Investigation of the solution for discontinuous contact problem between a functionally graded (FG) layer and homogeneous half-space

2020 ◽  
Vol 90 (12) ◽  
pp. 2799-2819
Author(s):  
Erdal Öner ◽  
Ahmet Birinci
Keyword(s):  
Contact Problem ◽  
Half Space ◽  
Functionally Graded ◽  
Homogeneous Half Space

Some Criteria for Coating Effectiveness in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts—Part II: Lubricated Contacts

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Ilya I. Kudish ◽  
Sergey S. Volkov ◽  
Andrey S. Vasiliev ◽  
Sergey M. Aizikovich
Keyword(s):  
Analytical Solution ◽  
Contact Problem ◽  
Half Space ◽  
Contact Problems ◽  
Functionally Graded ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Lubricated Contacts ◽  
Dry Contact ◽  
Appl Math

Contacts of indentors with functionally graded elastic solids may produce pressures significantly different from the results obtained for homogeneous elastic materials (Hertzian results). It is even more so for heavily loaded line elastohydrodynamically lubricated (EHL) contacts. The goal of the paper is to indicate two distinct ways the functionally graded elastic materials may alter the classic results for the heavily loaded line EHL contacts. Namely, besides pressure, the other two main characteristics which are influenced by the nonuniformity of the elastic properties of the contact materials are lubrication film thickness and frictional stress/friction force produced by lubricant flow. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods developed earlier by the authors such as Kudish (2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman & Hall/CRC Press, Boca Raton, FL), Kudish and Covitch (2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC Press, Boca Raton, FL), Aizikovich et al. (2002, “Analytical Solution of the Spherical Indentation Problem for a Half-Space With Gradients With the Depth Elastic Properties,” Int. J. Solids Struct., 39(10), pp. 2745–2772), Aizikovich et al. (2009, “Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth,” Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, p. 317), Aizikovich and Vasiliev (2013, “A Bilateral Asymptotic Method of Solving the Integral Equation of the Contact Problem for the Torsion of an Elastic Halfspace Inhomogeneous in Depth,” J. Appl. Math. Mech., 77(1), pp. 91–97), Volkov et al. (2013, “Analytical Solution of Axisymmetric Contact Problem About Indentation of a Circular Indenter Into a Soft Functionally Graded Elastic Layer,” Acta Mech. Sin., 29(2), pp. 196–201), Vasiliev et al. (2014, “Axisymmetric Contact Problems of the Theory of Elasticity for Inhomogeneous Layers,” Z. Angew. Math. Mech., 94(9), pp. 705–712), Aizikovich et al. (2008, “The Deformation of a Half-Space With a Gradient Elastic Coating Under Arbitrary Axisymmetric Loading,” J. Appl. Math. Mech., 72(4), pp. 461–467), and Aizikovich et al. (2010, “Inverse Analysis for Evaluation of the Shear Modulus of Inhomogeneous Media in Torsion Experiments,” Int. J. Eng. Sci., 48(10), pp. 936–942). More specifically, it is based on the analysis of contact problems for dry contacts of functionally graded elastic solids and the lubrication mechanisms in the inlet and exit zones as well as in the central region of heavily loaded lubricated contacts. The way the solution of the EHL problem for coated/functionally graded materials is obtained provides a very clear structure of the solution. The solution of the EHL problem in the Hertzian region is very close to the solution of the dry contact problem while in the inlet and exit zones the solutions of the EHL problem with the right asymptotes coming from the solution of the dry contact problem can be related to the solutions of the classic EHL problem for homogeneous materials.

scholarly journals MATHEMATICAL MODELING OF EXPERIMENT ON BERKOVICH NANOINDENTATION OF ZRN COATING ON STEEL SUBSTRATE

2020 ◽  
Vol 27 ◽  
pp. 18-21
Author(s):  
Evgeniy Sadyrin ◽  
Andrey Vasiliev ◽  
Sergei Volkov
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Analytical Solution ◽  
Contact Problem ◽  
Half Space ◽  
Steel Substrate ◽  
Elastic Half Space ◽  
Functionally Graded ◽  
Good Agreement

In the present paper the experiment on Berkovich nanoindentation of ZrN coating on steel substrate is modelled using the proposed effective mathematical model. The model is intended for describing the experiments on indentation of samples with coatings (layered or functionally graded). The model is based on approximated analytical solution of the contact problem on indentation of an elastic half-space with a coating by a punch. It is shown that the results of the model and the experiment are in good agreement.

