Prerequisite: ME 820, ME 821(or concurrent).

Instructor: T. J. Pence, 2452EB, 353-3889, pence@egr.msu.edu.

Time and Place: MW 4:10-5:30 in 105 Bessey Hall

Texts:  The course is based on the Professor's notes and it is really not necessary to purchase any texts.  The following other references are useful for us.  In particular, both the book by Atkin & Fox, and the book by Ogden, are avialable in  Dover paperback and so very inexpensive. 

References:

• An Introductin to the Theory of Elasticity, R. J. Atkin and N. Fox, Dover
  
• Nonlinear Continuum Mechanics for Finite Element Analysis, J. Bonet and R. D. Wood, Cambridge University Press.
• Non-Linear Elastic Deformations, R. W. Ogden, Dover.
• Nonlinear Theory of Elasticity: Applications in Biomechanics, L. Taber, World Scientific Publishing Company
  

Course Rationale: This course will provide an introduction to the theory of finite (nonlinear) elasticity governing large deformations for highly deformable solids. Both material and geometrical nonlinearities will be addressed. Since the use of nonlinear theories may require a new point of view on the part of the practitioner, it is natural to develop a bias to stick with the standard linear theories with which we are comfortable. The present state of science and technology, however, is being driven by advanced materials and novel structures that cannot always be described by linear models. Furthermore, as conventional materials (e.g. new alloys, polymers, biological materials) are pushed to their limit, the issue of nonlinear behavior must be addressed. The importance of this issue is evidenced by the fact that conventional FEA codes are now increasingly including nonlinear elastic response in their materials library. A quick scan indicates:

	ABAQUS	ANSYS	MARC	NIKE & DYNA
multilinear		X
neo-Hookean	X	X	X	X
Mooney-Rivlin	X	X	X	X
Janus-Green-Simpson			X
Ogden	X		X
Blatz-Ko		X		X
Frazier-Nash				X
other special foams	X-Storakers-Ogden			X-crushable

The goal of this course is to provide a basis for understanding the issues that must be confronted for nonlinear elastic response. This need not be painful. In fact, by a careful examination of some fairly simple problems, a number of interesting and useful ways in which nonlinear theories depart from the standard linear treatment are uncovered.

Course Description: We will follow the notation of Bonet and Wood and somewhat correlate our treatment with their text. The course will begin with a brief review of those concepts from continuum mechanics which are necessary for describing large strain for both analytical and numerical treatments. Governing equations for the equilibrium theory will be posed in the setting of virtual work. Constitutive relations for both compressible and incompressible elastic materials will then be developed, including those models that are currently in wide use for rubbers and foams. This will be followed by the solution of certain basic equilibrium problems and a consideration of how these solutions differ from those in the classical linear theory. This will be followed with a considertion of: various non-uniqueness results and their interpertation in terms of bifurcation and generalized buckling. We will then examine various issues that are important for numerical implementation, including formulation of the constitutive laws for incompressible materials in terms of nearly incompressible materials, and the superposition of an infinitesimal deformation on a finite deformation.

Grading: Grades will be assigned on the basis of homework (50%), an exam (25%) and a project (25%). You may expect perhaps half a dozen homework assignments. The exam will be given when the course is substantially complete. The projects will involve independent examination of a topic of interest and reporting of that topic to the class. Possible topics include: dynamic effects, exact solution of specific interesting boundary value problems, non-isotropic finite elasticity, applications to tissue mechanics.