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1 Lahore University of Management Sciences B.Sc. (Honours) Programme MS Programme Course Outline Course: Continuum Mechanics (Math 364) Pre-requisite: Linear Algebra (Math221), Multivariate Calculus (Math 103) Quarter: Summer Instructor: Dr. Wasiq Hussain E-mail: [email protected] Text: Nonlinear Solid Mechanics, A Continuum Approach for Engineering, by Gerhard A. Holzapfel (2000). (John Wiley & Sons) Reference book: Non-Linear Elastic Deformations by R.W. Ogden, 1997, Dover Publications (New York). Units: 4 Sessions: 5 lectures of 80 minutes per week 2 At the microscopic level matter is discontinuous. It is composed of molecules and atoms. However, in real life we are concerned with volumes of matter with dimensions large compared with these particles. CONTINUUM MECHANICS deals with the behaviour of bodies of solids and fluid matter on a macroscopic scale. It disregards the microscopic structure and treats matter as being smoothly distributed throughout the body under consideration. GOALS: 1. To provide an introduction to the basic theoretical ideas in continuum mechanics using the theory of CARTESIAN TENSORS. 2. Apply the theory in the solution of different types of problems in nonlinear elasticity. TEACHING PLAN: WEEKS TOPIC READ CHAPTER 1 Introduction to Vectors and Tensors IST (1) Algebra of vectors Pages: 1 - 9 (2) Transformation laws for Basis Vectors Pages: 28 - 32 and Components IST (3) Algebra of Tensors Pages: 9 - 20 (4) Eigenvalues, Eigenvectors of Tensors Pages: 24 – 27 3 CHAPTER 2 KINEMATICS 2ND (1) Configurations, and Motions of Pages: 56 – 60 Continuum Bodies. (2) Displacement, Velocity, Acceleration Fields. (3) Gradients and Related Operators (from Chapter 1). (4) Material, Spatial Derivatives. (5) Deformation Gradient 2ND (6) Strain Tensors (7) Rotation, Stretch Tensors with applications like SIMPLE SHEAR deformation. (8) Rates of Deformation Tensors. Pages: 61 – 64 Pages: 44 – 51 Pages: 64 – 69 Pages: 70 – 76 Pages: 76 – 84 Pages: 85 – 94 Pages: 95 - 105 CHAPTER 3 THE CONCEPT OF STRESS 3RD (1) Traction Vectors, and Stress Tensors Pages: 109 – 118 4 CHAPTER 4 BALANCE PRINCIPLES 3RD (2) Conservation of Mass Pages: 131 – 138 (3) Integral Theorems (from Chapter 1) Pages: 52 – 54 (4) Transport Formulae 3RD Mid-Term (5) Momentum Balance Principles 4TH Pages: 138 – 140 (1) Balance of Mechanical Energy Pages: 141 - 152 Pages: 152 - 160 CHAPTER 5 SOME ASPECTS OF OBJECTIVITY 4TH (2) Objective Tensor Fields Pages: 179 - 187 4TH (3) Invariance of Elastic Material Response Pages: 196 - 203 CHAPTER 6 HYPERELASTIC MATERIALS 4TH (4) General Remarks on Constitutive Equations Pages: 206 – 212 5TH (1) Isotropic Hyperelastic Materials Pages: 212 - 222 5TH (2) Incompressible Hyperelastic Materials Pages: 222 - 227 Grading: 40% Final. 30% Midterm. 30% Quizzes. 5