Southern Federal University
Milchakova str. 8a, Rostov-on-Don, 344090
Phone.: (863) 2975 111;
Fax: (863) 2975 113;
SYLLABUS
This course studies material nonlinearity as an important property of biological tissue models. The method of gathering material properties of the models is the main object of this course, namely, Saint-Venant semi-inverse method. It allows to determine and test various material parameters in set of simple experiments such as extension, torsion, disclination, and so on, of elastic canonic-shaped bodies. Along with that students get a deeper understanding of constitutive equation theory and approaches to solid media modeling. They also get a fully theoretical task of finding a square root of Cauchy deformation measure tensor, being left tensor of deformations.
Secondary aim of this course is to get use of Maple software and its freeware and opensource analogs as Maxima along with solvers Octave and SciLab. All tasks are intended to realization in two different ways using different software.
Linear algebra and differential equations courses are obligatory. Strong background in theoretical and/or continuum mechanics is also mandatory (tensor concepts and notations, stressstrain relations, theory of constitutive equations). Knowledge in computational algorithms is optional, as necessary algorithms such as shooting method will be discussed during the course.
After completing the course, the students are expected to be able to:
· Get Lame coefficients of any orthogonal coordinate system.
· Develop and solve boundary value problem (BVP) from semi-inverse deformation of canonic-shaped elastic bodies and unit strain energy function.
· Become an experienced user of advanced engineering environments such as Maple,
Maxima, SciLab, Octave.
The course will be taught mainly in lecture format along with class examples and demonstrations. Biological issues will be presented using demonstrations. Readings will be drawn from a text sources indicated below. Problems (Individual Tasks) will be assigned as follows: first and second one on second and third weeks respectively, and main third upon completion of first two (usually it takes three weeks). At the end of the term the qualification will take place. Upon the successful completion, the students will gain 3 credits.

№
Subject (theory) Subject (practice)
1
Computer algebra methods and definition. Software: history, capabilities, licensing, comparison.
2 3
Nonlinear biomechanics.
Nonlinear theory of elasticity in biomechanics. Theory of
Saint-Venant semi-inverse method.
4
5 6
Curvilinear coordinates.
Covariant derivative.
Christoffel symbols of second kind. Lame coefficients and gathering of them.
7
8
Automatization of gathering
Lame coefficients both in
Maple and Maxima. Gathering a
Christoffel symbols of second kind.
Maxima computer algebra system
Maple computer algebra system
Scilab package
Assignments
Individual problem #1:
«The orthogonality check and gathering of
Lame coefficients for a curvilinear coordinate system» in Maple and
Maxima.
Semi-inverse Saint-Venant method and its application to the problem of inflating of left heart ventricle
Deformation gradient and two methods of getting it from semi-inverse representation.
Algorithms.
Distortion and rotation tensors.
Cauchy deformation measure and its computer realization.
4
4
4
4
Individual problem #2:
«Computing of left distortion tensor using pre-defined Couchy deformation measure coefficient matrix».
4
Individual problem #3:
«Computer realization of semi-inverse method for solving the problems of nonlinear biomechanics»
4
Duration
(in hours)
4
8 Unit strain energy function for isotropic and transversal isotropic media. Governing equation for Piolla tensor.
Gathering of Piolla tensor.
Gathering deformation gradient

9
Equilibrium equation of solid media using Piolla stress tensor. Algorithms. Typical boundary conditions.
Gathering Piolla stress tensor and comparing it with known analytical one
10
Biomechanical applications of semi-inverse method.
Stretching and torsion of an artery with residual stresses.
Blastosphere with inner pressure. Stretching and torsion of muscular tissue.
Creating a program for getting divergence of second-rank tensor and getting equilibrium equations and BCs
Shooting method in Maple
Scilab, Octave and Maxima packets integration
Supersoft tissue modeling. An approach to in-vitro experiments with real tissue.
Creating a program for numeric experiment in CAS
Maple
Using SciLab for getting a numeric solution of BVP gathered. Coupling it with CAS
Maxima
Getting the resulting charts.
Completing the report.
The equilibrium stability of nonlinear elastic bodies. The method of applying small deformation to a finite one.
Final lesson.
Defending reports.
During the session students are required to
· attend class lectures;
· attend and conduct a research in the laboratory;
· write reports;
· present the report results in oral presentation at the end of course;
· be prepared to participate in discussions.
· Class participation - 30%
· Laboratory work - 45%
· Written report and its presentation – 30%
4
4
8
4
4
4

Core
1.
Biomechanics : principles and applications / edited by D. Schneck and J. D. Bronzino.
CRC Press, 2003. – 300 p.
2.
Fung Y.C. Biomechanics: Mechanical Properties of Living Tissues. – Springer, New
York. 1993.
3.
Sachse F. B. Computational Cardiology: Modeling of Anatomy, Electrophysiology, and
Mechanics. - Springer-Verlag, Berlin-Heidelberg, 2004. – 327 p.
4.
Taber L.A. Nonlinear theory of elasticity. Applications in Biomechanics. – Word
Scientific Publishing, 2004. – 399 p.
Internet Resources
1.
Maple User Contributed Applications http://maplesoft.com/applications/
2.
Maxima documentation http://maxima.sourceforge.net/documentation.html
3.
Scilab Online documentation http://www.scilab.org/support/documentation/manuals
Assistant Dmitry Sukhov, devitor@mail.ru