1. Introduction
Estimation of the real contact area between two contacting solids can be seen as an initial step towards investigating friction and wear of that contact, which are of uttermost interest in various engineering applications [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10]. In the context of metalworking, bulk forming operations such as hot rolling of
alloys are greatly dependent on the contact conditions: the real area of contact partly defines the friction forces that move the workpiece through the roll bite [
11], being thus a fundamental aspect of the process. Accurate prediction and control of friction in these processes are highly desirable, since they ultimately contribute to an optimised process in terms of energy consumption.
Although attempts to measure the real contact area have been explored in the literature [
12], it is generally impractical to directly observe the contact itself. Hence, researchers have often turned to
contact models to obtain information on the degree of contact (ratio between real contact area and nominal or apparent contact area). Contact model, in the sense used in this work, refers to a methodology for quantifying the real contact area in a contact between solid bodies. An appropriate contact model must take into account not only the topography of the surfaces and loading conditions but also the nature of the deformation of the materials involved. In metal forming, one of the major features of deformation of workpieces is the
thermo-viscoplastic behaviour of the material, i.e., the interdependence between stress, strain, plasticity, time, and temperature in the deformation process. Such interplay lies within the scope of
creep, defined as an inelastic time-dependent deformation when a material is subjected to a (constant or variable) stress at sufficiently high temperatures [
13]. thermo-viscoplasticity causes certain aluminium alloys such as 6061 [
14] to have significant changes in its flow stress
as the strain
, strain rate
, and temperature
T vary [
15]. In view of this, a proper contact model in the metal-forming context should account for such thermo-viscoplastic parameters.
In the literature,
hardness has been the primary translator between material and contact when plasticity is involved. In an indentation, it is well known that, when the mean pressure reaches a value of approximately 2.6–3 times the yield stress (
) of the softer material, the material around the region of contact has plastically deformed. Therefore, the mean contact pressure in a so-called fully plastic indentation, i.e., the
indentation hardness H, is expressed in terms of the yield stress of the deforming material as
, where
c, sometimes named the “constraint factor”, varies between 2.6 and 3 (for aluminium, the value 2.8 is reported by Tabor [
16]). The relation between hardness and contact area goes back to Bowden and Tabor [
17,
18], who assumed that the real contact area between two surfaces occurs at the tip of surface peaks, i.e., the tip of
asperities. At these tips, the contact pressure would be high enough so that plastic flow would occur at the softer material, resembling plastic indentation as studied by Brinell and thoroughly discussed in the book of Tabor [
16]. In this fully plastic scenario, the contact pressure equals the indentation hardness; consequently, the ratio of total normal load
L (carried by all asperities) over the hardness
H of the soft material results in the real contact area:
.
Fully plastic contact conditions can be reasonably assumed to occur in the metal forming context [
19,
20]. Nevertheless, such a model assumes a constant hardness value, which, albeit true for severely cold worked metal (because they essentially do not work harden), is not appropriate depending on the situation. In the case of work-hardening metals, Tabor writes about using a “representative strain”, which depends on the geometry of the indenter, to evaluate the flow stress and maintain the proportionality between hardness and yield stress at a value of
. With regards to temperature and strain rate effects, there seems to be no experimental agreement for the real dependence of hardness with the former [
21] or extensive investigation on the effects of the latter. Nonetheless, the relation
with
is frequently used in the metal forming tribology literature [
22,
23,
24], implying that
with
is assumed.
The indentation hardness, or mean contact pressure, of thermo-viscoplastic materials is not readily available in the literature, especially when indented by non-standard geometries. Analytical solutions for normal contact of single asperities are possible under simplifying assumptions and geometries [
25,
26], which can be extended to develop multi-asperity contact models, such as the well-known elastic Greenwood–Williamson model [
27]. Nevertheless, for complex situations such as those of metal-forming operations, numerical methods such as the Finite Element (FE) Method are better suited to deal with contact problems that would otherwise be very complicated or impossible to solve analytically [
28,
29,
30,
31,
32,
33]. The use of numerical methods assist in the development of new tribological models that may include features presently lacking further efforts, for example, the inclusion of morphology and plasticity [
2]. Recently, Shisode et al. [
34] proposed an approach consisting of performing FE analyses to calculate contact pressure in the flattening of multiple coated asperities. In this manner, the total force carried by each asperity is obtained from a database of FE simulations. Along with further assumptions, a contact model was developed in a more customised and presumably more accurate way than simply assuming asperity tips always support 2.8 to 3 times the flow stress.
