**INFORMATION ABOUT PROJECT,
SUPPORTED BY RUSSIAN SCIENCE FOUNDATION**

*The information is prepared on the basis of data from the information-analytical system RSF, informative part is represented in the author's edition. All rights belong to the authors, the use or reprinting of materials is permitted only with the prior consent of the authors. *

COMMON PART

Project Number22-19-00732

Project titleNanoindentation methods for determining the mechanical and physical properties of liquid-saturated poroelastic materials

Project LeadAizikovich Sergei

AffiliationFederal State-Funded Educational Institution of Higher Education Don State Technical University,

Implementation period | 2022 - 2024 |

Research area 09 - ENGINEERING SCIENCES, 09-106 - Mechanical problems in the development of new materials

Keywordsnanoindentation, liquid-saturated media, poroelastic materials and tissues, contact problem, contact stresses, contact pressure, layer, strip, quasi-static, mechanical properties, physical properties

PROJECT CONTENT

Annotation

Determination of the mechanical and physical properties of materials of natural and artificial origin is a scientific and technical task of vital importance in all the fields of human practical activity: industry, aviation and automobile construction, architecture, etc., as well as in the most important spheres of human life – medicine, biology, sports, culture, etc. The primary method for determining the mechanical and physical properties of materials today in the course of the scientific and technological development is nanoindentation. It began playing the role of the separate scientific and technical branch in the 1990s. The development of the industry is characterized by the improvement of the apparatus of the nanoindentation test unit, as well as the mechanical and mathematical support represented by a mechanical and mathematical model of the indentation process and by a mechanical and mathematical model of the material under study, taking into account its mechanical and physical properties, such as elasticity, viscoelasticity, plasticity, porosity, liquid saturation, etc. The mathematical model of the process of indentation of liquid-saturated materials of natural and artificial origin is based on the formulation and solution of quasi-static spatial contact problems on the penetration of a rigid indenter into a liquid-saturated poroelastic Biot medium in the form of a spatial layer rigidly coupled with a non-deformable substrate. Within the framework of the project, a spatial quasi-static contact problem on indentation of the flat impermeable to liquid indenter into a layer of liquid-saturated poroelastic material coupled with a non-deformable substrate is considered. The surface of the liquid-saturated material outside the contact area is free of stresses and liquid pressure. The problem is reduced to solving the corresponding planar quasi-static contact problem on the penetration of a rigid indenter with a flat substrate, impermeable for liquid, into a liquid-saturated poroelastic strip coupled with a non-deformable substrate. The problem under consideration is reduced to solving a system of two two-dimensional integral equations of the second kind with two-dimensional kernels in variables in coordinate and time using integral transformations, depending on two unknown functions - contact stresses of a liquid-saturated poroelastic medium and contact pressure of a liquid. The solution of the resulting system is reduced to the solution of an equivalent system of two one-dimensional integral equations of the second kind with respect to the Laplace transforms of unknown contact stresses and pressures. The solution to the system is constructed by the method of successive approximations with justification of its convergence. The zero approximation of the solution of the contact problem is obtained in analytical form, like all other characteristics of the contact - the contact force (all components), the surface displacement of the liquid-saturated medium (all components), the rate of fluid flow through the free surface outside the contact area, and other characteristics of the contact problem. This makes it possible to determine the elastic moduli of the skeleton of a liquid-saturated medium - Young's modulus and Poisson's ratio, as well as the coefficient of fluid permeability into the skeleton and other parameters of the medium using experimental indentation data. The proposed method for solving quasi-static contact problems on indentation is adapted to solving quasi-static spatial axisymmetric contact problems on indentation of a liquid-saturated poroelastic medium in the form of a half-space and a layer using a cylindrical indenter with a flat base. The developed method for solving quasi-static contact problems for complex media with two unknown functions at the contact is original.

Expected results

As the result of the project implementation the following tasks would be fulfilled:
- new original fundamental methods for modeling nanoindentation process would be developed based on the solution of contact problems ;
- new original fundamental methods for determining the mechanical and physical properties of both natural and artificial materials would be developed;
- the database of materials for which indentation methods are applicable in order to determine their mechanical and physical properties(including liquid-saturated and poroelastic ones) would be expanded.

