A novel multi-scale numerical model for prediction of texture-related impacts on fuel consumption.

: It is estimated that to overcome rolling resistance (RR) a typical vehicle on average consumes 4152 MJ/119 litres of fuel annually depending not only on vehicle-related factors but also pavement-related ones. A slight improvement in surface properties may thus decrease fuel consumption bringing substantial long-term socio-economic benefits per capita/country. This aligns with ever-tighter limits on CO 2 in the EU (95 g/km until 2021) fostering sustainable construction/exploitation of tyres and pavements. This paper outlines a newly developed multi-scale 3-D numerical methodology to quantify texture-dependent RR due to indentation of aggregates into visco-elastic tread compound. It consists of a micro-scale tread block - single aggregate model and a macro-scale car tyre finite element model, rolling in a steady state mode over a rigid smooth surface. Micro-scale interaction rates are deduced from the macro-scale model. Tread compound is simulated by application of a time-dependent linear visco-elastic model. The micro-scale simulations enabled quantification of RR induced by an arrangement of surface aggregates. The outlined texture-dependent RR estimates are based on contact force moment around the contact patch centre. The computed contact force results show a significant peak of normal force due to visco-elastic and inertia effects at the onset of the tyre-surface contact phase, followed by a gradually decreasing/relaxing stress region with a sudden release at the end of the interaction. The contact forces appear to be of a reasonable distribution and magnitude. The proposed approach allows prediction of RR losses due to compressive forces at the micro-scale. Macro-distortional RR (which is not the subject of this paper) would then have to be added to find the total tyre-related RR.


Introduction
In view of an escalating transportation demand, particularly in the USA and Europe, everstricter regulatory emission limits (95 g/km by 2021 in the EU) are being enforced by governments all over the world to minimize detrimental impact on the quality of life. Both vehicle/tyre and pavement research centers are therefore in constant search of ways to enable operation of tyres and roadways at an increased sustainability level. While vehicle and tyre makers, and now also highway institutes, have made considerable progress in enhancing fuel economy in terms of aerodynamic drag, drive-train and tyre/roadway rolling resistance (RR) [1,2], pavements represent a potential key to further reducing energy loss. It is estimated that to overcome RR a typical vehicle on average consumes 4152 MJ/119 litres of fuel yearly depending not only on tyre-related characteristics such as structural composition or material compound properties, but also on pavement-related factors, of which weak pavement structure and rough texture have been found to cause extra fuel consumption [3,4].
RR is a cumulative term that comprises a suite of energy loss mechanisms, whose total value typically ranges from 25 to 80 N per car tyre [19]. These energy losses vary in magnitude and simultaneously occur because of the following causes:  tyre macro-distortional, mainly visco-elastic dissipation (texture-independent)  micro-distortional visco-elastic and inertial dissipation (texture-dependent)  pavement macro-distortional dissipation (texture-independent) Texture-independent energy dissipation in the tyre is generally attributed to macrodistortional hysteresis that appears as a result of compressive, tangential (shearing/microslippage and shear-related inertia) and wave-induced inertial deformations. The deformation is caused by flattening of the contacting tread part leading to compression, shearing and bending of the tread part and bending of the crown, sidewall and bead elements. Shearinduced inertial forces are activated when the tread is in a slippage phase and are usually incorporated into a resultant tangential force. According to Michelin [5], tread compression and longitudinal shearing share the greatest portion of the energy consumption. Additionally, the losses could be amplified by standing waves, centrifugal and to lesser extent Corriolis forces.
Micro-distortional losses are triggered by aggregate indentations into elastomeric tread compound. These indentations generate local compression which includes a viscoelastic and an inertial component that affect the moment balance in the contact region. Figure 1 illustrates that this part of RR is dependent on penetration of individual stones and the distance between adjacent asperities. The presence of road protrusions could slightly modify tread-related tangential stresses/excitations, which were included in the macro-distortional term, permitting the tread block to rise and descend throughout the stick-slip phase. Aggregate texture, tread block shape and in general tyre profile all influence the forces contributing to RR.
Unlike the two previous terms, irrecoverable pavement energy dissipation is formed as a consequence of wave propagation along, across and inside the road structure that leads to an asymmetrical deflection bowl under the tyre. According to recent numerical predictions [4], stiffness of the uppermost pavement layer is much more influential than that of lower layers; thus, a stiffer upper pavement saves energy. In the scenario examined in [4], changing from a flexible to a rigid pavement may reduce pavement RR by approximately 3 N per HGV tyre at 40 kN. Other models (e.g. Akbarian [6]) have been developed, confirming the main findings.

