Patent Description:
Wear experiments at tire-level are complex and expensive. Several methods are described in the state of art for the prediction of wear performance. In particular, simulations are often used on this purpose, but proper inputs related to the actual contact conditions and the abrasion characteristics of materials have to be properly assessed in order to properly compute the energy leading to wear and achieve reliable results from the simulations.

In particular, a fundamental input required by Wear Simulations is the "Critical Slip Distance" (CSD), i.e., the distance starting from which the tread blocks show a constant wear rate, which cannot be simply and intuitively provided by standard friction and abrasion tests.

Currently, friction and abrasion characterization of the material are carried out by means of sliding tests on small rubber wheels, which does not provide any information about the actual contact conditions, or by using a linear friction tester with rubber blocks, which are tested in full sliding mode. Anyway, these kinds of tests are not able to provide any information about the CSD.

For this purpose, a new methodology has been specifically designed to determine the CSD by means of linear friction tests with rubber samples in different testing conditions.

<NPL> discloses tribological behavior of StyreneButadiene Rubber (SBR) compound reinforced by silica and carbon-black fillers on two different asphalt-like surfaces using theoretical and experimental approaches. Linear sliding friction and wear tests were performed on two different roofing asphalt papers. The surface roughness power spectra of asphalts and the master curves of the large-strain viscoelastic modulus of SBR compound were used to model the contact area and friction coefficient theoretically.

<NPL> discloses a numerical technique for predicting the wear amount of automobile tire. A power function wear model derived based on the laboratory testing of rubber abrasion is adopted to correlate the wear rate of tread rubber blocks with the frictional energy dissipation, and the driving and loading conditions of the tire in the outdoor wear test are extracted from the virtual driving simulation using ADAMS. The tire frictional energy rates produced in each driving mode are computed by the frictional dynamic rolling analysis of 3D patterned tire model.

According to a first aspect of the present invention, a method for determining a critical slip distance (CSD) for a sample is provided. The method comprises providing a force to the sample towards a sliding surface. The method further comprises sliding the sample n times along the sliding surface over a plurality of intervals, the plurality of intervals having different widths w, n being an integer ≥<NUM>, determining, for each width wi, a respective wear rate m(wi)=ΔM/(n * wi) of a plurality of wear rates m(w), wherein ΔM is a weight difference of the sample after sliding the sample over n intervals of width wi, and determining, based on the plurality of wear rates and the respective widths, the CSD as a width wo, wherein, for any w > wo, <MAT>.

According to an example of the first aspect, the widths wi of the plurality of intervals are integer dividers of a total length of the sliding surface.

According to another example of the first aspect, the method may further comprise sliding the sample over multiple intervals of each width wi, obtaining the weight difference ΔM by weighing the sample before and after sliding over n intervals of width wi, and determining the wear rate based on the weight difference ΔM and a total slidden distance.

According to another example of the first aspect, the wear rate m is a weight loss per total slidden distance.

According to another example of the first aspect, the total slidden distance is equal for each one of the widths wi.

According to another example of the first aspect, the force is lifted after sliding the sample along each one of the intervals and the force is reapplied before sliding the sample along the subsequent interval.

According to another example of the first aspect, the sample is slidden along the sliding surface at a substantially constant speed.

According to another example of the first aspect, the method further comprises identifying a first distance, along which a static friction force exceeds a sliding friction force, identifying a second distance, along which the sliding friction force exceeds the static friction force and determining the CSD further based on the first distance.

According to another example of the first aspect, the method further comprises providing a sliding surface having a total length, sliding the sample over the sliding surface multiple times in steps of each of the plurality of widths w and measuring the plurality of wear rates for each one of the plurality of widths w.

According to another example of the first aspect, the CSD is measured based on any one of a geometry of the sample, a material property of the sample, a texture of the sliding surface, an amount of the force, an amount of the sliding speed, an amount of the acceleration to reach the set sliding speed, and a temperature of the sample.

According to a second aspect of the present invention, a computer-implemented method for estimating the wear of a tire is provided. The method comprises providing a wear model of the tire configured to convert a frictional energy rate into a wear, wherein the wear model is based on moveable and non-moveable parts, with the moveable parts providing higher wear than the non-moveable parts, dividing a contact patch of the tire into a moveable part and a non-moveable part, wherein the non-moveable part exhibits substantially no movement relative to a substrate of the tire, and wherein the moveable part exhibits movement relative to a substrate of the tire; estimating the wear of the tire based on the wear model and the divided contact patch.

