F7AMBBB - Biomechanics and Biomaterials

The objective of the course is to provide students with a comprehensive foundation in the mathematical and mechanical principles of continuum mechanics and to demonstrate their application in describing the behavior of biological tissues. Students will learn to analyze kinematics, stress, and deformational response (elasticity, viscoelasticity, fluid flow) and will understand more complex models, such as mixture theory, which are key to understanding the functionality of materials, such as bones, tendons, blood, or cartilage.

Conditions for Credit Assessment

Required attendance in classes.

Achieving a minimum of 50% of points from the final credit test.

Exam Requirements The exam verifies the ability to independently apply knowledge. To pass, it is necessary to correctly solve:

2 practical examples (application of theory to specific problems).

1 theoretical derivation (based on topics covered in lectures).

BLOCK 1: Mathematical and mechanical foundations (Weeks 15)

Week 1: Introduction to continuum biomechanics and tensor algebra

Introduction to the course. Why continuum mechanics in biology?

The concept of a continuum vs. real tissues.

Fundamentals of tensor algebra: definitions of scalars, vectors, and tensors.

Tensor operations (addition, multiplication, transpose, dyadic product).

Week 2: Tensor analysis and kinematics

Tensor analysis: Gradient (vector, tensor), divergence (vector, tensor).

Tensor invariants.

Continuum kinematics: Description of motion (Lagrangian vs. Eulerian description).

Week 3: Kinematics Deformation

Deformation gradient and its decomposition.

Strain tensors: Green-Lagrange tensor (finite strain), Euler-Almansi tensor.

Linear strain tensor (small strain).

Rate of deformation and spin (vorticity) tensor.

Week 4: Stress analysis

Concept of stress, surface and body forces.

Cauchy stress tensor.

Equilibrium equations (without acceleration).

Principal stresses and principal directions, Mohr's circle.

Week 5: Conservation laws

Derivation of the general conservation equation (Reynolds transport theorem).

Conservation of mass (continuity equation).

Conservation of momentum (Cauchy's equation of motion).

Conservation of angular momentum (symmetry of the stress tensor).

BLOCK 2: Elasticity and Hyperelasticity (Weeks 68)

Week 6: General elasticity and hyperelasticity

Introduction to constitutive equations.

Definition of an elastic material.

General isotropic elasticity.

Hyperelasticity (finite strain): Strain energy density function.

Examples of models (Neo-Hookean, Mooney-Rivlin).

Week 7: Applications of hyperelasticity and introduction to linear elasticity

Modeling of biological tissues undergoing large deformations.

Application: Biomechanics of tendons and ligaments (nonlinear response).

Transition to small strains: Linear elasticity.

Week 8: Linear elasticity

Hooke's law for an isotropic material.

Material constants (Young's modulus, Poisson's ratio, shear modulus, bulk modulus) and relations between them.

Solution of classic problems (uniaxial tension/compression, torsion, bending).

Application: Mechanics of bone in the small strain region.

BLOCK 3: Linear viscoelasticity (Weeks 911)

Week 9: Introduction to viscoelasticity and differential models

Phenomenology of viscoelasticity: Creep and stress relaxation.

Difference from pure elasticity and pure viscosity.

Differential description: Basic spring and dashpot models.

Maxwell model (fluid) and Kelvin-Voigt model (solid).

Week 10: Advanced models and integral description

Standard Linear Solid (SLS) description of a solid exhibiting relaxation.

Integral description: Creep compliance and relaxation modulus functions.

Boltzmann's superposition principle.

Week 11: Dynamic analysis and correspondence principle

Analysis of dynamic (harmonic) loading.

Complex modulus (storage and loss modulus).

Correspondence principle: Relationship between elastic and viscoelastic solutions to a problem.

BLOCK 4: Coupled theories and mixture theories (Weeks 1214)

Week 12: Coupled theories

Introduction to multiphysics problems.

Brief overview of poroelasticity: Interaction of a porous solid matrix and a fluid (e.g., in bone).

Brief overview of thermoelasticity: Influence of temperature change on stress and strain.

Week 13: Mixture theory and biphasic theory

Introduction to mixture theory.

Detailed focus on biphasic theory solid and fluid phases.

Derivation of governing equations.

Application: Biomechanics of articular cartilage.

Week 14: Application of biphasic theory and summary

Solution of confined compression problems creep and relaxation.

Application: Biomechanics of the intervertebral disc.

Course summary, discussion, reserve.

Week 1: Introduction to continuum biomechanics and tensor algebra

Week 2: Tensor analysis and kinematics

Week 3: Kinematics Deformation

Week 4: Stress analysis

Week 5: Conservation laws

Week 6: General elasticity and hyperelasticity

Week 7: Applications of hyperelasticity and introduction to linear elasticity

Week 8: Linear elasticity

Week 9: Introduction to viscoelasticity and differential models

Week 10: Advanced models and integral description

Week 11: Dynamic analysis and correspondence principle

Week 12: Coupled theories

Week 13: Mixture theory and biphasic theory

Week 14: Final credit test.

To master the key mathematical (tensors) and mechanical (kinematics, stress) concepts of continuum mechanics.

To learn to apply theories of elasticity, hyperelasticity, and viscoelasticity to describe the behavior of biological materials, such as bones, tendons, and ligaments.

To become familiar with advanced coupled models (poroelasticity) and mixture theories (biphasic model) for the analysis of complex tissues, like cartilage

Athanasiou, K. A., & Natoli, R. M. (2012). Introduction to Continuum Biomechanics. Morgan & Claypool Publishers.

Fung, Y. C. (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag.

Fung, Y. C. (1977). A First Course in Continuum Mechanics. Prentice-Hall.

Ethier, C. R., & Simmons, C. A. (2007). Introductory Biomechanics: From cells to organisms. Cambridge University Press.

Mase, G. E., & Mase, G. T. (1999). Continuum Mechanics for Engineers. CRC Press.

• Prospectus - magisterský (!)
• Biomedical and Clinical Engineering (compulsory course)