Size dependent yield hardness induced by surface energy

Gang Feng Wang; Yue Ding

REVIEW 4 major objections 4 minor 28 references

Size dependent yield hardness induced by surface energy

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Surface energy, through the dimensionless ratio $YR/\gamma$, can explain why hardness rises as spherical indenters shrink to the nanoscale.

desk verdict Plausible surface-energy size effect for nanoindentation, but the universal scaling law rests on an undemonstrated one-parameter collapse; deserves revision, not desk rejection. read the letter →

arxiv 1908.08175 v1 pith:ZGA7POPD submitted 2019-08-22 physics.app-ph cond-mat.mtrl-sci

Yue Ding , Gang Feng Wang This is my paper · ORCID

classification physics.app-phcond-mat.mtrl-sci

keywords yieldhardnesssurfaceenergynanoindentationsize-dependentsphericalindentationdimensionalanalysisfiniteelementsimulationcontactmechanics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that surface energy, rather than strain-gradient plasticity, can explain why hardness measured by nanoindentation grows as the indenter shrinks to the nanoscale. It defines yield hardness as the contact pressure at first yield and shows that in classical continuum mechanics this quantity depends only on yield strength and Poisson's ratio. With surface energy added, dimensional analysis and finite element simulation reduce the problem to a single parameter $YR/\gamma$: both the yield hardness and the critical indent depth increase as $R$ decreases, following Eqs. (11) and (12). The result matters because it turns a long-standing empirical size effect into a quantitative prediction and suggests nanoindentation as a route to measure yield strength and surface energy.

What carries the argument

The load-bearing object is the dimensionless ratio $YR/\gamma$, the yield stress times the indenter radius divided by the surface energy density, which the paper's dimensional analysis identifies as the sole control parameter. The classical elastic contact solution (Eqs. (1)-(6)) supplies the baseline maximum von Mises stress and yield-location dependence on Poisson's ratio, and the finite element model adds surface energy through a surface traction. Equation (10), $H_y/H_{y0}=f(YR/\gamma)$, is the scaling assumption that lets the numerical results be summarized by the fitted power laws (11) and (12).

What would settle it

Simulate or indent two solids with the same value of $YR/\gamma$ but with different elastic-modulus-to-yield-strength ratios or different Poisson's ratios, and compare the normalized yield hardness $H_y/Y$; if the values differ, the collapse behind Eqs. (11) and (12) fails. Alternatively, indent the same material with spherical tips of radius $R$ and $2R$ and check whether the hardness ratio follows the predicted exponent $0.863$.

Watch

Extended reading notes

Core claim

The paper defines yield hardness $H_y$ as the mean contact pressure at the moment the von Mises stress first reaches the yield strength $Y$. In classical Hertzian contact this value is an intrinsic material property, $H_{y0}=Y/(1.134-0.674\nu)$, independent of indenter radius. When a constant surface energy density $\gamma$ is present, dimensional analysis and finite element simulations collapse the problem onto the single parameter $YR/\gamma$, and the paper finds $$H_y/Y = \frac{1+1.92(\gamma/(YR))^{0.863}}{1.134-0.674\nu}, \qquad d_s/d_0 = 1+0.213(\gamma/(YR))^{0.59},$$ where $d_0$ is the classical critical indent depth at yield onset. As $R$ decreases, both the yield hardness and the critical indent depth increase, so surface energy resists yield and supplies a quantitative scaling account of the indentation size effect seen at the nanoscale.

Load-bearing premise

The load-bearing premise is that the entire effect of surface energy on yield is captured by the single dimensionless group $YR/\gamma$; if other dimensionless groups such as $E/Y$ or $\nu$ also matter beyond the prefactor, the fitted equations are material-specific rather than general scaling laws.

Editorial extensions

If this is right

At fixed material properties, shrinking the spherical indenter radius $R$ increases $\gamma/(YR)$, so the yield hardness $H_y/Y$ rises monotonically; this constitutes a quantitative mechanism for the indentation size effect.
Surface energy delays the onset of yield: the critical indent depth $d_s$ exceeds the classical $d_0$, growing with $(\gamma/(YR))^{0.59}$ as the indenter gets smaller.
When the yield strength is much larger than $\gamma/R$, the surface effect becomes negligible and the classical values $H_{y0}$ and $d_0$ are recovered.
Because the exponents in Eqs. (11) and (12) differ, measuring yield hardness and yield-onset depth in nanoindentations at different tip radii offers a route to extract both the yield strength and the surface energy density of a solid.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the single-parameter collapse is exact, two nanoindentation measurements at different tip radii are enough to solve for both $Y$ and $\gamma$ from Eq. (11) alone; Eq. (12) then serves as a consistency check.
Beyond the paper: the same surface-energy mechanism should also shift the pop-in load commonly observed in nanoindentation, and separating it from strain-gradient effects is possible because the two mechanisms scale differently with tip radius.
Beyond the paper: replacing the sphere radius with an equivalent radius for a Berkovich or conical tip would test whether these scaling laws carry over to standard sharp-indenter hardness measurements.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The one-sentence take: this is a credible extension of the authors' earlier surface-tension contact work. They define a yield hardness (the contact pressure at first yield), recover the standard Hertzian result, and then use FEM to show that surface energy raises that hardness at small indenter radii. The scaling formulas in Eqs. (11) and (12) are new, but they are empirical fits to the authors' own simulations, and the paper has not demonstrated the one-parameter collapse these fits rely on.