According to an example of the second aspect, the non-moveable part provides substantially no wear.

According to another example of the second aspect, the non-moveable part has a first width, and a critical slip distance (CSD) is proportional to the first width.

According to another example of the second aspect, the CSD is determined experimentally prior to performing the computer-implemented method, and the CSD is used as an input variable for the computer-implemented method.

According to another example of the second aspect, the CSD is determined according to a method of the first aspect.

According to another example of the second aspect, the computer-implemented method is based on a finite element method, FEM.

The present invention provides a method for determining a CSD for a sample and a computer-implemented method for estimating the wear of a tire.

<FIG> illustrates a tire in the contact region, as segmented for a wear simulation based on a Finite Element Method (FEM), as well as the division of the contact patch of the tire in a sticking and sliding portion. In this embodiment, the tire is used as the sample, and the method is explained in the context for estimating the wear of a tire. It should, however, be understood, that the functionality described in this context may also be used for other samples, such as predicting the wear of shoe soles, rubber seals, especially rubber seals which are part of a rotary system, or any other rubber compound.

The tire <NUM> is segmented into multiple finite elements (= segments), for example segments <NUM> and <NUM>. At contact patch <NUM>, the tire <NUM> is in contact with the road surface <NUM>. As the tire <NUM> travels in the indicated travelling direction <NUM>, it begins rolling on the road surface <NUM>, such that after a time interval, the segment <NUM> will encounter the road surface <NUM> and will thus become part of the contact patch <NUM>. Similarly, segment <NUM> which at the beginning is in contact with the road surface <NUM>, will be lifted from the road and will thus move out of the contact patch <NUM>. Consequently, during rolling of the tire <NUM>, a tire segment will first encounter the road surface <NUM>, will then move through the contact patch <NUM> and will in the end be lifted from the road surface <NUM> and leave contact patch <NUM>.

When encountering the road surface <NUM>, i.e., in the frontmost portion of the contact patch <NUM>, a segment does not undergo substantial movement with respect to the road surface <NUM>. Only as the segment approaches the rearmost end of the contact patch <NUM>, a segment will start undergoing some movement with respect to the road surface <NUM>, i.e., the segment will start to slide over the road surface.

Thus, the contact patch can be divided into a sticking region <NUM> in which segments do not undergo movement with respect to the road surface, and a sliding region <NUM>, in which segments do undergo movement with respect to the road surface.

Generally, when the tire is not in movement, there is no movement with respect to the road surface, and therefore the complete contact patch consists of one sticking region <NUM> and no sliding region <NUM>. As the tire starts moving, a sliding region <NUM> begins to form at the contact patch and the sticking region <NUM> thus becomes smaller. Consequently, with increasing rolling speed, the sliding region <NUM> becomes larger and the sticking region <NUM> becomes smaller. The width of the sticking region <NUM> may then be characterized as the Critical Slip Distance (CSD), i.e., the distance starting from which the adhesive part of the sliding phenomenon becomes irrelevant to the computation of cumulative wear energy per unit of distance, and so after that threshold the sliding phenomenon shows a constant trend vs. sliding distance.

The CSD can be used as a new fundamental input for a phenomenological model which serves as a basis for tire wear simulations to increase the reliability level of the prediction provided by the simulations.

<FIG> illustrates a sliding surface <NUM> amongst which a sample is slidden in small intervals, in order to determine the CSD according to the present invention.

<FIG> depicts a sliding surface <NUM>. A sample <NUM> is provided on the sliding surface <NUM>, e.g., in a starting position <NUM>. In some aspects, the sample <NUM> may be a rubber sample. A force Fz <NUM> towards the sliding surface is provided to the sample. For example, the force can be provided by applying a load to the sample, such that a gravitational force Fz presses the sample <NUM> to the sliding surface <NUM>. In other examples, the force may be applied hydraulically, pneumatically or mechanically by means of a motor, for example an electric motor. In yet other examples, the force may be generated by the sample's own weight, such that not extrinsic force needs to be applied.

