Continuum Mechanics

Introduction

Continuum Mechanics is an introductory course in the analysis of the kinematic and mechanical behavior of materials modeled on the continuum assumption. The course will focus on the tools necessary to model soft tissues, including the essential mathematics, stress principles, kinematics of deformation and motion, elasticity, and viscoelasticity.

Course Rationale

The course is appropriate for anyone wishing to learn continuum mechanics, even those outside the field of biomechanics. The emphasis for applications will be on soft tissues, but the student will develop general skills for continuum stress and strain analysis.

Course Goals

After active participation in this course and an effort to learn the material, students will be able to:

1. Describe kinematics and deformation using Lagrangian and Euler descriptions.

2. Develop constitutive equations based on fundamental laws and equations.

3. Become a skilled user of advanced design tools such as nonlinear, explicit finite elements.

4. Apply linear elasticity to solve for stress distributions.

5. Apply nonlinear elasticity (based on a molecular and phenomenological development) to rubber-like materials.

6. Use linear (and quasi-linear) viscoelasticity to solve for stresses in modern polymers and soft tissues.

Course Materials

Continuum Mechanics for Engineers, 2nd Edition, by G. Thomas Mase and George E. Mase (1999)

Course Outline

Day 1:

Course Introduction ("prerequisites"), Policies,

Introduction to Blackboard Software,

Essential Mathematics (Chapter 1)

Weeks 2-3: Essential Mathematics (Chapter 1)

Scalars, Vectors, Tensors, Symbolic and Indicial Notation

Matrices & Determinants, Tensor Transformations, Eigenvalues and Eigenvectors

Tensor Fields & Tensor Calculus, Integral Theorems of Gauss and Stokes

Weeks 3-5: Stress Principles

Body/Surface Forces & Density, Cauchy Stress Principle, The Stress Tensor, Force and Moment Equilibrium, Stress Tensor Symmetry, Stress Transformation Laws, Principle Stresses & Directions, Maximum & Minimum Stresses, Mohr�s Circle for Stress, Plane Stress, Deviator and Spherical Stress, Octahedral Shear Stress

Weeks 5-7: Kinematics of Deformation & Motion

Particles and Configurations, Deformation and Motion, Material and Spatial Coordinates, Lagrangian & Eulerian Descriptions, Displacement Fields, Material Derivatives, Deformation Gradients, Finite Strain Tensors, Infinitesimal Deformation Theory, Stretch Ratios, Rotation & Stretch Tensors, Velocity Gradient, Rate of Deformation, Vorticity, Material Derivatives of Line Elements, Areas and Volumes

Weeks 8-9: Fundamental Laws and Equations

Balance Laws, Field and Constitutive Equations, Material Derivatives of Line, Surface, and Volume Integrals, Conservation of Mass and Continuity Equation, Linear Momentum Principle and Equations of Motion, Piola-Kirchhoff Stress Tensors, Lagrangian Equations of Motion, Angular Momentum Principle Conservation of Energy, Entropy, Material Restrictions due to 2nd Law, Invariance Constitutive Equation Restrictions Due to Invariance, Constitutive Equations

Week 10: Nonlinear Elasticity

Molecular Approach to Rubber, Strain Energy Theory, Specific Forms of Strain Energy, Neo-Hookean Material

Week 11-12: Linear Viscoelasticity

Viscoelastic Constitutive Equations, One-Dimensional Theory/Models, Creep and Relaxation, Superposition Principle, Heredity Integrals, Harmonic Loadings, Complex Modulus and Compliance, Three-Dimensional Problems, Correspondence Principle

Week 13-15: Quasi-Linear Viscoelasticity

Quasi-Linear Viscoelastic Constitutive Equations, Applications to soft tissues

Course Requirements and Grading

The grading scheme is subject to change. Current plans are as follows:

Activities (20%)

Midterm 1 (40%)

Final Exam (40%)