SSG 516 Continuum Mechanics

Lecture OneIntroduction

“Most engineering curricula teach momentum balance laws for an array of materials, often without informing the students that these laws are actually independent of the materials. Further, while courses do discuss the balance of energy, they often fail to

mention the second law of thermodynamics” – Gurtin et. Al.Bower pp 1-12

Assumes Gurtin et al. chapters 1-4 from last termOr Bower appendix A, B & C

Materials used in this course are largely based (not restricted to) on the books shown below.

Specific locations in the texts that you will need to read are also shown for your convenience.

Worked examples in the third material will form the bases of continuous assessment and examinations

Two important software are necessary for this course. They are available via the LG Laboratory

Course Materials

1. Bower, AF, Applied Mechanics of Solids, (2010) CRC Press, pp 711-764, 1-63, 65-93 in that order

2. Gurtin, ME, Eliot, F & Annand, L, (2010) The Mechanics and Thermodynamics of Continua, Cambridge 2010, Part I: Vector and Tensor Algebra, Part II: Vector and Tensor Analysis.

3. Fakinlede, OA, (2012) Nonlinear Mechanics, PDF, Solved problems pp 1-200

Textbooks

Mathematica for Symbolics, Numerics and Graphics– Wolfram Research. University of Lagos has a site License and you should get your registered copy from the LG Lab.

MSC Nastran for Simulation and Modeling. A student version is available for you.

Willie Aniebiet will tell you the procedure for obtaining both of these

Software

At this stage in your degree program, you have studied courses such as Fluid Mechanics, Strength of Materials, Thermodynamics, etc. and applied them to the behavior of materials without necessarily perceiving the relationship between the different issues.

Here we take as our starting point, the balance laws for mass, linear and angular momenta, energy and entropy imbalance

What is Continuum Mechanics?

This collection of physical laws will be augmented by specific constitutive models to fully describe the response of material bodies when they are subjected to Mechanical, Thermal or other kinds of external influences or loading.

All bodies are subject to exactly the same physical laws. Material constitutions differ. The latter is usually described by a model with specific parameters evaluated by experimental data.

Continuum Mechanics

The combination of physical laws and material constitutive models form the mathematical description of the body in question.

These laws are obeyed by the material as it responds to the influences of loading in the environment.

Everything is basically the same (Fluids, solids, mixtures, etc.) except for the empirical constitutive laws that tell how the particular material responds.

Mathematical Description

Only the simplest of these can be completely solved by the present mathematical tools available.

In typical problems, we are compelled to seek approximate solutions.

Several general algorithms have been devised over the years to do just that.

With the advent of powerful computational tools, some of these have grown in their range of applicability and sophistication

Solution Methods

We are therefore able to model and simulate the loading and responses in order to gain valuable insight that can be useful in design, failure prediction and prevention as well as using these materials in new ways such as biological implants, and other medical devices

Models and Simulations

Civil Structures. Buildings, roads and bridges, retaining walls and soil foundations.

Mechanical Design. Load bearing components for vehicles, engines, and other appliances. Selection of materials, dimensions and shapes appropriate to loading and service conditions

Manufacturing processes such as metal and polymer production, Computation of energy requirements and process design.

Design Examples

Biomechanics. Implants and medical devices for prosthetics, analysis of loads generated by biological processes such as heartbeats, blood flows and extraneous loads due to medical conditions such as blockages, etc

Other examples of applications are in Geomechanics, materials science, microelectronics, nanotechnology, etc

Application Examples

In your earlier courses such as strength of materials, you were able to compute the stresses and displacements in elastic materials subject to simple loading conditions. Furthermore, computations of the idealized “elastica” using the Euler-Bernoulli’s bending theory was used to predict displacements of members subjected to various kinds of loading and support systems.

Computations

Continuum Mechanics allow you to model under more general conditions. Materials are not necessarily elastic and may not be isotropic. Instead of focusing on the elastica, computer simulators allow you to model actual member dimensions directly. In addition, other computations that can be done include:

The forces that can cause prescribed shape changes.

Computations

Failure conditions due to multiaxial stresses; theories of failure. Extrapolations from uniaxial material properties

Computation of natural frequencies of vibration for dynamical problems; Avoidance of catastrophic resonance conditions

Critical fracture loads Failure under cyclic (fatigue) loading

Classical theoretical analysis model systems on dominant physical causes.

It is becoming increasing important to look at the interaction of several physical phenomena concurrently. The strains in a loaded material may be caused by applied forces as well as thermal loading concurrently.

This consideration has given rise to several simulation packages emphasizing the “multiphysics”

Multiphysics

The most successful constitutive model is that of homogeneous linear isotropic elasticity.

Several important materials are not accurately modeled under these assumptions.

The relaxation of any of these introduces great complexity to the mathematical formulation.

Even materials that are constitutively linear may have nonlinear strains as a result of large displacements or rotations.

Constitutive Models

The scope of Continuum Mechanics is wide In this course we will have the modest goals of

understanding the 1. Mathematical theory of shape changes that define

appropriate measure of displacement. This is also called kinematics

2. Theory of stress – a mathematical description of the internal forces in a solid.

3. Mechanical balance laws that every material is subjected to4. Work and energy concepts and conjugate analysis5. Basic constitutive models

Limitations

After understanding the kinematical relationships, Subjecting the material to balance laws (mass conservation, momenta, energy as well as entropy imbalance), formulating the required empirical constitutive relationships (cause and effect for the particular materials), we shall have had the required mathematical model that can be solved.

The obtained equations are partial differential equations that can be solved in closed form for only the simplest cases

In more complex cases, we must recourse to approximate methods

Solution Methods

Some of the most popular methods in practical use today are based on Finite Element Analysis

The MSC Nastran Simulation package is one of the most popular solution packages that we will look at.

University of Lagos Engineering Faculty also has two other packages: Siemens NASTRAN, Comsol Multiphysics. There are popular packages such as ANSYS, ABAQUS and ProEngineer we do not support here but good to know about because they are very popular in the Industry.

Solution Methods

We also are developing our own Finite Element Package here. It is based on Mathematica and we are extending some

toolboxes called ACE Fem. Students who are successful in this course may have the opportunity at a later date to join this effort.

Mathematica is also useful in understanding the Mathematics of Kinematics. Several of the tensor operations concrete examples are easily programmed in Mathematica. 20 percent of this course will be judged on your programming experience.

This should not scare you as we will help you through the process. Only lazy students need to be scared.

Mathematica