Contents:

Overview and history, Discussion on inertial effects. Mathematical preliminaries. Basic linear elastodynamics, Waves in Periodic Structures, Causality Principle. One-dimensional Models. Static Cracks in a Linearly Elastic Body, Stress Intensity Factors and Crack Tip Singularity, Energy Release, General Crack System, Cohesive Zone Model. Elastodynamic solutions for a stationary crack, Scattering of a pulse and a time harmonic waves, Fracture initiation due to dynamic loading. Elastodynamic crack growth, the asymptotic crack tip field, Dynamic energy release rate. Onedimensional Discrete Models, Mode III fracture in square cell elastic lattice. Scattering of lattice waves. Instabilites in dynamic fracture. Modern topics and challenges in dynamic fracture.

Lecture-wise Breakup (50/75min)

I. Overview of the course. Historical origins. Discussion on inertial effects in fracture mechanics. (1/1 Lecture)

II. Mathematical preliminaries: Fourier and Laplace Transform, Green’s functions, Asymptotics. (3/2 Lectures)

III. Basic linear elastodynamics, Longitudinal and Shear Waves, Rayleigh Wave, Waves in PeriodicStructures, Energy Flux in a Wave, Causality Principle, Brittle vs Ductile behavior. (3/2 Lectures)

IV. One-dimensional Models, Difference Between Crack Initiation and Propagation Criteria. (3/2 Lectures)

V. Static Cracks in a Linearly Elastic Body, Kolosov-Muskhelishvili Representation, PapkovichRepresentation, Stress Intensity Factors and Crack Tip Singularity, Energy Release, Nonlinear elasticity effects, J-integral, Crack Opening and Stresses on the Crack Line, Integral Equations for a General Crack System, Cohesive Zone Model. (6/4 Lectures)

VI. Basic elastodynamic solutions for a stationary crack, Scattering of a pulse and a time harmonicwave by a mode III crack tip, Scattering of a pulse and a time harmonic wave by a mode I/II crack tip, scattering by a finite crack and several cracks, Fracture initiation due to dynamic loading. (6/4 Lectures)

VII. Dynamic Fracture in a Homogeneous Elastic Medium: Elastodynamic crack growth, the asymptotic crack tip field, Steady crack growth in a strip and in unbounded medium, Dynamic energy release rate, Crack growth due to time-dependent loading. (6/4 Lectures)

VIII. One-dimensional Discrete Models, Mode III fracture in square cell elastic lattice, Local energyrelease, Global energy release, Scattering of lattice waves, Dynamic Amplification Factor in Fracture. (6/4 Lectures)

IX. Crack growth at nonuniform speed, Fracture mode transition, Crack branching and instabilitesin dynamic fracture. (4/3 Lectures)

X. Modern topics and challenges in dynamic fracture: the role of material inelasticity, ElasticPlastic Fracture, rate effects, Cracks at material interfaces, Micromechanisms of fracture, Inverse problems in dynamic fracture, Intersonic and supersonic fracture, Earthquakes. (2/2 Lectures)

Text and References:

1. Freund, L. B., 1990. Dynamic Fracture Mechanics. Cambridge University Press (Textbook).

2. Slepyan, L. I., 2002. Models and Phenomena in Fracture Mechanics, Springer.

Aim:

Information such as energy and momentum is communicated through space and time via waves. Their study in elastic solids constitutes the subject of elastodynamics. This course presents the formulation and solution of elastodynamic problems in one, two and three dimensions. The notion of waveguides — structures that guide waves — is introduced through several examples, specially plates. Waves in anisotropic elastic media and crystals are also discussed. Experimental characterization is demonstrated.

Pre-requisites:

Introductory graduate courses on (a) theory of elasticity, (b) applied mathematics covering ordinary- and partial- differential equations. Exposure to complex analysis is recommended.

Course contents:

Waves in 1-d; Method of characteristics; Three-dimensional waves; Plane and harmonic waves; Reflection and transmission; Half-space problems; Waveguides: Dispersion, 1-d, Rods, Plates; Anisotropic media; Crystals; Experimental characterization; Advanced topics.

