Abstract
The present chapter contains hierarchical models for elastic prismatic shells, beams with rectangular variable cross-sections, fluids and elastic solid–fluid structures occupying prismatic domains. To this end, a dimension reduction method based on Fourier–Legendre expansions is mostly applied to basic equations of linear theory of elasticity of homogeneous isotropic bodies and Newtonian fluids. The governing equations and systems of hierarchical models are constructed with respect to so-called mathematical moments of stress and strain tensors and displacement vector components.
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Notes
- 1.
In the following sense: if \(X_{ij}, \) \(e_{ij}, \) and \(u_{i}\) satisfy the relations (3.2)–(3.4), then constructed by (3.1) functions \(X_{ijr}, \) \(e_{ijr}, \) \(u_{ir}\) will satisfy the infinite relations (3.6)–(3.8) and, vice versa, if \(X_{ijr}, \) \(e_{ijr}, \) \(u_{ir}\) satisfy the infinite relations (3.6)–(3.8), then constructed by means of (3.5) functions \(X_{ij}, \) \(e_{ij}, \) \(u_{i}\) will satisfy the relations (3.2)–(3.4).
- 2.
We remind that \(C^2(\Upomega)\) denotes a class of functions twice continuously differentiable with respect to the variables \(x_{1}, x_{2}, \) \(x_{3}, \) \((x_{1},x_{2},x_{3})\in\Upomega. \) Note that for the uniform convergence of the above Fourier–Legendre expansions it suffices to demand this property only with respect to \(x_3\in\Big[{\mathop h\limits^{(-)}}(x_{1},x_{2}), {\mathop h\limits^{(+)}}(x_1,x_2)\Big]\) and \(x_i\in[{\mathop {h_i}\limits^{(-)}}(x_1), {\mathop {h_i}\limits^{(+)}}(x_1)], i=2,3, \) respectively.
References
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Jaiani, G. (2011). Hierarchical Models. In: Cusped Shell-Like Structures. SpringerBriefs in Applied Sciences and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22101-9_3
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DOI: https://doi.org/10.1007/978-3-642-22101-9_3
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