A finite element method is proposed for investigating the general elastic multi-structure problem, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and ...rotational angles on rods are discretized using conforming linear elements, transverse displacements on plates and rods are discretized respectively using TRUNC elements and Hermite elements of third order, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The unique solvability of the method is verified by the Lax–Milgram lemma after deriving generalized Korn’s inequalities in some nonconforming element spaces on elastic multi-structures. The quasi-optimal error estimate in the energy norm is also established. Some numerical results are presented at the end.
We investigate the behavior of the deformations of a thin shell, whose thickness
δ
tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a ...deformation
v
are estimated in terms of the
L
2
-norm of the distance from
∇
v
to
SO
(3). This permits in particular to derive accurate nonlinear Korn’s inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant’s strain tensor when the “
strain energy
” is of order less than
δ
3/2
.
Saint Venant's and Donati's theorems constitute two classical characterizations of smooth matrix fields as linearized strain tensor fields. Donati's characterization has been extended to matrix ...fields with components in
L
2
by T.W. Ting in 1974 and by J.J. Moreau in 1979, and Saint Venant's characterization has been extended likewise by the second author and P. Ciarlet, Jr. in 2005. The first objective of this paper is to further extend both characterizations, notably to matrix fields whose components are only in
H
−1
, by means of different, and to a large extent simpler and more natural, proofs. The second objective is to show that some of our extensions of Donati's theorem allow to reformulate in a novel fashion the pure traction and pure displacement problems of linearized three-dimensional elasticity as quadratic minimization problems with the strains as the primary unknowns. The third objective is to demonstrate that, when properly interpreted, such characterizations are “matrix analogs” of well-known characterizations of vector fields. In particular, it is shown that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma.
Les théorèmes de Saint Venant et de Donati constituent deux caractérisations classiques de champs de matrices réguliers comme des champs de tenseurs de déformation linéarisés. La caractérisation de Donati a été étendue aux champs de matrices dont les composantes sont dans
L
2
par T.W. Ting en 1974 et par J.J. Moreau en 1979. La caractérisation de Saint Venant a été pareillement étendue par le second auteur et P. Ciarlet, Jr. en 2005. Le premier objectif de cet article est de montrer que l'on peut généraliser encore davantage ces caractérisations, en particulier à des champs de matrices dont les composantes sont seulement dans
H
−1
, au moyen de démonstrations différentes, et dans une large mesure plus simples et plus naturelles. Le second objectif est de montrer que certaines de nos généralisations du théorème de Donati conduisent à de nouvelles façons de poser les problèmes de traction pure et de déplacement pur de l'élasticité linéarisée tridimensionnelle, sous la forme de problèmes de minimisation quadratique où les déformations deviennent les inconnues principales. Le troisième objectif est de montrer que, une fois convenablement interprétées, ces caractérisations apparaissent comme les « analogues matriciels » de caractérisations bien connues de champs de vecteurs. En particulier, on montre que le théorème de Saint Venant n'est autre que l'analogue matriciel du lemme de Poincaré.
In this work we extend the well-posedness for infinitesimal dislocation-based gradient viscoplasticity with linear kinematic hardening from the subdifferential case to general nonassociative monotone ...plastic flows. We assume an additive split of the displacement gradient into nonsymmetric elastic distortion and nonsymmetric plastic distortion. The thermodynamic potential is augmented with a term taking the dislocation density tensor Curl p into account. The constitutive equations in the models we study are assumed to be only of monotone type. Based on the generalized version of Korn's inequality for incompatible tensor fields (the nonsymmetric plastic distortion) due to Neff et al. the existence of solutions of quasistatic initial-boundary value problems under consideration is shown using a time-discretization technique and a monotone operator method.
We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in
. We assume that the stress tensor is generated by a potential of ...the form
,
denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions
h
having the property that
h
′ (
t
)/
t
increases (shear thickening case).
In a bounded domain
Ω
⊂
R
3
we consider a discrete network of a large number of concentrated masses (particles) connected by elastic springs. We provide sufficient conditions on the geometry of the ...array of particles, under which the network admits a rigorous continuum limit. Our proof is based on the discrete Korn's inequality. Proof of this inequality is the key point of our consideration. In particular, we derive an explicit upper bound on the Korn's constant. For generic non-periodic arrays of particles we describe the continuum limit in terms of the local energy characteristic on the mesoscale (intermediate scale between the interparticle distances (small scale) and the domain sizes (large scale)), which represents local energy in the neighborhood of a point. For a periodic array of particles we compute coefficients in the limiting continuum problems in terms of the elastic constants of the springs.
The paper is dedicated to the asymptotic behavior of
ε
-periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as
ε
→
0
. In case of plate-like or beam-like structures ...the asymptotic reduction of dimension from
3
D
to
2
D
or
1
D
respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as
ε
→
0
we use the periodic unfolding method.
A method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold ...subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.
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