# Solid Mechanics and Dynamics

## Solid Mechanics and Dynamics

Mechanics is the oldest branch of physics, but also the foundation for many burning research topics. Mechanics is the study of the motion and deformation of bodies under the action of forces. Even in ancient times this issue has occupied and fascinated scientists for very practical reason: heavy loads had to be lifted and carried, buildings were erected or military equipment designed. Classical mechanics, which is taught today, is based on the three Newton’s laws, which describe the principle of inertia, the relationship between acceleration and force and the effect of force and counterforce. The development of mechanics has always been closely linked to developments in mathematics, e.g. the integral and differential calculus, vector calculus or the calculus of variations.

In particular, the development of numerical methods for solving differential equations and the rapid improvement of the computing power have opened up new fields of application for Computational Mechanics. Today mechanical simulations are irreplaceable in industrial design and manufacturing processes: the crash behavior of entire cars is modeled on the computer, as well as laser welding of pipes or the formation of micro-cracks in composite materials. The constantly increasing demands on materials, components and structures create also new challenges in computational mechanics, such as the development of problem-specific numerical methods and algorithms or the development of mathematical models to describe complex material behavior. This requires both a sound knowledge of the basics of mechanics and mathematics, as well as the ability to implement the resulting models. For this reason, the TAF Mechanics and Dynamics originated: we are convinced that you as CE students bring the best conditions and prior knowledge to successfully collaborate in the field of computational mechanics. In your Bachelor’s study you will learn at first the fundamentals of mechanics: Statics, Elastostatics and Dynamics of Rigid Bodies. You will also learn the basics of an important numerical method in engineering, the Finite Element Method. These courses are supplemented by lectures on fluid mechanics or vibration theory. In the master program advanced courses on computational mechanics (nonlinear finite element method, material modeling and simulation) and continuum mechanics are offered. Courses on particular fields of mechanics (biomechanics, fracture mechanics, damage mechanics, …) complete the study plan.

The later career fields for graduates with a good knowledge in mechanics are manifold: mechanical engineering, automotive engineering, aerospace engineering, environmental engineering, chemical engineering, civil engineering, medical engineering, materials engineering, process engineering, …

## Content of the Bachelor Studies

### Statics, Elastostatics and Strength of Materials

The module *Statics, Elastostatics and Strength of Material* deals with the statics of rigid bodies (stereo statics) and the statics of deformable bodies (elastostatics). After the mechanical fundamentals and definitions, planar and stereo structures are regarded, reaction forces and momentums as well as internal force variables are introduced. Besides friction and the principle of virtual work, the description of characteristic area properties (center of gravity, moment of inertia) is regarded, providing a basis for the following elastostatics.Introducing the local loads, stresses and strains, the lecture leads to the stress and deformation of straight slender beams, loaded by tension, bending, torsion and shear forces. Finally, basic energy methods are shown and an introduction to the theory of strength of materials is given. The module *Dynamics* deals with kinematic and kinetic of mass points, systems of mass points and rigid bodies. Based on the principles of linear and angular momentum, the conservation equations are derived and discussed. The lecture ends with an introduction to the theory of vibrations for systems with one degree of freedom.

### Mechanical Vibrations

The module *Mechanical Vibrations* deals with technological relevant mechanical vibration problems. To this end, firstly the equations of motion have to be formulated based on an appropriate physical/mathematical modelling. Thereby the focus is on discrete systems with one and multiple degrees of freedom. The solution of the resulting (ordinary) differential equations allows the analysis and assessment of technological systems that are susceptible to mechanical vibrations. The module *Multi body Dynamics* extends these treatments to the case of multi body systems.

### Finite Element Methods

*The module Finite Element Methods* provides a general framework for the computational solution of boundary value problems from various different engineering applications. In the present lecture, we introduce its theoretical background, discuss its characteristic features and illustrate its algorithmic realization. Starting with a one dimensional model problem, we discuss the strong form of the governing equations, the derivation of the related weak form and its computational solution. We then turn to the evaluation of more complex problems, such as heat conduction, bending problems or the classical elasticity problem. In addition, finite element specific aspects like the isoparametric concept or numerical integration will be addressed. Computational examples and related MATLAB codes will be provided throughout to illustrate the algorithmic realization of the Finite Element Method.

