ME 921 Nonlinear Elasticity - Fall 2017 

Prerequisite: The suggested prerequisites are ME 820 (continuum mechanics) and ME 821 (linear elasticity).  Students with alternative preparation may also enroll provided that they have an appropriate background.  The appropriate background is an understanding of how  boundary value problems are formulated in solid mechanics so as to involve both displacement and traction surface conditions.  

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

Time and Place: MW 4:10-5:30 in 2245 EB

Texts:  The course is based on the Professor's notes.   It is not necessary to purchase any texts.  The  references listed below are useful for us and will be put on reserve in the engineering library.  However, both the book by Atkin & Fox, and the book by Ogden, are avialable in  Dover paperback and so very inexpensive. 

References:

                    

Course Overview: This course will provide an introduction to the theory of finite (nonlinear) elasticity governing large deformations for highly deformable solids, including soft materials such as gels and hydrated soft tissue.  The course material should be of interest to mechanical engineers, civil engineers, chemical engineers, materials scientists, polymer chemists,  applied physicists, applied mathematicians, theoretical biologists and specialists in biomechanics.

Course Rationale: This course will provide an introduction to the theory of finite (nonlinear) elasticity governing large deformations for highly deformable solids: polymers, rubbers, gels, rheological materials with solid-like behavior, and liquid saturated soft tissue. 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 a notation that is similar both to that of Bonet & Wood, and of Holzapfel.   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. We shall then consider 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 situations where more than one solution is possible and its interpertation in terms of bifurcation (generalized buckling) and material stability.  The course will conclude with topics drawn from the following: 

 

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, anisotropic finite elasticity, fluid saturation and swelling for very soft materials, applications to tissue mechanics.