[edit]

\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

ENGN2210 Continuum Mechanics

Notation

• Basic notation
• Einstein Summation Convention
  • Kronecker-delta symbol
  • Einstein Summation Convention: examples
• Components of Vectors
• Basic Mathematical Notation and Definition

Mathematical preliminaries

• (real) Vector space ($\mathbb{V}$)
• Dimension and basis. Finite dimensional vector space $\mathbb{V}^n$
• (real) Inner product space
• (real) Euclidean vector space $(\mathbb{E}^n)$
• Affine point space $(\mathcal{E})$
• (real) Euclidean space ($(\mathcal{E}^n,\mathbb{E}^n)$)
• Functions and mappings
• Maps between Euclidean spaces
• Linear maps between Euclidean spaces and Tensors

Tensors

• Functions and mappings
• Components/Matrix form
• Identity Tensor
• Null Tensor
• Dyadic products
• Representation of tensors as a sum of dyadics products
• Transpose, Symmetric and Skew-Symmetric Tensors, Symmetric and Skew-symmetric parts of a tensor
• Trace of a Tensor
• Product of Tensors

Kinematics

Kinematics is the study of motion, without regard to the forces that produce it

• Lagrangian co-ordinates
• Eulerian co-ordinates

Deformation Mapping

• Material manifold, and reference and deformed bodies
• Deformation mapping

Deformation Gradient

• Deformation Gradient

Metric Changes:

• Infinitesimal length
• Volume Changes
• Area Changes
• Length Changes
• Velocity and acceleration fields
  • Material velocity field
  • Material acceleration field
  • Spatial velocity field
  • Spatial acceleration field
  • Example:The shrinking sphere

Linear Algebra and Analysis

Linearization

Multivariable

Time

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Reference

Rotation Tensors:

• Rotation tensors in 3D
• Transpose, Symmetric, Skew Symmetric parts of R
• Finding Angle and Axis of Rotation given a Rotation Tensor
• Orthogonal Tensor
• Orthogonal and Proper Orthogonal transformation
• Rotations as Proper Orthogonal transformations
• Co-rotational basis
• Time derivative of the Co-rotational basis

Non-dimensional kinematics

• Non-dimensionalization
• Non-dimensional rigid body kinematics

Euler angles

• Euler angles
• Components of the rotation tensor in terms of Euler angles

Angular velocities

• Velocity field
• Axial vectors
• Angular velocity tensor and vectors
• Time derivative of angular velocity
• Composition of rotations
• Relative angular velocity vector