REPORTS

Annotation of the results obtained in 2022

In the reporting period, a theoretical and experimental technique was developed for indenting liquid-saturated biomedical tissue-materials with a rigid strip indenter to determine the mechanical and physical characteristics of this material. Knowledge of the mechanical parameters of biomedical tissue-materials makes it possible to select a biocompatible material of artificial or natural origin to create an implant (substitute) to replace the lost biomedical tissue. When creating the theoretical part of such a technique, an important role is played by the presence of two circumstances - a mathematical model of a liquid-saturated poroelastic material that simulates the behavior of biomedical liquid-containing tissue materials under load, and a mathematical model of the indentation process of such a liquid-saturated poroelastic material. As a mathematical model simulating liquid-containing biomedical tissues, the model of a liquid-saturated poroelastic Biot medium was considered. As a mathematical model of the indentation process, a spatial quasi-static contact problem of a rigid strip indenter displacement into a liquid-saturated poroelastic layer adhered to a non-deformable base was considered. Such a quasi-static contact problem was reduced to solving a system of the two two-dimensional integral equations of the first kind of the repeated Laplace-Fourier convolution type with respect to two unknown functions - the contact stresses of the elastic skeleton of a liquid-saturated poroelastic medium and the contact pressures of the pore fluid. To solve a system of two-dimensional integral equations, a method was developed to reduce it to the solution of an equivalent one-dimensional system of two integral equations of the second kind of a triangular form with difference kernels in the coordinate variable with respect to the unknown Laplace transforms of the skeleton contact stresses and pore fluid contact pressures.To solve the latter, the method of successive approximations was used, which made it possible to determine the zero approximation of the solution in analytical form, and after its Laplace inversion, the zero approximation of the original, initial system of two two-dimensional integral equations as well, and hence also the solution of a quasi-static contact problem that simulates the indentation of a liquid-saturated poroelastic material. The obtained solutions made it possible for the first time to determine the function classes of the obtained solutions, as well as other characteristics of the contact problem - the pore fluid velocity components, their features at the edges of the contact area, the contact force on the indenter, etc. The experimental component of the general technique for indentation of liquid-containing tissue materials consisted of sample preparation liquid-containing tissue-materials, modernization of the robotic nano-microindentation test unitNanotest 600 (Platform 3) to expand its capabilities, as well as indentation of liquid-containing biomedical materials on a modernized nanoindentation test unit. The third stage of the methodology for determining the mechanical and physical characteristics of liquid-saturated porous-elastic materials consisted of comparing theoretical and experimental data, as a result of which the effective Young's moduli of liquid-saturated porous-elastic materials were determined. To determine the filtration coefficient of a liquid-saturated porous-elastic material, a laboratory setup was created based on a system of communicating vessels, a Vesta VM313M analytical scales, and other auxiliary circuit elements. In the reporting period, all the main elements of the methodology for determining the mechanical and physical properties of liquid-containing biomedical tissue materials were implemented.

Publications

**1.** *Aizikovich S.M., Lapina P.A., Volkov S.S.* **Analysis of equivalence conditions of model of an inhomogeneous elastic half-space and model of an inhomogeneous elastic layer on the elastic foundation** Advanced Structured Materials: Solid Mechanics, Theory of Elasticity and Creep. – Springer, Magdeburg, - (year - 2023)

**2.** *Sadyrin E.V., Yogina D.V., Volkov, S.S., Aizikovich S.M.* **Оценка плотности и микрогеометрических характеристик пломб из стеклоиономерного цемента и композитного материала: биомеханическое ex vivo исследование** Российский журнал биомеханики, Т. 26, №. 2, С. 67–73 (year - 2022) https://doi.org/10.15593/RZhBiomeh/2022.2.06

**3.** *Zelentsov V. B., Lapina P.A.* **Pore fluid filtration by squeezing a fluid-saturated poroelastic medium** Advanced Structured Materials: Mechanics of Heterogeneous Materials. – Springer: Magdeburg., - (year - 2023)

**4.** *Aizikovich S.M., Lapina P.A.* **Две приближённые модели функционально-градиентного основания и современные асимптотические методы решения контактных задач для этих моделей** Математика, Компьютерные науки и Цифровые технологии в преподавании и в научных исследования: сб. трудов Международной научной и научно-методической конференции, г. Уральск, Казахстан, 7-10 декабря 2022 г., - (year - 2022)

**5.** *Aizikovich S.M., Lednov A.S.* **Математическое моделирование индентирования роговицы глаза плоским штампом с учётом порового давления в слоях** Математика в медицине: тез. докл. Всероссийской конф. с междунар. участием, г. Владивосток, 10-15 окт. 2022 г., С.10 (year - 2022)

