## 1. Introduction

- When elastic deformations around the contact region are in the same order as plastic ones; then the hardness is dependent on residual stresses.
- In case of equi-biaxial surface stresses, correlation with the Johnson parameter is accurate and produces a general relation with corresponding stress-free results when residual stresses are appropriately accounted for.
- In case of uniaxial surface stresses, correlation with the Johnson parameter is not good and in such a situation; the hardness variation is not a good tool for an experimental determination of residual stresses.

## 2. Theoretical Background

^{2})σ

_{rep})

_{rep}is the material flow stress at a representative value of the equivalent (accumulated) plastic strain ε

_{p}. Based on the parameter Λ three regions or levels of contact behavior can be defined. These levels, as shown in Figure 2, are:

- Level I: Dominating elastic deformations, i.e., low indentation load, where an elastic contact analysis is sufficient.
- Level II: Elastic and plastic deformations are of equal magnitude.
- Level III: Plastic deformations are dominating in the contact region.

_{rep}

_{p}≈ 0.08 (Tabor [2]) and for a cone indenter C ≈ 2.54 and ε

_{p}≈ 0.11 (Atkins and Tabor [22]). The latter case is specified for a cone indenter with an angle of 22° between the indenter and the undeformed surface being at issue here, see Figure 1. In Equations (2) and (3), H is the material hardness defined as the average contact pressure between indenter and material.

^{2}= A/A

_{nom}

_{nom}(nominal contact area) are projected areas (Figure 1). This quantity can indeed be related to an equi-biaxial residual stress σ

_{res}according to:

^{2}= c

^{2}(σ

_{res}= 0) − 0.35ln(1 + (Fσ

_{res}/σ

_{y}))

^{2}(σ

_{res}= 0) is the value on the relative contact area at indentation of a virgin (unstressed material) and F is a constant that takes on different values at tension and compression. For simplicity but not for necessity ideal plasticity, with a yield stress σ

_{y}, is here assumed. The Equation (5) was originally proposed by Rydin and Larsson [5] based on the similar mechanical behavior between contact induced stresses in an unstressed material or contact induced stresses in a material with a properly chosen apparent initial yield stress. Rydin and Larsson [5] suggested an apparent yield stress:

_{y,apparent}= σ

_{y}+ Fσ

_{res}

_{y}being the initial yield stress of the material) in Λ in Equation (1), according to:

_{y,apparent}(1 − ν

^{2}))

^{2}-curve, shown schematically in Figure 2, regardless if residual stresses are present or not. This curve can be used to determine σ

_{res}when c

^{2}(σ

_{res}= 0) is known.

_{y}, in Figure 2. It was recently shown by Larsson [13] that it is possible to correlate the influence from residual stresses on hardness values, in essentially the same way as for the relative contact area c

^{2}, when the residual stresses are equi-biaxial. This correlation is, again as for c

^{2}, based on Equations (6) and (7) and is explicitly shown in Figure 3, and obviously very good agreement with a universal curve is achieved. Corresponding results [14] were then presented for a uniaxial stress state and unfortunately, correlation with a universal curve was not good.

## 3. Numerical Analysis

_{e}= σ

_{y}

_{1}and σ

_{2}. For a general surface stress state, the contact area will become ellipsoidal with semi-axes a

_{1}≠ a

_{2}. Accordingly, a three-dimensional finite element analysis is required at uniaxial residual loading. Clearly, since surface residual stresses are at issue the principal stress σ

_{3}= 0.

_{1}and σ

_{2}. Accordingly, these stresses are given by Hooke’s law in a homogeneous situation. Indentation loading was applied by controlling the transversal displacement of the rigid indenter.

## 4. Results and Discussion

_{2}= 0) and equi-biaxial residual stresses (σ

_{1}= σ

_{2}) are presented. Accordingly, these results will be compared with the corresponding ones for other values on the ratio σ

_{1}/σ

_{2}. The present results are, unless otherwise stated, derived for the case of:

_{1}/σ

_{y}and σ

_{2}/σ

_{y}, where again σ

_{1}and σ

_{2}are the principal residual surface stresses with σ

_{3}= 0. It should also be mentioned that in the present situation a direct comparison between uniaxial and equi-biaxial results is rather straightforward based on the equivalent stress σ

_{e}, see [14]. In short, this is due to the fact that the value on the equivalent stress σ

_{e}is the same (σ

_{e}= σ

_{res}) for both these residual stress systems (obviously, this refers to a situation prior to indentation). In order to relate the residual stress fields, in other biaxial cases, to the material yield stress also ((σ

_{res})

_{e}/σ

_{y}) is used to describe the present results. In this case, (σ

_{res})

_{e}is the Mises equivalent stress derived based solely on the residual surface stress field.

_{e}. As already stated above, this is due to the fact that the value on the equivalent stress (σ

_{res})

_{e}is the same ((σ

_{res})

_{e}= σ

_{res}) for both these residual stress systems (obviously, this refers to a situation prior to indentation). The explicit results are shown in Figure 5 where the non-dimensional hardness is depicted as function of the stress ratio (σ

_{res}/σ

_{y}). Note that in Figure 5 and below, H

_{0}is the hardness for the residual stress-free case. It is very clear from what is shown in Figure 5 that the hardness is far more influenced by an equi-biaxial residual stress σ

_{res}than a corresponding uniaxial one. For example, at (σ

_{res}/σ

_{y}) = 1, the hardness value is reduced (compared to the stress-free hardness H

_{0}) with around 18% in the equi-biaxial case but with only approximately 6% in the uniaxial one. In practice, it would be very hard, if not experimentally impossible, to accurately determine σ

_{res}based on the small variation, in the uniaxial case, from the stress-free results.

