Advanced Mechanics of Solids

Syllabus and Lecture Notes

Course Goals: on completing EN1750, you will:

Understand the mathematical and physical foundations of the continuum mechanics of solids, including deformation and stress measures, elastic and plastic stress-strain relations, and failure criteria; have the ability to pose and solve boundary value problems involving deformable solids; be able to analyze wave propagation and vibrations in elastic solids and understand the theoretical basis for finite element analysis of elastic solids.
Relates to ABET outcome (1)
Be proficient in the use of a modern finite element analysis program (Abaqus/CAE) for analyzing stress, deformation and failure in components, assemblies and structures.
Possess the ability to apply the principles of solid mechanics to solve engineering problems and to design systems or components to meet desired needs, including (a) to idealize a system or component for the purposes of stress analysis; (b) to use appropriate numerical and analytical techniques to model the system; (c) to interpret and draw appropriate conclusions from the results; and (d) present results and conclusions clearly.
Relates to ABET outcome (1), (2), (3)

Workload Expectation

Lectures: 39 hours
Review lecture notes, background reading: 30 mins per scheduled class (20 hrs)
Homework assignments: 10 at 8 hours each (one ungraded)
Final Project: 15 hours
Midterm exam: 1 hour, in class, plus 10 hours preparation
Final exam, 3 hours (as scheduled by Registrar) plus 15 hrs preparation

TOTAL: 183 hours.

A note on downloading matlab codes: if you download a code more than once, the second time the file name may be stored as something like matlab_file (2).m or matlab_file (3).m. If you run these, they will crash. That is because matlab is confused by the (2) and (3) in the file name. If you rename the file to remove it they will work.

Notes from Lecture (10:30am Tu/Th).

Detailed notes (electronic text)