Knowledge gaps and objectives
Most of the studies on aggregate indentation effects have emphasised structural vibration, noise radiation, wear resistance and friction [7,8]. Meanwhile the majority of semi-empirical, analytical and numerical models have focused on macro-distortional RR, ignoring road irregularities and associated small interfacial deformations [9,10,11]. Some have developed tread-asperity indentation models studying contact stress distribution, but have omitted energy loss calculations [12,13]. Only a few have produced estimates of energy losses due to stone indentation [14,15,16,17,18], and not all of these have clearly distinguished microdistortional loss from tyre macro-distortional loss.
Due to the lack of detailed numerical RR predictions, pavement-tyre interaction, which is believed to account for 7% -18% of total vehicle energy consumption in Europe [8,19], is still very much of interest. Specifically, the key factor that has been ignored is the individual stone effect (micro-distortion) with both tread inertia and visco-elastic components. None of the existing models has attempted to examine these individual components, often representing protrusions as a series of 2-D/3-D linear or non-linear spring-dampers or through spectral representation derived from scanned real-life surfaces. There is a need to fill this knowledge gap and propose an approach that would, in the long-term, assist in optimising tyre treads and road surface topography for minimal RR without compromising skid resistance and drainage function.
The objective here is to present a computationally efficient numerical multi-scale methodology in order to compute the micro-distortional RR induced by single particles of macro-texture (0.5mm -50 mm bandwidth). The model involves macro-scale (whole tyre) and micro-scale (tread-asperity interaction) analyses as explained below and has been developed using Abaqus commercial software.

Macro-Scale Model
The macro-scale model is of a 175 SR14 3-D slick radial Yokohama tyre adapted from Abaqus open source [25]. In accordance with the ABAQUS documentation, the first step is to generate a half axi-symmetric model for 2-D inflation analysis. The adapted tyre crosssection consists of rubber-based components such as the tread and sidewalls, and two belts and the carcass, which are made up of fiber-reinforced composites. The tread and sidewalls have been discretised with CGAX4H and CGAX3H hybrid elements with twist and modelled as a viscoelastic material, the properties of which have been supplied from the tyre manufacture. The application of this material model to a tyre construction, in which it is not actually used, is considered to be justifiable since the micro-distortional RR methodology is being proposed in generic.
In turn, the belt and carcass fiber parts have been represented as a linear elastic material with elastic modulus, Poisson's ratio and density, respectively, being equal to 172.2 GPa, 0.3, 5900 kg/m 3 and 9.87 GPa, 0.3, 1500 kg/m 3 [25]. The reinforcement belts and carcass have been meshed with a rebar layer in SFMGAX1 surface elements being embedded in host continuum elements CGAX4H. In order to prevent an offset of the embedded element nodes from the host element edges resulting from numerical roundoff, a roundoff tolerance is specified. The introduction of a roundoff technique enhances the performance by adjusting the positions of embedded elements to lie precisely on the host elements and thus minimizes the number of constraint equations used to embed surface elements.
A half 3-D tyre is produced by revolving the half cross section model about the rotation axis and then reflecting it to generate a full 3-D tyre. The analysis comprises three stages: static, braking/traction and steady-state rolling are conducted sequentially. The results from each analysis are transferred to the subsequent models using a transfer capability. The static tyre analysis provides the boundary conditions for the braking/traction analysis, and the braking/traction model is the basis of the steady-state tyre rolling simulation. The main purpose of the macro-scale tyre model for this research is to provide loading, stationary and unloading rates for input into the micro-scale simulation.
Evaluation of tyre distortion has been carried out applying Abaqus/Standard Steady-State Transport Analysis in a mixed Arbitrary Lagrangian Eulerian (ALE) formulation travelling against a smooth rigid road. In this framework, the deformation is described by a Lagrangian method whilst the rigid body rotation is characterised by an Eulerian framework. To overcome local material instability, the STABILIZATION option was used to introduce artificial viscous forces. To account for nonlinearity effects that arise due to large deformations, the NLGEOM function was activated. Inclusion of centrifugal forces into the analysis has been done using the INERTIA command. Slip tolerance value has been set to 0.02 as more relaxed (larger) values generally prevent convergence. Efficient computational speed has been achieved by incorporating a finer mesh in the footprint region covering 40°of arc and a coarser mesh covering the remaining 320°, both being discretised with general linear hybrid elements (C3D8H and C3D6H) suitable for incompressible material ( Figure 2).
The total numbers of nodes and elements in the model were 13548 and 7255, respectively, as illustrated in Figure 2. Steady-state tyre analysis was conducted under 3300 N ( 1 P ) loading condition at an inflation pressure of 200 kPa.