The approach of [
34] inspired the development of this work. Here, instead of
flattening of coated substrate, the FE models are
indentations of asperities on a thermo-viscoplastic material. In this manner, Tabor’s concept of hardness as an average pressure is revisited but with the flexibility of allowing complex plastic flow not only due to different sizes and shapes of “indenters” but also due to the nonlinear interplay between strain, strain rate, and temperature and to the displacement of non-contacting regions evidenced by the sink-in and pile-up of material around the contact. The main goal is to investigate how the thermo-viscoplastic flow stress nature of a deformable body influences the load–area relation of the contact while building an appropriate contact model for a metal-forming situation. To achieve this goal, a metal-forming situation is analysed in which a thermo-viscoplastic smooth surface (representing a workpiece) is indented by a rough surface (representing a tool). The choice for a rough tool was motivated by the fact that rolled products are often imprinted and thus defined by the topography of the roll [
11]. The workpiece material represents a 6061 aluminium alloy, for which multiple accurate constitutive relations are available. The rough surface of the tool is represented in such a way to consider coalescence of contact and formation of specific geometries of contact patches, which are treated as indenting asperities of different shapes in the FE model. In this sense, the hardness due to an indentation of a single asperity is viewed as a contact pressure derived from an FE simulation for that particular asperity. The database of FE simulations along with a surface representation algorithm allow for the development of a thermo-viscoplastic contact model.
3. Results and Discussion
The FE simulations allowed us to build a database from which the influence of the thermo-viscoplastic behaviour of the deforming material in the load–area relation was investigated. The results of the FE simulations are analysed from a single asperity perspective and from the rough surface perspective, i.e., the toll–workpiece contact. In order to investigate how strain, strain rate, and temperature affect the load–area relation of the contact, the indenting velocity of the asperities and the temperature of the substrate were varied. The effects from changing the material model was also investigated. The analysis is focused on four different aspects of the contact evaluated from the results database: the contact load, the average pressure supported by the substrate, the real contact area, and the displacement of the non-contacting area.
3.1. Asperity
3.1.1. Strain, Strain Rate, and Temperature
A single indenting asperity is analysed under velocities of , , and , combined with temperatures of , , and . The strain and strain rate effects are evidenced by changes in the indenting velocity, since it modifies strain rate fields in the deforming material and, consequently, how the deformation develops. Similarly, temperature effects on the material are reflected by changes in the substrate domain temperature. The asperity with the biggest is chosen since its indentation depth allows for an extensive visualisation of the effects of the variables. This “asperity k” has height and radius .
Figure 6 shows the vertical contact force
using the GA model. The contact load is obtained as the z-component of the total contact force from the FE simulation and is shown as a function of its indentation depth
d normalised by
. Thus,
means the asperity has penetrated a distance equivalent to its height
into the substrate as measured from the original substrate surface level. The results show how the increase in temperature (causing softening of the substrate material) leads to a smaller load carrying capacity for the same indentation depths. Analogously, increasing the indenting velocity results in strain rate hardening of the material, leading to higher contact loads. Interestingly, despite the nonlinear nature of the material model, the load–indentation relation throughout the height of the asperity exhibits a nearly linear behaviour, which is quantified by the slope triangles in the figure displaying the rate at which the load increases per unit indentation depth in each case.
Figure 7 shows the contact area as a function of
for the same cases as
Figure 6. The contact area is calculated by integrating a Boolean equation on the contact pressure from the FE results. The area–indentation relation could also be reasonably modelled in a linear manner, but a slight quadratic behaviour is more evident. In general, all cases present similar values of contact area, but higher temperatures yielded smaller contact area at the end of indentation in all cases. An increase in the indenting velocity also seems to indicate a slight decrease in the contact area. It is worth highlighting the reason behind such a result, which has to do with piling-up of the material around the indentation.
Figure 8 shows the surface profile of the substrate along the distance from the center of the indentation, normalised by
and
, respectively. The profile is shown for
, i.e., at full indentation. Visibly, a higher pile-up occurs at the lowest indenting velocity, which consequently creates a higher probability that the surrounding material will contact the extended edge of the asperity (detailed in
Figure 4), resulting, thus, in more contact area. On the other hand, a softer substrate (higher temperature) results in less pile-up, which lessens the contact area.
Figure 8 also suggests that strain hardening and temperature affect pile-up more than strain rate hardening. A increase in the indenting velocity did not increase the pile-up; in fact, it even decreased for the
between
and
. Meanwhile, in the
and
case, the highest pile-up was observed at a nearly 20% rise relative to
. Interestingly, despite the varied heights and shapes of the piled-up surface, the material returns to the original level at practically the same position in all cases, which is at a distance of about
; this may suggest that such a distance depends only on the geometry of the indenting asperity, although the general pile-up profile clearly depends on the thermo-viscoplastic conditions. It can also be noticed that, for
, the substrate undergoes a small reduction in height in some cases, which is linked to the lack of mechanical constraint for lateral displacement in the FE model.
