46E35 Sobolev spaces and other spaces of \smooth" functions, embedding theorems, trace theorems
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- Galerkin-Methode (1)
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This work aims to study textile structures in the frame of linear elasticity to understand how
the structure and material parameters influence the macroscopic homogenized model. More
precisely, we are interested in how the textile design parameters, such as the ratio between
fibers’ distance and cross-section width, the strength of the contact sliding between yarns,
and the partial clamp on the textile boundaries determine the phenomena that one can see in
shear experiments with textiles. Among others, when the warp and weft yarns change their
in-plane angles first and, after reaching some critical shear angle, the textile plate comes out
of the plane, and its folding starts.
The textile structure under consideration is a woven square, partially clamped on the left
and bottom boundary, made of long thin fibers that cross each other in a periodic pattern.
The fibers cannot penetrate each other, and in-plane sliding is allowed. This last assumption,
together with the partial clamp, adds new levels of complexity to the problem due to
the anisotropy in the yarn’s behavior in the unclamped subdomains of the textile.
The limiting behavior and macroscopic strain fields are found by passing to the limit with
respect to the yarn’s thickness r and the distance between them e, parameters that are asymptotically
related. The homogenization and dimension reduction are done via the unfolding
method, which separates the macroscopic scale from the periodicity cell. In addition to the
homogenization, a dimension reduction from a 3D to a 2D problem is applied. Adapting
the classical unfolding results to both the anisotropic context and to lattice grids (which are
constructed starting from the center lines of the rods crossing each other) are the main tools
we developed to tackle this type of model. They represent the first part of the thesis and are
published in Falconi, Griso, and Orlik, 2022b and Falconi, Griso, and Orlik, 2022a.
Given the parameters mentioned above, we then proceed to classify different textile problems,
incorporating the results from other works on the topic and thoroughly investigating
some others. After the study is conducted, we draw conclusions and give a mathematical
explanation concerning the expected approximation of the displacements, the expected solvability
of the limit problems, and the phenomena mentioned above. The results can be found
in “Asymptotic behavior for textiles with loose contact”, which has been recently submitted.
We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.