**6.** *Aizikovich S.M., Sadyrin E.V., Nikolaev A.L., Vasiliev A.S.* **Математическое моделирование процесса индентирования покрытий ZrN на различных подложках** Математика в медицине: тез. докл. Всероссийской конф. с междунар. участием, г. Владивосток, 10-15 окт. 2022 г., С.5 (year - 2022)

**7.** *Karotkiyan R.V., Nikolaev A.L.* **Laboratory filtration factor determination technique** Abstracts of the II International conference «Modern Modeling of Materials for Mechanical, Medical and Biological Applications», Divnomorskoe, September 26-30, 2022. – Rostov-on-Don: DSTU, 2022., С.12 (year - 2022)

**8.** *Nikolaev A.L., Sadyrin E.V., Aizikovich S.M., Krenev L.I.* **Теоретико – экспериментальное исследование покрытия ZnO, полученного импульсным лазерным напылением** Математика в медицине: тез. докл. Всероссийской конф. с междунар. участием, г. Владивосток, 10-15 окт. 2022 г., С.12 (year - 2022)

Annotation of the results obtained in 2023

During the reporting period of the project, a spatial axisymmetric quasi-static contact problem of the displacement of a rigid cylindrical indenter with a flat base shape into a liquid-saturated poroelastic medium in the form of a half-space was considered. It is the fundamental theoretical basis for mathematical modeling of the process of identification of natural and artificial, soft and hard biomedical tissues. The Biot model was considered as a mathematical model of a liquid-saturated poroelastic medium. A set of differential equations describing the behavior of the Biot medium and mixed boundary conditions of the spatial axisymmetric quasi-static contact problem, together with the initial conditions and conditions at infinity, represents the formulation of the spatial axisymmetric quasi-static contact problem of the settlement of a rigid cylindrical indenter into a liquid-saturated poroelastic medium in the form of a half-space. The solution to the formulated contact problem is constructed in integral form using the integral Laplace (with respect to time) and Bessel (with respect to the radial variable) transforms. Such an approach allows one to reduce the formulated contact problem to the solution of a system of two two-dimensional integral equations of the first kind of the Laplace-Bessel convolution type with respect to two unknown functions: the distribution of contact stresses from a liquid-saturated poroelastic Biot medium at the base of the indenter and the distribution of contact pressures of the pore fluid. In order to reduce the multiplicity of integrals in the resulting system of two-dimensional integral equations, the Laplace integral transform is applied, which leads the system of two two-dimensional integral equations to an equivalent system of two one-dimensional integral equations of the first kind, such as Bessel convolutions of the Laplace transforms of the desired contact stresses and contact pressures of the pore fluid. By using integral operators over the radial variable and integrals connecting Bessel functions with trigonometric functions, a system of one-dimensional integral equations of the first kind of the Bessel convolution type is reduced to an equivalent system of integral equations of the first kind of the Fourier convolution type with difference kernels of unknowns in the form of integral operators of the original Laplace transforms contact stresses and contact pressures of pore fluid. Analysis of the difference kernels of the system led to the need to represent them as a sum of irregular and regular parts, the latter of which were transferred to the right side of the equations. Application of the method of eliminating unknowns leads the system of integral equations to a triangular form. Assuming the existence of inverse operators for the irregular parts of integral operators, after their inversion, a triangular system of two one-dimensional integral equations of the second kind of the Fourier convolution type with difference kernels is obtained. The solution of a system of the second kind is carried out by the method of successive approximations. The zero approximation of the solutions is determined from a triangular system of integral equations, in the absence of regular operators on the right side, starting with the inversion of the singular operator in the second equation. To obtain solutions that are effective for short and medium-sized times of the indentation process, the Wiener-Hopf method to isolate features in analytical form was used. Since the found analytical solutions are expressed through the operators of the Abel equation, then after applying the inverse Abel operator to them we obtain in analytical form the Laplace transforms of the solutions to the problem posed, and after their inversion using the inverse Laplace transform we obtain a zero approximation of the solutions of the considered spatial axisymmetric quasi-static contact problem: contact stresses and contact pressures of the liquid during the indenter displacement into the liquid-saturated poroelastic Biot medium. It should be noted that the contact stresses arising at the base of the indenter are determined in the class of integrable functions with a root singularity at the edges of the contact area, and the contact pressures of the liquid are determined in the class of functions limited at the edges of the contact area. The contact force of the indenter on a liquid-saturated poroelastic medium (the integral characteristic of the solutions obtained) consists of the algebraic sum of the force on the elastic skeleton of the medium and the force on the pore fluid. The resulting formulas make it possible to analyze the process of consolidation of a liquid-saturated porous-elastic Biot medium, as well as the influence of drainage of pore fluid, both through the free surface outside the contact area and through the base of the indenter, on contact stresses and indenter displacement.

