MAE 256 Course Overview

Linear Elasticity
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Description of the CourseChris Lynch

Lecture, four hours; outside study, eight hours. Linear elastostatics. Cartesian tensors; infinitesimal strain tensor; Cauchy stress tensor; strain energy; equilibrium equations; linear constitutive relations; plane elastostatic problems, holes, corners, inclusions, cracks; three-dimensional problems of Kelvin, Boussinesq, and Cerruti. Introduction to boundary integral equation method. Letter grading.

Background students will need
Although prerequisites are not enforced for graduate students, it is strongly recommended that each student has taken courses equivalent to “Advanced Strength of Materials” or “Analysis of Flight Structures”.

About the Instructor
Christopher Lynch
Research Interest: Ferroelectric materials including experimental characterization of constitutive behavior under multiaxial loading.
Lecture 1: Review of matrices, direction cosines, vector transformation, index notation, tensors, tensor products, tensor transformations
Lecture 2: Tensor calculus: gradient, div, curl, divergence thm, stokes thm
Lecture 3: Cylindrical coordinates: grad, div, curl, laplacian Spherical coordinates: grad, div, curl, laplacian
Lecture 4: Stress,strain, equilibrium, Hooke’s law, boundary conditions (traction vector), thermal strain
Lecture 5: Field equations: Equilibrium, constitutive, compatibility strain displacement, boundary conditions, cylindrical, spherical, Navier’s equations, reciprocity
Lecture 6: Cylindrical pressure vessel, rotating shaft, cylindrical inclusion, thermal stress
Lecture 7: Spherical shell, gravitating sphere, thermal stress in sphere, further discussion of compatibility, anisotropic materials
Lecture 8: More about anisotropic materials, Laplace’s equation for plane stress
Lecture 9: Laplace’s equation for plane strain, Generalized plane stress, Airy’s stress function
Lecture 10: Midterm I
Lecture 11: Solutions in Cartesian coordinates, beam examples, integration to obtain strain
Lecture 12: Dam problem, Airys’s stress function in polar, welded ring, arch problem
Lecture 13: Line load on semi infinite solid, wedges with vertex loads, circular hole in a plate
Lecture 14: Place stress, plane strain, compatibility, polynomial solutions
Lecture 15: Complex variable based solutions, 3-D problems
Lecture 16: Circular home in a plate, the inclusion problem, Complex variable problems: circular disc, concentrated force
Lecture 17: Brazil nut, stress intensity factor, mode III fracture
Lecture 18: Combined mode I and mode II fracture solutions, strains and displacements
Lecture 19: Torsion of cylinders, general solution of equilibrium equations. Concentrated force in an infinite elastic medium, Boussinesq’s problem
Lecture 20: Review
Final Exam
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