What is genuinely good: the classical part is clean. Eq. (7), H_y0 = Y/(1.134 − 0.674ν), is a correct reformulation of the known Hertzian yield-onset criterion. The FEM stress distributions reproduce the analytical no-surface-energy solution, which gives confidence in the numerical setup. The central qualitative trend—surface energy makes yield harder as R shrinks—is physically sensible and consistent with earlier work on surface tension in contact mechanics.

The soft spots: the more serious one is the dimensional analysis. The paper writes H_y/H_y0 = f(YR/γ) (Eq. 10) and then fits Eqs. (11) and (12). A full dimensional analysis of an elastic contact with surface tension gives H_y/Y as a function of ν, E/Y, and γ/(ER) separately. Reducing to YR/γ alone requires an argument that the stress field at yield depends on the contact radius a only through a combined parameter—or, equivalently, that the surface-tension correction enters via γ/(Ea) and that a/R is set by Y/E. The paper doesn't give that argument, and it doesn't report the FEM parameter ranges or the scatter around the fits. So the claimed universality of Eqs. (11) and (12) is not yet established; they are material-specific fits until proven otherwise. This is fixable, but it's load-bearing. There is also a clear typo in Section 3: "For cases with the ratio γ/R much larger than Y, the surface energy effect is negligible" should be "much smaller"—Eq. (11) shows the correction grows as γ/R grows relative to Y.

The paper includes no code or data, which limits reproducibility, though that's normal for an FEM study.

Who this is for: researchers working on nanoindentation size effects and surface-tension contact. They will find the trend and the potential route to measure Y and γ from multi-size indentation interesting, but they should not use Eqs. (11) and (12) as general laws until the collapse is validated.

My recommendation: give it a serious peer review. It's a legitimate, useful contribution with a real gap. A conditional acceptance should request revision that supplies the scaling argument and the parameter ranges.

Referee Report

4 major / 4 minor

Summary. The paper studies spherical indentation of an elastic half-space with surface energy, defining a 'yield hardness' H_y as the mean contact pressure at the onset of yielding. In the classical (surface-energy-free) case, the authors use the Hertz stress solution to derive H_y0 = Y / (1.134 – 0.674ν), which depends only on yield strength and Poisson's ratio. They then incorporate surface energy through finite element simulations with user-defined surface elements, and via dimensional analysis propose the one-parameter scaling law H_y/H_y0 = f(YR/γ). Fitting FEM data yields explicit power-law forms: Eq. (11) for H_y/Y and Eq. (12) for the critical indentation depth ratio d_s/d0. The paper concludes that when Y is comparable to or smaller than γ/R, surface energy increases the yield hardness and the critical depth, providing a possible explanation for nanoindentation size effects and suggesting a route to extract yield strength and surface energy from nanoindentation data.

Significance. If the proposed scaling laws are valid, the paper offers a concise and potentially useful interpretation of nanoindentation size effects in terms of surface energy, and it gives explicit formulas (Eqs. (11) and (12)) that could be used for inverse characterization. The classical derivation leading to Eq. (7) is clean and correct. The FEM implementation using user-defined surface elements follows established methodology, and the paper honestly identifies that surface energy makes an appreciable difference only when YR/γ is of order unity or smaller. However, the central deliverable—the universal collapse in Eq. (10) and the subsequent fits—is asserted rather than demonstrated. The evidence currently available in the manuscript is insufficient to establish that Eqs. (11) and (12) are general scaling laws rather than material-specific empirical fits. This is a load-bearing gap that can likely be addressed with additional parametric simulations and analytical argument, so the paper merits revision rather than rejection.

major comments (4)