Publications

**1.** *Aizikovich S.M., Lapina P.A., Volkov S.S.* **Advance approximate analytical solutions of the contact problem for an inhomogeneous layer** Advanced Structured Materials: Sixty Shades of Generalized Continua. Dedicated to the 60th Birthday of Prof. Victor A. Eremeyev. Springer, Nature Switzerland AG, Chapter 2. Vol. 170. Р. 13-20 (year - 2023) https://doi.org/10.1007/978-3-031-26186-2_2

**2.** *Kuznetsova T., Lapitskaya V., Khabarava A., Trukhan R., Chizhik S., Torskaya E., Fyodorov S., Aizikovich S., Sadyrin E., Warcholinski B.* **Features of wear of DLC-Si coating under microcontact conditions during the formation of secondary structures** Composite Structures, Vol. 316. Article number 117039 (year - 2023) https://doi.org/10.1016/j.compstruct.2023.117039

**3.** *McHugh J.V., Slipinsky A., Nabozhenko M.V., Perkovsky E.E., Sadyrin E.V.* **A new species of Cerylonidae (Insecta: Coleoptera) described from Baltic amber using X-Ray microtomography** Historical Biology, P.1-5 (year - 2023) https://doi.org/10.1080/08912963.2023.2220001

**4.** *Sadyrin E.V.* **Моделирование механизма снижения плотности минерализации эмали в окрестности вершины фиссуры** Российский журнал биомеханики, № 1. С. 31-39 (year - 2023) https://doi.org/10.15593/RZhBiomech/2023.1.03

**5.** *Zelentsov V.B., Sadyrin E.V., Mitrin B.I., Swain M.V.* **Mathematical tools for recovery of the load on the fissure according to the micro-CT results** Journal of the Mechanical Behavior of Biomedical Materials, Vol. 138. Article number 105625 (year - 2023) https://doi.org/10.1016/j.jmbbm.2022.105625

**6.** *Aizikovich S.M., Lapina P.A., Zelentsov V.B., Volkov S.S.* **Современный метод исследования эквивалентности различных моделей неоднородного упругого полупространства и неоднородного упругого слоя ** Тезисы докладов XXIII Зимней школы по механике сплошных сред, Пермь, 13-17 февраля 2023 г. – Пермь: ПФИЦ УрО РАН, 2023г., С.21 (year - 2023)

**7.** *Aizikovich S.M., Lapina P.А.* **Метод исследования эквивалентности различных моделей неоднородного упругого основания** XXVIII Всерос. конф. по численным методам решения задач теории упругости и пластичности, г. Красноярск, 10-15 июля 2023 г., С.1 (year - 2023) https://doi.org/-

**8.** *Aizikovich S.M., Vasiliev A.S.* **Simplified analytical solution of the indentation problem and its application for nanoindentation analysis of coated solids** Proceedings of the Third African Congress In Tribology (ACT-2023), April, 24–27, 2023, Yamoussoukro, Côte d’Ivoire, Р.1 (year - 2023)

**9.** *Aizikovich S.M., Vasiliev A.S.* **Упрощенные аналитические выражения контактных характеристик при индентировании однородных и функционально-градиентных покрытий** XIII Всероссийский съезд по фундаментальным проблемам теоретической и прикладной механике, Санкт-Петербург, 21-25 августа 2023 г.: сб. тез. докл. в 4 т. Политех-Пресс, 2023, С.1-2 (year - 2023) https://doi.org/-

**10.** *Karotkiyan R.V.* **Моделирование индентирования роговицы** Математическое моделирование и биомеханика в современном университете: тез. докл. XVII Всерос. шк.-семинара, с. Дивноморское, 28 мая-1 июня 2023 г. – Ростов-на-Дону; Таганрог: Издательство Южного федерального университета, С.48 (year - 2023)

**11.** *Karotkiyan R.V., Sadyrin E.V.* **Mechanical behavior of scaffolds with different wall thickness values evaluated using in situ compression testing and microtomography** Modern Problems in Modeling Materials for Mechanical, Medical and Biological Applications (MPMM&A-2023): Theses of the reports of the Third International Conference, Rostov-on-Don, November 27-30, 2023. – Rostov-on-Don: DSTU, Р. 9 (year - 2023)

