By G I Pshenichnov

This quantity provides the speculation of partial differential equations (PDEs) from a latest geometric viewpoint in order that PDEs could be characterised through the use of both means of differential geometry or algebraic geometry. this enables us to acknowledge the richness of the constitution of PDEs. It provides, for the 1st time, a geometrical concept of non-commutative (quantum) PDEs and offers a normal program of this thought to quantum box concept and quantum supergravity

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Extra resources for A theory of latticed plates and shells

Sample text

30) referring to the given lattice. 48) 22 Chapter 1. ' be an arbitrary oblique-angled system of coordinates at the middle surface relative to the system of coordinates in lines of curvature a, /? as shown in Fig. 6. Positive directions of linear forces and moments in the oblique-angled system of coordinates are shown in Fig. 7. The following formulae are derived from the equilibrium condition of an element of the calculation model's middle surface (S' = S[ = S'2): N[sinx N[ sin x == N1sm20 9 + N2 cosN22cos e 20-Ssin20, 22 2 = Nis\n N = JVisinXA + + JV2cos A + Ssin2A, 5sin2A, Ni&inx 2smx S'sinx = 5'sinx — AfjSinx = M2 sin x = tfjsinx X = tfjsinx = H'2s'mx = Ni sin Asin# — —N N22cos\cos0 cos A cos 5++ Ssm(0 Ssin(0— —X), A), Nism\sm0 {M\ cos A + Hi sin A)sin0 + (M2 sin A + H2cos A)\)cos6, [Mi cos 6, H\ sin 0) 6) sin A + (M2 sin 06 — H2 cos 0) 6) cos A, (Mi cos 0 — Hi {M2 - Mi) sin 0cos 0 + Hi sin2 0[M 6- H2 cos2 0, 2 2 (M2 — Mi)s'm A cos A — Hi sinsin A —cos H22cos2 A.

Now we obtain another version of t h e calculation model's constitutive equation. Let us assume t h a t coordinates a and /? coincide with t h e main lines of curvature of the calculation model's middle surface. 0. 50) 1-S2 Now we introduce small terms to constitutive Eq. 24) making it possible to strictly fulfil Eq. 50): Ni = Cuet S\ = CeiBi Ceeu + CeiEi + Ce C&2e2e22 -f Ce^ui + C12£2 e2 + Cl6u, +[(Dn S2 = = N2 = C 2 i£i + C22 2 2e£22 + C 22 6^, 6^, + / $> ) « , + (D62 - K$)K2')*» + (Dee - *£')■K^)r}k2, Cei£i + t/6 2 £ 2 + Ceeu Ceew + +[{Dn + 4 !

Coincide with t h e main lines of curvature of the calculation model's middle surface. 0. 50) 1-S2 Now we introduce small terms to constitutive Eq. 24) making it possible to strictly fulfil Eq. 50): Ni = Cuet S\ = CeiBi Ceeu + CeiEi + Ce C&2e2e22 -f Ce^ui + C12£2 e2 + Cl6u, +[(Dn S2 = = N2 = C 2 i£i + C22 2 2e£22 + C 22 6^, 6^, + / $> ) « , + (D62 - K$)K2')*» + (Dee - *£')■K^)r}k2, Cei£i + t/6 2 £ 2 + Ceeu Ceew + +[{Dn + 4 ! J ) « i + (D62 + K\$)K2')*2 + (Dee + K^)r]h. K^)r]h. 52) with a symmetrical matrix T (asterisk means a transposition).