2. Contact 17
2.2. Load and Deformation
As equation (2.4) states, the pressure at the center of the apparent area of contact is 1.5-times the average pressure. Since the tribometer brings a ceramics ball into contact with a flat steel specimen (see figure 1.7 on page 15), in this thesis the contact of a sphere and a flat is considered. Focusing on the maximum pressure, the above equations transform for a sphere in contact with a flat (one radius is infinite) into the following:
pmax = 1 π
3 6FN r2 ·
1−ν12
E1 +1−ν22 E2
−2
(2.5) FN = π3
6 r2
1−ν12
E1 +1−ν22 E2
2
p3max (2.6)
The question now is, whether the contact of a spot within the apparent area of contact is direct or not. Even without direct contact, there is interaction of the bodies, but this is not necessarily the most relevant interaction regarding the resulting friction. Where there is direct contact, the resulting friction is higher: The interlocking of asperities comprehensibly gives rise to friction, so the direct contact indeed is of interest. Nonethe-less, the Hertz theory gives a good representation of the average pressure in the area of contact. Additionally Mo and Szlufarska state in their Physical Review B publication [12] that elaborates more detailed on their work about the friction laws at the nanoscale published in a 2009 Nature article [13] that received great attention (121 citations at the 5th of November 2012 according to ISI web of knowledge): „Friction and contact area directly measured by surface force apparatus (SFA) frequently follow predictions of the Hertz model.“
One single junction will be the first direct contact formed upon bringing two bodies closely together. Consequently, this one single spot will bear all the load, and the resulting stress will be immensely high, because the area of a single junction is minuscule. Resultantly the consideration of the stress and the elastic or plastic deformation of asperities is significant in all contacts, which is the topic of the next section.
2.2. Load and Deformation
The distinguishing between the apparent area of contact, the bottom surface area of the smaller body lying on the larger, and the real area of contact, the area where interactions take place, and especially the fact that there is a huge difference in magnitude dates back to Bowden and Tabor in their monographs on “The Friction and Lubrication of Solids”
published in the 1950s [14, 15]. They considered the interactions of bodies as completely
plastic. On the other hand, interactions can also be elastic, but only if the force is small compared to the area.
Any real material has a finite elasticity. The assumption made by Hertz, that all the contact interactions are elastic, is limited to very small contact stresses for very smooth surfaces. In any real case, the surface roughness as well as “regular” macroscopic loads will make this assumption break down. Using the symbol Areal for the real area of contact, instead of the distribution of the normal force FN over the apparent area of contactA0, the normal force will be distributed only over the real area of contact. The resulting stressσ on the surfaces will accordingly be:
σ= FN
Areal (2.7)
The threshold value between the regime of lower stress, resulting in elastic deformation of the body, and the regime of high stress, resulting in plastic deformation, is the so-called yield stressσc, which is the same as the penetration hardness of a material. It is called yield stress, because if the stress on the body exceeds the yield stress, the body yields plastically to the interaction. In the theory of von Mises [16], which agrees well with experiments [2], the yield stress is related to the von Mises yield criterions0 via:
s0 = 8σ2c
3 (2.8)
Resulting from the definition, the energetic interpretation of equation (2.8) is, that if the elastic energy per unit volume that stems from shear deformations overcomes the threshold, the body flows plastically.
Now, considering the contradicting theories of Hertz on the one hand and Bowden and Tabor on the other hand, observations and logics contradict both theories. If there would be only elastic deformations, as assumed by Hertz, the friction force would not depend linearly on the normal force, because that is also predicted by Hertz theory.
