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Usually, some severe efforts are required to obtain tribological parameters like Archard’s wear depth parameter _{d}. Complex tribological experiments have to be performed and analyzed. The paper features an approach where such parameters are extracted from effective interaction potentials in combination with more physical-oriented measurements, such as Nanoindentation and physical scratch. Thereby, the effective potentials are built up and fed from such tests. By using effective material potentials one can derive critical loading situations leading to failure (decomposition strength) for any contact situation. A subsequent connection of these decomposition or failure states with the corresponding stress or strain distributions allows the development of rather comprehensive tribological parameter models, applicable in wear and fatigue simulations, as demonstrated in this work. From this, a new relatively general wear model has been developed on the basis of the effective indenter concept by using the extended Hertzian approach for a great variety of loading situations. The models do not only allow to analyze certain tribological experiments, such as the well known pin-on disk test or the more recently developed nano-fretting test, but also to forward simulate such tests and even give hints for structured optimization or result in better component life-time prediction. The work will show how the procedure has to be applied in general and a small selection of practical examples will be presented.

In order to achieve the goal set in the headline, namely, to analytically optimize surface and coating structures, which includes the extraction of generic tribological or wear parameters, it is necessary to combine quite a few fields and/or concepts of material science. Therefore, this introduction needs to cover the following issues:

first principle based effective interatomic potential description of mechanical material behavior [

the effective indenter concept [

the extension of the Oliver and Pharr method to analyze nanoindentation data to layered materials and time dependent mechanical behavior [

the physical scratch and/or tribological test and its analysis [

It was shown in [_{Morse} = ε[e^{−2p(r−r0)} − 2e^{−p(r−r0)}]

A contact problem can be evaluated using the mechanical parameters derived from such a potential. Here, _{0} usually denotes the equilibrium bond length. In such a case, the potential would define the pair interaction. Here, however, as in [_{0} denoting the lattice constant (see also [

“Surface” usually is not “bulk” and, so, subsequently, the behavior of surfaces, especially their mechanical behavior can be dramatically different from what one might expect by just applying bulk concepts to surface problems, such as contact situations with very surface-located, surface-dominated stress and strain fields. In tribo and wear problems, this very often compromises our ability to simulate and understand the physical processes taking place in certain tests (e.g., [

The “effective indenter concept” itself can mathematically be described or understood as some kind of quasi conform coordinate transformation transforming the difficult problem of a curved surface contacted by a well defined indenter (like a cone) into a flat surface loaded with a complexly formed indenter. Already, in 1994, Bolshakov, Oliver, and Pharr [

The extension of this concept is simply performed by substituting the homogenous half space model describing the loaded sample body by a layered half space model [

As it is a well established fact, that the classical Oliver and Pharr method [^{®} [

In an indentation test we always find complex three-dimensional stress states with usually all stress components being non-zero. Thus, as Fischer-Cripps put it, “the nature of the loading is a complex mixture of hydrostatic compression, tension, and shear” [

The standard scratch test is a widely used method to test the mechanical stability of coatings on different types of substrates and has become a sensitive technique to control the reliability of the manufacturing process. It is based on various standards [

Unfortunately, these standards do not allow to design and/or perform the tests in a truly physical way, meaning to extract generic physical parameters being later applicable in life-time prediction and optimization. For this, a more sophisticated approach is required, which assures physical parameter identification at each and every step towards physical tribological tests. The procedure is outlined in

Following the flowchart in

The reader will find illustrative examples and a much more comprehensive elaboration of the method elsewhere [

A flow chart of the procedure of mechanical characterization and optimization of arbitrary structured surfaces with respect to wear and fretting. See references about features and tools [

As shown in [_{0} (_{0},

As estimates for the critical _{00} = _{0}) to extract the critical _{m} for maximum _{m}_{00}) − 3e^{r00(cm−1)}(2 + _{m}_{00}) = 0

As a purely mathematically based measure for the critical bond length or in our case of an effective potential the lattice distance, the inflexion point for _{ifp} > _{m} could be used. This can be numerically obtained for the Morse potential via:
_{ifp}_{00}(4 + _{ifp}_{00}) − 2e^{r00(1−cifp)}(3 + 2_{ifp}_{00}(2 + _{ifp}_{00})) = 0

As shown in [

One also needs to check for temperature gradients caused by external and internal friction due to the tribological shear. It was shown in [

With a known principle structure for the temperature distribution function deduced from the shear stress distribution (adiabatic approach) such an approach seems to be manageable by the means of iterations. With a typical dependency for the temperature dependent relaxation time like [_{A} and the _{0} – viscosity _{0} and the Boltzmann-constant _{B}) one can evaluate the mechanical contact situations applying the techniques elaborated in [

In order to have a sufficiently great variability for the definition of differently shaped “effective indenters”, we apply the extended Hertzian approach as shown in [

Together with lateral loads (occurring in all scratch-, tribotesters, or the next generation of nanoindenters and their applications, see, e.g., [

These stresses can for example occur when the indenter shaft is dragged over the surface. As the shaft itself is elastic and, thus, would be bent during the lateral loading, an unavoidable tilting moment results and acts on the contacted surface. In addition, curved surfaces (e.g., due to roughness) can lead to such tilting moments.

