# Applied Mathematics Colloquium: Dr Scott Hansen

## Iowa State University

Friday, April 8, 2022

2:00 PM – 3:00 PM

2:00 PM – 3:00 PM

Title Modeling and Analysis of Layered Elastic Systems

Abstract: Beams and plates equations are approximations of 3-dimensional elasticity that describe motions of a centerline (for a beam) or a mid-surface (for a plate). For layered composite structures, in some

cases classical beam and plate theory and be successfully applied by averaging through the thickness to obtain elastic properties. However, when composite structures are made of materials with greatly different

elastic properties, a preferable approach is to treat each later as part of a geometrically coupled system of independent beams or plates. Three-layer models of this type (Yan and Dowell model (1972), Mead and

Markus (1969) and Rao and Nakra (1974)) have been used numerous engineering applications.

In this talk I’ll describe some multilayer generalizations of these models consisting of alternating stiff and compliant layers. The resulting system of equations vary in complexity depending on how many energy terms

are retained in the model formulation. I’ll describe some of the modeling assumptions and approximation choices that can be made to produce more simplified models. In addition, for many applications it is important to include damping. In constrained-layer damping, for example, the goal is to introduce damping in the system to reduce vibrations. Typically, a viscoelastic layer is sandwiched between two much stiffer elastic layers. This promotes shear motions in the viscoelastic layer and hence improves damping. In active constrained layer damping controlled feedback is applied on an outer layer to promote damping. Well-posedness and stability results will be described and compared. For example, when shear dampingis included in the compliant layers, under some minimal assumptions, exponential stability holds for the mul-tilayer Rao-Nakra system and in more restrictive cases, analyticity of the semigroup holds for the multilayer Mead-Markus system.

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