Abstract
In the present chapter prismatic and cusped prismatic shells are exposed. Relation of the prismatic shells to the standard shells and plates are analyzed. Cusped beams are defined. The Lipschitz boundaries are defined. In a lot of figures 3D illustrations of the cusped prismatic shells and beams are given. Typical cross-sections of cusped prismatic shells and longitudinal sections of the cusped beams are illustrated. Moments of functions and their derivatives are introduced and their relations clarified.
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Notes
- 1.
The boundary \(\partial\Upomega\) of an open bounded subset \(\Upomega\) of \({{\mathbb{R}}}^{n},\) \(n\ge 2,\) is said to be Lipschitz (Lipschitz-continuous) boundary Lipschitz boundary if the following conditions are simultaneously satisfied [see e.g. Ciarlet (1988)]: there exist constants \(\alpha>0,\) \(\beta>0,\) \(\gamma>0\) and a finite number of local coordinate systems with origins \(O_r,\) \(r=\overline{1,R},\) coordinates \(\zeta'_r:=(\xi_1^r,\xi_2^{r},\ldots,\xi_{n-1}^r),\) and \(\zeta_r:=\xi_n^r,\) and corresponding functions \(a_r,\) \(r=\overline{1,R},\) such that:
$$ \partial\Upomega={\mathop\cup\limits_{r=1}^R}\{(\zeta^{\prime}_r,\zeta_r):\zeta_r=a_r(\zeta^{\prime}_r), \;\;\vert\zeta^{\prime}_r\vert<\alpha\}; $$$$ \begin{aligned} & \{(\zeta^{\prime}_r,\zeta_r):a_r(\zeta^{\prime}_r)<\zeta_r<a_r(\zeta^{\prime}_r)+\beta,\;\;\vert\zeta^{\prime} _r\vert\le\alpha\}\subset\Upomega,\;\;\; r=\overline{1,R}; \\& \{(\zeta^{\prime}_r,\zeta_r):a_r(\zeta^{\prime}_r)-\beta<\zeta_r<a_r(\zeta^{\prime}_r),\;\;\vert\zeta^{\prime} _r\vert\le\alpha\}\subset{{\mathbb{R}}}^{n}- \overline{\Upomega},\;\;\; r=\overline{1,R}; \\& \vert a_r(\zeta^{\prime}_r)-a_r(\eta^{\prime}_r)\vert\le\gamma\vert\zeta^{\prime}_r-\eta^{\prime}_r\vert\;\;\; {\hbox{for all}}\;\;\; \vert\zeta^{\prime}_r\vert\le\alpha,\;\;\vert\eta^{\prime}_r\vert\le\alpha,\;\;r=\overline{1,R}, \end{aligned} $$where the last inequalities express the Lipschitz-continuity of the functions \(a_r,\) \(r=\overline{1,R}.\) The Lipschitz boundary \(\partial\Upomega\) is necessarily bounded, while this is not necessarily true of the set \(\Upomega\) which can be interchanged with the set \({{\mathbb{R}}}^{n}\backslash\overline{\Upomega}\) in the definition. Such a set is called a Lipschitz set.
- 2.
\(C(\overline{\omega})\) denotes a class of functions continuous on \(\overline{\omega};\) \(C^{2}(\omega)\) denotes a class of twice continuously differentiable functions with respect to the variables \(x_{1}\) and \(x_{2}\) with \((x_{1},x_{2})\in\omega.\)
References
P.G. Ciarlet, Mathematical Elasticity, Vol I: Three-dimensional elasticity (North-Holland Amsterdam, 1988)
G. Jaiani, Mathematical Models of Mechanics of Continua (in Georgian) (Tbilisi University Press, Tbilisi, 2004)
I.N. Vekua, On one method of calculating of prismatic shells (Russian). Trudy Tbilis Mat. Inst. 21, 191–259 (1955)
I.N. Vekua, Shell Theory: General Methods of Construction (Pitman Advanced Publishing Program, Boston, 1985)
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Jaiani, G. (2011). Preliminary Topics. In: Cusped Shell-Like Structures. SpringerBriefs in Applied Sciences and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22101-9_2
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DOI: https://doi.org/10.1007/978-3-642-22101-9_2
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