Analytic solution of a system of differential equations describing the axisymmetric deformation of an orthotropic hollow spherical segment. (Q1588990)

In the paper a precise solution of the Timoshenko type equations is constructed for an orthotropic spherical segment under a load with principal vector \({\mathbf P}\), which is uniformly distributed along the parallel \(\alpha= \alpha_0\) and directed normally to the shell middle surface. The mathematical analogy to the solution of a similar problem in the Kirchhoff-Love theory [\textit{N. G. Gur'yanov}, A shallow orthotropic spherical segment under an axially symmetric local load, Proc. 17th Int. Conf. in the theory of shells and plates, Saratov, Vol. 3, 65--70 (1997)] is employed. Let \(R\) and \(H\) be the sphere radius and the thickness of the segment, respectively; \(Z\) -- the intensity of the transverse load with principal vector \({\mathbf P}\), distributed uniformly along the parallel \(\alpha= \alpha_0\); \(E_\alpha\) and \(E_\beta\) -- the elasticity moduli of the material in the meridional and parallel directions, respectively; \(\upsilon_\alpha\) and \(\upsilon_\beta\) -- the Poisson coefficients, \(G\) -- the transverse shear modulus, \(\omega\) -- the lag of the shell, \(\Phi\) -- the potential part of the transverse shear function: \(L\equiv {d^4\over d\alpha^4}+ {2\over \alpha} {d^3\over d\alpha^3}- {q^2\over \alpha^2} {d^2\over d\alpha^2} + {q^2\over \alpha^3} {d\over d\alpha}\), \(\nabla^2\equiv {d^2\over d\alpha^2}+ {1\over \alpha} {d\over d\alpha}\), \(q^2= {E_\beta\over E_\alpha}\), \(\upsilon_\beta= \upsilon_\alpha q^2\), \(D_\alpha= {E_\alpha H^3\over 12(1- \upsilon_\alpha \upsilon_\beta)}\), \(D_\beta= D_\alpha q^2\), \(c^2= {5\over 6} GH\). The system of equilibrium equations in terms of stresses and moments takes the following form [\((*)\) \textit{S. A. Ambartsumyan}, General theory of anisotropic shells (Russian), Nauka, Moscow (1974)(1964; Zbl 0114.16602)]: \({1\over\alpha} {d\over d\alpha} (\alpha T_\alpha)- {1\over\alpha} T_\beta= 0\), \({1\over\alpha} {d\over d\alpha} (\alpha M_\alpha)- {1\over\alpha} M_\beta- RN_\alpha= 0\), \({1\over \alpha} {d\over d\alpha}(\alpha N_\alpha)- (T_\alpha+ T_\beta)+ RZ= 0\). The stress function \(\varphi\) is introduced such that \(T_\alpha= {1\over R^2\alpha} {d\varphi\over d\alpha}\), \(T_\beta= {1\over R^2} {d^2\varphi\over d\alpha^2}\). Using the relations for the moments and the cross cutting force [\((*)\)]: \(M_\alpha= {D_\alpha\over R^2} ({d^2\over d\alpha^2}+ {\upsilon_\beta\over \alpha} {d\over d\alpha})(\Phi- \omega)\), \(M_\beta= {D_\beta\over R^2} ({1\over \alpha} {d\over d\alpha}+ \upsilon_\alpha {d^2\over d\alpha^2})(\Phi- \omega)\), \(N_\beta= {c^2\over R} {d\Phi\over d\alpha}\), the following system of the solving equations is obtained \[ D_\alpha L(\Phi- \omega)- R\nabla^2 \varphi+ R^4Z= 0,\;L(\varphi)- RHE_\beta\nabla^2 \overline\omega= 0,\;\nabla^2\Biggl(\Phi- {1\over c^2R} \varphi\Biggr)+ {R^2Z\over c^2}= 0. \] It is proved that this system is equivalent to the system of both equations \[ L(\Omega)- f^2 e^{2i\psi} \nabla^2\Omega- {R^4Z\over D_\alpha} e^{4i\psi}= 0,\;\nabla^2\psi+ {R^2 Z\over c^2}= 0, \] where \[ \Omega=\Phi- \omega- r^2 e^{2i\psi} \varphi,\;\psi= \Phi- {1\over c^2 R}\varphi, \] \[ r^2= {f^2\over E_\beta HR},\;\cos(2\psi)= {3E_\beta\over 5Gf^2},\;f^2= {qR\over H}\sqrt{12(1- \upsilon_\alpha \upsilon_\beta)}, \] as its general solution depends on eight arbitrary real constants.