*1.1. Surface Texturing in Tribological Applications*

In most dynamic rubber applications, low friction is desired to increase the energy efficiency of the tribological system. Therefore, improved friction behaviour is an important design objective in the development process of components like dynamic seals. Within the industrial and scientific communities, significant improvements in dynamic seal performance have been achieved through a series of technological advances, such as the introduction of low-friction polymers [1]. However, this approach is limited by the operating conditions, such as temperatures or the chemical resistance of the new materials.

Surface texturing is an effective method to modify the friction level without changing materials or the lubricant in the dynamic seal contact. An early method of surface texturing, used to improve the tribological performance of mechanical components, is cylinder honing in internal-combustion engines [2]. Further experimental and theoretical studies revealed a significant reduction in friction due to grooves [3] and, in particular, micro dimples in the reciprocating contact of piston rings and cylinder bores of combustion engines [4,5].

The analysis of laser surface textured (LST) piston rings have been extended to modelbased and experimental investigations of mechanical seals, identifying optimised dimple

**Citation:** Zambrano, V.; Brase, M.; Hernández-Gascón, B.; Wangenheim, M.; Gracia, L.A.; Viejo, I.; Izquierdo, S.; Valdés, J.R. A Digital Twin for Friction Prediction in Dynamic Rubber Applications with Surface Textures. *Lubricants* **2021**, *9*, 57. https://doi.org/10.3390/ lubricants9050057

Received: 30 March 2021 Accepted: 16 May 2021 Published: 20 May 2021

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dimensions with lowest friction for specific operating conditions [6,7]. In this context, the dimples are advantageous as they serve as hydrodynamic bearings and exhibit the ability to store lubricant [8]. Moreover, the surface textures reduce the real area of contact and trap wear particles [9,10].

Other authors even highlighted the positive effect of surface textured rubber seals in soft elasto-hydrodynamic lubrication (SEHL), resulting in a significant friction reduction compared to the untextured references [11–13]. However, it has also been found that an inappropriate selection of the surface texture dimensions leads to a detrimental increase in friction due to excessive enlargement of cavitation zones in the lubricant, causing a reduced local film thickness and load carrying capacity [14,15]. Furthermore, the influence of the real dimple shape of textured mechanical seals has been identified as an important factor that needs to be considered for valid friction determination [16].

#### *1.2. Reduced Order Modelling*

The introduction of extended methods, based on artificial intelligence (AI), which go further than usual tribological measurements, is beneficial for friction predictions. The selected AI technique is based on the digital twin (DT) paradigm, i.e., a virtual representation of reality [17–19]. Several techniques belong within the DT paradigm, the most popular being Reduced Order Modelling (ROM) [20,21] and machine learning (ML) [22]. Both techniques are based on mathematical models for real-time simulations. In particular, ROM consists of a set of numerical strategies for multi-variable problem simplification to solve complex numerical systems and it aims to describe and, hence, predict a system's behaviour through a mathematical approximation, by preserving its main characteristics, as described in [23]. Tensor rank decomposition (TRD) approach is a non-intrusive, i.e., completely data-driven, method that allows to describe a complex system's behaviour, where its variables influence each other, as a simplified mathematical function that describes each variable's effect separately. As previously detailed in [23], TRD is based on the assumption that a problem of N, not necessarily independent, variables can be rewritten as the product of N one-dimensional functions, one for each of the variables of the system, as shown in Equation (1):

$$F(v\_1, \dots, v\_N) = \sum\_{m=1}^{M} \alpha\_m \prod\_{n=1}^{N} f\_{m,n}(v\_n) \tag{1}$$

where *M* is the order of approximation of the ROM model and *αm*, *m* = 1, ... , *M* are weighting coefficients. The functions *fm*,*n*, in their most simple form, are piecewise linear functions; hence the adjustment parameters are the positions and the values at which the functions change slope. The adjustment of all these parameters is carried out through a least square optimisation.

The first term of the sum in Equation (1) represents a first approximation of the system, being its corresponding coefficient *α*<sup>1</sup> the largest one, while the following terms would be corrections to it and will generally have lower coefficient values unless the correction only applies to a specific outliers population and does not affect the general trend of data.

