On the Variation of Hardness Due to Uniaxial and EquiBiaxial Residual Surface Stresses at Elastic–Plastic Indentation
Abstract
It is established long since that the material hardness is independent of residual stresses at predominantly plastic deformation close to the contact region at indentation. Recently though, it has been shown that when elastic and plastic deformations are of equal magnitude this invariance is lost. For materials such as ceramics and polymers, this will complicate residual stress determination but can also, if properly understood, provide additional important information for performing such a task. Indeed, when the residual stresses are equibiaxial, the situation is quite well understood, but additional efforts have to be made to understand the mechanical behavior in other loading states. Presently therefore, the variation of hardness, due to residual stresses, is examined at a uniaxial stress state. Correlation with global indentation quantities is analyzed, discussed and compared to corresponding equibiaxial results. Cone indentation of elastic–perfectly plastic materials is considered.
Keywords
correlation equibiaxial stresses residual stresses sharp indentation uniaxial stressesIntroduction
Residual stresses can have a very destructive influence on the loadcarrying capacity of engineering structures. The best way to avoid this feature is of course to ensure that such stresses are negligible to start with, but this can in many situations be a difficult task to achieve. Another alternative is to determine such stresses and account for them during the design process. In this context, indentation has been during recent years emerged as a very attractive experimental method for determining residual stresses, especially at smaller length scales.

At sharp indentation in a situation with dominating plastic deformation, pertinent to most metals and alloys, the hardness H (here and below defined as the average contact pressure) is invariant of residual stresses, and determination of these stresses has to be attempted based on the size of the contact area between the material and the indenter, cf. Ref 15.

Explicit relations for the correlation of residual stresses with the size of the contact area have been presented, cf. Ref 35, 14, for the case above with dominating plastic deformation. This correlation is relying upon the socalled Johnson parameter (Ref 17, 18) to be discussed in some detail below.

In a sharp indentation situation where elastic and plastic deformations are of comparable magnitude, pertinent to many ceramics and polymers, the invariance of hardness is lost, cf. Ref 14, 15.

