1.Introduction to Composite Materials 2. Macrochemical Behavior of a Lamina 3.Micromechanical Behavior of a Lamina 4.Macromechanical Behavior of a Laminate 5.Bending, Buckling, and Vibration of Laminated Plates 6.Other Analysis and Behavior Topics 7.Introduction to Design of Composite Structures Appendix A.Matrices and Tensors Appendix B.Maxima and Minima of Functions of a Single Variable Appendix C.Typical Stress-Strain Curves Appendix D.Governing Equations for Beam Equilibrium and Plate Equilibrium, Buckling, and Vibration Index

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Mechanics of composite materials

Nonlinear superposition operators: Preface

If $- \infty < \alpha < \beta < \infty $ and $f \in C^{3} \left( [ \alpha , \beta ] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${\phi } \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ for which ${\phi }^{(k)}( \alpha ) = {\phi }^{(k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {\phi }^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {\phi }^{3} \right) dx$ $\geq \int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {\phi }^{2} \phi ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } \phi \phi " + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } \phi ' {\phi }^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } \phi {\phi '}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$

Variational inequalities

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters.

A Lagrange multiplier method for a Stokes–Biot fluid–poroelastic structure interaction model

The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Shape Optimization of an Electric Motor Subject to Nonlinear Magnetostatics

We present a modified finite element method that is able to approximate interface problems with high accuracy. We consider interface problems where the solution is continuous; its derivatives, however, may be discontinuous across interface curves within the domain. The proposed discretization is based on a local modification of the finite element basis functions using a fixed quadrilateral mesh. Instead of moving mesh nodes, we resolve the interface locally by an adapted parametric approach. All modifications are applied locally and in an implicit fashion. The scheme is easy to implement and is well suited for time-dependent moving interface problems. We show optimal order of convergence for elliptic problems, and further, we give a bound on the condition number of the system matrix. Both estimates do not depend on the interface location relative to the mesh.

https://www.acmac.uoc.gr/EFEF2013/get_talk.php?id=25

A Locally Modified Parametric Finite Element Method for Interface Problems

We derive a Nitsche-based formulation for fluid-structure interaction (FSI) problems with contact. The approach is based on the work of Chouly and Hild [SIAM Journal on Numerical Analysis. 2013;51(2):1295–1307] for contact problems in solid mechanics. We present two numerical approaches, both of them formulating the FSI interface and the contact conditions simultaneously in equation form on a joint interface-contact surface. The first approach uses a relaxation of the contact conditions to allow for a small mesh-dependent gap between solid and wall. The second alternative introduces an artificial fluid below the contact surface. The resulting systems of equations can be included in a consistent fashion within a monolithic variational formulation, which prevents the so-called “chattering” phenomenon. To deal with the topology changes in the fluid domain at the time of impact, we use a fully Eulerian approach for the FSI problem. We compare the effect of slip and no-slip interface conditions and study the performance of the method by means of numerical examples.

A Nitsche-based formulation for fluid-structure interactions with contact

Long-term simulation of large deformation, mechano-chemical fluid-structure interactions in ALE and fully Eulerian coordinates

In this thesis, we develop an accurate and robust numerical framework for interface problems
involving moving interfaces. In particular, we are interested in the simulation of
fluid-structure interaction problems in Eulerian coordinates.

Our numerical model for fluid-structure interactions (FSI) is based on the monolithic "Fully Eulerian"
approach. With this approach we can handle both strongly-coupled problems and
large structural displacements up to contact.

We introduce modified discretisation schemes of second order for both space and time discretisation.
The basic concept of both schemes is to resolve the interface locally within the discretisation.
For spatial discretisation, we present a locally modified finite element scheme that is based on a fixed patch mesh
and a local resolution of the interface within each patch. It does
neither require any remeshing nor the introduction of additional degrees of freedom. 
For discretisation in time, we use a modified continuous Galerkin scheme. Instead of polynomials
in direction of time, we define polynomial functions on space-time trajectories that do not cross the interface.

Furthermore, we introduce a pressure stabilisation technique based on "Continuous Interior Penalty" method
and a simple stabilisation technique for the structural equation that increases the robustness of the 
Fully Eulerian approach considerably.
We give a detailed convergence analysis for all proposed discretisation and stabilisation schemes and
test the methods with numerical examples.

In the final part of the thesis, we apply the numerical framework to different FSI applications.
First, we validate the approach with the help of established numerical benchmarks. Second, we investigate its
capabilities in the context of contact problems and large deformations.
We study contact problems of a falling elastic ball with the ground, an inclined plane and some stairs
including the subsequent bouncing. For the case that no fluid layer
remains between ball and ground, we use a simple contact algorithm.

Furthermore, we study plaque growth in blood vessels up to a complete clogging of the vessel.
Therefore, we use a monolithic mechano-chemical fluid-structure-interaction model and include
the fast pulsating flow dynamics by means of a temporal two-scale scheme.
We present detailed numerical studies for all three applications including a numerical 
convergence analysis in space and time, as well as an investigation of the influence of
different material parameters.

Eulerian finite element methods for interface problems and fluid-structure interactions

We present a second order time-stepping scheme for parabolic problems on moving domains and interfaces. The diffusion coefficient is discontinuous and jumps across an interior interface. This causes the solution to have discontinuous derivatives in space and time. Without special treatment of the interface, both spatial and temporal discretization will be sub-optimal. For such problems, we develop a time-stepping method, based on a cG(1) Eulerian space-time Galerkin approach. We show −both analytically and numerically− second order convergence in time. Key to gaining the optimal order of convergence is the use of space-time test- and trial-functions, that are aligned with the moving interface. Possible applications are multiphase flow or fluid-structure interaction problems.

Stefan Frei

Papers

A Locally Modified Parametric Finite Element Method for Interface Problems

A Nitsche-based formulation for fluid-structure interactions with contact

Long-term simulation of large deformation, mechano-chemical fluid-structure interactions in ALE and fully Eulerian coordinates

Eulerian finite element methods for interface problems and fluid-structure interactions

A second order time-stepping scheme for parabolic interface problems with moving interfaces