Contact Mechanics is of huge importance for numerous applications in Nature and technology. We at Multiscale Consulting have worked out a unique and powerful approach to adress contact mechanics and understand what happens when two solids come into contact with each other. The theory has now been tested numerically and experimentally usually resulting in good agreement and thus showing that the basic theory predicts properties correctly.

In the past we have worked out some applications for our multiscale contact mechanics approach. In thes section we give a short introduction to the applications we have studied so far. In case you are interested in our software this and our validation section give you strong arguments for why it is useful to get our software.



A seal is a device for closing a gap or making a joint fluid tight. Seals play a crucial role in many modern engineering devices, and the failure of seals may result in catastrophic events, such as the Challenger disaster. In spite of its apparent simplicity, it is not easy to predict the leak rate and (for dynamic seals) the friction forces. The main problem is the influence of surface roughness on the contact mechanics at the seal-substrate interface. Most surfaces of engineering interest have surface roughness on a wide range of length scales, e.g., from centimetre to nanometre, which will influence the leak rate and friction of seals, and accounting for the whole range of surface roughness is impossible using standard numerical methods, such as the Finite Element Method.
The leakage of hydraulic systems is both an environmental and an economical problem. Despite of its huge importance, the mechanisms were not fully understood up to now. We at Multiscale Consulting have developed a physical model based on our contact mechanics approach to learn more about the leakage of fluids through the contacting interface of a seal.


Rubber seal (schematic). The liquid on the left-hand side is under the hydrostatic pressure Pa and the liquid to the right under the pressure Pb (usually, Pb is the atmospheric pressure). The pressure difference delta P = Pa - Pb results in liquid flow at the interface between the rubber seal and the rough substrate surface. The volume of liquid flow per unit time is denoted bẏ dQ/dt, and depends on the squeezing pressure P0 acting on the rubber seal.


Consider the fluid leakage through a rubber seal, from a high fluid pressure Pa region, to a low fluid pressure Pb region, as in the figure above. We study the contact region between the rubber seal and the counter surface as we change the magnification zeta. Here zeta is defined as length of the contact region divided by the resolution lamda (zeta  =L / lamda). We study how the apparent contact area (projected on the xy-plane), A(zeta), between the two solids depends on the magnification zeta. At the lowest magnification we cannot observe any surface roughness, and the contact between the solids appears to be complete i.e., A(1) = A0. As we increase the magnification we will observe some interfacial roughness, and the (apparent) contact area will decrease. At high enough magnification, say zeta = zeta_crit, a percolating path of non-contact area will be observed for the first time, see figure below. We denote the most narrow constriction along this percolation path as the critical constriction. The critical constriction will have the lateral size lamda_crit = L / zeta_crit and the surface separation at this point is denoted by u_crit. We can calculate u_crit using a recently developed contact mechanics theory. As we continue to increase the magnification we will find more percolating channels between the surfaces, but these will have more narrow constrictions than the first channel which appears at zeta = zeta_crit, and as a first approximation one may neglect the contribution to the leak rate from these channels.


The contact region at different magnifications zeta = 3, 9, 12 and 648, is shown in (a)–(d), respectively. When the magnification increases from 9 to 12 the non-contact region percolates. At the lowest magnification zeta  = 1 : A(1) = A0. The figure is the result of Molecular Dynamics simulations of the contact between elastic solids with randomly rough surfaces


A first rough estimate of the leak rate is obtained by assuming that all the leakage occurs through the critical percolation channel, and that the whole pressure drop delta P = Pa - Pb (where Pa and Pb is the pressure to the left and right of the seal) occurs over the critical constriction (of width and length lamda_crit of order L / zeta_crit and height u_crit). We will refer to this theory as the “critical-junction” theory. If we approximate the critical constriction as a pore with rectangular cross-section (width and length lamda_crit and height crit << lamda_crit) one can calculate the volume flow per unit time through the critical constriction assuming an incompressible Newtonia fluid and Poiseuille flow.

For calculating the fluid flow rate, we must calculate the separation u_crit of the surfaces at the critical constriction. We first determine the critical magnification zeta_crit by assuming that the apparent relative contact area at this point is given by percolation theory. Thus, the relative contact area  A(zeta)/A0 of order 1- p_crit, where p_crit is the so-called percolation threshold. The (apparent) relative contact area A(zeta)/A0 at the magnification zeta can be obtained using the contact mechanics formalism we have developed earlier, where the system is studied at different magnifications zeta. With the help of our contact mechanics theory we can also obtain the separation between the two solids in the critical constriction. Hence we have all information necessary to calculate the leak-rate of seals. More information can be found in the related publication.