Topics with suggested number of lectures in parenthesis

I. Review of elasticity: Navier’s equation of motion, Boundary and initial conditions. (1)

II. Longitudinal and torsional waves in 1-D. D’Alembert’s solution. (1)

III. Method of characteristics; Radiation conditions; Wave packets; Group velocity. (3)

IV. Three-dimensional waves: Helmholtz decomposition, Dilatational and shear waves. (2)

V. Plane waves. Harmonic waves. Slowness diagrams. (3)

VI. Reflection and transmission of P, SV, SH waves across interface; continuity conditions; Snell’s law; Reflection and refraction at interfaces. (5)

VII. Half-space problems: Rayleigh waves; Suddenly applied uniform normal pressure with zero body force; Cagniard de Hoop method; Buried load problem; Scattering from crack tips in mode III. (8)

VIII. Waveguides: 1-d waves; Dispersion; String on elastic foundation; Cut-off frequency; 2-d waves; Thin plates (Kirchhoff’s theory); Lamb waves; Love waves; Rods; Pochammer-Chree equation. (10)

X. Waves in anisotropic media and crystals. (3)

XI. Experimental characterization: Kolsky bar. (2)

XII. Advanced topics. One from: Plastic waves/Layered media/Visco-elastic waves/Shock waves/ Nonlinear waves/Thermal waves/Waves in discrete media/Scattering from mode I and II cracks. (4)

Textbooks and references:

1. Wave Propagation in Solids, J.D. Achenbach, Elsevier Science Publishers,1975.

2. Wave Motion in Elastic Solids, K. F. Graff, Dover Publications, 1991.

3. Mechanics of Continua and Wave Dynamics, L. Brekhovskikh and V. Goncharov, Springer-Verlag, 1985.

4. The Theory of Elastic Waves and Waveguides, J. Miklowitz, North-Holland Publishing Company, 1978.

Prepared by :

Ishan Sharma
N N Kishore
B L Sharma
S S Gupta
P M Dixit

Organization of animal cells; Structure and function of cell membrane; Role of fluid lipid bilayers in cell functionality; Experimental methods to study membranes; Self assembly of lipid bilayer, Brief review of differential geometry concepts; Development of elasticity models of membranes; Stable equilibrium shapes of red blood cells; shapes of phase separated fluid lipid bilayer vesicles; ; Adsorption of proteins to lipid membrane; Special topics from current research.

Lecturewise Breakup

I. Introduction (4 Lecture):

• Organization of animal cells (1 lecture)

• Structure and function of cell membranes (1 lecture)

• Role of fluid lipid bilayer in cell functionality and Experimental methods to study membranes (2 lecture)

II. Self assembly of lipid bilayer (5 lectures):

• Thermodynamics of self assembly (3 lectures)

• Self assembly of lipids – shape and aggregate structure (2 lecture) 

III. Review of differential geometry (8 Lecture):

• Introduction to membrane elasticity – stretching and bending (1 lecture)

• Differential geometry – curvilinear coordinates (1 lecture)

• Curvilinear coordinates on a surface (1 lectures)

• Covariant and contravariant derivatives (1 lecture)

• Mainardi-Codazzi relations (1 lecture)

• Ricci’s lemma, Surface divergence, Green’s theorem (3 lectures)

IV. Elasticity models for lipid bilayer (9 lectures):

• Development of elasticity models for lipid bilayer (3 lectures)

• Derivation of stress equilibrium equations of a lipid bilayer (1 lecture)

• Shape analysis of a red blood cell (3 lectures)

• Shape analysis of a two phase lipid bilayer vesicle (2 lectures)   

V. Adsorption of proteins to membranes (7 lectures):

• Review of relevant thermodynamic concepts (2 lectures)

• Adsorption of proteins to flat surface using van der Waals and Bragg Willimas gas models (3 lectures)

• Adsorption of proteins to cylindrical and spherical membrane tubes (2 lectures)

VI. Additional topics – a few topics to be selected from below (7-9 lectures):

• Monge parameterization of surface and membrane fluctuations 

• Adhesion of cells and vesicles to substrates

• Development of more refined fluid models for proteins based on protein-protein interactions and proteins shapes.

• Diffusion of proteins and domain in the membrane surface.

References:

1. B. Alberts et al., Molecular Biology of the Cell, Garland Science, NY. 

2. J. N. Israelachvili, Intermolecular and Surface Forces: With Applications to Colloidal and Biological Systems, Academic Press, 1992 (second edition). 

3. R. Lipowsky and E. Sackmann, Structure and Dynamics of Membranes, Handbook of Biological Physics Vol. 1, Elsevier, Amsterdam, 1995. 

4. S. A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes Westview Press, 2003. 

5. Relevant journal articles.