**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:

- An Introduction 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
- Nonlinear Solid Mechanics: A Continuum Approach for Engineering, G. A.
Holzapfel

**
**

**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:

- the consideration of directional reinforcement (e.g., fiber stiffening),
- generalizations so as to treat swelling and other physical phenomena,
- issues that are important for numerical implementation, (including formulation of the constitutive laws for incompressible materials in terms of nearly incompressible materials).

**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.