1. Brief introduction to the objectives and methods of solid mechanics

1.1 Fundamental Postulates of Solid Mechanics

1.2 Defining a Problem in Solid Mechanics

2. Introduction to Finite Element Analysis in Solid Mechanics (pdf version)

2.1 Introduction

2.2 Finite Element Mesh

2.3 Material Behavior

2.4 Boundary Conditions, Constraints, Interfaces and Contact

2.5 Initial Conditions and External Fields

2.6 Solution Procedures and Time Increments

2.7 Output

2.8 Units in FEA simulations, Using Dimensional Analysis

3. Mathematical prerequisites

3.1 Vectors and Vector Operations

3.2 Vector Fields and Vector Calculus

3.3 Matrices

3.4 Brief Introduction to Tensors

3.5 Vectors and Tensors in Spherical Polar Coordinates

3.6 Vectors and Tensors in Cylindrical Polar Coordinates

3.7 Index Notation for Vector and Tensor Operations

4. Analyzing Deformation of a Solid

4.1 The Displacement and Velocity Fields

4.2 Displacement Gradient and Deformation Gradient

4.3 Deformation Gradient Resulting From Successive Deformations

4.4 Jacobian of the Deformation Gradient

4.5 Lagrange Strain

4.6 Infinitesimal Strain

4.7 Engineering Shear Strain

4.8 Decomposition of Infinitesimal Strain into Volumetric and Deviatoric Parts

4.9 Infinitesimal Rotation Tensor

4.10 Principal Values of Infinitesimal Strain

4.11 Cauchy-Green Deformation Tensors

4.12 Rotation Tensor and Stretch Tensors

4.13 Principal Stretches

4.14 Generalized Strain Measures

4.15 Velocity Gradient

4.16 Stretch Rate and Spin

4.17 Infinitesimal Strain Rate and Rotation Rate

4.18 Other Deformation Rate Measures

4.19 Strain Equations of Compatibility

5. Analyzing Internal Forces in a Solid

5.1 Surface Traction and Body Force

5.2 Traction Acting on Planes Within a Solid

5.3 Cauchy (True) Stress Tensor

5.4 Kirchhoff, Nominal and Material Stress

5.5 Stress Measures for Infinitesimal Deformations

5.6 Principal Stresses

5.7 Hydrostatic and Deviatoric Stress, Von-Mises Effective Stress

5.8 Stresses Near an External Surface: Boundary Conditions for Stresses

6. Equations of motion and equilibrium for deformable solids

6.1 Linear and Angular Momentum Balance Relations for Deformable Solids

6.2 Work Done by Stresses

6.3 The Principle of Virtual Work

7. Stress-Strain Relations for Linear Elastic Materials

7.1 Isotropic, Linear Elastic Material Behavior

7.2 Stress-Strain Relations for Isotropic Elastic Materials: Youngs Modulus and Poissons Ratio

7.3 Reduced Stress-Strain Relations for Plane Deformations

7.4 Representative Values for Isotropic Elastic Materials

7.5 Other Elastic Constants

7.6 Physical Interpretation of Elastic Constants for Isotropic Materials

7.7 Strain Energy Density for Isotropic Solids

8. Solutions to Static Problems for Linear Elastic Solids

8.1 Summary of the Governing Equations of Linear Elasticity

8.2 Superposition and Linearity of Solutions

8.3 Uniqueness and Existence of Solutions for Elastic Solids

8.4 Saint-Venants Principle

8.5 Simplified Equations for Spherically Symmetric Linear Elastic Solids

8.6 General Solution to the Spherically Symmetric Linear Elasticity Problem

8.7 Examples of Solutions to Spherically Symmetric Elasticity Problems

8.8 Simplified Equations for Axially Symmetric Linear Elastic Solids

8.9 General Solution to the Axially Symmetric Linear Elasticity Problem

8.10 Examples of Solutions to Spherically Symmetric Elasticity Problems

8.11 Airy Function Solution to Plane Problems in Linear Elasticity

8.12 Examples of Airy Function Solutions

9. Energy Methods for Elastic Solids

9.1 Kinematically Admissible Displacement Fields

9.2 Definition of Potential Energy for Elastic Solids

9.3 The Principle of Minimum Potential Energy

9.4 Useful Formulas for Potential Energy (strings, rods, beams, membranes and plates)

9.5 Uniaxial Compession of a Cylinder Solved by Energy Methods

9.6 Energy Method for Calculating Stiffness

9.7 Rayleigh_Ritz Method for Calculating Approximate Static Solutions for Elastic Solids

10. Implementing the Finite Element Method for Linear Elastic Solids

10.1 The Finite Element Mesh and Element Connectivity

10.2 The Global Displacement Vector

10.3 Element Interpolation Functions

10.4 Element Strains, Stresses, and Strain Energy Density

10.5 The Element Stiffness Matrix

10.6 The Global Stiffness Matrix

10.7 Boundary Loading

10.8 Global Force Vector

10.9 Minimizing the Potential Energy

10.10 Eliminating Prescribed Displacements

10.11 Solution

10.12 Post Processing

10.13 Example FEA Code

Matlab Code

Input file for 2 element test

Input file for hole in a plate

11. Solutions for Solids with Special Shapes: Strings, Beams, Membranes and Plates

11.1 Analyzing Motion and Deformation of Straight Beams and Strings

11.2 Analyzing Motion and Deformation of Flat Plates and Membranes

12. Dynamics: wave propagation and vibrations in elastic solids

12.1 Wave Propagation in a String

12.2 Pressure and Shear Plane Waves in 3D Elastic Solids

12.3 Rayleigh Waves

12.4 Travelling Waves in Beams

12.5 Modelling Transient Dynamics with FEA

12.6 Free Vibration of Strings and Beams

12.7 Calculating Natural Frequencies and Mode Shapes with ABAQUS

13. Plasticity

13.1 Features of the Inelastic Response of Metals

12.3 Decomposition of Strain into Elastic and Plastic Parts

13.3 Yield Criteria

13.4 Graphical Repesentation of the Yield Surface

13.5 Strain Hardening Laws

13.6 The Plastic Flow Law

13.7 The Elastic Unloading Condition

13.8 Complete Incremental Stress-Strain Relations for Rate Independent Plasticity

13.9 Typical Values for Yield Stress of Polycrystalline Metals

13.10 Thin Walled Tube under Combined Tension and Torsion

13.11 Elastic-Perfectly Plastic Hollow Sphere Under Internal Pressure

14. Failure

14.1 Failure by Geometric Instability: Buckling

14.2 Summary of Mechanisms of Failure Under Static and Cyclic Loading

14.3 Stress Based Failure Criteria for Brittle Materials

14.4 Strain Based Failure Criteria for Ductile Materials

14.5 Criteria for Failure Under Cyclic Loading