In order to simulate the internal complex material structure of porous or composite materials, certain defect fields must be developed and combined with the external loads. Such fields can be extracted from the mathematical apparatus originally developed for the effective indenter contact by a simple generalization procedure. By developing such a defect model, one also obtains a very comprehensive tool for the construction of relatively general intrinsic stress distributions caused by internal inhomogeneities. Circular, disc-like inclusions could for example be simulated by the use of plane defects within the layered half space. Thus, introducing circular defects of radii _{i}_{i}_{i}_{i}

Finally, we need to take into account the curvature of the surfaces in order to its effect on the resulting contact pressure distribution [

Oliver and Pharr [_{0})^{m}

Applying now the concept of the effective indenter, as shown in ^{n}

The indices

The effective indenter concept (see: [

For this paper, we need to reformulate the basic contact equation incorporating the plastically deformed surface of the sample with the locally defined shape function (_{s}^{n}

Together with the following approach for the indenter part being in contact with the sample surface of the kind:
_{I}^{n}_{I}_{s}^{n}_{I}_{S}

Formulating the contact equation

Similarly, also more general effective indenter concepts can be derived, such as this one:
^{0}, ^{2}, ^{4}, ^{6}, and ^{8}. The reader might easily recognize the extended Hertzian character [

Some of the following examples and discussions will be based on these more general approaches. However, due to the wide use of the power law fit given above we will explicitly concentrate on the power law approach in connection with our practical examples.

In [

The next extension of need now is a time dependent analysis method for ordinary quasi-static nanoindentation tests (e.g., [_{0})^{m}_{0}(^{m(t)}

The next step is the introduction of a time dependent material model. Here, for the reason of simplicity, we at first resort to the well known three parameter approach given by a Young’s modulus of the following kind _{0} + _{1}exp(−

Now we substitute the time dependent Young’s modulus into the function

Please note that in cases of substantial preloading situations integration with respect to the time

This makes the classical fit of the three constants _{0}, _{0}, _{0}, _{1} and τ. It is clear that so many parameters automatically lead to numerical difficulties and instabilities if applied on data not providing sufficient linear independence. Thus, even in the pure visco-plastic case it is strongly suggested to use a variety of unloading curves (3 might be good number) obtained with different unloading speeds or at different maximum loads with similar unloading times.

In cases of time dependent inelastic behavior (like visco-plasticity) or more complex constitutive laws also _{0} have to be taken as _{0}(_{0} when there are great differences between unloading time and τ (comparable strain rates). The parameters n and B on the other hand are then only geometrical parameters and do not explicitly depend on time. A more detailed derivation and discussion of this extension can be found in [

In recent years more and more experimental concepts have been introduced trying to generalize the classical normal indentation process into a multiaxial, combined tilted or twisted contact test [_{i}

Here we have used the complex presentation of the lateral displacements with ^{c} = _{i}_{i}_{i}_{i}_{i}_{I}_{i}_{i}_{i}_{i}

The stress components in the case of linear elasticity can be found using the following identities:

In the visco-elastic case applying a 3-parameter Standard Linear Solid model given by a Young’s modulus of the following kind _{0} + _{1}exp(−_{jkl}_{jklm}

This results in

In some cases it is more convenient to use a continuous relaxation function, which results in:

With the assumption of adiabatic temperature fields following the shear field, such an approach appears to be manageable, but its solution strongly depends on the actual shear distribution (

We start as follows:

We see that by performing the following substitution underneath the time integrals:

Further on, in order to have maximum flexibility for the construction of a great variety of contact situations combined with sufficient simplicity to keep the model analytical, we introduce the method of time dependent Hertzian load dots as outlined in the

The approach has the advantage, that the evaluation of the complete elastic field for the whole body would only be a question of summing up a series of potentially time dependent Hertzian fields. In addition, the solution can easily be extended to layered materials [_{i} time-dependent. This, of course, requires reevaluation of the Integral in Equation (33) but caused by the distribution structure (properties of the Dirac delta function) in the integrand any such additional time-dependency would be very easy to handle. In addition one could go for extended Hertzian load dots instead of Hertzian ones (e.g., [

The governing contact equation and stress distributions for more general loading conditions as occurring during tests like scratch and pin on disc have already been given in the Section “the effective indenter concept”. The evaluation of the complex stress and strain fields for these tests is elaborated in the references given there. However, as these evaluations are rather cumbersome and lengthy, the reader is also referred to a software package performing such calculations in an automated and quick manner [

