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. 2022 Mar 17;14(6):1225.
doi: 10.3390/polym14061225.

Elastoplastic Indentation Response of Sigmoid/Power Functionally Graded Ceramics Structures

Mohamed A Eltaher  1   2 , Ahmed Wagih  2 , Ammar Melaibari  1 , Ghazi S Alsoruji  1 , Mohamed A Attia  2
Affiliations

Elastoplastic Indentation Response of Sigmoid/Power Functionally Graded Ceramics Structures

Mohamed A Eltaher et al. Polymers (Basel). .

Abstract

Due to the applicability of new advanced functionally graded materials (FGMs) in numerous tribological systems, this manuscript aims to present computational and empirical indentation models to investigate the elastoplastic response of FG substrate under an indention process with spherical rigid punch. The spatial variation of the ceramic volume fraction through the specimen thickness is portrayed using the power law and sigmoid functions. The effective properties of two-constituent FGM are evaluated by employing a modified Tamura-Tomota-Ozawa (TTO) model. Bilinear hardening behavior is considered in the analysis. The finite element procedure is developed to predict the contact pressure, horizontal displacement, vertical deformation, and permanent deformation of FG structure under the rigid cylindrical indentation. The empirical forms for permanent deformation were evaluated and assigned. Model validation with experimental works was considered. The convergence of the mesh and solution procedure was checked. Numerical studies were performed to illustrate the influence of gradation function, gradation index, and indentation parameters on the contact pressure, von Mises stresses, horizontal/vertical displacements, and permanent plastic deformation. The present model can help engineers and designers in the selection of an optimum gradation function and gradation index based on their applications.

Keywords: bilinear TTO elastoplastic; empirical indentation forms; finite element method; homogenization; sigmoid/power FG substrate.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Figure 1
Figure 1
Typical stress–strain curve for a bilinear elastoplastic material FGM based on TTO model. (a) Coordinate system of material gradation, (b) Effective constitutive relation for FGM.
Figure 2
Figure 2
Variation in modulus of elasticity, yield strength, and the tangent modulus of the elastoplastic FGM Ti/TiB specimen along the thickness direction for the sigmoid function (S-FGM) and power law function (P-FGM). (a) Variation in modulus of elasticity of S-FGM, (b) Variation in modulus of elasticity of P-FGM, (c) Variation in yield strength of S-FGM, (d) Variation in yield strength of P-FGM, (e) Variation in tangent modulus of S-FGM, (f) Variation in tangent modulus of P-FGM.
Figure 2
Figure 2
Variation in modulus of elasticity, yield strength, and the tangent modulus of the elastoplastic FGM Ti/TiB specimen along the thickness direction for the sigmoid function (S-FGM) and power law function (P-FGM). (a) Variation in modulus of elasticity of S-FGM, (b) Variation in modulus of elasticity of P-FGM, (c) Variation in yield strength of S-FGM, (d) Variation in yield strength of P-FGM, (e) Variation in tangent modulus of S-FGM, (f) Variation in tangent modulus of P-FGM.
Figure 3
Figure 3
Schematic representation of the spherical indentation of an elastoplastic FGM specimen. (a) Representation of the spherical indentation problem (b) Presentation of boundary condition and meshing.
Figure 4
Figure 4
Validation of the present FE model with the experimental results of elastoplastic composite [36].
Figure 5
Figure 5
Influence of the FE mesh size on the distribution of the indentation pressure along the half length of indentation for an elastoplastic P-FGM Ti/TiB specimen at k = 0.5.
Figure 6
Figure 6
Influence of the FE mesh size on displacement profiles along the upper surface of an elastoplastic P-FGM Ti/TiB specimen at k = 0.5; (a) horizontal, (b) vertical, (c) permanent horizontal, and (d) permanent vertical displacements.
Figure 6
Figure 6
Influence of the FE mesh size on displacement profiles along the upper surface of an elastoplastic P-FGM Ti/TiB specimen at k = 0.5; (a) horizontal, (b) vertical, (c) permanent horizontal, and (d) permanent vertical displacements.
Figure 7
Figure 7
Indentation force–penetration depth curves for the elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 8
Figure 8
Distributions of the indentation pressure (at full load) along the half length of indentation for the elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 9
Figure 9
Distributions of the von Mises equivalent stress (at full load) within the elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM with K = 0.5, and (b) P-FGM with K = 0.5, (c) S-FGM with K = 2, (d) P-FGM with K = 2, (e) S-FGM with K = 4 and (f) P-FGM with K = 4.
Figure 10
Figure 10
Distributions of the equivalent plastic strain (at full load) within the elastoplastic Ti/TiB specimen at different gradient indices; (a) Equivalent plastic strain of S-FGM at K = 0.5, and (b) Equivalent plastic strain of P-FGM at K = 0.5. (c) Equivalent plastic strain of S-FGM at K = 2, and (d) Equivalent plastic strain of P-FGM at K = 2. (e) Equivalent plastic strain of S-FGM at K = 4, and (f) Equivalent plastic strain of P-FGM at K = 4.
Figure 11
Figure 11
Distributions of the residual von Mises equivalent stress (after unloading) within the elastoplastic Ti/TiB specimen at different gradient indices; (a) the residual von Mises equivalent stress of S-FGM at K = 0.5, and (b) the residual von Mises equivalent stress of P-FGM at K = 0.5. (c) the residual von Mises equivalent stress of S-FGM at K = 2, and (d) the residual von Mises equivalent stress of P-FGM at K = 2. (e) the residual von Mises equivalent stress of S-FGM at K = 4, and (f) the residual von Mises equivalent stress of P-FGM at K = 4.
Figure 12
Figure 12
Profiles of the horizontal displacement (at full load) along the upper surface of an elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 13
Figure 13
Profiles of the vertical displacement (at full load) along the upper surface of an elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 14
Figure 14
Profiles of the permanent horizontal displacement (after unloading) along the upper surface of an elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 15
Figure 15
Profiles of the permanent vertical displacement (after unloading) along the upper surface of an elastoplastic Ti/TiB specimen at different gradient indices; (a) S-FGM, and (b) P-FGM.
Figure 16
Figure 16
Prediction of the permanent indentation depth as a function of the gradient index for both S-FGM and P-FGM functions of elastoplastic Ti/TiB specimen, accounting for the gradation in all material properties.
Figure 17
Figure 17
Normalized max permanent penetration depth (empirical) vs. normalized max permanent penetration depth (numerical).
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