j =, 2, 3) () σ i j and e i j are te mixed components, respectively, of stress and strain tensors, θ is te cubical dilatation wic will be written as wen j = 3 from () we ave from (3) θ = e i i = θ + e 3 3, θ = e α α, (α =, 2) (2) σ α 3 = 2µe α 3, σ 3 3 = θ + 2µe 3 3 = θ + ( + 2µ)e 3 3. (3) By inserting (4) into (2) we obtain e α 3 = 2µ σα 3, e 3 3 = + 2µ θ + + 2µ σ3 3. (4) = 2µ + 2µ. Substituting expression (5) into () we get σ i j = T i j + Q i j = θ = θ + + 2µ σ3 3, (5) ( θ + + 2µ σ3 3 ) g i j + 2µe i j T α β = θ g α β + 2µe α β, Q α β = σ σ 3 3g α β, T i 3 = 0, Q i 3 = σ i 3, and σ = + 2µ. Te vector T α satisfies te condition nt α = 0 and is terefore called te tangential stress field and te vector Q i will be called te transverse stress field.

2 Closed Convex Sells 7 Te vectorial equation of equilibrium as te form i ( σ i ) + Φ = 0, ( = aϑ, ϑ = 2Hx 3 + Kx 2 3) H and K are respectively, middle and principal curvatures of te surface S, may be written as [ α T α ) + i ( Q i ) + Φ = 0 (6) Let te surface Ŝ : x3 = const be te neutral surface of a non-sallow sell. Ten T α = 0, i.e. T αβ = 0 (on Ŝ), and equation (6) becomes α ( aϑq α ) + 3 (ϑσ 3 ) + ϑp i = 0, or [ α (ϑq α ) + 3 (ϑσ 3 ) + ϑφ x 3 =c = 0; ( x3 = x 3 ) 2 is te tickness of sell and Q α = σ σ 3 3r α + σ α 3 n, α ( ) = a α ( a(.)). Denoting te stress forces acting on te face surfaces S + and S by (+) P we ave (+) P = (σ 3 ) x 3 =, P = (σ 3 ) x 3 =, (7) If we approximately represent σ 3 by te formula From (7) we get σ 3 (x, x 2, x 3 ) = (0) σ (x, x 2 ) + x 3 () σ (x, x 2 ). or [ (+) P P + x3 σ 3 (x, x 2, x 3 ) = 2 = [ + x 3 2 σ 3 (x, x 2, x 3 ) = 2 [ + x 3 ((+) P + P ) ( (+) P P ) + 2x3 (p α r α + p 3 n) + 2x3 p α = P (+) α P (+) α, p = P 3 P 3. P P,

3 72 Meunargia T. Ten to define te vector field (+) P we ave te equation { α (σ A α βpr β + Ap α n) + B(pn + p α r α ) + Φ} x 3 =c = 0 (8) A α β = + c [aα β + c(b α β 2Ha α β), A = + c ϑ(c) B = [ 2H + 2(K 2H)c + 3kc2, Φ = 2ϑ(c)Φ(c) + α {σ 2c [aα β + c(b α β 2Ha α β) From (8) we ave + 2 [ϑ(c) + 2(kc ) p. p 3 r β + 2c ϑ(c) p α n} σ α (A α βp) + (Ba αβ Ab αβ )p α + Φ β = 0, Φβ = φr β, (9) α (Ap α ) + (σ A α βb β α + B)p + Φ 3 = 0, ( Φ 3 = Φn). (0) From te system of equation (9) we ave p α = P (+) α P α = d αβ [ γ (A γ β p) + Φ (+) β, p = P 3 P 3 () ˆd αβ = [(B 2AH)aαβ + Ab αβ ˆFβ = [ Φ β + α (A α βp), = B 2 2ABH + A 2 K. Inserting expressions () into (0) we obtain te equation σ α [A d αβ γ (A γ β p) (B + σ A α βb β α)p + Φ = 0. (2) It is easily seen tat equation (2) is of te elliptic type. Tus, if te surface x 3 = c is neutral ten te stresses (+) P applied to te face surfaces, must satisfy te vector equations (9) and (0). Tis means tat te stresses (+) P cannot be prescribed arbitrarily bot at te same time. However tere are problems wen tis does not occur. For example, in aircraft or submarine apparatus te force P acting on te inner face surface S may be assumed to be prescribed, but te force (+) P acting on te external face surface S + is not, in general, assigned beforeand. Te same situation occurs on dams. One face surface of te dam is free from stresses and te oter is under te ydrodynamic load, a variable wic is generally difficult to define exactly at any moment of time.