[Section 3, Eq. (10)] The one-parameter collapse H_y/H_y0 = f(YR/γ) is asserted without proof. A complete dimensional analysis of the elastic-contact-with-surface-tension problem gives H_y/E = F(ν, Y/E, γ/(ER)); reducing this to a function of YR/γ alone requires the additional argument that the yield condition forces E to drop out of H_y/Y. The manuscript does not supply this argument, and the numerical study does not show independent variation of E, Y, γ, and R at fixed YR/γ. If the collapse is not universal, Eqs. (11) and (12) are fits for the particular material (Ag) and parameter range simulated, and the claimed generality of the inversion method to extract Y and γ from nanoindentation is not established.
[Section 3, Eq. (11)] The ν-dependence in Eq. (11) appears only in the prefactor 1/(1.134 – 0.674ν), while the correction term 1.92(γ/(YR))^0.863 is independent of ν. Yet the surface-modified stress field and the yield location themselves depend on ν, as seen in Eqs. (5) and (6). The paper does not report simulations that vary ν at fixed YR/γ, nor does it provide error bars or confidence intervals for the fitted constants 1.92, 0.863, 0.213, and 0.59 in Eqs. (11) and (12). Without this information, the universality of the exponents in ν is unsupported.
[Section 2 and Figs. 7–8] The FEM parameter ranges are not reported. The text specifies one material (Ag, E=83 GPa, ν=0.37, γ=7.2 J/m²) for the stress profiles in Figs. 4–6, but Figs. 7 and 8 are described as covering 'various indenter radii and elastic materials with different elastic moduli, Poisson’s ratios and surface energy densities' without stating the ranges of Y, E, ν, γ, and R, the number of simulations, or the scatter of the data around the fitted curves. Also, the FEM model details (mesh density, model size, number of surface elements, verification of convergence) are delegated to reference [26]; for a self-contained claim of a general scaling law, these details should be summarized in the present paper.
[Introduction and Section 3 (terminology)] The paper calls H_y an 'intrinsic material property depending only on the yield strength and Poisson's ratio,' but Eq. (11) shows that H_y also depends on surface energy density and indenter radius. This is not inconsistent—the intrinsic statement refers to the macroscopic limit—but the text should state this distinction explicitly to avoid confusion, particularly because the central message is that H_y becomes size-dependent at the nanoscale.

minor comments (4)

[Abstract and text] There are typographical errors such as 'har dness' in the abstract and inconsistent spacing in equations; the manuscript would benefit from a careful proofreading pass.
[Eqs. (5)–(6) and Figs. 3–8] The fitted expressions in Eqs. (5), (6), (11), and (12) are given without any measure of fit quality (e.g., R², RMS error, or confidence bounds). Adding error estimates or at least stating the number of data points and their scatter would strengthen the reliability of the reported constants.
[Figure captions] Captions for Figs. 7 and 8 do not specify the materials, parameter ranges, or symbol definitions used in the plots; a reader cannot determine from the captions which data points correspond to which combinations of E, ν, γ, and R.
[Section 2, FEM model] The paper states that a constant surface energy density γ is assumed, but does not explain whether the surface energy is treated as a surface tension (with a corresponding surface stress) or merely as an energy term in the variational formulation. A sentence clarifying the constitutive treatment would help the reader interpret the boundary condition.

Circularity Check

1 steps flagged · score 6.0 of 10

Eqs. (11)-(12) are curve fits to the FEM data they are then claimed to explain and to invert for Y and gamma; the one-parameter collapse in Eq. (10) is asserted rather than demonstrated.

fitted input called prediction [Section 3, Eqs. (10)-(12) and Figs. 7-8]
"For various indenter radii and elastic materials with different elastic moduli, Poisson's ratios and surface energy densities, finite element simulations are performed to achieve the relation between yield hardness and yield strength. Fig. 7 displays the dependence of normalized yield hardness on the ratio YR/γ, and the relation can be well reproduced by the function [Eq. (11)] ... The relation between critical indent depth ds and ratio YR/γ can be formulated as [Eq. (12)]."

The constants 1.92, 0.863, 0.213, and 0.59 are free parameters chosen to reproduce the same FEM dataset that Eqs. (11) and (12) are then presented as 'achieved' expressions. No independent derivation or validation subset is provided, so the agreement is by construction: a curve fitted to a dataset will reproduce that dataset. The paper further proposes to invert Eqs. (11)-(12) to determine Y and γ from nanoindentation, but that inversion is just fitting the same empirical law back to data, not testing a first-principles prediction. The dimensional-analysis reduction to the single variable YR/γ in Eq. (10) is asserted without proof, so if that collapse fails for other ν, E/Y, or γ/(ER), the formulas are material-specific fits rather than universal scaling laws.

full rationale

The underlying FEM observation that surface energy reduces subsurface stresses and therefore delays yield is a genuine numerical result and is not circular; the zero-surface-energy limits in Eqs. (7) and (9) follow from classical Hertz contact mechanics. However, the central deliverable of the paper is the scaling laws (11) and (12), and these are empirical fits to the same simulation data used to construct them, with no derivation of the exponents or demonstration that the one-parameter collapse of Eq. (10) is universal. Thus the 'predictions' of size-dependent yield hardness and critical depth are statistically forced by the fit, producing partial circularity. No load-bearing self-citation chain is present, so the score is 6 rather than higher.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper's central result rests on the Hertzian yield-onset picture plus an FEM surface-energy model carried over from the authors' prior work. The new scaling laws contribute four fitted constants (1.92, 0.863, 0.213, 0.59) and an unproven one-parameter collapse. No new physical entities are postulated.

