Some relevant definitions and results in the theory of differentiable manifolds, smooth vector fields, differential forms, (exterior) calculus (differentiation and integration using differential forms), differential equations and their associated flow maps, Symplectic manifolds; Brief review of Hamiltonian mechanics (Lagrange’s vs Hamilton’s Equations), Canonical Transformation, Legendre Transformation, Symplectic Transformations, Some definitions and results in the theory of Continuous Groups for Symmetries and Conserved quantities, Poincar´e-Cartan invariant, The Hamilton-Jacobi Partial Differential Equation. Integrable systems (simple examples); Some basic notions of numerical algorithms (order conditions etc). Examples of Numerical methods, Symplectic Integrators, and Geometric integrators. Applications to simple problems in particle dynamics and a two body problem; Symplectic Runge-Kutta Methods, Generating Function for Symplectic Runge-Kutta Methods and Symplectic Methods Based on it. Variational Integrators. Introduction to Hamiltonian Perturbation theory (if time permits). Discussion on some open problems in symplectic algorithms and a brief discussion on geometric numerical integration with some applications to mechanical systems.

Lecture wise Breakup

I. Lecture 1–2 (2 Lectures): (Part 0: Introduction)

• Overview of the course. Discussion on some applications, to mechanical systems, of geometric numericalintegration.

• Some applications of symplectic algorithms in understanding dynamics (of conservative systems). Discussion on constrained mechanical systems.

II. Lecture 3–12 (10 Lectures): (Part 1a: Differential Forms, Vector Fields and Manifolds)

• Sets, Mappings, Structure of Rn, Differentiation and Integration in Rn, Diffeomorphisms in Rn.

• Dual Vector Space, Exterior Forms in Rn (1-form, 2-form, k-form).

• Exterior Product.

• Differentiable Manifolds, Charts, Tangent Vectors, Vector fields.

• Tangent Bundle, Cotangent bundle, Differential Forms.

• Behavior of Differential Forms Under Mappings.

• Exterior Derivative.

• Lie Derivative, Lie algebra of vector fields.

• Chains (1-chain, 2-chain, k-chain), Boundary of Chains, Integration of differential forms, Stokes’ theorem.

III. Lecture 13–23 (11 Lectures): (Part 1b: Hamiltonian mechanics)

• Newtonian dynamics of n interacting particles (potential based interaction). Newton’s, Lagrange’s and Hamilton’s equations for n interacting particles. Legendre Transformation.

• Configuration space and Phase space.

• Hamiltonian function and vector fields, Lie algebra of Hamiltonians.

• Differential equations and their associated flow maps, Phase flows.

• Canonical Transformation.

• Symplectic Transformations, Symplectic maps and Hamiltonian flow maps.

• Invariants.

• Poincar´e-Cartan invariant.

• Some definitions and results in the theory of Continuous Groups for Symmetries and Conserved quantities.

• The Hamilton-Jacobi Partial Differential Equation.

IV. Lecture 24–30 (7 Lectures): (Part 2a: Examples of Numerical Algorithms)

• Some basic notions of numerical algorithms (order conditions etc).

• Examples of Numerical methods.

• Symplectic Integrators and Geometric integrators.

• Applications to simple problems in particle dynamics and a two body problem.

• One-step methods, Numerical example. Higher-order methods, Numerical example.

• Runge-Kutta methods.

• Forward vs backward error analysis.

IV. Lecture 31–40 (10 Lectures): (Part 2b: Symplectic Algorithms)

• Construction of symplectic methods by Hamiltonian splitting. Case studies.

• Symplectic Runge-Kutta Methods (1).

• Symplectic Runge-Kutta Methods (2).

• Generating Function for Symplectic Runge-Kutta Methods (1).

• Generating Function for Symplectic Runge-Kutta Methods (2).

• Symplectic Methods Based on Generating Functions.

• Numerical experiments.

• Variational Integrators.

• Introduction to Hamiltonian Perturbation theory (if time permits).

• Discussion on some open problems in symplectic algorithms.

References:

1. Arnold, V. I., 1989. Mathematical Methods of Classical Mechanics. Springer. Second edition. [Textbook: for part 1 only sections 18, 32–41, 44–48, for part 2 only sections 13–17, 19].

2. Leimkuhler, B., Reich, S., 2004. Simulating Hamiltonian Dynamics. Cambridge University Press [Textbook: for part 2 chapters 1, 2, 4–7, 9].

3. Hairer, E., Lubich, C., Wanner, G., 2006. Geometric Numerical Integration: Structure-PreservingAlgorithms for Ordinary Differential Equations. Springer.

Introduction to types of composites: metal matrix, ceramic matrix, polymer matrix and carbon-carbon composites; Characteristics of polymer matrices, Method of preparation of fibres (glass and carbon), characteristics of different types of fibers; Processing of fibre reinforced polymer matrix composites. Micromechanics and prediction of elastic constants of continuous and short fiber composites; Strength of composites; Constitutive relations, failure modes and failure theories for an orthotropic lamina; Behavior of laminated composites, classical laminate theory (CLT); Analysis of Laminates for first ply failure, progressive failure and for hygro-thermal loads using CLT. Interlaminar stresses and their significance, Test methods for characterization of composite elastic constants and strength; Strength of notched laminates.

