MATH35021 - 2007/2008

 

Specification

Aims

The course unit introduces students to the mathematical theory of linear elasticity. The general theory is developed and then applied to a number of practical problems in solid mechanics. Analytical techniques will be introduced to solve the resulting differential equations.

Brief Description of the unit

This course unit gives an introduction to the linearised theory of elasticity. A typical problem of the subject is as follows: Suppose an elastic body (e.g. an underground oil pipe) is subjected to some loading on its outer surface. What is the stress distribution which is generated throughout the body? Does this stress distribution have unexpectedly large values which might lead to failure? The subject is developed, and particular problems solved, from a mathematical standpoint.

Learning Outcomes

On successful completion of this course unit students will be able to

• calculate stresses, strains and tractions, and formulate boundary value problems;
• understand constitutive relations for elastic solids and compatibility constraints;
• solve various two-dimensional problems (plane strain) using the Airy stress function.

Future topics requiring this course unit

None.

Syllabus

1. Analysis of strain. The infinitesimal strain tensor; maximum normal strain. Equations of compatibility of strain.
2. Analysis of stress. The traction vector and the stress tensor; maximum normal stress. Stress equations of motion and their linearisation.
3. Stress-strain relations. Elastic and linearly elastic materials; isotropic materials. Equations of compatibility of stress for an isotropic materials in equilibrium (Beltrami-Michell equations). Navier's equation of motion for the displacement vector. Formulation of boundary value problems of linear elastostatics.
4. One-dimensional problems. A selection of soluble problems (which are effectively one-dimensional) in Cartesian, cylindrical polar or spherical polar coordinates. St. Venant's principle.
5. Plane strain problems. Theory of plane strain, Airy stress function. A selection of soluble two-dimensional problems using plane-strain theory.

Textbooks

The course does not follow one particular book. A good book, covering most of the course, is

• P.L. Gould, Introduction to Linear Elasticity, 2nd Edition, Springer, 1994.

This book, and many others on the theory of elasticity can be found at 531.38 in the John Rylands University Library.

Teaching and learning methods

Two lectures and an examples class each week. In addition students should expect to spend at least four hours each week on private study for this course unit..

Assessment

Coursework: weighting 20%

End of semester examination: two hours weighting 80%

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Arrangements