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**.

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**.

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**.

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. |

**publication**.

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. |

**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.

**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.

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

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. |

**publication1**and

**publication2**.