Surface roughness is the main reason why macroscopic solids usually do not adhere to each other with any measurable strength; even a root-mean-square roughness amplitude of approx. 1 micrometre is enough to completely remove the adhesion between normal rubber (with an elastic modulus E of order 1 MPa) and a hard nominally flat substrate. Strong adhesion between solids with rough surfaces is only possible if at least one of the solids is elastically very soft. Biological adhesive systems used by insects, tree frogs and some lizards for locomotion are built from a relatively stiff material (keratin-like protein with E of order 1 GPa). Nevertheless, strong adhesion is possible even to very rough substrate surfaces by using non-compact solid structures consisting of thin fibers, plates and walls as show in the figures below. In order to optimize the bonding to rough surfaces while simultaneously avoiding elastic instabilities, e.g., lateral bundling (or clumping) of fibers, Nature uses a hierarchical building principle, where the thickness of the fibers (or walls) decreases as one approaches the outer surface of the attachment pad. Some lizards and spiders are able to utilize dry adhesion to move on rough vertical surfaces, which is possible due to the very compliant surface layers on their attachment pads. Flies, bugs, grasshoppers and tree frogs have less compliant pad surface layers, and in these cases adhesion to rough surfaces is only possible because the animals inject a wetting liquid in the pad-substrate contact area, which generates a relative long-range attractive interaction due to the formation of capillary bridges.
The hierarchical nature of the lizard adhesive system is compliant on all relevant length scales, and deforms elastically to optimize the contact area and the bonding to the rough substrate.
Fiber arrays of a beetle attachment pad. The corrugations of the fiber surfaces prevent bundling Beetle terminal plate adhering to a (relatively) smooth substrate. Note the sharp structures on the upper side of the plate, which inhibit plates bonding to each other. The scale bar corresponds to 1 micrometre.


More information can be found in these two related publication1 and publication2.


The heat transfer between solids is a topic of great importance. Classical applications include topics such as cooling of microelectronic devices, spacecraft structures, satellite bolted joints, nuclear engineering, ball bearings, tires and heat exchangers. Other potential applications involve microelectromechanical systems (MEMS). Heat transfer is also of crucial importance in friction and wear processes, e.g., rubber friction on hard and rough substrates depends crucially on the temperature increase in the rubber-counter surface asperity contact regions. A large number of papers have been published on the heat transfer between randomly rough surfaces. However, most of these studies are based on asperity contact models such as the model of Greenwood and Williamson (GW). Recent studies have shown that the GW-model (and other asperity contact models) are very inaccurate, mainly because of the neglect of the long-range elastic coupling.
When two solids touch each other and the temperature of these two objects is not the same, there will be a heat current through the contacting interface in order to equilibrate the temperature. The heat transfer through the interface can be calculated using the contact mechanics approach we have developed at Multiscale Consulting.
In this applications we consider the heat transfer between (macroscopic-sized) solids in the light of recent advances in contact mechanics. We study the contribution to the heat transfer not just from the area of real contact (observed at atomic resolution), but also the heat transfer across the area of non-contact, in particular the contribution from the fluctuating electromagnetic field, which surrounds all solid objects. For high-resistivity materials and for hard and very flat surfaces, such as those involved in many modern applications, e.g., MEMS applications, this non-contact radiative heat transfer may in fact dominate in the total heat transfer (at least under vacuum condition). We note that for flat surfaces (in vacuum) separated by a distance d larger than the thermal length, the non-contact heat transfer is given by the classical Stefan-Boltzman law, and is independent of d. However, for very short distances the contribution from the evanescent electromagnetic waves to the heat transfer will be many orders of magnitude larger than the contribution from propagating electromagnetic waves (as given by the Stefan-Boltzman law). In most applications (but not in spacecraft applications) one is interested in the heat transfer between solid objects located in the normal atmosphere and sometimes in a fluid. Most solid objects in the normal atmosphere have organic and water contamination layers, which may influence the heat transfer for at least two reasons: (a) Thin (nanometer) contamination layers may occur at the interface in the asperity contact regions, which will affect the acoustic impedance of the contact junctions, and hence the propagation of phonons between the solids (which usually is the origin of the heat transfer, at least for most non-metallic systems). (b) In addition, capillary bridges may form in the asperity contact regions and effectively increase the size of the contact regions and increase the heat transfer. In the normal atmosphere heat can also be transferred between the non-contact regions via heat diffusion or (at short separation) ballistic processes in the surrounding gas. For larger separations convective processes may also be important.
Our approach is based on studying the heat flow and contact mechanics in wave vector space rather than real space which has the advantage that we do not need to consider the very complex fractal-like shape of the contact regions in real space. We have included both the heat flow in the area of real contact as well as the heat flow across the non-contact surface region. For the latter contribution we have included the heat transfer both from the fluctuating electromagnetic field (which surrounds all material objects), and the heat flow via the surrounding gas or liquid. We have also studied the contribution to the heat transfer from capillary bridges, which form spontaneously in a humid atmosphere (e.g., as a result of organic and water contamination films which occur on most solid surfaces in the normal atmosphere). We have presented an illustrative application relevant for MEMS applications involving very smooth amorphous silicon dioxide surfaces. In this case we find that all the mentioned heat transfer processes may be roughly of equal importance. More information can be found in the related publication.



