Inequalities of Korn’s Type and Existence Results in the Theory of Cosserat Elastic Shells

Bîrsan, Mircea

doi:10.1007/s10659-007-9140-2

Cited by 18 publications

(14 citation statements)

References 11 publications

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“…In the same manner as in the classical theory of elasticity or in the shell theory, see e.g. (Ciarlet, 2000;Bîrsan, 2008), we can then use this Korn-type inequality to obtain existence results both in the statical and in the dynamical theory of porous rods. The existence of solutions have been proven by this procedure, but these results are not presented here.…”

Section: Uniqueness Of Solution In the Linear Theorymentioning

confidence: 97%

On the theory of porous elastic rods

Bîrsan

Altenbach

2011

International Journal of Solids and Structures

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a b s t r a c tWe consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamical nonlinear field equations of the model. Then, in the framework of linear theory, we prove the uniqueness of the solution to the associated boundary-initial-value problem. We identify the relevant field quantities from the theory of directed curves by comparison with the three-dimensional equations of straight porous rods. Finally, for orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches.

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Section: Uniqueness Of Solution In the Linear Theorymentioning

confidence: 97%

On the theory of porous elastic rods

Bîrsan

Altenbach

2011

International Journal of Solids and Structures

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“…The proof is analogue to the proof of the corresponding Korn-type inequality for general surfaces or for the Cosserat surfaces (see [30,Theorem 2]). …”

Section: Theorem 63mentioning

confidence: 99%

“…The existence of a solution (u, u) ∈ H 1 0 ( ) of the system (73) can be derived from the general theory of elliptic systems (see e.g. [30]). We emphasize that the Korn-type inequality established by Theorem 6.3 plays here a crucial role once again.…”

Section: Proofmentioning

confidence: 99%

A mathematical study of the linear theory for orthotropic elastic simple shells

Bîrsan

Altenbach

2009

Math. Meth. Appl. Sci.

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The theory of simple shells is a surface-related Cosserat model for thin elastic shells. In this direct approach, each material point is connected with a triad of rigidly rotating directors. This paper presents a study of the governing equations for orthotropic elastic simple shells in the framework of the linearized theory. We establish the uniqueness of classical solutions, without any restrictive assumption on the strain energy function. The continuous dependence of solutions on the body loads and initial data is proved. Also, the existence of weak solutions to the equations of simple shells is proved by means of an inequality of Korn's type established for such directed surfaces.

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“…In the next section, we show the existence and uniqueness of (weak) solutions for the dynamic deformation of porous thermoelastic shells, based on the inequality of Korn's type presented in [13].…”

Section: Introductionmentioning

confidence: 99%

“…In the isothermal and non-porous case, the existence of solutions to the governing equations has been proved in [13]. We mention that, for the equilibrium theory, the existence of weak solutions has been investigated in [14] and [15], but these results are valid on some special (more restrictive) functional spaces.…”

Section: Introductionmentioning

confidence: 99%