Application of free rolling conditions
As a first step, free rolling conditions have to be reached, and for this it is necessary to separately (and correctly) specify the wheel velocity ( x V ) and its angular velocity ( 2  ). One of the possible techniques is to determine the effective rolling radius eff R of a tyre, spinning on a frictionless surface, based on the average tread surface horizontal velocity and then to calculate the required angular velocity for steady-state rolling: Alternatively, the free rolling angular velocity can also be found through trial and error [20] such that the moment balance about the rim centre becomes virtually zero ( Nm 05 . 0  , where achievable). This second approach has been applied for approximate determination of free rolling conditions. After experimentation the time step used in computation was 1.0 sec; the initial and analysis time increments were taken as 0.001 and 0.01 secs respectively. These were found to give good convergence.
The adopted procedure was as follows:  estimate the angular velocity value for the first run  increase/decrease the angular velocity ( 2  ) depending on RR moment about the hub  select 2  such that the RR moment (RM2 in Figure 2) ~ 0, output the reaction force in the longitudinal direction (RF1), which quantitatively equates to contact force, i.e.
the micro-distortional RR Despite slight numerical instability and imperfect convergence of the RR moment around the hub point, the average RR magnitudes for the Abaqus tyre matched the outcomes reported in the literature for this tyre [21].

Rubber Properties
It is well established that rubber behaviour is strongly contingent on temperature (an increase softens the compound), frequency of cyclic loading (an increase stiffens the compound) as well as strain level (an increase softens the compound) [5].
where  g is the long-term dimensionless modulus for the rubbery region and i g and i  are material constants. Stiff and soft (higher and lower long-term modulus as compared with soft rubber) rubber compound properties were used, with 12 and 14 Prony terms respectively, corresponding to 55°C operating temperature. The modular ratio of soft to stiff rubber amounted to 0.92. To simplify the modelling problem, strain-dependent nonlinearity (Payne and Mullins effects) was excluded from the analysis.
All tyre rubber-like parts have been assumed to have the same properties. As can be seen in Figure 3, the tyre with soft compound dissipates less energy than the stiff compound tyre.

Friction Effect
The surface friction coefficient is known to be dependent on slip velocity, pavement surface, rubber properties, temperature and contact pressure. Taking account of this, three friction models have been studied and compared, namely a simple Coulomb coefficient (equation 4), the slip-velocity-dependent coefficient (equation 5) and direct usage of test data that included contact pressure.
The parameters for the Eqn. 5 model have been extrapolated based on measurement data in Guo et al (2004) to match measured pressures inside the contact patch, which is a function of the inflation pressure and bulging of the tyre belt [12,22].

Micro-Scale Model
A 3-D FE model utlizing the Abaqus/Explicit solver has been used to investigate contact forces between a tread block and a stone in order to deduce micro-distortional RR. In the first instance, Abaqus/Implicit was applied to study contact forces. However, it was abandoned due to inability to achieve a converged solution after trying various time increments, an adaptive meshing function, using a finer mesh discretisation and adaptation of contact softening/damping controls. It was therefore concluded that its applicability for this specific analysis would be difficult and computationally expensive compared to Abaqus/Explicit. The ) with no micro-texture  asperity penetration takes place in the vertical direction; no tangential movement is applied  pre-stress from macro-distortion (lateral stretching and twisting) of a tread block prior to stone-induced micro-distortion is neglected  no adhesion forces are present  temperature is constant

Pavement Surface Properties and Tread Pattern Assumptions
There are a wide range of asphalt mixtures all having their own surface macro-texture layout (positive/negative) and amplitude (texture depth). Aggregate gradation is a primary factor influencing road surface properties in addition to shape (angular, cubic etc) and maximum aggregate size. Overall, based on their topography, pavements could be categorised into smooth (e.g. 0/8mm Stone Mastic Asphalt) and rough (e.g. surface dressing/chip seal) types.
The advantage of the model is that it allows both surface classes to be covered by considering a range of hemispherical diameters. To examine different road textures, spacing between stones can be varied along and across the contact. To simplify the problem, a hypothetical mixture is considered of a single-sized gradation making all surface stones identical. Even though the assumption of hemispherical asperities is unrealistic, it is still, as indicated by Greenwood and others [12,24], a justifiable approximation of road chippings. In the context of RR, tyre tread designs are also important. These involve the arrangement of continuous ribs, uncoupled tread blocks, circumferential and lateral grooves as well as moulded sipes, which in part share the functions of road texture. Treads are generally classified into four groups: symmetric, asymmetric, directional and those that feature both asymmetric and directional tread patterns. It is believed that the tread pattern impacts are particularly influential for local interaction mechanics as well as tyre dynamics. Since at the micro-scale level the interaction is established between a single block and a single stone and ALE formulation does not take account of it at the macro-scale level, the tread patterns could be implicitly incorporated via void area ratio which is the fraction of the tyre outer plane that is  To reduce the number of time increments required, mass scaling is introduced. Excessive mass scaling was observed to cause substantial oscillations compared to zero mass scaling case. After checking various levels of mass scaling, an optimum factor was adopted to enable increased computational speed without generating unwanted noise.
To effectively transmit deformations of nodes through the tread-sphere interface, a contact stiffness between them needed to be established, preventing numerical overlap between contacting bodies. Having experimented with a wide range of contact stiffnesses, an optimised high, but computationally stable, level was selected for all analysis steps. Friction was assumed to be constant, but was not found to significantly affect computed forces.