The variation of hardness can be correlated based on the Johnson parameter (Ref 17, 18), cf Ref 16 at equibiaxial stress states.
Accordingly, quite a lot of important knowledge has been gained about the mechanics of sharp indentation and residual stresses. In this, most of the results, however, are pertinent to the case of an equibiaxial residual stress state. This is certainly natural as, for example, stresses due to a distributed temperature often become equibiaxial. Admittedly, the assumption of equibiaxiality also simplifies the analysis as, for example, the ratio between principal stresses does not enter the governing equation and axisymmetry of all the field variables can be quietly assumed.
In particular when it comes to the situation with a varying hardness, the case mentioned above where elastic and plastic deformations are of comparable magnitude, very little is known about the effect of a stress state that is not equibiaxial. It is therefore the intention of the present investigation to remedy this shortcoming.
In doing so, finite element simulations for the case of a uniaxial residual stress state will be performed, and correlation with global indentation quantities is analyzed, discussed and compared to corresponding equibiaxial results first presented in Ref 16. The investigation is restricted to cone indentation of elastic–perfectly plastic materials, but generality of results, for example, to a contact situation with other indenter geometries and other types of plastic strain hardening can be expected, cf. Ref 4, 5.
Theoretical Background
The theoretical foundation as to a large extent presented in for example (Ref 16) will be used for background. This background includes residual stress determination at rigidplastic conditions, based on the size of the contact area between the material and the indenter, as well as the new features associated with the loss of hardness invariance in case of indentation where elastic and plastic deformation are of comparable magnitude. In this background section, it is assumed that equibiaxiality of field variables prevails.
The results in Fig. 4 are pertinent to an equibiaxial residual stress state, which is an important special case to be expected, for example, at temperature loading of thin film/substrate structures. A more general approach to the problem is of course of substantial interest, and for this reason, the analysis below will also include uniaxial residual, or applied, stress states encountered in a practical situation at for example unidirectional bending.
Numerical Analysis
In this section, the present finite element simulations of the cone indentation problem are described. It is then important to emphasize that due to the fact that uniaxial residual stress states are considered, axisymmetric conditions are lost and a full threedimensional solution has to be sought for. Similar analyses have been performed previously, for example, in Ref 15), and according to concerning details of the numerical approach, this article is referred to.
When elastic loading, or unloading, is at issue, a hypoelastic formulation in Hooke’s law is used.
Results and Discussion
It should be emphasized that compressive stresses will increase Λ, when again defined according to Eq 4 and 5, leading to a more pronounced level III (rigidplastic) situation pertinent to the material hardness, which is not of direct interest presently. However, compressive residual stresses are included in the analysis for clarity and of course also to analyze the hardness invariance at ln Λ > 3, approximately (Λ defined according to Eq 4 and 5).
Before presenting explicit results, it should also be mentioned that in the present situation a direct comparison between uniaxial and equibiaxial results is rather straightforward based on the Mises effective stress σ_{e}. In short, this is due to the fact that the value on the effective stress σ_{e} is the same (σ_{e} = σ_{res}) for both the residual stress systems in 10 and 11. (Obviously this refers to a situation prior to indentation.) Clearly, since surface residual stresses are at issue, the principal stress σ_{3} = 0.
To summarize the results presented in Fig. 7, 8 and 9, it can be concluded that when residual stresses are uniaxial the hardness variation is not an appropriate feature for stress determination. In this context, it should be mentioned, however, that it has been shown previously that this can be achieved using the relative contact area c^{2}, cf. Ref 23 where also correlation with the Johnson (Ref 17, 18) parameter Λ is confirmed. Without going into details, the reason for this is essentially that c^{2} is much more sensitive to the uniaxial stresses (elastic deformation) than the material hardness.
Having said this it is certainly of significant interest to discuss alternatives to the present approach. While the major advantage with the method at issue is simplicity (residual stresses can be explicitly determined from a single simple relation), the limited variation of the hardness due to residual stresses (in particular then in the presently investigated uniaxial case) makes it impossible to achieve acceptable accuracy at determination of such quantities. An attractive and more advanced alternative in such a case could be to rely on indentation experiments in combination with inverse analysis (Ref 911). This can be particularly advantageous if indentation is combined with other standard methods for residual stress determination such as hole drilling (Ref 11). Inverse modeling will, however, require substantial efforts but ways to handle this issue have been presented in for example (Ref 24).
There are of course also important features that are not included in the present theory. Such features include, for example, indentation creep (Ref 25), scale effects (Ref 26) and influence from the substrate at thin film characterization (Ref 27). Indentation creep concerns the constitutive behavior of the material and cannot be accounted for without another material description and also properly determining creep material quantities. The other effects can, however, be minimized by an appropriate choice on the value of the indentation depth h. Another feature that is not included in this analysis is any possible effect from the residual stresses on the material properties. This is certainly a very interesting issue and could, for example, be addressed by incorporating the relevant results by Ma et al. (Ref 28) into the present approach for residual stress determination. Such a matter is, however, left for future considerations.
As a final comment, it should be emphasized that the present results have bearing also for other types of contact problems. In particular, this is related to scratching and scratch testing, where correlation using the Johnson (Ref 17, 18) parameter Λ also is a major issue, cf. Ref 2933, but also material characterization by inverse modeling, cf. Ref 34 and using hardness testing to analyze multifields, cf. Ref 35.
Conclusions

The material hardness is much less influenced by uniaxial residual stresses than by corresponding equibiaxial ones.

Correlation with the Johnson (Ref 17, 18) parameter Λ is not accurate in case of uniaxial residual stresses. In the equibiaxial case, such correlation can produce a universal curve together with stressfree hardness results.

From a practical point of view, the hardness variation is not an appropriate quantity to be used experimentally for uniaxial residual stress determination in, for example, ceramics and polymers. A better alternative for this purpose is to use the relative contact area, here denoted c^{2}.
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