free parameters (6)

Equation (11) prefactor = 1.92
Calibrated to FEM data in Fig. 7; scaling-law prefactor.
Equation (11) exponent = 0.863
Best-fit power-law exponent for the gamma/(YR) dependence.
Equation (12) prefactor = 0.213
Calibrated to FEM data in Fig. 8.
Equation (12) exponent = 0.59
Best-fit power-law exponent.
Equation (5) intercept = 1.134
Fitted to numerical evaluation of the classical Hertzian Mises-stress maximum as a function of Poisson's ratio; affects H_y0.
Equation (5) slope = -0.674
Same fit; affects H_y0.

assumptions (5)

standard math Hertzian contact theory for frictionless spherical indentation of an elastic half-space (Equations 1-4)
Used to derive the classical yield-hardness H_y0 and critical depth d0.
domain assumption Yield initiates when the maximum von Mises stress reaches the yield strength Y
Definition of yield hardness; standard von Mises criterion.
domain assumption Surface energy is represented by a constant energy density gamma acting through user-defined surface elements as described in reference [26]
The surface-energy boundary condition is not stated in this paper; all FEM results depend on that implementation.
ad hoc to paper The dimensionless yield hardness depends only on YR/gamma (Equation 10), independent of E/Y and gamma/(ER)
This collapse is asserted via dimensional analysis but not proven; it is the load-bearing scaling assumption.
domain assumption The material is isotropic and elastic (no plastic flow is simulated before yield), and the indenter is rigid
Standard assumptions in the FEM model.

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Cite this review

Pith. "Pith review of Size dependent yield hardness induced by surface energy." pith.science (2026). https://pith.science/paper/ZGA7POPD

@misc{pith2026190808175,
  author       = {Pith},
  title        = {Pith review of: Size dependent yield hardness induced by surface energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGA7POPD}},
  note         = {Machine review of arXiv:1908.08175}
}

read the original abstract

Size dependent hardness has long been reported in nanosized indentations, however the corresponding explanation is still in exploration. In this paper, we examine the influence of surface energy on the hardness of materials under spherical indentation. To evaluate the ability of materials to resist indentation, a yield hardness is defined here as the contact pressure at the inception of material yield. It is found that this defined hardness is an intrinsic material property depending only on the yield strength and Poisson ratio in conventional continuum mechanics. Then, the impact of surface energy on the yield hardness is analyzed through finite element simulations. By using the dimensional analysis, the dependences of the yield hardness and critical indent depth at yield initiation on surface energy have been achieved. When the yield strength is comparable to the ratio of surface energy density to indenter radius, surface energy will alter the yield hardness and the critical indent depth. As the size of indenter decreases to nanoscale, both the yield hardness and the indent depth will increase significantly. This study provides a possible clarification to the size dependence of hardness and a potential approach to measure the yield strength and surface energy of solids through nanosized indentations.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [26]

Niu, and G

Ding, Y ., X.-R. Niu, and G. -F. Wang, Elastic compression of nanoparticles with surface energy. Journal of Physics D: Applied Physics, 2015. 48(48): p. 485303

work page 2015

[2] [1]

Tribology Letters, 2017

Broitman, E., Indentation Hardness Measurements at Macro -, Micro -, and Nanoscale: A Critical Overview. Tribology Letters, 2017. 65(1)

work page 2017

[3] [2]

Mechanics of Materials,

Akatsu, T., et al., Representative indentation yield stress evaluated by behavior of nanoindentations made with a point sharp indenter. Mechanics of Materials,

work page

[4] [3]

Applied Mechanics Reviews, 2017

Ghaednia, H., et al., A Review of Elastic -Plastic Contact Mechanics. Applied Mechanics Reviews, 2017. 69(6)

work page 2017

[5] [4]

Oliver, W.C. and G.M. Pharr, Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. Journal of Materials Research, 2004. 19(1): p. 3-20

work page 2004

[6] [5]

Ishlinsky, A.Y ., Axially symmetric problem in plasticity and Brinell’s hardness test. Prikl. Mat. Mekh., 1944. 8: p. 201-224

work page 1944

[7] [6]

Storakers, and A.B

Hill, R., B. Storakers, and A.B. Zdunek, A THEORETICAL -STUDY OF THE BRINELL HARDNESS TEST. Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 1989. 423(1865): p. 301-330

work page 1989

[8] [7]

Biwa, S. and B. Storakers, AN ANA LYSIS OF FULLY PLASTIC BRINELL INDENTATION. Journal of the Mechanics and Physics of Solids, 1995. 43(8): p. 1303-1333

work page 1995