Now, a horizontal force Fx <NUM> may be provided to the sample, such that the sample <NUM> is moved along the sliding surface <NUM> over a first interval of small width w<NUM>, until it reaches the second position <NUM>. In the example depicted herein, the small width w<NUM> is <NUM>. However, other widths can be used. As illustrated in <FIG>, the lower portion <NUM> of the sample <NUM> undergoes deformation due to the horizontal force <NUM> as well as the friction of the sample <NUM> on the sliding surface <NUM>, while at the same time the upper portion <NUM> of sample <NUM> remains substantially undeformed. The deformation increases as long as the static friction force exceeds the sliding friction force. At the point of maximum deformation, the sliding friction force starts exceeding the static friction force, such that the sample <NUM> starts sliding along the sliding surface <NUM>. The sample <NUM> is then slidden until it reaches the second position <NUM>. Here, the sample may be stopped, the force <NUM> may be lifted to resolve the deformation of the sample and the force <NUM> may then be reapplied before the same procedure may be repeated, and the sample <NUM> may be slidden over a second interval of equal width w<NUM>, until it reaches a third position <NUM>. This procedure may be repeated multiple times until the sample <NUM> reaches an end position <NUM>.

In the above example, the sample may be slidden over the total sliding surface in multiple intervals. In some examples, the sample may only be slidden over a portion of the sliding surface in one or more steps. In some examples, the sample may be slidden over the same portion of the sliding surface multiple times.

<FIG> illustrates a graph <NUM> showing the development of a friction coefficient over sliding time for sliding over small intervals w<NUM> as depicted in <FIG>.

The abscissa of graph <NUM> shows the sliding time from one position to the next over an interval w<NUM>, as depicted in <FIG>. The ordinate of graph <NUM> shows the respective friction coefficient µ for the respective times. The friction coefficient may be defined as the ratio between a horizontal force Fx and a vertical force Fz, such that µ = Fx/Fz. Each line in the graph shows the development of the friction for one of the intervals having width w<NUM>. The lines show increasing friction over time until a break-off-point <NUM>. Beyond the break-off-point <NUM>, the friction coefficient drops slightly and then shows substantially constant behavior. Thus, before the break-off-point, there is an adhesion area <NUM>, at which the adhesion contribution exceeds the friction contribution, and a slippage area <NUM>, at which the friction contribution exceeds the adhesion contribution.

<FIG> illustrates a sliding surface <NUM> amongst which a sample is slidden in large intervals, in order to determine the CSD according to the present invention.

<FIG> depicts sliding surface <NUM>. A sample <NUM> is provided on the sliding surface <NUM>, e.g., in a starting position <NUM>. In some aspects, the sample <NUM> may be a rubber sample. A force Fz <NUM> towards the sliding surface <NUM> is provided to the sample <NUM>. For example, the force <NUM> can be provided by applying a load to the sample <NUM>, such that a gravitational force Fz presses the sample <NUM> to the sliding surface <NUM>.

Now, again, a horizontal force Fx <NUM> may be provided to the sample, such that the sample <NUM> is slidden along the sliding surface <NUM> over a first interval of larger width w<NUM>, until it reaches the second position <NUM>. In the example depicted herein, the larger width w<NUM> is <NUM>. However, other widths can also be used. As illustrated in <FIG>, the lower portion <NUM> of the sample <NUM> undergoes deformation due to the horizontal force <NUM> as well as the friction of the sample <NUM> on the sliding surface <NUM>, while at the same time the upper portion <NUM> of sample <NUM> remains substantially undeformed. The deformation increases as long as the static friction force exceeds the sliding friction force. At the point of maximum deformation, the sliding friction force starts exceeding the static friction force, such that the sample <NUM> starts sliding. The sample <NUM> is then slidden until it reaches the second position <NUM>. Here, the sample may be stopped, the force <NUM> may be lifted to resolve the deformation of the sample and the force <NUM> may then then reapplied before the same procedure may be repeated, and the sample <NUM> may be slidden over a second interval of equal width w<NUM>, until it reaches a third position <NUM>. This may be repeated multiple times until the sample <NUM> reaches the end position <NUM>.

<FIG> illustrates a graph <NUM> showing the development of a friction coefficient over sliding time for sliding over large intervals as depicted in <FIG>.