#### *1.3. Objectives*

Although many experimental and model-based studies have been accomplished on the subject of friction reduction of surface textured components, all investigations require extensive series of experiments or complex contact- or fluid-mechanic simulations. Therefore, this paper aims to introduce a novel approach, in which a limited number of friction measurements of surface textured rubber samples are combined with ROM to identify optimal surface textures as a function of the prevailing operating conditions in real-time. Besides the nominal dimple texture parameters, the real dimple dimensions, defined by diameter, distance, and depth, are taken into account for the friction values

computed by the ROM. Therefore, ROM technique is further used to determine nominal surface texture values uncertainties for valid friction prediction.

#### **2. Materials and Methods**

The following Sections are dedicated to the description of the experimental studies and the used software, data analysis (dataset publicly available under [24]), and statistical tools.

#### *2.1. Rubber Specimen Geometry and Surface Texture Parameters*

The objective of the experimental testing procedure is to measure the friction between surface textured rubber specimens and a rotating counter surface by utilising a pin-on-disc tribometer. The grinded steel counter surface exhibits a surface roughness of *Ra* = 0.50 μm, but no further texturing. Due to the variety of seal geometries available on the market, a simplified rubber sample geometry was chosen for the experiments, in order to test the surface texture-induced friction variation independent of a specific seal geometry. The corresponding geometry of the rubber samples is shown in Figure 1a.

**Figure 1.** (**a**) Geometry of the rubber specimens including the most relevant dimensions, (**b**) picture of the rubber specimen with focus on the dimple texture and (**c**) positioning of the dimples in relation to the relative velocity vector.

A 2 mm thick layer of a fluorelastomer with a shore hardness of 80A (FKM 80A) is vulcanised onto a blue anodised aluminium specimen holder. Furthermore, the contact zone of the 30 mm diameter rubber sample has a spherical shape to avoid edge effects in the dynamic contact.

The surface textures are applied to the contact areas of the rubber samples in the form of deterministic positioned dimples, see Figure 1b. The geometry of the circular dimples is defined by the diameter, the distance and the depth. The corresponding parameters and the alignment of the dimples are shown in Figure 1c. During the test procedure, the positioning of the dimples, within a squared area, was rotated by an angle of *φ* = 45◦, since preliminary experiments showed about 20% greater friction reduction in this arrangement compared to *φ* = 0◦. The reason for this is the avoidance of continuous flow channels without dimples in the direction of relative motion for *φ* = 45◦. The angle *φ* between the square shaped texture arrangement and the relative velocity vector is visualised in Figure 1c.

In order to perform the experiments, test specimens with eight different surface textures were manufactured by texturing during moulding (TDM). The associated sample number *i* and the nominal texture parameters are listed in Table 1.


**Table 1.** Nominal dimple texture parameters defined by diameter, distance and depth.

The parameter range of the dimple dimensions was selected to ensure a transferability of the textures to dynamic seal applications. For example, if the diameter of a dimple is larger than the contact width of a seal, the system would leak. Since the contact width of many lip seals is only 0.80 mm or less [25], the maximum size of the dimple diameter and distance was set to 300 μm. In contrast, e.g., the lower limit of the dimple diameter of 100 μm was determined by the accuracy of the LST process of the TDM manufacturing method. Based on the texturing results of the rubber samples, the transfer of the textures to real seals has already been successfully realised, but will not be discussed within the scope of this work.

The design of experiment (DoE) on the variation of the surface texture parameters is chosen according to the requirements of the ROM, described in Section 2.4. While seven specimens exhibit a dimple texture, one specimen was produced without dimples to serve as a reference for the dimple-induced friction variation of the other textures. The surface roughness of all eight samples was adjusted to an identical value of R*<sup>a</sup>* = 0.50 μm by laser surface processing of the mould. Therefore, the influence of dimple textures on friction is investigated independently of different surface roughnesses.

#### *2.2. Test Rig Setup and Experimental Procedure for Determining the Coefficient of Friction*

A pin-on-disc tribometer is utilised to measure the friction forces between the rubber specimens and the rotating steel counter surface. The design of the test rig is shown in Figure 2a.

**Figure 2.** (**a**) Pin-on-disc tribometer design, (**b**) picture of the tribometer with focus on the rubber specimen and the rotating counter surface, and (**c**) contact conditions between the rubber sample and the counter surface.

