Linear Elasticity

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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.

**Syllabus**

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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