## Content of the Master Studies

### Continuum Mechanics I

Continuum mechanics provides the basic framework for the solution of a number of mechanically oriented engineering problems, e.g. the characterization of relations between loading and deformation or between stress and strain. This lecture deals with the main aspects of geometrically linear continuum mechanics, formulated in terms of a modern tensorial notation. As such, the lecture can be understood as a conceptual supplementation of the basic lectures on mechanics. Moreover, it provides a profound basis for more enhanced lectures, e.g. the lecture *Linear Finite Element Methods*.

### Continuum Mechanics II

Nonlinear continuum mechanics provides the basic framework for the solution of a number of mechanically oriented engineering problems, e.g. the buckling and failure of structural elements. This lecture deals with the main aspects of geometrically nonlinear continuum mechanics, formulated in terms of a modern tensorial notation. As such, the lecture can be understood as a conceptual supplementation of the lecture on linear continuum mechanics. Moreover, it provides a profound basis for more enhanced lectures, e.g. the lecture *Nonlinear Finite Element Methods*.

### Nonlinear Finite Element Methods

Based on the lecture *Linear Finite Element Methods*, nonlinear methods are to be discussed in the lecture *Nonlinear Finite Element Methods*. Hence, the basics of nonlinear continuum mechanics are considered at first. For the special case of a geometrically nonlinear rod the main ideas of linearization, finite element discretization, geometric and material part of the tangential stiffness matrix, and the iterative solution with a Newton-Raphson scheme are introduced. Analogously to the lecture *Linear Finite Element Methods* the numerical implementation is realized with MATLAB. Furthermore, stability analyses of truss frameworks are considered as well as appropriate numeric algorithms, for example, the arc length method. Following the detailed investigation of the one-dimensional rod element, the nonlinear finite element method is generalized towards continuum elements resulting from nonlinear continuum mechanics. The goal is to extend the isoparametric triangular and rectangular elements which where first introduced and implemented during the lecture * Linear Finite Element Methods* in the frame of a linear theory to the fully nonlinear case.

### Computational Dynamics

The lecture Computational Dynamics deals with the formulation and numerical treatment of the equations of motion for mechanical systems based on the principles of analytical mechanics. Thereby the focus is on the Hamiltonian principle, which leads to the Lagrange equations, and the Hamiltonian canonical equations. Related numerical time integration methods are outlined in connection with spatially discrete systems (e.g. particle systems, rigid bodies or FE-discretised elastic bodies). Thereby the relevant conservation properties of these algorithms are analysed.

### Computational Plasticity

Plasto-mechanics provides the basic framework for the solution of a number of mechanically oriented engineering problems, e.g. the formation of plastic hinges/zones in structural elements or slip lines. This lecture deals with the main aspects of geometrically linear plasto-mechanics, i.e. the modelling and the suited algorithmic treatment of plastic materials behaviour. As such, the lecture can be understood as a conceptual supplementation of the basic lectures on mechanics and a continuation of the lecture on linear continuum mechanics.

### Biomechanics

In past years biomechanical problems had a strong impact on the research done in the field of computational and continuum mechanics. To name but a few, fields of application are the numerical simulation of blood flow running through veins, adaptation of bone structures or modeling of tissues. A numerical implementation of appropriate phenomenological models within the finite element method, for example, leads to illuminative insights into biological processes and is supposed to complete detailed medical examinations or even replace them in the future. The lecture gives an overview of computational and continuum mechanical treatments of biomechanical problems and discusses their numerical implementation based on selected topics.

## TAF Adviser

### Prof. Dr.-Ing. Julia Mergheim

Department of Mechanical Engineering

Institute of Applied Mechanics (Prof. Dr. Steinmann)

- Phone number: +49 9131 85-28505
- Email: julia.mergheim@fau.de