It should explicitly be pointed out that knowledge of the complete stress and strain field is essential for a proper failure characterization (scratch) or wear mechanism analysis (tribo). An example is presented in the figures below. There,

The evolution of von-Mises stress during the scratch test shown at three measurement positions: (_{C} failure, and (

The evolution of normal stress in scratch direction illustrated at three measurement positions: (_{C} failure, and (

Hence, only such a physical analysis of mechanical contact measurements like instrumented indentations, scratch, and tribo-tests enable one to find out why a surface structure fails under certain loading conditions. These results provide indications on how the investigated coating or surface structure can be improved (structurally be optimized of adapted with respect to the material selection).

Illustrative scheme of the failure mechanism (_{C} position is marked by the red dashed line [

Now we need to establish the relationship between quasi-static characteristics, like hardness, yield strength, Young’s modulus,

First of all, it must be understood that within the concept of this approach the process of wear, fretting or a general tribological process is a to be considered as a multi-physical, multi body (asperity) ensemble of contact situations (called load dots,

In addition to that, the connection between non-physical parameters (like hardness) towards wear is not the intention of the paper. In fact, there is no such connection—strictly speaking. Something not generic, like hardness, simply cannot, not generally, be extended or applied to a dynamic process like wear. Hardness, after all, is a mixed parameter. There are so many things contributing to hardness that, in general, it is not clear what effects affecting the hardness are also influencing wear (and in what manner), for instance. Thus, yield strength, critical fracture stresses, fracture toughness (energy loss caused by fracture propagation), in short, parameters, which could clearly be connected to certain stress fields and stress components, are much better suited to also interpret and discuss tribological effects.

We are now concentrating on some questions that often pop up in connection with tribological tests and observations.

Where is the role of debris particles?

They are just adding up to the complex jumble of contact situations mentioned above.

How are strain-rate or even more general time-dependent effects taken into account?

This is usually been done by applying time-functional dependencies for the mechanical parameters. However, as shown in [

How are intrinsic stresses been taken into account?

Intrinsic stresses are just adding to the external stresses (and strain) fields and lead to a subsequent shift of onset of inelastic behavior, be it fracture, plastic flow, or phase transition. Therefore it is very important to know and consider intrinsic stresses as accurately as possible. Here, “accurately” explicitly means that intrinsic stresses are usually not homogeneously distributed but also present a complex field adding up to the deformation field coming from external loading situations.

Due to their importance we here refer to a more comprehensive elaboration about the modeling of these stresses especially in connection with external loads (e.g., [

How can the many faces of wear and tribo effects be incorporated into one theoretical apparatus?

The straight forward answer would be: By decomposition limits extracted from first principle approaches. Hereby, the most general way would be to extract decomposition limits from the first principle approaches as described in [

At first, we explicitly point out, that wear or any other tribological effect cannot be connected by a simple _{d}

It should be made clear, that the sensitivity of a given tribological pairing towards certain components of the deformation field strongly depends on the initial failure mechanisms driving, respectively dominating the process. Thus, we will probably have a higher dependency on the deviatoric stress field (given by the second invariant or the von Mises stress) where the tribology is mainly determined by plastic flow. The examples given in the paper are apparently governed by these effects. In those cases however, where we expect fracture to dominate the tribological process, wear components coupling to the normal stresses or directed shear stresses will increase over the other components of the tribo- or wear tensor. Here, we can easily deduce that mode I fracture would lead to higher components connected with the normal stresses, while mode II and mode III fracture driven tribology requires shear dominated wear tensors. In some cases also pressure driven phase transitions could be responsible for the observed tribology. Then the tribo- or wear tensor will have dominant terms coupling to the first invariant of the stress field, because this gives the pressure field multiplied by three.

Here we will concentrate mainly on linear dependencies [first line in Equation (34)], where we used the following denotations: ^{xx}_{ijkl}^{kl}^{kl}^{i}_{n}_{ij}_{ij}_{ij}_{ijkl}^{kl}_{w}_{w}_{ij }^{i }^{j}^{i}

We have to point out explicitly, that all the stresses, strains, vectors, energies _{vM}_{k}_{dvM}

Using the quantum mechanical marking to note operators we could use the following generalization for an operational form of (34):

For the wear-example given above we can formulate this as:

As we can see, the Archard’s law given with a scalar wear coefficient _{d}_{w}_{d}^{33} is nothing but a rather dramatic simplification of Equation (35) being possible wherever either the stress is dominated by its normal component in the surface normal direction (here, we named it σ^{33}) or where the coefficient tensor _{ijkl}^{33}, which would then read:
_{w}_{ij} n^{i} n^{j}_{ijkl} σ^{kl} n^{i} n^{j}_{33kl }^{kl}_{3333}^{33} ≡ _{d}σ^{33}

