Course detail
Constitutive Equations for BIO
FSI-RKB-A Acad. year: 2021/2022 Winter semester
The course provides a comprehensive overview od constitutive dependencies and constitutive models of matters, not only of solids (i.e. strructural materials) but also of liquids and gases. It deals also with time dependence of stress-strain response of materials and describes it using different viscoelastic models. It introduces the theory of finite strains and applies it in description of non-linear elastic as well as poroelastic and non-elastic behavour of soft biological tissues, also with taking their anisotropy caused by their fibrous structure into consideration. Models accounting for waviness and directional dispersion of collagen fibres in the tissues are adressed. Also other specific properties of biological tissues absent at technical materials are presented, including their impact on procedures of mechanical testing and ways how to take them into consideration in constitutive models of soft tissues. For each of the presented models basic constitutive equations are formulated, on the basis of which the response of the tissue under load is derived using both analytical and numerical (FEM) methods, including applications of the models in ANSYS software.
Supervisor
Learning outcomes of the course unit
Students get an overview of mechanical properties and behaviour of matters and of possibilities of their modelling, especially under large strains. They will have a clear idea of sophisticated application of computational modelling in biomechanical problems with soft tissues. Within the framework of capabilities of the used FE programme systems, they will be made familiar with the practical use of some of the more complex constitutive models (hyperelastic and non-elastic, isotropic and anisotropic) in stress-strain analyses.
Prerequisites
Students are expected to have knowledge of basic terms of theory of elasticity (stress, strain, general Hooke's law), as well as some basic terms of hydrodynamics (ideal, Newtonian and non-Newtonian liquids) and thermodynamics (state equation of ideal gas, thermodynamic equilibrium). Fundamentals of FEM and basic skills in ANSYS program system are required as well.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical application of topics presented in lectures using ANSYS software.
Assesment methods and criteria linked to learning outcomes
The course-unit credit is awarded on condition of having actively participated in seminars and submitted an individual semester project. The exam is based on a written test of basic knowledge and defense of the individual semester project.
Language of instruction
English
Aims
The objective of the course is to provide students a comprehensive and systematic overview of constitutive dependencies of various types of matters, to interconnect their knowledge acquainted in various courses and fields (solid mechanics, hydromechanics, thermomechanics) and, at the same time, to make students familiar with practical applications of some of the constitutive models (in FE program system ANSYS) useful in modelling of soft tissues.
Specification of controlled education, way of implementation and compensation for absences
Attendance at practical training is obligatory. An apologized absence can be compensed by individual works controlled by the tutor.
The study programmes with the given course
Programme N-IMB-P: Engineering Mechanics and Biomechanics, Master's
branch BIO: Biomechanics, 6 credits, compulsory
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
1. Definition of the term constitutive model. Overview of constitutive models in mechanics, constitutive models for individual states of matter.
2. Hooke's law and Newton's law of viscosity in general tensor notation. Introduction to linear theory of viscoelasticity.
3. Models of linear viscoelasticity – response under static and dynamic loads.
4. Complex modulus of elasticity, relaxation and creep functions.
5. Stress and deformation tensors under large strain conditions. Basic tensor operations. Definition of hyperelasticity.
6. Decomposition of deformation tensor and its invariants. Hyperelastic isotropic polynomial models, their applicability for soft tissues.
7. Mechanical tests of hyperelastic materials. Predictiive capability of constitutive models.
8. Structure based hyperelastic models. Models of very compressible elastomers (foams).
9. Models describing inelastic effects of soft tissues, large strain viscoelasticity.
10. Models of shape memory alloys.
11. Anisotropic hyperelastic models of soft tissues with reinforcing fibers. Pseudoinvariants of deformation tensor.
12. Models accounting for dispersion and waviness of fibres.
13. Models of active behaviour of biological tissues: muscle contraction and remodelation of tissues.
Computer-assisted exercise
13 hours, compulsory
Teacher / Lecturer
Syllabus
1.-2. Revision of applications of linear elastic constitutive model. Matrix and tensor forms of Hooke’s law and Newton's law of viscosity.
3.-4. Linear viskoelasticity – behaviour of simple rheological mdoels.
5.-6. Introducing experimental data into FE models of viscosleasticity and temperature dependence of viscoelastic parameters.
7.-8. Hyperelastic models in ANSYS – testing of soft tissues and their input into the constitutive model.
9.-10. Choice of a suitable constitutive model of an incompressible soft tissue, predictive capability of the model.
11.-12. Anisotropic hyperelastic models, models of non-elastic behaviour.
13. Semester project, course-unit credit.