_{1}= σ

_{res}= 2σ

_{2}

_{res})

_{e}= 3

^{1/2}σ

_{res}/2

_{res}= σ

_{y}and σ

_{res}= σ

_{y}/2 are investigated.

_{2/}σ

_{1}

_{1}= σ

_{res}= σ

_{y}. It can be seen that at k = 0.25, the influence from residual stresses on the hardness value is small and it seems appropriate to conclude that k = 0.5 constitutes an approximate lower bound for the practical applicability of the present approach. For k-values smaller than 0.5, the influence from residual stresses is so small that it would be difficult to measure any relevant changes of the material hardness in an accurate manner.

_{1}= σ

_{res}= σ

_{y}/2 are shown in Figure 8. As could be expected, the practical applicability of the present approach is then very doubtful. Indeed, it can be argued that only in the case of equi-biaxial stresses any degree of accuracy of results can be expected.

^{2}in Equation (4). Corresponding results for equi-biaxial stresses, to the hardness results in Figure 9, for this quantity for ln Λ = 5.7, are shown in Figure 10. Clearly, the Λ-dependence is strong, which is encouraging. Such dependence has been shown previously [3,4,5,6,7,8], but here it is directly compared with the corresponding situation for the material hardness and established for a particular value on Λ.

## 5. Conclusions

- The material hardness dependence on residual stresses is highest for equi-biaxial stresses and less for uniaxial ones. Other values on the principal stress ratio yield results that lie between these two extremes.
- At residual stresses well below the material yield stress, it can be argued that only in the case of equi-biaxial stresses any degree of accuracy of results can be expected with the present approach.
- For values on the Johnson parameter Λ higher than the one presently investigated (ln Λ = 3.4), the hardness dependence on residual stresses vanishes rapidly. A better alternative for this purpose is then to use the relative contact area, here denoted c
^{2}.

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic of the geometry of cone indentation where a represents the true contact radius. In the present investigation β = 22°. The nominal contact area A

_{nom}= πh

^{2}/(tanβ)

^{2}where h is the indentation depth.

**Figure 2.**Normalized hardness, $\overline{H}$ = H/σ

_{y}, and area ratio, c

^{2}, as functions of lnΛ, defined by Equation (1). Schematic of the correlation of sharp indentation testing of elastic-ideally plastic materials. The three indentation levels, I, II and III, are also indicated. Approximately, level II contact initiates at Λ = 3 level III contact at Λ = 900. The $\overline{H}$-curve flattens out at (approximately) Λ = 30.

**Figure 3.**Results for equi-biaxial residual stress fields. Normalized hardness, H/σ

_{y}, as function of lnΛ, defined by Equation (7) with the yield stress σ

_{y}replaced by σ

_{y,apparent}in Equation (6). The straight line represents Equation (7). (○), stress-free results taken from Larsson [17]. (●), hardness values taken from [12] with and without residual stresses. (⋆), hardness values taken from [14] with and without residual stresses.

**Figure 4.**Finite element mesh, close to the region of contact, used in the numerical simulations. The coordinate Y corresponds to X

_{2}in Figure 1. The finite element mesh, modelling a quarter of the material accounting for symmetries, is shown observed obliquely from above.

**Figure 5.**Nondimensionalized hardness, H/H

_{0}, as a function of the residual stress ratio, σ

_{res}/σ

_{y}. H

_{0}is the residual stress-free hardness. (- - -), H/H

_{0}= 1. Results from [14]. (○), numerical results for uniaxial residual stresses, σ

_{1}= σ

_{res}and σ

_{2}= 0. (●), numerical results for equi-biaxial residual stresses, σ

_{1}= σ

_{2}= σ

_{res}.

**Figure 6.**Nondimensionalized hardness, H/H

_{0}, as a function of the residual stress ratio, σ

_{res}/σ

_{y}. H

_{0}is the residual stress free hardness. (- - -), H/H

_{0}= 1. (○), numerical results from [14] for uniaxial residual stresses, σ

_{1}= σ

_{res}and σ

_{2}= 0. (▲), present numerical results for the case σ

_{1}= σ

_{res}= 2σ

_{2}. (●), numerical results from [14] for equi-biaxial residual stresses, σ

_{1}= σ

_{2}= σ

_{res}.

**Figure 7.**Nondimensionalized hardness, H/H

_{0}, as a function of the stress ratio k in Equation (13). H

_{0}is the residual stress-free hardness. (- - -), H/H

_{0}= 1. (●), present numerical results for the case σ

_{1}= σ

_{res}= σ

_{y}.

**Figure 8.**Nondimensionalized hardness, H/H

_{0}, as a function of the stress ratio k in Equation (13). H

_{0}is the residual stress-free hardness. (- - -), H/H

_{0}= 1. (●), present numerical results for the case σ

_{1}= σ

_{res}= σ

_{y}/2.

**Figure 9.**Nondimensionalized hardness, H/H

_{0}, as a function of the residual stress ratio, σ

_{res}/σ

_{y}for the case ln Λ = 5.7. H

_{0}is the residual stress free hardness. (- - -), H/H

_{0}= 1. (○), numerical results for uniaxial residual stresses, σ

_{1}= σ

_{res}and σ

_{2}= 0. (●), numerical results for equi-biaxial residual stresses, σ

_{1}= σ

_{2}= σ

_{res}.

**Figure 10.**Normalized relative contact area, RCA = c

^{2}/ c

_{0}

^{2}, as a function of the residual stress ratio, σ

_{res}/σ

_{y}for the case ln Λ = 5.7. c

_{0}

^{2}is the value on the relative contact area at residual stress-free conditions. (- - -), c

^{2}/ c

_{0}

^{2}= 1. (○), numerical results for equi-biaxial residual stresses, σ

_{1}= σ

_{2}= σ

_{res}.

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