Similar as in <FIG>, in <FIG> each line in the graph <NUM> shows the development of the friction for one of the intervals having width w<NUM>. The lines show increasing friction over time until a break-off-point <NUM>. Beyond the break-off-point <NUM>, the friction coefficient drops slightly and shows substantially constant behavior. Thus, before the break-off-point <NUM>, there is an adhesion area <NUM>, at which the adhesion contribution exceeds the friction contribution, and a slippage are <NUM>, at which the friction contribution exceeds the adhesion contribution.

As compared to graph <NUM> of <FIG>, graph <NUM> shows a break-off-point at approximately the same time as graph <NUM>. However, as the sample <NUM> in the case illustrated in <FIG> moves much longer than in the case illustrated in <FIG>, the slippage area <NUM> of <FIG> is extensive as compared to the slippage area <NUM> of <FIG>. Consequently, this means that the ratio of adhesion vs. slippage for a sample <NUM>/<NUM> is a function of the sliding distance: A longer sliding distance will lead to a lower contribution of the adhesion area <NUM>/<NUM> compared to the contribution of the slippage area <NUM>/<NUM>.

Since abrasion occurs only when there is a relative movement between the bodies in contact (slippage), this means that the wear rate (= total mass loss per unit of sliding distance) is as well a function of the interval width: for shorter interval width, the wear rate is lower due the larger contribution of the adhesive part; while, by increasing the interval width, the adhesive part becomes less important and the wear rate increases up to a substantially constant value. From this phenomenon, the CSD can be defined as the interval width beyond which the adhesion part becomes negligible, i.e., when the wear rate becomes constant. Consequently, the CSD can be determined by a method as described in the following.

<FIG> illustrates a flow chart of a method <NUM> for determining a CSD for a sample, according to the present invention.

At <NUM>, a force is provided to a sample towards a sliding surface. In some aspects, the sample may be a rubber sample. For example, the force can be provided by applying a load to the sample, such that a gravitational force Fz presses the sample towards a sliding surface. In other examples, the force may be applied hydraulically, pneumatically or mechanically by means of a motor, for example an electric motor, or in some other way. In some examples, the force may be applied solely by the sample's own gravitational force, without any extrinsic additional force. The force towards the sliding surface ensures that there is adhesion between the sample and the sliding surface. In some examples, the force may be varied over the experiment, in other example the force may be kept constant over the course of the experiment. Since the vertical force also generates a contact pressure proportional to the force, the magnitude of the force may be expressed by the magnitude of the contact pressure generated by the vertical force.

At <NUM>, the sample is slidden n times along the sliding surface along a plurality of intervals, the plurality of intervals having different widths w, n being an integer greater than or equal to <NUM>. As discussed elsewhere herein, the sample may be slidden over the sliding surface in intervals of a small width w<NUM> (as discussed with reference to <FIG> and <FIG>) and then the sample may be slidden over the sliding surface in intervals of larger width w<NUM> (as discussed with reference to <FIG> and <FIG>). For each interval width, the sample may be slidden in n steps over a total sliding distance, the total sliding distance for the i-th interval width thus being (n * wi).

In some examples, the sample may be slidden over the total sliding surface in multiple intervals. In some examples, the sample may only be slidden over a portion of the sliding surface in one or more steps. In some examples, the sample may be slidden over the same portion of the sliding surface multiple times.

In a preferred embodiment, the total sliding distance (n * wi) may be constant for all interval widths wi. For example, the sample may be slidden n=<NUM> times along intervals with width w<NUM> = <NUM>, then n=<NUM> times along intervals with width w<NUM> = <NUM>, then n=<NUM> times along intervals with w<NUM> = <NUM>, etc., such that (n * wi) is constant for all wi at (n * wi) = <NUM>. This setup may increase comparability of the results by providing similar sliding conditions.

In some aspects, the widths wi of the plurality of intervals may be integer dividers of a total length of the sliding surface.

In some aspects, the method <NUM> may further comprise sliding the sample over multiple intervals of each width wi, obtaining the weight difference ΔM by weighing the sample before and after sliding over n intervals of width wi, and determining the wear rate based on the weight difference ΔM and a total slidden distance.

In some aspects, the force may be lifted after sliding the sample along one of the intervals and the force may be reapplied before sliding the sample along the subsequent interval. This allows for the deformation of the sample to resolve between subsequent sliding steps.

