Continuum Mechanics

Continuum mechanics is a fundamental branch of mechanics that analyzes the kinematic and mechanical behavior of materials modeled as a continuous mass, or continuum, rather than as a collection of discrete particles. This idealization allows engineers to use differential equations to describe the macroscopic response of solids, liquids, and gases to applied forces, temperature changes, and other external effects. By applying principles of conservation of mass, momentum, and energy, and defining concepts like stress and strain, this field provides the governing theoretical framework for essential engineering disciplines such as solid mechanics, fluid dynamics, and heat transfer, enabling the analysis of material deformation, flow, and failure.

1. Introduction to Continuum Mechanics
   1. Continuum Hypothesis
      1. Definition of Continuum
      2. Assumptions and Limitations
      3. Discrete vs. Continuous Models
      4. Scale Separation Principle
   2. Macroscopic vs. Microscopic Scales
      1. Atomic and Molecular Structure
      2. Length Scale Hierarchy
      3. Validity of Continuum Approximation
      4. Breakdown Conditions
   3. Representative Volume Element
      1. Definition and Criteria
      2. Size Requirements
      3. Homogenization Concepts
      4. Application to Heterogeneous Materials
   4. Field Variables and Descriptions
      1. Definition of Fields
      2. Scalar Fields
         1. Temperature
         2. Pressure
         3. Density
      3. Vector Fields
         1. Displacement
         2. Velocity
         3. Acceleration
         4. Force per Unit Volume
      4. Tensor Fields
         1. Stress
         2. Strain
         3. Deformation Gradient
   5. Scope and Applications
      1. Solid Mechanics
      2. Fluid Mechanics
      3. Heat Transfer
      4. Multiphysics Problems