An obstacle problem for elliptic membrane shells

2018 ◽  
Vol 24 (5) ◽  
pp. 1503-1529 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare ◽  
Paolo Piersanti
Keyword(s):  
Vector Field ◽  
Asymptotic Analysis ◽  
Obstacle Problem ◽  
Displacement Vector ◽  
Middle Surface ◽  
Two Dimensional ◽  
Displacement Vector Field ◽  
Membrane Shells ◽  
Signorini Condition ◽  
Confinement Condition

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic elliptic membrane shells, all sharing the same middle surface [Formula: see text], where [Formula: see text] is a domain in [Formula: see text] and [Formula: see text] is a smooth enough immersion, all subjected to this confinement condition, and whose thickness [Formula: see text] is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as [Formula: see text] approaches zero, the corresponding “limit” two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space [Formula: see text]. The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the “lower face” of the shell is required to remain above the “horizontal” plane. Such a confinement condition renders the asymptotic analysis substantially more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

An obstacle problem for Koiter’s shells

2019 ◽  
Vol 24 (10) ◽  
pp. 3061-3079 ◽  
Author(s):  
Philippe G Ciarlet ◽  
Paolo Piersanti
Keyword(s):  
Asymptotic Analysis ◽  
A Priori ◽  
Elastic Shell ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Three Dimensional Model ◽  
Limit Model ◽  
Membrane Shells ◽  
Point Of Departure ◽  
Confinement Condition

In this paper, we define, a priori, a natural two-dimensional Koiter’s model of a ‘general’ linearly elastic shell subject to a confinement condition. As expected, this model takes the form of variational inequalities posed over a non-empty closed convex subset of the function space used for the ‘unconstrained’ Koiter’s model. We then perform a rigorous asymptotic analysis as the thickness of the shell, considered a ‘small’ parameter, approaches zero, when the shell belongs to one of the three main classes of linearly elastic shells, namely elliptic membrane shells, generalized membrane shells and flexural shells. To illustrate the soundness of this model, we consider elliptic membrane shells to fix ideas. We then show that, in this case, the ‘limit’ model obtained in this fashion coincides with the two-dimensional ‘limit’ model obtained by means of another rigorous asymptotic analysis, but this time with the three-dimensional model of a ‘general’ linearly elastic shell subject to a confinement condition as a point of departure. In this fashion, our proposed Koiter’s model of a linearly elastic shell subject to a confinement condition is fully justified in this case, even though it is not itself a ‘limit’ model.

scholarly journals A distance to dose difference tool for estimating the required spatial accuracy of a displacement vector field

2011 ◽  
Vol 38 (5) ◽  
pp. 2318-2323 ◽  
Author(s):  
Nahla K. Saleh-Sayah ◽  
Elisabeth Weiss ◽  
Francisco J. Salguero ◽  
Jeffrey V. Siebers
Keyword(s):  
Vector Field ◽  
Displacement Vector ◽  
Spatial Accuracy ◽  
Displacement Vector Field ◽  
Dose Difference

scholarly journals COSMOLOGICAL MODELS WITH A VARYING Λ-TERM IN LYRA'S GEOMETRY

2012 ◽  
Vol 27 (29) ◽  
pp. 1250164 ◽  
Author(s):  
V. K. SHCHIGOLEV
Keyword(s):  
Vector Field ◽  
Exact Solutions ◽  
Equations Of State ◽  
Displacement Vector ◽  
Cosmological Models ◽  
Displacement Vector Field ◽  
Varying Λ ◽  
Lyra’S Geometry ◽  
Model Equations ◽  
The Universe

Cosmological models in Lyra's geometry are constructed and investigated with the assumption of a minimal interaction of matter with the displacement vector field and the dynamical Λ-term. Exact solutions of the model equations are obtained for the different equations of state of the matter, that fills the universe, and for the certain assumptions on the decaying law for Λ.

scholarly journals Strategy for Crack Width Measurement of Multiple Crack Patterns in Civil Engineering Material Testing Using a Monocular Image Sequence Analysis

2020 ◽  
Vol 88 (3-4) ◽  
pp. 219-238 ◽  
Author(s):  
F. Liebold ◽  
H.-G. Maas
Keyword(s):  
Vector Field ◽  
Sequence Analysis ◽  
Least Squares ◽  
Crack Width ◽  
Displacement Vector ◽  
Image Sequence ◽  
Multiple Crack ◽  
Displacement Vector Field ◽  
Monocular Image ◽  
Monocular Image Sequence

Abstract An image sequence analysis procedure is developed to quantitatively analyze complex multiple crack patterns in tension tests of fiber-reinforced composite specimens. Planar textured surfaces of such specimens can be observed with a monocular image sequence using a camera of suitable spatial and temporal resolution. Due to the narrow crack paths, a dense high-precision displacement vector field is computed applying least-squares image matching techniques. Some uniformly distributed matching points are triangulated into a mesh. To measure deformations, principal strains and crack widths are computed for each face. Stretched triangles presumably containing one or multiple cracks are subdivided into three new triangles to densify the mesh in critical regions. The subdivision is repeated for some iterations. The crack width computation of the triangles requires at least three vertices and its displacements. Due to the dense displacement vector field, there are more points available. In this paper, an algorithm for the crack width computation in a least-squares fit is presented.

scholarly journals A NEW APPROACH TO LINEAR SHELL THEORY

2005 ◽  
Vol 15 (08) ◽  
pp. 1181-1202 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE
Keyword(s):  
Vector Field ◽  
Minimization Problem ◽  
Shell Theory ◽  
Displacement Vector ◽  
Constrained Minimization ◽  
New Approach ◽  
Quadratic Minimization ◽  
Displacement Vector Field ◽  
Constrained Minimization Problem ◽  
Linear Shell Theory

We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

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