An evaluation of the average contact pressure (contact load over contact area) at full indentation,
, also exposes the effects of
and
T, as shown in the bar graph of
Figure 9. The graph on the right of the figure shows that, in reality,
decreases its value throughout the indentation (after quickly reaching its highest value at the start of the contact), which means that the contact area increases faster than the contact load. Thus, while the load caused by the indenting asperity continuously increases with depth, the deformation of the substrate material causes the average pressure to decrease. The tendencies shown in the figure were also observed to a lesser extent for other indenting velocities.
3.1.2. Material Model
The previous analyses were also performed using the nJC and HS material models. The results are compared in terms of average pressure in
Figure 10.
As with the GA model, the overall increase in average pressure with decreasing temperature and increasing indenting speed is also observed in the HS model. On the other hand, the nJC model practically does not show differences between
and
. The reason lies in the fact that, at such indenting speeds, the range of strain rates in the substrate are generally below the nJC model’s
threshold for consideration in the flow stress. Hence, from the point of view of the nJC model,
and
are practically the same. Additionally, since the nJC model considers that
for
, there is an overestimation of
for
and
in comparison to the other models. This result reveals how the correct description of the flow stress at small values of strain and strain rate can have significant influence in the development of the strain field and, thus, contact pressure caused by an indenting asperity. A visual inspection of the von Mises stress fields, as shown in
Figure 11, show the complex stress field in such an indentation of the thermo-viscoplastic material. During this process, the highest stresses (and also highest plastic strain and strain rates) always occurred at the most recent contact location, with the field propagating towards the interior of the substrate at lower values.
Another interesting aspect is to evaluate the pile-up/sink-in behaviour in each case through the contact area. From the surface algorithm of
Section 2, the base area of an asperity
i was obtained as
, which was then used to find
. With the volume
, the height
was defined. A circular paraboloid defined by
and
results in a surface area that can be calculated by revolving its parabolic profile and by calculating the surface of revolution. The resulting expression for the surface area
(not including the circular base) is given by the following:
In the FE model, pile-up and/or sink-in of the substrate material surrounding the indentation, which is not taken into account in the surface algorithm, may cause the contact area of asperity
i from the FE simulation,
, to deviate from
(calculated according to Equation (6)). Since the geometry of the asperity in the FE model is built with an extended edge (see
Figure 4), pile-up of the substrate material is likely to contact the asperity at that region, which is identified by comparing
to
.
Figure 12 compares the relative difference of the contact area in the indentation of asperity
k using the nJC, HS, and GA models.
The GA model results in very small differences, whereas the nJC and HS models result in contact areas clearly larger than that calculated from the surface algorithm, which means a general piling-up of the substrate occurs and contacts the asperity at the extended edge region lying above the original level of the substrate surface. As with the average pressures, the nJC model practically shows the same results for
and
, and values greater than the HS and GA models. The reasons can be again attributed to the formulation of the nJC model, as previously discussed for the average pressure. The similarity betwen the results for the HS and GA models in the average pressure does not repeat in
Figure 12, where evident differences are visible; this shows how slightly different mathematical descriptions of the material model may result in significantly different deformation patterns. It was also verified that the size of the FE mesh had a significant effect on the contact area at the near-edge region of the contact, since the deformation is inherently linked to the size of the FE elements; a mesh convergence was performed to ensure that the results are minimally affected by the size of the mesh.
Figure 13 details the complex material flow creating the differences in
Figure 12. The plastic strain fields show that the maximum equivalent plastic strain
for the GA model was the smallest among the three models but that the strain field is more spread out, as evidenced by the contour of the plastic regions. In the nJC case, the plastic strains are more concentrated, which is likely because of the lack of strain rate hardening for
, causing the substrate to initially accumulate the plastic strain before the stress can propagate throughout the material. An evaluation of the solution fields throughout the indentation revealed that strain rate ranges from 0 to
for the GA model,
for the HS model, and
for the nJC model. It is important to recall that the material models result in noticeably different predictions at small strain and strain rate values, as detailed in
Figure 14. Clearly, the pronounced strain hardening at the beginning of the flow curves occurs at a relative wide range of strains (up to
) for the GA model, whereas the HS and nJC models reach a nearly constant value in a much smaller range. The strain and stress fields in the initial moments of a deformation process greatly define the subsequent deformation of the body. In this sense, different predictions at small strains and strain rates are believed to be the main cause of different FE results in
Figure 13. Evidently, such results are a consequence not only of the material model but also of the asperity geometry and type of mechanical constraint of the substrate.
