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Journal of Elasticity

Erschienen in:

20.08.2015

Decay Estimates for Elastic Cylinders with Mixed Boundary Conditions

verfasst von: Vincenzo Coscia, Antonio Russo

Erschienen in: Journal of Elasticity | Ausgabe 2/2016

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Abstract

The classical Saint-Venant’s problem in cylinders is considered under various boundary conditions and some estimates are proven. In particular, if the cylinder is force free on the mantle and is subject to assigned displacements on the bases, satisfying exact compatibility conditions, then a unique elementary Saint-Venant’s solution can be selected in such a way that it agrees with the rigorous solution to the mixed problem away from the bases.

Vorheriger Artikel Kinking Instability in the Torsion of Stretched Anisotropic Elastomeric Filaments

Nächster Artikel Submacroscopic Disarrangements Induce a Unique, Additive and Universal Decomposition of Continuum Fluxes

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For the main notation we follow the classical monograph [7]. In particular, italic light-face letters, small upper-case letters and capital upper-case letters indicate scalars, vectors in ${\Bbb{R}}^{3}$ and second-order tensors (linear maps from ${\Bbb{R}}^{3}$ into itself), respectively; Lin is the linear space of second order tensors and Sym, Skw are the sets of its symmetric and skew elements, respectively. If $\boldsymbol{u}$ is a regular vector field, then $\nabla\boldsymbol{u}$ is the second order tensor with components $(\nabla\boldsymbol{u})_{ij}=\partial_{j}u_{i}$ in a orthonormal base $\{\boldsymbol{e}_{i}\}_{i=1,2,3}$; $\operatorname{div}\boldsymbol {u}=\operatorname{tr}\nabla\boldsymbol{u}$, $\Delta\boldsymbol {u}=\operatorname{div}\nabla\boldsymbol{u}$, $\nabla\boldsymbol{u}^{\top}$ is the transpose of $\nabla\boldsymbol {u}$ and $\hat{\nabla}\boldsymbol{u}={\frac{1}{2}}(\nabla\boldsymbol {u}+\nabla\boldsymbol{u}^{\top})$, $\tilde{\nabla}\boldsymbol{u}={\frac{1}{2}}(\nabla\boldsymbol {u}-\nabla\boldsymbol{u}^{\top})$. The cylinder (1) has length $2h$ and axis $\boldsymbol{e}_{3}$, passing through the centroids of the sections $\mathcal{C}_{x_{3}}\equiv \mathcal{C}$ of the cylinder at $(0,x_{3})$.

$W^{1,2}(\varOmega)$ is the ordinary Sobolev space [3], $W^{1/2,2}(\partial\varOmega)$ is its trace space and $W^{-1/2,2}(\partial\varOmega)$ is the dual space of $W^{1/2,2}(\partial\varOmega)$, $W^{-1/2,2}(\partial\varOmega)= [W^{1/2,2}(\partial\varOmega)]^{*}$. If $\varSigma$ is a subsurface of $\partial\varOmega$, $W^{1/2,2}(\varSigma)=\{\varphi_{\vert\varSigma}, \varphi\in W^{1/2,2}(\partial\varOmega)\}$ and $W^{-1/2,2}(\varSigma)=\{\varphi\in W^{1/2,2}(\partial\varOmega ):\operatorname{supp} \varphi\subset\varSigma\}^{*}$. To keep notation to the minimum, if $\psi\in W^{-1/2,2}(\partial\varOmega)$ and $\varphi\in W^{1/2,2}(\partial\varOmega)$, we use the integral $\int_{\partial\varOmega}\psi\varphi$ to denote the value of the functional $\psi$ at $\varphi$. It will be clear from the context when the integral keeps its usual meaning.

More recently, Theorem 1 has been extended to cover the case of concentrated forces $\boldsymbol{s}(\boldsymbol{u})\in W^{-1,q}(\partial\varOmega)$ for some $q<2$ [16]. In such a case, (12) writes

$$\int_{\varOmega_{R}}|\hat{\nabla}\boldsymbol{u}|^{2}\le c e^{c_{0}(R- h)}\bigl\| \boldsymbol{s}(\boldsymbol{u})\bigr\| _{W^{-1,q}(\partial\varOmega)}^{2}. $$

Clearly, for $\beta_{1}=\beta_{3}=0$ we recover the classical Saint-Venant problem.

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Flavin, J.N., Knops, R.J., Payne, L.E.: Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason, J., Ogden, R.W. (eds.) Elasticity: Mathematical Methods and Applications, pp. 101–111. Ellis Horwood, Chichester (1989)

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Horgan, C.O., Knowles, J.K.: Recent developments concerning Saint-Venant’s principle. In: Wu, T.Y., Hutchinson, J.W. (eds.) Advances in Applied Mechanics, vol. 23, pp. 179–269. Academic Press, San Diego (1983)

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Horgan, C.O., Payne, L.E.: Saint-Venant’s principle in linear isotropic elasticity for incompressible or nearly incompressible materials. J. Elast. 46, 43–52 (1997) MATHMathSciNetCrossRef

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Kondratév, A., Oleinik, O.A.: Boundary-value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Russ. Math. Surv. 43, 65–117 (1988) CrossRef

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Titel: Decay Estimates for Elastic Cylinders with Mixed Boundary Conditions
verfasst von: Vincenzo Coscia
Antonio Russo
Publikationsdatum: 20.08.2015
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 2/2016
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-015-9541-6

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