We can deduce now that for complex contact conditions, where the stress tensor is fully set and no component is dominant against all the others, one should be rather careful with the assumption of having the wear-tensor being of the most simple, Archard’s-law-like kind

One should also take the other stresses into account and investigate their possible influence regarding the resulting global wear, which, in this case, has to be taken as the sum over all stress-components in connection with the wear-tensor:
_{w}_{33}_{kl} σ^{kl}_{3311}σ^{11} + _{3322}^{22} + _{3333}^{33} + 2(_{3312}^{12}_{3313}^{13} + _{3323}^{23})

Here, we have made use of the symmetry of the stress tensor, also requiring a symmetric wear-tensor.

We also point out, that in the general law as given above in Equation (35) the hydrostatic (sphere) and deviatoric stress parts are distinguishable. Such a simplified law might read:
_{ij}_{ij}_{dvM}σ_{vM}_{dH}σ_{H}_{vM}_{H}

Within a variety of examples the procedure or parts of it have already been applied [

It was shown in [

We have demonstrated this on the wear track example presented in [

Still, there had been deviations from the experiment which could not been explained in a satisfactory manner. However, it was already mentioned (

Simply assuming such an influence and incorporating it into the wear simulation as described in [

We hope to result into a better fit of our wear or tribo model to the experimental data. Using a similar dependency for the yield strength

Here, we need to pay attention to the fact, that with having a temperature dependent Young’s modulus by the means of Equations (29) and (45) we also result in a temperature dependency for the von Mises stress maximum and its position. This makes the evaluation of Equation (46) relatively complicated and we therefore restrict ourselves here to the task of finding a temperature function _{0} × tanh(_{1} ×

Comparison of various wear laws with the experimental data (

Comparison of various wear laws with the experimental data (

It should be pointed out that, especially in scratch and wear tests, often, effects like work hardening, grain refinement, intrinsic stress, and general defect accumulation lead to significant material parameter changes during the test. In principle the model allows to account for such effects simply by making the material parameters which are affected by such effects dependent on the evolution of the test. Such a dependency could either be steered via time, wear cycle or any other evolutional parameter counting the test progress. Within the examples given in

More examples including better illustration of the intermediate steps as well as a more comprehensive description of the experiments are published elsewhere (e.g., [

By combining the global incremental wear model with the concept of the effective indenter, the extended Hertzian approach and a layered half space solution for contact problems with rather arbitrary combinations of normal, lateral and tilting loads a general, quick and powerful wear model has been created.

Inversion of the model makes it fit for parameter identification and optimization problems.

Further, adding first principle models, like effective interatomic interaction potentials allows a deeper understanding of those failure mechanisms being responsible for wear results observable in practical tests.

Temperature effects had been taken into account by allowing the temperature to couple into the mechanical properties Young’s modulus and yield strength.

Possible subsequent time dependent material behavior leads to a relatively complex inhomogeneous Young’s modulus distribution which must be taken into account if one aims for a better simulation of wear experiments.

The analytical method as outlined here has the advantage of delivering quasi “in-sight-views” about the deformation fields, strains, stresses, and so on, of the surfaces subjected to such tribology tests. Due to the completely analytical character of the tools used the calculation usually is extremely fast and robust. This way weak spots within the possibly layered material composition can be found more easily and efficiently. Subsequently, also optimizing surface systems towards a better tribo-performance could be done quicker and still more accurately.

This work was partially funded through the European Metrology Research Programme (EMRP) Project IND05 MeProVisc. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.

By introducing Hertzian load-dots as:

Applying an exponential approach for the Λ_{i}_{i}

Now we derive the situation for a stress driven deformation state. Again applying the 3-parameter model together with our simple load-dot approach we have:

Generalized with respect to field dependent mechanical properties and sticking to a 3-parameter potential interaction the Equation (A7) would read:

Extension to discrete or continuous retardation gives:

For reasons of simplicity however, we will here proceed with the gradient free simple SLS-model.

Together with the normalization condition and the total normal load _{i}(_{i}

Both cases are relatively easy whenever there will be no change of the contact area during the evolution of the deformation time being considered. In most practical applications however, this is not the case. Concentrating on contact experiments we can often assume that they are load controlled. However, due to the complex mixed stress states underneath the contact zone, there are also strain effects involved and it is not clear yet how to take these into account. Therefore Equations (A6)–(A11) and (A16) are applicable but need to be extended with respect to strain driving effects (e.g., [_{i}_{i}_{i}

For many cases in contact applications we are facing mixed stress-strain driven conditions requiring a combination of Equations (A4) and (A7). Choosing a sufficiently high number of load dots

The author declares no conflict of interest.