In some aspects, the sample may be slidden along the sliding surface at substantially constant speed. This allows for increased comparability of the results. Furthermore, this allows performing a speed series to examine the influence of speed on the CSD.

In some aspects, the method <NUM> may further comprise identifying a first distance along which a static friction force exceeds a sliding friction force, identifying a second distance, along which the sliding friction force exceeds the static friction force, and determining the CSD further based on the first distance.

At <NUM>, a respective wear rate is determined for each interval width. The wear rate for the i-th interval width is defined as m(wi) = ΔM/(n * wi), with ΔM being the weight difference of the sample after sliding the sample n times over interval width wi as compared to before the sliding.

In the preferred embodiment of para. [<NUM>], with (n * wi) = const. , the wear rate m(w) is directly proportional to the (absolute) weight difference ΔM.

In some aspects, the method <NUM> may further comprise providing a sliding surface having a total length, sliding the sample over the sliding surface multiple times in steps of each of the plurality of widths w, and measuring the plurality of wear rates for each one of the plurality of widths w.

At <NUM>, the CSD is determined based on the plurality of wear rates m(w) and the respective width. In detail, the CSD is determined as a width wo, wherein for any w greater than wo: <MAT> In other words, the CSD is determined as the interval width, above which the wear rate remains constant with increasing interval width.

In some aspects the CSD may be determined based on any one of: a geometry of the sample, a material property of the sample, a texture of the sliding surface, an amount of the force, an amount of the sliding speed, and an amount of the acceleration to reach the set sliding speed, and a temperature of the sample. This allows for a more precise prediction of tire wear by using the respectively determined CSD as an input for a tire wear simulation model.

In some examples, different rubber samples may be used to determine different CSD values. In some other examples, different substrates may be used to determine a substrate dependent CSD. In yet some other examples, different sliding speeds may be used to determine a speed dependency of the CSD. Preferably, sliding speeds between <NUM>/s and <NUM>/s may be used. More preferably, sliding speeds between <NUM>/s and <NUM>/s may be used. Most preferably, a sliding speed of about <NUM>/s may be used.

In some examples, different contact pressures may be used to determine a load dependency of the CSD. As indicated above, the magnitude of the contact pressure is proportional to the magnitude of the vertical force. Preferably, contact pressures between <NUM> bar and <NUM> bar may be used. More preferably, contact pressures between <NUM> bar and <NUM> bar may be used. Most preferably, a contact pressure of about <NUM> bar may be used.

<FIG> illustrates a graph <NUM> showing a wear rate m(w) for different withs wi, including a knee characterizing the CSD.

The abscissa of the graph <NUM> illustrated in <FIG> shows the different interval widths in units of [mm]. The respective ordinate shows weight loss in units of [mg] over a constant sliding distance of (n * wi). The graph shows two lines, both of which are equal except for a shift in horizontal direction. The first graph, denoted as "Nominal distance", indicates the interval width corresponding to the movement of the upper portion <NUM>/<NUM> of the sample <NUM>/<NUM>. The second graph, denoted as "Actual distance", is shifted by the maximum deformation of the sample in horizontal direction. Thus, the second graph corresponds to the movement of the contact surface on the lower portion <NUM>/<NUM> of the sample <NUM>/<NUM> over the sliding surface. The CSD is determined by the "Actual Distance", i.e., by the movement of the contact surface of the sample.

It can be seen from graph <NUM> that the lines show bi-linear behavior with a well-defined knee <NUM> at a certain value of the interval width, and above that knee, the mass loss is substantially constant with increasing interval width. Thus, the width at which the knee <NUM> is located corresponds to the CSD.

<FIG> illustrates a graph <NUM> showing wear rate m(w) for different withs wi, including the knee characterizing the CSD and for different rubber compounds A, B, C and D.

The abscissa of the graph <NUM> illustrated in <FIG> shows the different interval widths in units of [mm]. The respective ordinate shows weight loss in units of [mg] over a constant sliding distance of (n * wi). The graph <NUM> includes four lines, which are denoted A, B, C and D respectively. Each line represents measurement with a different rubber compound having different physical properties. The respective measurements have been corresponding to the methods described in <FIG>. The different compounds differ from one another in terms of stiffness, friction and abrasion features and have been tested by keeping the same sample geometry (rounded blocks). It can be seen that the initial slope of the curves differs with different materials and that different CSDs can thus be determined for different rubber compound. For rubber component A, knee <NUM> corresponds to the CSD at approximately <NUM>. For rubber component B, knee <NUM> corresponds to the CSD at approximately <NUM>. For rubber component C, knee <NUM> corresponds to the CSD at approximately <NUM>. For rubber component D, knee <NUM> corresponds to the CSD at approximately <NUM>.

