Advanced Lecture Course:

Indentation testing of biological tissues

by Ivan Argatov

Technische Universität Berlin

Course modalities:
in person & on-line

Tuesday 9th through Friday 12th & Monday 15th August 2022, 12:00 - 14:00 h, Classroom 206, Edificio Principal del Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Ciudad Universitaria, CDMX.


Upon successful completion, students will be issued a certificate with curricular validity. The course has no cost.


Due to COVID restrictions, Professor Argatov has generously allowed access to the lectures and materials of the course to the students unable to take the course in person. For the online modality, please fill out the:

About the course

Artificial biological materials are used increasingly in bioengineering practice and yet a standard curriculum of mechanical engineering courses overlooks these complex and fascinating materials. The unique material properties of living tissues arise due to complicated multi-scale micro structure that manifests itself in macroscopic properties. Understanding the underlying mechanisms of deformation and failure in biological materials under dynamic loading and being able to describe the mechanical response of living tissues to indentation is a growing challenge. 


The course “Indentation testing of biological tissues” develops a mathematical modeling approach to capture the indentation phenomena in biomedical materials and applies this to the analysis of dynamic and impact deformation of biological tissues. The methods and ideas developed here find wide use in the design of indentation testing techniques for viability identification of living tissues. In particular, the course includes a discussion of methods of novel dynamic indentation tests. Also, the course provides the student with the basic knowledge of modeling of mechanical material behavior, more specifically linear visco-elasticity, linear biphasic theory and damage mechanics.


The intensive course "Indentation testing of biological tissues" contains 5 lectures and 5 tutorials. Students are encouraged to seek additional help from the original research papers as well as the following literature: 

  • "Indentation Testing of Biological Materials" by I. Argatov and G. Mishuris.
  • "Contact Mechanics and Friction" by V.L. Popov.
  • "Contact Mechanics" by K.L. Johnson.
  • "Nanoindentation" by A.C. Fischer-Cripps.

Also, the books "Biomechanics: Mechanical Properties of Living Tissues" by Y.C. Fung and "Contact mechanics of articular cartilage layers" by I. Argatov and G. Mishuris are recommended for reading.

Lecture 1

Introduction. Elastic material; Viscoelastic material; Cylindrical (flat-ended) indenter; Spherical indenter; Frictionless contact; Indentation stiffness. 

Lecture 2

Frictionless spherical indentation. Unilateral contact; Hertz’s theory of axisymmetric contact; Force-displacement relationship; Bulychev–Alekhin–Shorshorov relation; Oliver-Pharr method. 

Lecture 3

Thickness effect in indentation. Contact problem for an elastic layer; Indentation scaling factor; Hayes’ formula; Asymptotic solution for relatively thick layers. 

Lecture 4

Rebound indentation test for a viscoelastic layer. 

Lecture 5

Dynamic indentation and impact testing. Vibration indentation test; Sinusoidally-driven indentation test; Incomplete storage modulus. Impact testing; Pulsatile dynamic modulus. 


Tutorial 1

Governing equations of the linear biphasic theory. Biphasic stress-relaxation and creep compression tests. Confined and unconfined compression tests. 

Tutorial 2

Governing equations of the Hertzian theory. Incremental indentation stiffness. General force-displacement relationship in the frictionless axisymmetric indentation. 

Tutorial 3

Indentation of relatively thin elastic layers (coatings) deposited on a rigid substrate. 

Tutorial 4

Stress-relaxation and creep indentation tests for a viscoelastic layer. 

Tutorial 5

Vibration indentation test; Sinusoidally-driven indentation test.