**12.** *Karotkiyan R.V., Soloviev A.N.* **Numerical modeling of corneal indentation** Modern Problems in Modeling Materials for Mechanical, Medical and Biological Applications (MPMM&A-2023): Theses of the reports of the Third International Conference, Rostov-on-Don, November 27-30, 2023. – Rostov-on-Don: DSTU, P. 10 (year - 2023)

**13.** *Lapina P.A., Zelentsov V.B.* **Indenter settlement in the water-saturated soil** X International Scientific Conference «Actual problems of solid state physics»: book of abstracts, 22-26 May 2023, Minsk, Р.431 (year - 2023)

**14.** *Lednov A.S.* **Конечно-элементный расчёт внедрения штампа в пороупругую водонасыщенную среду** Математическое моделирование и биомеханика в современном университете: тез. докл. XVII Всерос. шк.-семинара, с. Дивноморское, 28 мая-1 июня 2023 г. – Ростов-на-Дону; Таганрог: Издательство Южного федерального университета, С.63 (year - 2023)

**15.** *Lednov A.S., Solovyov A.N.* **Finite element analysis of flat punch indentation into poroelastic water-saturated medium using ANSYS** Modern Problems in Modeling Materials for Mechanical, Medical and Biological Applications (MPMM&A-2023): Theses of the reports of the Third International Conference, Rostov-on-Don, November 27-30, 2023. – Rostov-on-Don: DSTU, P. 14 (year - 2023) https://doi.org/-

**16.** *Sadyrin E.V.* **Biomechanical characterization of tooth tissues altered by early caries** Brazil-India-Cuba-China-United Kingdom (BIC²UK) Conference on Nanomaterials & Machine Learning - Proceedings Book Rio de Janeiro, May 29 to June 1, 2023., P. 136-139 (year - 2023)

**17.** *Sadyrin E.V., Nikolaev A.L., Yogina D.V., Swain M.V.* **Characterization of dental materials and pathologically altered tissues in their vicinity** 4th International Conference on Nanomaterials, Nanofabrication and Nanocharacterization (NANOMACH): Book of Abstracts, April 13-19, 2023, Oludeniz, Turkey, P.23-24 (year - 2023)

**18.** *Sadyrin E.V., Yogina D.V.* **Mechanical behavior, microstructure and microgeometry of the pathologically altered tooth tissues and dental materials** International Conference on «Physics and Mechanics of New Materials and Their Applications» (PHENMA 2023): Abstracts and Schedule (Surabaya, Indonesia, October 3–8, 2023). – Rostov-on-Don; Taganrog: Southern Federal University Press, 2023. – 369 p., Р. 248 (year - 2023)

**19.** *Sadyrin E.V., Yogina D.V., Evsyukov A.P., Zabiyaka I.Yu., Swain M.V.* **Механические свойства стоматологических материалов и окружающих их тканей при лечении кариеса в стадии белого пятна** Математическое моделирование и биомеханика в современном университете: тез. докл. XVII Всерос. шк.-семинара, с. Дивноморское, 28 мая-1 июня 2023 г. – Ростов-на-Дону; Таганрог: Издательство Южного федерального университета, С.99 (year - 2023)

**20.** *Zelentsov V.B., Lapina P.A.* **Моделирование процесса индентирования биомедицинских тканей посредством модели Био** Современные проблемы механики сплошной среды (МСС 2023): тез. докладов XXI Междунар. конф., г. Ростов-на-Дону, 11-13 октября 2023 г. – Ростов-на-Дону; Таганрог: Изд-во Южного федерального университета, 2023., С.46 (year - 2023) https://doi.org/-

**21.** *Zelentsov V.B., Lapina P.A.* **Mathematical modeling of the process of indentation of Biot medium** Modern Problems in Modeling Materials for Mechanical, Medical and Biological Applications (MPMM&A-2023): Theses of the reports of the Third International Conference, Rostov-on-Don, November 27-30, 2023. – Rostov-on-Don: DSTU, P. 40 (year - 2023) https://doi.org/-

**22.** *Садырин Е.В., Свэйн М.В.* **Nanoindentation derived mechanical properties of dental materials and tissues in their vicinity in the treatment of early caries** X International Scientific Conference «Actual problems of solid state physics»: book of abstracts, 22-26 May 2023, Minsk., Р.406 (year - 2023)

**23.** *Karotkiyan R.V., Nikolaev A.L., Aizikovich S.M.* **Установка для определения коэффициента фильтрации пористых биологических материалов** Федеральная служба по интеллектуальной собственности, Патент на изобретение № 2801785. Дата регистрации 15.08.2023 г. (year - 2023)