83 research outputs found
Asymptotic behavior of structures made of straight rods
This paper is devoted to describe the deformations and the elastic energy for
structures made of straight rods of thickness when tends to
0. This analysis relies on the decomposition of the large deformation of a
single rod introduced in [6] and on the extension of this technique to a
multi-structure. We characterize the asymptotic behavior of the infimum of the
total elastic energy as the minimum of a limit functional for an energy of
order ()
Junction between a plate and a rod of comparable thickness in nonlinear elasticity. Part II
We analyze the asymptotic behavior of a junction problem between a plate and
a perpendicular rod made of a nonlinear elastic material. The two parts of this
multi-structure have small thicknesses of the same order . We use the
decomposition techniques obtained for the large deformations and the
displacements in order to derive the limit energy as tends to 0.Comment: arXiv admin note: substantial text overlap with arXiv:1107.528
Asymptotic behavior of a structure made by a plate and a straight rod
This paper is devoted to describe the asymptotic behavior of a structure made
by a thin plate and a thin rod in the framework of nonlinear elasticity. We
scale the applied forces in such a way that the level of the total elastic
energy leads to the Von-K\'arm\'an's equations (or the linear model for smaller
forces) in the plate and to a one dimensional rod-model at the limit. The
junction conditions include in particular the continuity of the bending in the
plate and the stretching in the rod at the junction
Straight rod with different order of thickness
In this paper, we consider rods whose thickness vary linearly between \eps
and \eps^2. Our aim is to study the asymptotic behavior of these rods in the
framework of the linear elasticity. We use a decomposition method of the
displacement fields of the form , where stands for the
translation-rotations of the cross-sections and is related to their
deformations. We establish a priori estimates. Passing to the limit in a fixed
domain gives the problems satisfied by the bending, the stretching and the
torsion limit fields which are ordinary differential equations depending on
weights.Comment: in Asymptotic Analysis, IOS Press, 201
Homogenization via unfolding in periodic layer with contact
In this work we consider the elasticity problem for two domains separated by
a heterogeneous layer. The layer has an periodic structure,
, including a multiple micro-contact between the structural
components. The components are surrounded by cracks and can have rigid
displacements. The contacts are described by the Signorini and Tresca-friction
conditions. In order to obtain preliminary estimates modification of the Korn
inequality for the dependent periodic layer is performed.
An asymptotic analysis with respect to is provided and
the limit problem is obtained, which consists of the elasticity problem
together with the transmission condition across the interface. The periodic
unfolding method is used to study the limit behavior.Comment: 20 pages, 1 figur
Asymptotic behavior of structures made of curved rods
In this paper we study the asymptotic behavior of a structure made of curved rods of thickness 2δ when δ → 0. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We show that any displacement of a structure is the sum of an elementary rods-structure displacement (e.r.s.d.) concerning the rods cross sections and a residual one related to the deformation of the cross-section. The e.r.s.d. coincide with rigid body displacements in the junctions. Any e.r.s.d. is given by two functions belonging to H1 (S;R3) where S is the skeleton structure (i.e. the set of the rods middle lines). One of this function U is the skeleton displacement, the other R gives the cross-sections rotation. We show that U is the sum of an extensional displacement and an inextensional one. We establish a priori estimates and then we characterize the unfolded limits of the rods-structure displacements. Eventually we pass to the limit in the linearized elasticity system and using all results in [5], on the one hand we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacement and the limit of the rods torsion angles
Asymptotic behavior of structures made of curved rods
In this paper we study the asymptotic behavior of a structure made of curved
rods of thickness 2\delta when \delta rightarrow 0. This study is carried on
within the frame of linear elasticity by using the unfolding method. It is
based on several decompositions of the structure displacements and on the
passing to the limit in fixed domains. We show that any displacement of a
structure is the sum of an elementary rods-structure displacement (e.r.s.d.)
concerning the rods cross sections and a residual one related to the
deformation of the cross-section. The e.r.s.d. coincide with rigid body
displacements in the junctions. Any e.r.s.d. is given by two functions
belonging to H1 (S;R3) where S is the skeleton structure (i.e. the set of the
rods middle lines). One of this function U is the skeleton displacement, the
other R gives the cross-sections rotation. We show that U is the sum of an
extensional displacement and an inextensional one. We establish a priori
estimates and then we characterize the unfolded limits of the rods-structure
displacements. Eventually we pass to the limit in the linearized elasticity
system and using all results in [5], on the one hand we obtain a variational
problem that is satisfied by the limit extensional displacement, and on the
other hand, a variational problem coupling the limit of inextensional
displacement and the limit of the rods torsion angles
Interior error estimate for periodic homogenization
In a previous article about the homogenization of the classical problem of
diff usion in a bounded domain with su ciently smooth boundary we proved that
the error is of order . Now, for an open set with su ciently
smooth boundary and homogeneous Dirichlet or Neuman limits conditions
we show that in any open set strongly included in the error is of order
. If the open set is of polygonal (n=2) or
polyhedral (n=3) boundary we also give the global and interrior error
estimates
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