Derivation of Loading and Unloading Rates for Micro-Scale Model
The indentation and release rates ( Figure 5) have been extracted from the macro-scale model surface nodes as they make contact with the surface at the leading edge and leave at the trailing edge. Figure 5 shows a vertical velocity distribution for a series of nodes before and after the velocity becomes zero in the contact area. Linear integration of vertical velocity between nodes with respect to time allowed the indentation/release velocities to be determined for specific distances from the smooth surface.
Assuming these distances correspond to indentations experienced by protruding surface texture, loading/unloading rate boundary conditions have been formulated.
It was found that similar rates applied regardless of rubber compound rigidity. The duration of the contact phase was also derived from the macro-scale data. More specifically, indentation/release durations were determined based on the length between nodes, which was constant. loading and unloading at a given indentation level was calculated. As shown in Figure 5, the indentation rates were always predicted to be higher than the release rates. This can be ascribed to rubber viscosity effects and inertia.

Micro-Distortional RR Force Estimation
The micro-distortional RR was derived from the moment (RRM) given by contact force distribution, as shown in Figure 6, and the number of stones in the contact at a given time.
The RRM about the point on the contact patch directly below the centre of the wheel is quantified. It is then divided by the loaded tyre radius (measured in the macro-scale model) to determine the texture-dependent RR for the whole tyre. An advantage of this numerical technique is that by altering stone size/shape/spacing, the influence of different texture patterns could be explored.

Results and Discussion
The computational technique described above has been applied to evaluate the compressive contact forces between a tread compound block and a hemispherical stone of 5mm radius  As an example, the results for 0.5 mm indentation ( Figure 6) clearly demonstrate that the tread-stone model captures the expected visco-elastic behavior caused by a single indenter. In particular, Figure 6 shows an asymmetrical contact force distribution: peak normal force at the beginning of the contact region, followed by a gradual decrease of the force until the beginning of unloading phase where the indenter snaps out from the rubber surface. It can be seen that due to stiffening of the compound at higher loading rates (40, 60, 80, 100 kph), the average contact forces at these velocities are greater in magnitude by around 27% compared with 20 kph, at which speed the rubber has a longer time to relax. The example in Figure 7 illustrates the computed micro-distortional RR for a slick tyre, which rises linearly as velocity increases, ranging from 28 N at 20 kph to 32 N at 100 kph for a hypothetical road surface with 103 hemispherical stones in contact with the tyre, at an average contact force of about 19.5 N/ind, represents a tyre load of about 2 kN. As can be seen the computed RR is at a similar level to the macro-distortional RR depicted in Figure 3, although this maybe rather higher than that for a real surface, due to the idealised hemispherical pavement topography (perhaps comparable to a surface dressing [15]) and the assumption that each stone indents at the centre of a tread block ignoring edge effects. The vertical projection of the wheel centre on the contact patch also carries a degree of uncertainty. The relatively small effect of velocity, this is reasonable since at a lower loading rate a slightly larger indentation would be expected due to visco-elastic effect. Taking account of tread pattern with 0.8 void ratio, typical for car tyres [27], micro-distortional RR was found to increase quite considerably. Figure 7 shows that the RR of a treaded tyre is, on average, 20 % higher than that of a slick tyre. It is expected to grow since number of contact indenters reduces to 82 (103 0.8) indenters resulting in a higher loading per stone and indentation.
A range of hemispherical radii has been examined, all with closely packed stones. At identical speeds, contact forces per indenter increase as the radius (and therefore centrecentre spacing) grows, and consequently, each indenter in the contact patch will take more force and indent deeper into the tread compound. The increase of force together with higher indentation will evidently dissipate a higher amount of energy per stone. However, when the effects are summed, a dense packing with a finer indenter will tend to induce an approximately equivalent texture-dependent RR compared to a densely packed coarser indenter.

Conclusion
A novel multi-scale approach for micro-distortional RR computation has been presented. It is capable of quantifying the visco-elastic response of a tread compound block indented by a single stone. The forces, computed in the contact patch, appear to be of a reasonable distribution and magnitude, although the predicted RR is probably greater than for most real road textures. The model, though it allows useful comparison of different vehicle speeds, loads, rubber properties and texture patterns, still has to be calibrated, for real pavement surfaces by means of empirical techniques due to the hemispherical assumption.