The rotation of the counter surface is realised by a servomotor. The force sensors measure both the force *FN* in the normal direction of the sample, as well as the friction force *FF* in the circumferential direction of the rotating counter surface, compare Figure 2c. The rotational speed *n* of the counter surface and the normal force *FN* are kept constant during each measurement, in order to measure quasi-stationary friction values under steady operating conditions. Each relative velocity *vr*, specified in Table 2a, is tested at each of the 3 different contact pressure levels *pc*,*max*, given in Table 2b.

The relative velocity *vr* between the rubber specimen and the counter surface is equal to the circumferential velocity *vc* at the point of contact: *vr* = *vc*. The velocity *vr* is calculated from the rotational speed *n* of the counter surface and the distance *r* = 100 mm between the shaft centre and the contact point, see Figure 2c, as defined in Equation (2).

$$
\omega v\_r = \omega r = 2\pi nr \tag{2}
$$

The focus of this paper is on dynamic friction, so static friction is not investigated. Thus, the rotational speed *n* of the servomotor was varied between 0.6 and 24.0 min−1, resulting in relative velocities *vr* of 6 to 251 mm/s.

**Table 2.** Operating parameters that are examined during the test procedure. (**a**) Rotational speeds *n* of the servomotor and corresponding relative velocities *vr* between the rubber sample and the counter surface (2), (**b**) together with the normal force *FN*, the related maximum contact pressure *pc*,*max*, the contact diameter *dc* between the rubber sample and the counter surface, as well as the nominal contact area *Anominal* (2).


The normal forces *FN* were selected to achieve maximum contact pressures *pc*,*max* of 0.5, 0.7, and 0.9 MPa, which are typical values in pneumatic seal applications [26]. Because of the spherical shape of the rubber specimen, the variation of the normal force *FN* influences not only the magnitude of the parabolic contact pressure distribution *pc*, but also the dimensions of the nominal contact area between the rubber sample and counter surface, see Figure 3b. Therefore, not only the maximum of the contact pressure distribution *pc*,*max* is specified in Table 2b, but also the corresponding contact diameter *dc* and the respective circular nominal contact area *Anominal*. The contact pressure distribution *pc* and the contact diameter *dc* are computed by finite element analyses, taking into account the rubber thickness of 2 mm. Within the static simulations, the rubber sample is pressed against the counter surface with the defined normal force *FN*, see Figure 3a. At this, the coloured spherical contact area of the rubber specimen is brought into contact with the grey flat counter surface, resulting in the specified contact area and contact pressure *pc*.

All components are modelled as 2D axisymmetric parts. A hyper elastic Mooney-Rivlin material behaviour is assigned to the 2 mm layer of the FKM80A rubber material, which is specified by the temperature-dependent material parameters *C*<sup>10</sup> = 1,442,425.12, *<sup>C</sup>*<sup>01</sup> = 208,308.34, *<sup>D</sup>* = 6.059933161 ×10−10, considering a Poisson's Ratio of 0.4995. For both measurements and simulations, the temperature is equal to 20 ◦C. The material of the anodised aluminium sample holder is modelled as pure elastic part with a Young's modulus of 70 GPa and a Poisson's Ratio of 0.34. The counter surface is defined as rigid part.

All experiments are performed with an adherent silicone grease OKS 1155 in the dynamic contact, which exhibits a base oil viscosity of 100 mm2/s at 25 ◦C [27]. The same lubrication and conditioning procedure was applied to every rubber specimen before the actual friction measurements to ensure comparability between the results. Every single measurement lasts for 5 s and each measurement was repeated 5 times to ensure a statistical certainty. In order to generate the quasi-stationary friction values *μuntext* for rubber specimen 8 and *μtext*,*<sup>i</sup>* for rubber samples *i* = 1–7, the mean value of each measurement is calculated over the entire measuring period of 5 s. The friction coefficients *μuntext* and *μtext*,*<sup>i</sup>* are evaluated to identify dimple textures with the highest friction reduction potential. In addition, the results are further processed to generate the ROM model.

**Figure 3.** (**a**) Assembly of the finite element method (FEM). The coloured spherical contact area of the rubber specimen is brought into contact with the grey flat counter surface. (**b**) Parabolic contact pressure distribution *pc* as a function of the contact diameter *dc*.
