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On the relationships between hardness and the elastic and plastic properties of transversely isotropic power-law hardening materials

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Abstract

Using a combination of dimensional analysis and large deformation finite element simulations of indentations of power-law hardening model materials, a framework for capturing the hardness characteristics of transversely isotropic materials is developed. By considering 4800 combinations of material properties, relationships that predict the hardness of transversely isotropic materials are formulated for both longitudinal and transverse indentations. It is found that hardness tends to be higher for materials with plastic anisotropy than those that exhibit equivalent elastic anisotropy. For perfectly plastic materials, the hardness to the yield stress ratio (R0) can vary from 2.5 to 3.5, while for materials with strain hardening, R0 can be as high as 10. A tighter relationship between the hardness and the Tabor’s representative stress is observed with the ratio of the hardness to the Tabor’s representative stress (Rt) ranging from 2.2 to 4, depending on the elastic and plastic properties and the degree of anisotropy.

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Funding

The present study was supported in part by a National Science Foundation Grant DMR-2004944.

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Authors and Affiliations

  1. Department of Materials Science and Chemical Engineering, Stony Brook University, New York, NY, 11794, USA

    T. S. Bhat & T. A. Venkatesh

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  1. T. S. Bhat
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Correspondence to T. A. Venkatesh.

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Appendix A

Appendix A

As described in “Computational modeling” section, the entire material property database was divided into six domains. For each of the material domains, dimensionless functions that capture the relationships between true projected contact area (Am), the indentation hardness (Hm) and the five elastic and plastic properties of transversely isotropic materials (E0, σ0, EL/ET, σL/σT, n) are identified. Here, of the 24 equations, a list of 6 dimensionless functions for longitudinal indentation hardness is provided. The equations are listed as Пjkl. Here, j varies from 1 to 2 and signifies the indenter orientation—1 for longitudinal, and 2 for parallel indentation; k varies from 4 to 5, and conveys the type of dimensionless function—4 for Am and 5 for Hm; and l represents the part number and varies from 1 to 6. X, Y, and Z represent the properties E0, EL/ET, and σL/σT, respectively.