However, macroscopic experiments find such a linear relationship between friction and normal force. If there would be onlyplastic deformations, then the fatigue, the gradual degradation of material integrity, would lead to machine failure very fast. For example metals exhibit the so-called “work-hardening”, which changes the material structure in highly stressed areas to a more resilient but less elastic one, leading to fatigue cracks and finally failure. Engines would not surpass a few minutes of running if this would be the case. The reconciliation of both theories was published by Greenwood and Williamson [17] in 1966. The Greenwood-Williamson contact theory assumes a gaussian distribution of heights on the average surface level, which was corroborated by experiments. The resulting equations stated an independence of the friction of the apparent area of contact,
Chapter 2. Contact 2.2. Load and Deformation
while the friction force was to depend almost linearly on the normal force. Following that, Johnson, Kendall and Roberts [18] developed a model of an elastic contact, which focuses on the compensation of the loss of surface energy by deformation and the increase in stored elastic energy. Utilizing both results, Fuller and Tabor [19] connected the energy change as well as the gaussian height distribution to the result, that the higher asperities try to separate the surfaces, while the lower asperities try to maintain contact. Further augmenting the previous results on spheres in contact with flats, Volmer and Nattermann in 1997 [20] stated, that viewing the surfaces of a cuboid on a surface as self-affine on small length scales gives a COF, which depends on the ratio between elastic and repulsive forces. The resulting equations based upon the elasticity of the majority of asperities in contact directly result in the previously stated friction laws (cf. section 1.5).
Summarizing the theories, upon contact, at first so few junctions occur, that there is plastic deformation of these junctions, which results in further junctions to be formed.
The resulting increase in real area of contact reduces the average contact stress per junction, up to the point where only elastic deformations occur. At this point, the junctions support the load, the junction interactions are the load-bearing interactions.
The vast majority of the interactions will be elastically, and can be modeled by the theory of elasticity, if a gaussian height distribution as surface roughnesses is assumed, to arrive at the macroscopically observed friction laws.
Finally, the material characteristics determine the behavior of the real area of contact with increasing stationary time. Considering the ideal elastoplastic behavior of a mate-rial, the real area of contact should not change over time at all. However, real materials exhibit creep [2], the slow rearrangement of atoms, resulting in the reduction of elastic energy in favor of rearranging atoms, thus creating plastic deformation over time. The relevance of this behavior depends on the time scale of the contact period.
Considering a rectangular block of material under the constant uniaxial tension in x-direction, the development of the tension over time depends on the length of the block l in the following way
˙ = l˙
l(t) (2.9)
There is an initial response of the system to the applied tension, which takes the relax-ation time τ. If the contact time is less or equal to τ, there will be no creep. Once the contact time exceeds the relaxation time, the stress distribution has changed accordingly and depends on the von Mises yield criterion. Consequently the length of the block increases over time:
l(t) =l0et˙ (2.10)
Since the volume of the block is constant, the cross-sectional area decreases over time, thereby reducing the acting force. The strain rate tensor˙ij for uniaxial tension has the form:
˙ ij =
⎛
⎜⎝
˙
0 0
0 −˙/2 0 0 0 −˙/2
⎞
⎟⎠ (2.11)
The resulting stress is uniaxial in x-direction. The creep of a material needs a temperature activation, which allows us to write the uniaxial stress in terms of the activation energy EA, the temperature T and the Boltzmann constant kB:
σ =σc
1 +kBT EA ln
˙
˙ 0
(2.12)
where the constant ˙0 is defined as follows, using the Poisson number at zero stress ν0 and the elastic modulus E:
˙
0 = 8ν0(1 +ν)σckBT
3E EA (2.13)
If we set the stress to zero, meaning complete relaxation of the stress, we arrive at:
0 = 1 + kBT EA ln
˙
˙ 0
(2.14)
where ˙ is the creep rate. Persson [2] states, that for a typical material the activation energy is of the order of 5 eV as determined experimentally and thus the diameter of a stress block is about 3 nm, or an area of about 7 nm2.
To obtain the time dependence of the real area of contact, we start from the variation of the block length l˙and the requirement, that the block volume is constant. In this case we arrive at:
˙ =−A˙
A (2.15)
The acting stress is defined as:
σ=−F
A (2.16)
and if the area is the initial area A0 the yield stress is obtained. Using the above in equation (2.12) deduced result, the yield stress can be expressed as:
σ
σc =−1−kBT EA ln
˙ ˙0
(2.17)