The influence of surface roughness on fluid flow at the interface between solids in stationary or sliding contact is a topic of great importance both in Nature and technology. Technological applications include leakage of seals, mixed lubrication and removal of water from the tire–road footprint (see figures below). In Nature fluid removal (squeeze-out) is important for adhesion and grip between the adhesive toe pads of a tree frog or a gecko and the countersurface during rain, as well as for cell adhesion.
A tire rolling on a wet surface. In order to make contact the tire has to squeeze away the water first. This needs to be done on different length scales also. It might be, depending on the fluid viscosity, the normal pressure and the time scales involved that not all of the fluid gets squeeze-out of the contact region but that a small layer of fluid stays in the contact, separating the two solids. Water can also remain in sealed off regions within the contact, hence changing the frictional properties. Fluid flowing at the interface through a network of narrow channels. Here dark color is contact region between two solids while the colored region is where the fluid flows. The more narrow the channels get, the higher the volume per time as indicated by the color.
In the context of fluid flow at the interface between closely spaced solids with surface roughness, Patir and Cheng have shown how the Navier–Stokes equations of fluid dynamics can be reduced to effective equations of motion involving locally averaged fluid pressures and flow velocities. In the effective equations occur so-called flow factors, which are functions of the locally averaged interfacial surface separation. They showed how the flow factors can be determined by solving numerically the fluid flow in small rectangular units with linear sizes of the order of (or larger than) the roll-off wavelength lamda_0. We at Multiscale Consulting have developed an analytical theory for the pressure flow factors based on the Persson contact mechanics model and the Bruggeman effective medium theory to take into account the topography disorder resulting from the random roughness.
This analytical theory based on the contact mechanics theory of Persson and the Bruggeman effective medium theory uses fluid flow factor to calculate the removal of the fluid from the contacting interface of two solids. When studying the fluid flow between two macroscopic solids, one may replace the microscopic equations of fluid dynamics with effective equations describing the average fluid flow on length scales much larger than lamda_0 and which can be used to study, for example, the lubrication of the cylinder in an engine. This approach of eliminating or integrating out short length scale degrees of freedom to obtain effective equations of motion, describing the long distance (or slow) behavior, is a very general and powerful concept often used in physics.
This approach on decribing fluid flow at interfaces squeezed into contact can also be applied to study macroscopic, elastically soft solids in sliding contact with randomly rough surfaces in the presence of a fluid. Here we can describe the transition from boundary lubrication, prevailing at low sliding velocity where direct solid-solid contact occurs, to hydrodynamic lubrication at high sliding velocity where the solids are separated by a thin fluid film. During this transition the solid also experiences the mixed lubrication regime where the contact is partially still dry (the fluid can still be squeeze-out so that dry contact appears) and partially already separated by a thin fluid layer. Here we have developed cutting-edge numerical models coupled to our analytical approaches. Describing the transition we can also take into account viscoelasticity, plasticity or cavitation.

More information can be found in these two related publication1 and publication2.


The tribology of the human skin is a fascinating but very complex topic. This is partly due to the layered nature of skin, to its complex viscoelastic nature, and to its great sensitivity to the humidity. Thus, it is known that the effective elastic modulus of the skin top-layer (the of order 20 micrometre thick stratum corneum) may change by a factor of 100-1000 as the humidity increases from zero to 100% relative humidity.
Due to its importance for living objects, it is likely that the skin has been highly optimized during million of years of evolution. This probably also reflect its tribology properties and it is likely that many of its mechanical and structural properties reflect this fact. Thus, the pattern of channels on the skin surface on the hands may facility the squeeze-out of fluids (similar to the tread pattern on tires) and may increase the friction and grip between a fluid-contaminated object and the human hand. Similarly, the strong reduction in the elastic modulus of the stratum corneum in humid condition results in a large increase in the contact area and the friction during sweating, which again may increase the friction or grip between the hand and an object, which may be particularly important in emergency situations.
Skin friction is of great importance in many things we are doing. Companies selling sport articles, e.g., tennis rackets or golf clubs, want to increase the grip for better handling. In Syringes friction is a comfort issue as the user should not need to apply too high forces in order to inject the fluid. Also should the frictional properties be stable over the product lifetime. Another topic here is leackage of the fluid through the rubber stopper.
Another topic we have worked on recently is the breakloose friction and the leakage of syringes. Here we have applied our theories to predict the leakage of the fluid through the glas or PMMA barrel and the rubber stopper. We have also calculated the size of the non-contact channels at the sealing interface to study if bacteria or viruses can penetrate from the atmosphere into the syringe container thus contaminating the drug. During actuation of the syringe the user has to first overcome what is called the breaklosse friction, a barrier when going from static to dynamic configuration where the rubber stopper is sliding relative to the glas or PMMA barrel hence pushing the drug through the needel. This is a topic of great importance as the syringe should still be easy and comfortable to use even at the end of the shelf live.
More information can be found in these two related publication1 and publication2.
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