3.2. Surface
In this section, we investigate the total load–area relation of the selected separation in
Section 2.2, i.e., considering the
asperities of case
(
Figure 3). For this goal, the database of the GA model is used. As discussed previously,
may deviate from
consequently causing the total contact area from the FE database to deviate from that calculated by the surface algorithm. In
Figure 15, the database contact area of each asperity,
, is compared to that given by Equation (6),
in terms of a relative difference plotted against the aspect ratio
.
The figure shows that is generally smaller than that predicted by Equation (6). The rightmost marker in the figure refers to asperity k, which evidently is not a representative behaviour of the asperities of the surface; asperities were more likely to cause sink-in of the material surrounding the indentation. As the substrate temperature decreases and the material becomes harder, the sink-in make room for pile-up, subsequently reducing the difference but likely leading to a positive relative difference with further decrease in temperature. The distribution in the figure reveals an approximate quadratic tendency in the relative difference of the contact area with respect to the aspect ratio , meaning that asperities of higher aspect ratios were less likely to cause sink-in. In a rather general way, asperities with bigger contact area (displayed by the marker size) were also less likely to sink-in in comparison to asperities of smaller areas.
With regards to contact load, a nearly quadratic relation was found between contact load and radii, as shown in
Figure 16. As expected, bigger asperities carry higher contact loads.
In terms of average pressure, the distribution also as a function of the radii is shown in
Figure 17. Visibly, the values of average pressure, or hardness, show a decreasing tendency with increasing radius and area, with the effects being more pronounced with colder and, thus, harder substrate. The results suggest that, in general, smaller asperities carry more pressure than larger ones in such a thermo-viscoplastic material.
Table 3 sums up the overall results in terms of total contact load and area, which can be said to be the results of the thermo-viscoplastic contact model. The superscript
is omitted from the total variables for improved readability. The total normal load
L carried by contacting asperities between toll and workpiece, and the total contact area
(or real contact area) can be calculated as the sum of the contribution of all asperities, i.e.,
and
.
Table 3 also shows the total degree of contact (
) for different studied cases.
It is important to highlight that the values of
Table 3 assume that a superposition of the load and area of each asperity is valid for calculation of the total load
L and total contact area
. The deterministic contact patch approach used (
Section 2) indeed attempts to account for the interaction between surface heights, which was done by identifying connected surface heights and, thus, coalescence. Nevertheless, two separated contact patches may still be “close-enough” to each other in such a way that their stress fields may affect one another, consequently affecting the load they support and contact area. Such effects, explored for example by [
52], were not investigated in this work.
Finally, with the results of
Table 3, a parallel to the fully plastic model can be drawn. If the contact model was developed using a fully plastic approach, i.e., a model such that
, where
is the nominal pressure between toll and workpiece, one may write
to calculate
, which would be an equivalent hardness value to obtain the same results in terms of contact area and contact load of
Table 3. It is interesting to verify how the values of
compare to the constraint factor times the flow stress of the material, since this is often used to express hardness of a material. For such purpose,
Figure 18 shows
with
for different values of strain and strain rate at the temperatures investigated. Dashed contour lines represent values equal to
according to
Table 3.
Figure 18 reveals that, in order to obtain the same results as the thermo-viscoplastic contact model performed in this work, in terms of contact area and carried load, a fully plastic contact model in the format
should consider flow stress
evaluated at certain nonzero values of strain and strain rate, denoted by the contour line in
Figure 18. Although the values of
and
are relatively small, the flow stress gradient at small strains and strain rates is evidently very pronounced for such materials, which causes the material to harden significantly fast. Consequently, the advantages of correctly describing the material behaviour at low strains and strain rates is evidenced again. Another remark to be made is that the displayed contour line is valid for a constraint factor of
and for the separation studied in this section. If these conditions are changed, the value of
is also expected to change. Cases
and
show that the decrease in temperature (and thus hardening of the material) causes the contour line to move towards higher
and
, which is also expected to occur similarly for higher indenting velocities. When the material shows less pronounced variations at low strains and strain rates, such as in the
case
, the contour line may also move towards higher
and
.
The approach using a FE database has the advantage of automatically resulting in an equivalent hardness through quantification of load and area while accurately and simultaneously considering the shape and sizes of the contacting asperities. The contact load quantification may be directly related to the external load through equilibrium conditions, while the contact area in combination with custom-shaped asperities allow for the development of physically based investigations on friction.