MEG 805: Continuum Mechanics

About Course

Continuum Mechanics can be thought of as the grand unifying theory of engineering science. Many of the courses taught in an engineering curriculum are closely related and can be obtained as special cases of the general framework of continuum mechanics. The balance laws of mass, momentum and energy that are derived in the context of specific material constitutions are natural laws that are natural laws and independent of these contexts. This fact is easily lost on most undergraduate and even some graduate students.

The language of Continuum Mechanics is called Tensor Analysis, you need Software to practice and engage more challenging problems; then Simulation will help you deploy the knowledge gained to design virtually in order to save prototyping costs. In this set of courses, we take you through all these stages to enrich your knowledge. The approach here is to optimize your time so to learn things the shortest way and remain focused on doing engineering with your knowledge.

Engineering is the application of Science to create technology products and services. It is rooted in theory. If you do not organize the learning of theory very well, you end up with full heads and no products as we have been doing. If you leave theory and simply do “practicals”, you end up with half-baked crafts trade – again, no serious products. We are offering you an approach to avoid both extremes and learn, in order to do engineering correctly.

What you will learn here will alter your view about some of the other courses you will take on your way to a degree in engineering. If you do your part, you will be given skills, tools and knowledge that are directed at making you productive people that can change the narrative of dependency and hopelessness that has been Africa’s story.

Course Content

Pre-Lecture Guide

Virtual Interaction Session

01:37:57
Learning Management Registration

Vectors: Elementary Principles & Computations Practicum
“God made the integers, all else is the work of man.” – Leopold Kronecker

Vectors: Elementary Principles & Computation Practicum – PG 2021 Videos

Vectors: Elementary Principles & Computation Practicum – PG 2023 Videos

Tensor Algebra: Properties of a Tensor
“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

Tensor Algebra: Properties of a Tensor PG – 2023 Videos
“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

Tensor Analysis: Differential and Integral Calculus with Tensors
“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.

Tensor Analysis: Differential and Integral Calculus with Tensors PG- 2023 Videos
“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.

Kinematics
The various possible types of motion leaving aside the causes to which the initiation of motion may be ascribed. - E.T. Whittaker

Week Thirteen: Introduction
Week Fourteen: The Deformation Gradient
Week Fifteen: Polar Decomposition Theorem
Week Sixteen: Mathematica Coding Session
Week Seventeen: The Strain Tensors
Week Seventeen: Tutorial Kinematics 1
Week Eighteen: Fusion 360 Tutorial – 1 of 2

00:00
Week Eighteen: Fusion 360 Tutorial – 2 of 2

00:00
Week Eighteen: Differential & Integral Calculus on Deforming Bodies
Week Nineteen: Velocity Gradient & other Tensors of Motion
Week Twenty: Kinematic Tutorial II

00:00
Week Twenty: Online Mathematical Coding Class 4

00:00

Student Ratings & Reviews

No Review Yet