$$\Pi_{151} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} {(17}{{.92 - 3}}{{.7y}}^{{2}} + {0}{{.2yz - 2}}{{.1 z}}^{{2}} {)(241}{{.1 - 92}}{{.88Log[x]}} + {9}{{.01Log[x]}}^{{2}} {)} + \hfill \\ {{n( - 0}}{.13 - 0}{{.66y}}^{{2}} + {6}{{.18yz - 0}}{{.07z}}^{{2}} {)( - 35}{{.98}} + {13}{{.65Log[x] - 1}}{{.26Log[x]}}^{{2}} {)} + \hfill \\ { (2}{{.6 - 0}}{{.13y}}^{{2}} + {0}{{.17yz}} + {0}{{.17z}}^{{2}} {)(16 - 5}{{.68Log[x]}} + {0}{{.54Log[x]}}^{{2}} {)} + \hfill \\ \end{gathered} \right)$$
(A.1)
$$\Pi_{152} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} {(39}{{.6}} + {{13y}}^{{2}} {{ - 34yz}} + {17}{{.85z}}^{{2}} {)(255}{{.37 - 95}}{{.3Log[x]}} + {8}{{.9Log[x]}}^{{2}} {)} + \hfill \\ {{n( - 9}} + {4}{{.27y}}^{{2}} { - 9}{{.79yz}} + {3}{{.47z}}^{{2}} {)(263}{{.75 - 98Log[x]}} + {9}{{.1Log[x]}}^{{2}} {)} + \hfill \\ {(3}{{.6 - 0}}{{.09y}}^{{2}} + {0}{{.13yz}} + {0}{{.24z}}^{{2}} {)(11}{{.9 - 4}}{{.17Log[x]}} + {0}{{.4Log[x]}}^{{2}} {)} \hfill \\ \end{gathered} \right)$$
(A.2)
$$\Pi_{153} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} {(21}{{.5}} + {16}{{.3y}}^{{}} { - 41}{{.7yz}} + {19}{{.15z}}^{{2}} {)( - 334}{{.5}} + {127}{{.6Log[x] - 12}}{{.13Log[x]}}^{{2}} {)} \hfill \\ + {{n(1}}{.2 - 7}{{.6y}}^{{2}} + {19}{{.1yz - 7}}{{.27z}}^{{2}} {)(65}{{.3 - 25}}{{.3Log[x]}} + {2}{{.5Log[x]}}^{{2}} {)} + \hfill \\ {(1}{{.85}} + {0}{{.02y}}^{{2}} { - 0}{{.11yz}} + {0}{{.23z}}^{{2}} {)(12}{{.8 - 4}}{{.5 Log[x]}} + {0}{{.45Log[x]}}^{{2}} {)} \hfill \\ \end{gathered} \right)$$
(A.3)
$$\Pi_{154} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} { (20}{{.4}} + {5}{{.46y}}^{{2}} { - 18}{{.28yz}} + {6}{{.34z}}^{{2}} {)( - 229}{{.7}} + {87}{{.4Log[x] - 8}}{{.25Log[x]}}^{{2}} {)} \hfill \\ + {{n(1}}{.9 - 4}{{.97y}}^{{2}} + {14}{{.46yz - 3}}{{.97z}}^{{2}} {)( - 107}{{.4}} + {40}{{.9Log[x] - 3}}{{.9Log[x]}}^{{2}} {)} + \hfill \\ {(5}{{.4 - 0}}{{.12y}}^{{2}} + {0}{{.14yz}} + {0}{{.36z}}^{{2}} {)(12}{{.42 - 4}}{{.6Log[x]}} + {0}{{.45Log[x]}}^{{2}} {)} \hfill \\ \end{gathered} \right)$$
(A.4)
$$\Pi_{155} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} {(494}{{.4 - 37y}}^{{2}} + {90}{{.58yz - 13}}{{.6z}}^{{2}} {)( - 40}{{.54}} + {15}{{.9Log[x] - 1}}{{.56Log[x]}}^{{2}} {)} \hfill \\ + {{n(117}}{.7 - 9}{{.45y}}^{{2}} + {24}{{.6yz}} + {9}{{.04z}}^{{2}} {)(42}{{.1 - 16}}{{.54Log[x]}} + {1}{{.63Log[x]}}^{{2}} {)} + \hfill \\ {(5 - 0}{{.52y}}^{{2}} + {0}{{.94yz}} + {0}{{.28z}}^{{2}} {)(19}{{.1 - 7}}{{.42Log[x]}} + {0}{{.74Log[x]}}^{{2}} {)} \hfill \\ \end{gathered} \right)$$
(A.5)
$$\Pi_{156} = \frac{H}{{\sigma_{0} }} = \left( \begin{gathered} {{n}}^{{2}} {(208}{{.26 - 11}}{{.76y}}^{{2}} + {20}{{.3yz}} + {2}{{.9z}}^{{2}} {)( - 143}{{.3}} + {56}{{.35Log[x] - 5}}{{.5Log[x]}}^{2} {)} \hfill \\ + {{n(172}}{.35 - 11}{{.64y}}^{{2}} + {19}{{.7yz}} + {4}{{.7z}}^{{2}} {)(40}{{.5 - 15}}{{.9Log[x]}} + {1}{{.56Log[x]}}^{{2}} {)} + \hfill \\ {(11 - 0}{{.72y}}^{{2}} + {1}{{.28yz}} + {0}{{.5z}}^{{2}} {)(20}{{.4 - 8Log[x]}} + {0}{{.8Log[x]}}^{{2}} {)} \hfill \\ \end{gathered} \right)$$
(A.6)

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Bhat, T.S., Venkatesh, T.A. On the relationships between hardness and the elastic and plastic properties of transversely isotropic power-law hardening materials. Journal of Materials Research 37, 3599–3616 (2022). https://doi.org/10.1557/s43578-022-00730-y

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