Accordingly, different wear performance can be obtained. Generally, a rubber compound showing longer CSD will result in lower wear rates. At the same time, the level of saturation influences the overall wear of the tire as it corresponds to the wear rate that occurs beyond the respective CSD. Thus, rubber compound A, while having the largest CSD, also has the largest wear rate beyond the CSD.

<FIG> illustrates a graph <NUM> showing wear rate m(w) for different withs wi, including the knee characterizing the CSD and for different sample shapes from the same rubber compound.

The abscissa of the graph <NUM> illustrated in <FIG> shows the different interval widths in units of [mm]. The respective ordinate shows weight loss in units of [mg] over a constant sliding distance of (n * wi). The graph <NUM> shows four lines, each line corresponding to a different sample shape, while all shapes are formed from the same rubber compound. The respective measurements with the differently shaped samples have been corresponding to the methods described in <FIG>. The line for the shape of a wheel <NUM> shows a knee <NUM> corresponding to a CSD of approximately <NUM>. The line for the shape of a plain <NUM> shows a knee <NUM> corresponding to a CSD of approximately <NUM>. The line for the shape of a lug <NUM> shows a knee <NUM> corresponding to a CSD of approximately <NUM>. The line for the shape of a sipe <NUM> shows a knee <NUM> corresponding to a CSD of approximately <NUM>.

Accordingly, different wear performance can be obtained. Generally, a sample shape showing longer CSD will result in lower wear rates. At the same time, the level of saturation influences the overall wear of the tire as it corresponds to the wear rate that occurs beyond the respective CSD. Thus, the sipe shape <NUM>, while having the largest CSD, also has the largest wear rate beyond the CSD.

<FIG> illustrates a flow chart <NUM> of a computer-implemented method for estimating the wear of a tire, according to the present invention.

At <NUM>, a wear model is provided. The wear model may be configured to convert a frictional energy rate into a wear. The wear model may be based on moveable and non-moveable parts. The moveable parts may provide higher wear then the non-moveable parts.

In some examples, the non-moveable parts provide substantially no wear. This example provides for increased simplicity of the computer-implemented method and thus allows for saving of computational resources while at the same time providing an accurate representation of the realistic physical conditions.

At <NUM>, a contact patch of the tire is divided into a moveable part and a non-moveable part. The non-moveable part therein exhibits substantially no movement relative to a substrate of the tire. The moveable part in contrast may exhibit movement relative to a substrate of the tire.

The non-moveable part has a first width, and a CSD is proportional to the first width. This allows for increased accuracy of the wear prediction provided by the computer-implemented method.

In some examples, the CSD may be determined prior to performing the computer-implemented method. In some examples, the CSD may be used as an input variable for the computer-implemented method.

The CSD is determined according to the method as described in <FIG>.

In some examples, multiple CSD values may be used as input variables for the computer model. For example, different CSD values for different amounts of velocities may be used as input variables. In other examples, different CSD values for different amounts of accelerations, different amounts of vertical force or different temperature of the sample may be used.

Using the CSD as an input value for the computer-implemented method increases the quality of prediction for the tire wear. This is achieved by providing a more accurate model of how tire wear occurs in reality.

At <NUM>, the wear of the tire is estimated based on the wear model and the divided contact patch.

In the following, an exemplary algorithm for estimating the wear of a tire is described. The algorithm may be based on the Finite Element Method (FEM), which may include geometrical and/or material data for a simulated tire. To conduct the method, the simulated tire may be discretized in multiple finite elements as described with regards to <FIG>. Each finite element may be characterized by a set of nodes, which represents the degrees of freedom in the model.

The following experimental data may be used as input parameters for the algorithm:.

The following quantities may be required as output from the simulation for each node:.