4 Closed Convex Sells 73 For closed convex sells wen x 3 = c is te middle surface (i.e. c = 0 x 3 = 0) te omogenous equation (2) may be written in te form α (d αβ β u) d 2 u = 0, (3) d αβ = [a αβ ( 2H) + b αβ 2H + K 2 + 4H(2H ), d 2 = σ [ 2( σ )H > 0, ( σ = ) + 2µ a αβ = r α r β, r α r β = a α β = σ α β, 2H = b + b 2 2, K = b b 2 2 b 2 b 2, I = a αβ dx α dx β, a αβ = r α r β, II = b αβ dx α dx β, b αβ = n α r β. Let u be te regular solution (3) on S(x 3 = 0), i.e. u is te continuous function of te point of te surface S and as continuous partial derivatives wit respect to Gausian coordinates of tis surface. We represent te surface S as S = S S 2, S and S 2 are parts of te surface wit no common points S S 2 = = Ø. Let L be te common boundary of S and S 2. Denote te tangential normal to L by l directed to S. Multiplying bot sides of equation (3) by u, we may rewrite it as α (ud αβ β u) d αβ α u β u d 2 u 2 = 0. Integrating tis equality wit respect to te surfaces S and S 2, and ten applying Green s formula, we ave ul α d αβ β uds (d αβ α u β u + d 2 u 2 )ds = 0, L S L ul α d αβ β uds S 2 (d αβ α u β u + d 2 u 2 )ds 2 = 0. By adding tese equalities we obtain (d αβ α u β u + d 2 u)ds = 0. (4) s Since d αβ α u β u 0, d 2 > 0 from (4) is follows tat u = 0, wic was to be proved. Te problem under consideration is tus reduced to te determination of te globally regular particular solution of te non-omogeneous equation σ α (d αβ β p) [ 2( σ )Hp + Φ = 0. (5)

5 74 Meunargia T. It remains to sow tat if equation (5) as globally regular solution, ten te middle surface S : x 3 = 0 of te sell is neutral. To do tis we ave to sow first tat te tangential stress field vanises on S, i.e. it sould be sown tat te equation α ( at α ) α ( at αβ r β ) = 0 (6) a a as no globall solution, except trivial T αβ = 0. It is evident since, wit respect to isometric-conjugate coordinates x, y, equation (6) is equivalent to te omogeneous generalized Caucy-Riemann equation z w B w = 0, (z = x + iy) w = 2 ak 4 (T T 22 2iT 2 ), T + T 22 = 0. (7) Te complex stress function w is continuous on te wole plane E of te complex variable z = x + iy and at infinity satisfies te condition w = 0( z 4 ). Tis implies, in view of te generalized Liouville teorem, tat w = 0. ten from (7) it follows tat T αβ = 0, wic was to be proved. Acknowledgment. Te designated project as been fulfilled by a financial support of Sota Rustaveli National Science Foundation (Grant SRNSF/FR /358/5-09/4). R E F E R E N C E S. Vekua I. N. Sell Teory: General Metods of Construction, Pitman Advanced Publising Program. Boston-London-Melburne, 985. Received ; revised ; accepted Autor s address: T. Meunargia I. Vekua Institute of Applied Matematics of I. Javakisvili Tbilisi State University 2, University St., Tbilisi 086 Georgia tengiz.meunargia@viam.sci.tsu.ge