Lecture wise Breakup

I. Introduction (1 Lecture):

• Need for composites, Types of composites, Metal matrix, Ceramic matrix and Carbon-Carbon composites; Polymer matrix composites

II. Constituent materials and fabrication methods (6 Lectures):

• Characteristics of thermosetting and thermoplastic resins, 

• Characteristics of Glass, Carbon and Kevlar Fibers, method of making and properties, types of fiber mats.

• Manufacturing of fiber composites: Hand layup, Pressure bag, Vacuum Bag and Autoclave processes, Pultrusion, Filament Winding, Bulk and Sheet molding compounds, Prepregs etc, including a video demonstration of a hand layup process. 

III. Micromechanics of continuous unidirectional fiber composites (8 Lectures):

• Prediction of elastic properties using strength of materials approach

• Introduction to elasticity based approach for prediction of elastic constants (concentric cylinder model)

• Empirical relations (Halpin-Tsai) for elastic property prediction

• Comparison of different approaches with examples, 

• Prediction of strength and discussion on failure modes

• Prediction of thermal and diffusion properties

IV. Short fiber composites (3 Lectures):

• Load transfer length, Prediction of elastic properties

• Elastic property calculation for random fiber composites

V. Analysis of orthotropic lamina (8 Lectures):

• Generalized Hooke’s law, Material symmetry

• Orthotropic materials and transversely isotropic materials

• Transformation of stress and strain, 

• Stress-strain relations for transversely isotropic lamina under plane stress in material axis and off-axis

• Failure theories (Maximum stress, strain, Tsai-Hill and Tsai-Wu)

VI. Analysis of laminated composites (12 Lectures):

• Description of laminate sequence and type of laminates (UD, Symmetric and Asymmetric, Balanced, Quasi-Isotropic) etc.

• Classical laminate theory (CLT)

• Failure analysis of laminates using CLT: First ply failure, progressive failure analysis

• Hygro-thermal stresses in laminates

• Discussion on interlaminar stresses

VII. Additional topics (4 Lectures):

• Characterization methods- Test methods for determining elastic constants and strength

• Fracture oriented failure- Strength of notched composite laminates

References:

1. Analysis and performance of fiber composites, B. D. agarwal, L. J. Broutman & K. Chandrashekhara

2. Engineering Mechanics of Composite Materials, I. M. Daniel & O. Ishai

3. Mechanics of Composite Materials, Autar K. Kaw

Fracture: Energy release rate, Linear elastic fracture mechanics: crack tip stress and deformation fields, Stress intensity factor (SIF) for plane and penny shaped crack, First order estimate of plastic zone using Irwin’s and Dugdale approach; Elasto-plastic fracture: HRR fields, J-integral and CTOD, Mixed mode fracture; Evaluation of SIF from experimental measurements, numerical simulations. Determination of fracture toughness under small scale yielding and elasto-plastic situations. Fatigue fracture: Crack nucleation and growth, Fatigue life prediction, Advanced topics.

Lecture wise Breakup

I. Introduction (4):

• Background; Griffith theory of fracture, energy release rate (ERR), conditions for stable and unstable crack growth, crack arrest

II. Linear elastic fracture mechanics (14):

• Williams analysis of stress field at the tip of a crack

• Solution of stress and displacement field for plane cracks using complex methods in plane elasticity (Westergaards or Kolosov-Muskhelishvili approach)

• Stress intensity factor (SIF) for plane and penny shaped cracks

• Equivalence of SIF and ERR, fracture toughness

III. Elasto-plastic fracture mechanics (10):

• First order estimate of crack tip plastic zone using Irwin’s and Dugdle’s approach

• Plastic zone for plane stress and plane strain situation and effect on fracture toughness

• Review of small strain plasticity

• Crack tip fields in an elasto-plastic material (Discussion on HRR fields)

• J-integral as a fracture parameter and crack tip opening displacement

IV. Mixed mode fracture (3):

• Prediction of crack path and critical condition for crack extension under mixed mode loading using Maximum tensile stress, Minimum strain energy density and Maximum energy release rate criteria

V. Experimental measurement of SIF and fracture toughness (3):

• SIF measurement using strain gages, optical techniques

• Evaluation of fracture toughness

VI. Fatigue crack growth (4):

• Mechanism of crack nucleation and growth under cyclic loading

• Determination of life of a cracked solid using Paris-Erdogan law and its variants

VII. Advanced topics (one from the following) (4):

• Computational fracture mechanics, Dynamic fracture, Bi-material fracture

References:

1. Fracture Mechanics, C.T. Sun and Z.H. Jin

2. Fracture Mechanics, T.L. Anderson

3. Fracture Mechanics, An Introduction, E.E. Gdoutos