The finite elements may be represented by a finite element mesh <NUM> as illustrated in <FIG>. The finite element mesh <NUM> comprises multiple contact elements which are represented by quadrilaterals. In the present example, at a given time tn, node <NUM> and <NUM> are not in contact with the road surface. Nodes <NUM> to <NUM> are in contact with the road surface and do not undergo movement relative to the road surface, i.e., they stick to the road surface. Nodes <NUM>, <NUM> and <NUM> are also in contact with the road surface and do undergo movement relative to the road surface, i.e., they slip over the road surface. Consequently, they show nodal pressure p (p><NUM>) and slip (γ><NUM>).

Each node <NUM> to <NUM> in contact with the road surface has a corresponding nodal area. Exemplary shown in <FIG> are nodal area <NUM> for node <NUM> and nodal area <NUM> for node <NUM>. However, it is to be understood that all other nodes in contact with the road also have a corresponding nodal area An.

From the nodal data, a corresponding frictional stress τ can be defined. For the i-th node, the frictional stress τi may be defined as <MAT> with µi(pi,γ̇i) being the respective friction coefficient, dependent on the nodal pressure p and slip velocity γ̇. Then, the frictional energy rate (friction energy per unit of time) can be defined by <MAT> Ai being the nodal area of the i-th node.

For the nodes in sticking condition, γ̇ = <NUM>, such that they do not contribute to the frictional energy rate. However, this formula is independent of the nodal slip γ and thus no contribution of the critical slip distance is included therein. Thus, according to the above formula, the result will be characterized by an unrealistically high wear rate.

In a transient framework, the nodal slip γ for each node may be a direct output of the simulation framework and can be used to determine which part of the bi-linear model applies for a respective node. In the following, the CSD may also be regarded as γcr. This way, the frictional energy rate may be reduced by a scaling factor fi, resulting in the wear energy rate Ėw as <MAT> with <MAT> or <MAT> otherwise.

From the wear energy rate, the mass loss rate q̇ per node can be calculated as <MAT> Therein, k and n are material-specific abradability parameters, which have been discussed above as input parameters for the algorithm. From the mass loss rate, the nodal mass loss q may be calculated as <MAT> tr being the rolling time. The rolling time is the difference between the current time of analysis tn and the previous time step tn-<NUM>.

To generalize this for the total width of the tire, it may be assumed that the total abrasion is the sum of all mass loss computed at the nodes.

Consequently, the energy rate to be used for the wear phenomenon may be obtained by means of a factor fi(γ) to be computed based on the nodal slip and the critical slip distance obtained from experiments.

Finally, the wear energy rate may be applied to the wear model in order to compute a mass loss rate (or volume loss rate, depending on the units used) to be applied at the node. , the output of the framework allows to evaluate the amount of mass which is abraded per unit time at every node in contact. This data can be further used, amongst others, for the estimation of milage, evaluation of irregular wear in the tread, comparison of performance between geometries, materials and rolling conditions.

<FIG> illustrates a graph <NUM>, comparing the accuracy of standard simulations without consideration of the CSD and the accuracy of simulations which consider CSD.

The ordinate of graph <NUM> shows the rank of wear rate performance for the different rubber compounds A, B, C and D, as discussed with reference to <FIG>. The wear rate performances in graph <NUM> are given in units of [%] relative to compound C, such that compound C corresponds to the value of <NUM>%. The abscissa of graph <NUM> shows the wear rate performance obtained in prior abrasion tests in tires performed by the manufacturer, also in units of [%] relative to compound C. Data shown by circles represent the relation between the experimental ranking and the simulation without the use of CSD. The dotted line <NUM> indicates a square of the Pearson correlation coefficient R<NUM>=<NUM>. It can be seen that there is a considerable discrepancy between the data points <NUM> and <NUM>, which correspond to compounds B and D, and the respective simulation data, as well as a low quality of the correlation.

Claim 1:
A method for determining a critical slip distance, CSD, for a sample, the method comprising:
providing a force to the sample towards a sliding surface;
sliding the sample n times along the sliding surface over a plurality of intervals,
the plurality of intervals having different widths w, n being an integer ≥<NUM>;
determining, for each width wi, a respective wear rate m(wi)=ΔM/(n * wi) of a plurality of wear rates m(w), wherein ΔM is a weight difference of the sample after sliding the sample over n intervals of width wi as compared to before the sliding; and
determining, based on the plurality of wear rates and the respective widths, the CSD as a width wo, wherein, for any w > wo, <MAT>