Investigation of solutions, positive with respect to curvature, of a system of equations of the equilibrium of a closed cylindrical shell. (Q1603131)

The convex forms of equilibrium of a closed infinitely long elastic cylindrical shell are investigated in the case when the bending stiffness tends to zero, external loads are tangent, outer moments are absent, the median surface is non stretched and the initial form of the shell is circular. Under these assumptions the equations of static equilibrium of the cylindrical shell are [see \textit{K. Z. Galimov}, Foundations of the nonlinear theory of thin shells, Kazan Univ. Press, (1975)]: \[ {dT\over ds}+ k(s) N(s)= 0,\;{dN\over ds}- k(s) T(s)- f(s)= 0,\;{dM\over ds}- N(s)= 0,\tag{1} \] where \(T(s)\), \(N(s)\) are the tangent and cutting forces, respectively, at the point of the contour of the deformed cylinder with arc coordinate \(s\); \(f(s)\) is the linear density of the normally acting load; in the motion along the increase of \(s\) the external normal is directed to the right; \(M(s)\) is the bending moment connected with the curvature of the directrix \(k(s)\) via the Loeve formula \[ M(s)= D(k(s)- k_u (s)),\tag{2} \] \(D\) is the bending stiffness, \(k_u(s)\) -- the curvature of the cylinder contour in the nondeformed state. Let \(s\) vary within the limits of the interval \([0,2]\), then \(k_u (s)= \pi\). In addition to (1) and (2) the conditions of closeness \[ \int^2_0 \cos\Biggl(\int^s_0 k(\xi)\,d\xi\Biggr)\,ds= \int^2_0 \sin\Biggl(\int^s_0 k(\xi)\,d\xi\Biggr)\,ds= 0,\tag{3} \] and the condition upon the rotation angle of the tangent within the complete path-tracing \[ \int^2_0 k(\xi)\,d\xi= 2\pi\tag{4} \] are imposed. Only classical solutions of the problem (1), (2) are considered and it is assumed that \(k(s)\in C_2[0,2]\). From the last equation in (1), by excluding \(M\) by means of (2), it follows that \(N(s)= Dk' (s)\), and then from the equation in (1) it is obtained that the value \[ \omega= {T(s)\over D}+{1\over 2} k^2(s)\tag{5} \] is a constant independent of \(s\). The second equation of system (1) is transformed into the equation \[ k^2(s)+ {1\over 2} k^3(s)- \omega k(s)= {1\over D} f(s).\tag{6} \] Further the load \(f(s)\) is supposed to be symmetric with respect to \(s= 1\). For example it is considered the deformation under symmetric flow-around of an elastic cylinder which is detained in the flow by a rigid restraint at the point \(s= 0\). It is underlined that the assumption is introduced for a simplification of proofs -- the extending of the obtained results to a more general case has no any difficulties. For a symmetric with respect to \(s=1\) solution \(k(s)\) conditions (3), (4) take the form \[ \int^1_0\cos\Biggl(\int^s_1 k(\xi)\,d\xi\Biggr)\,ds= 0,\quad k^1(1)= 0,\quad \int^1_0 k(s)\,ds= \pi.\tag{7} \] With application of the Leray-Schauder theory it is proved the following Theorem. Let \(f(s)\leq 0\), \(f\in C^1[0,1]\). Then the problem (6), (7) has a nonnegative solution.