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

2017 ◽  
Vol 33 (2) ◽  
pp. 406-414 ◽  
Author(s):  
Jie Su ◽  
Liao-Liang Ke ◽  
Yue-Sheng Wang ◽  
Yang Xiang
Keyword(s):  
Contact Problem ◽  
Half Space ◽  
Functionally Graded ◽  
Functionally Graded Piezoelectric

scholarly journals Simplified Analytical Solution of the Contact Problem on Indentation of a Coated Half-Space by a Conical Punch

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 983
Author(s):  
Andrey S. Vasiliev ◽  
Sergey S. Volkov ◽  
Evgeniy V. Sadyrin ◽  
Sergei M. Aizikovich
Keyword(s):  
Analytical Solution ◽  
Contact Problem ◽  
Half Space ◽  
Functionally Graded ◽  
Quadratic Functions ◽  
Contact Radius ◽  
Geometrical Parameters ◽  
Indentation Force ◽  
Coated Materials ◽  
Explicit Analytical Solution

The contact problem on indentation of an elastic coated half-space by a conical punch is considered. To obtain an explicit analytical solution suitable for applications, the bilateral asymptotic method is used in a simplified form. For that purpose, kernel transform of the integral equation is approximated by a ratio of two quadratic functions containing only one parameter. Such an approach allows us to obtain explicit analytical expressions for the distribution of contact stresses and relations between the indentation force, depth, stiffness and contact radius. The obtained solution is suitable both for homogeneous and functionally graded coatings. The dependence of the characteristics of contact interaction on a relative Young’s modulus of the coating and relative coating thickness is analyzed and illustrated by the numerical examples. Ranges of values of elastic and geometrical parameters are obtained, for which the presence of a coating sufficiently changes the contact characteristics. The accuracy of the obtained simplified expressions is studied in detail. Results of the paper sufficiently simplify engineering calculations and are suitable for inverse analysis, e.g., analysis of indentation experiments of coated materials using either a conical or a pyramidal (Berkovich) indenter.

scholarly journals Group and Phase Velocity of Love Waves Propagating in Elastic Functionally Graded Materials

2015 ◽  
Vol 40 (2) ◽  
pp. 273-281 ◽  
Author(s):  
Piotr Kiełczyński ◽  
Marek Szalewski ◽  
Andrzej Balcerzak ◽  
Krzysztof Wieja
Keyword(s):  
Group Velocity ◽  
Half Space ◽  
Functionally Graded Materials ◽  
Integral Formula ◽  
Liouville Problem ◽  
Elastic Half Space ◽  
Functionally Graded ◽  
Love Waves ◽  
Graded Materials ◽  
Sturm Liouville

AbstractThis paper presents a theoretical study of the propagation behaviour of surface Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in acoustics. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). Two Love wave waveguide structures are analyzed: 1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and 2) a semi-infinite nonhomogeneous elastic half-space. The Direct Sturm-Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved 1) analytically in the case of the step profile, exponential profile and 1cosh2type profile, and 2) numerically in the case of the power type profiles (i.e. linear and quadratic), by using two numerical methods: i.e. a) Finite Difference Method, and b) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The results obtained in this paper can give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials.

Axisymmetric indentation of an electroelastic piezoelectric half-space with functionally graded piezoelectric coating by a circular punch

2017 ◽  
Vol 230 (4) ◽  
pp. 1289-1302 ◽  
Author(s):  
S. S. Volkov ◽  
A. S. Vasiliev ◽  
S. M. Aizikovich ◽  
B. I. Mitrin
Keyword(s):  
Half Space ◽  
Functionally Graded ◽  
Circular Punch ◽  
Functionally Graded Piezoelectric
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