Analytical model of friction behavior during polymer scratching with conical tip

To investigate the effects of the contact geometry, interfacial friction, and substrate recovery on the behavior of polymer scratching using a conical tip, an analytical model is proposed. The normal stress acting on the contact surface between the tip and the substrate is described as a function of the included angle θ, representing the angle between two planes across the axis of the conical tip, and the attack angle β, representing the angle between the conical surface and the substrate material surface. The effects of the rear contact geometry on the scratch friction between the tip and substrate, represented by recovery angle φ, owing to the instantaneous elastic recovery of the polymer substrate, are also introduced. Validated by the experimental and numerical results from the literature, the proposed analytical model can describe well the scratch coefficient of friction (SCOF), which is defined as the ratio of the tangential force to the normal force. Meaningful guidance is provided to understand the scratch friction behavior.


Introduction
Owing to their relatively poor mechanical properties, polymeric materials can easily be scratched, and the surface damage from such scratches will change/weaken the morphology, functionality, and aesthetics of the original surface. It is well known that the adhesive behavior between the tip and substrate significantly affect the polymer scratch behavior. To evaluate the scratch resistance of a polymer, the single-pass test with a hard scratch tip has been widely adopted [1][2][3][4][5][6]. Many scratch experiments have been conducted for different polymers and their micro-/nano-composites [7−14]. Various scratch damage patterns have also been observed [2, 8−13].
A finite element (FE) analysis was also conducted to study the scratch damage mechanisms, as well as the influence of the material properties and/or contact conditions on the scratch behavior [15−24]. For instance, Jiang et al. [15] found that the scratch performance of polypropylene (PP) improves with a decrease in interfacial friction when using the FE method. Pelletier et al. [16] numerically demonstrated that the interfacial friction significantly affects the real contact area between the scratch tip and polymethylmethacrylate (PMMA) substrate. Feng et al. [23,24] considered the combined Coulomb and plastic friction effect to study the impact of the yield strength on the scratch friction behavior.
Considering the complex contact condition between the scratch tip and substrate, necessary assumptions are needed when establishing an analytical model. For scratch friction with a conical tip, Subhash and Zhang [25] assumed the normal stress applied to a contact surface as a constant value, which was an oversimplification. Lafaye et al. [26−28] established an analytical model and explored three forms of normal stress distribution. Komvopoulos [29] established an analytical model for a conical tip with a spherical extremity sliding on a rigid-perfectly-plastic material. Jardret et al. [30] studied the use of a Berkovich tip. Tayebi et al. [31] investigated a spherical tip, and Briscoe et al. [32] studied a conical tip. Most of these works assumed that the contact only exists between the frontal half of the scratch tip and the substrate. Based on an in-situ observation of the scratch experiment, Gauthier et al. [33,34] found that the substrate instantaneously recovers after the tip passes by. The contact between the rear part of the scratch tip and recovered substrate is observed, which should be considered to establish a comprehensive analytical model of the scratch behavior. Bucaille et al. [35] extended the model of Goddard and Willman [36] to describe the rear contact between a conical tip and an elastic-perfect plastic material as a frictionless case. Lafaye et al. [37,38] further developed this using a conical tip, a spherical tip, and a conical tip with a blunted spherical extremity. They found that the effects of the instantaneous elastic recovery of the substrate on the scratch behavior cannot be neglected.
In this paper, an analytical model is proposed to study the scratch friction behavior of a conical tip-polymer substrate system. Because of the viscoelastic nature of a polymer, a theoretical description of the scratch friction is not easy to achieve. To analyze the local interaction between the tip and substrate, the substrate material is assumed as an elastic-plastic material without strain hardening. First, the normal stress distribution on the contact surface between the tip and substrate is explicitly defined as a function of the included angle  and attack angle  . The elastic recovery of the substrate is also included. After validation based on experimental and numerical results from the literature [21,25,35,39], the proposed analytical model is utilized to thoroughly investigate the influences of the contact geometry, interfacial friction, and substrate recovery on the scratch friction.

Theoretical model
The scratch coefficient of friction (SCOF), also called the apparent friction coefficient [26−28], or overall friction coefficient [25,39], is defined in Eq. (1) as the ratio between the tangential force T F and the normal force N F exerted on the scratch tip, as schematically shown in Fig. 1(a).

Subhash and Zhang (S&Z) model
Subhash and Zhang [25] proposed an analytical model where  is the attack angle, which represents the angle between the conical surface and the substrate material surface,  is a parameter to characterize the material flows on the contact surface, and s  is the local interfacial friction coefficient. All variables are illustrated in Fig. 1. Assuming that the distribution of normal contact stress is independent of the included angle  , they obtained a constant value of the normal stress  along any arbitrary cross-section on the contact surface, as schematically shown in Fig. 2  contact stress are not included in the above model. In addition, their work assumed that contact only exists between the frontal half of the scratch tip and the substrate, i.e., [0, / 2]    , which completely neglects the elastic recovery of the polymer substrate at the rear contact.

Proposed model
The effects of the contact geometry and the elastic substrate recovery on the contact normal stress between the scratch tip and underneath part of the substrate are taken into consideration in the present study.

Effects of geometric factors on normal contact stress
The normal stress on the contact surface between the tip and substrate is correlated with the tip geometry, the mechanical properties of the substrate, and the adhesive behavior of the contact pair. For conical tip scratching, the normal stress distribution on the horizontal cross-section, with different heights, of the contact surface should follow the same tendency. In this work, two geometrical factors, i.e., the included angle  and the attack angle  , were separated and explicitly considered to study the scratch friction behavior. Without a loss of generality, the normal stress acting on the contact surface line of one horizontal cross-section of the conical tip, as a function of many r , may be simplified as Here, ( , ) r   , written as 0  [25], may change with the location of the horizontal cross-section but remain constant at the same cross-section (see Fig. 2

(b)). In addition, ( , )
f   is a function of the included angle  and attack angle  .
Because the substrate material flowing on one horizontal cross-section of the contact surface is phenomenologically similar to the fluid uniformly flowing around the circular cylinder, ( , ) f   can be presented in a form similar to the distribution of the fluid pressure on the cylinder surface [40], that is, The parameter  is discussed in Section 2.2.2.
Whereas the S&Z model assumes that the normal contact stress is independent of  and  , Eqs. (3) and (4) explicitly include the effects of  and  on the normal stress distribution, which is described in Figs. 2(c) and 3. As illustrated in Fig. 3(a), the tangential and normal projection contact areas depend on  . When  = 0°, the conical tip converts into a flat punch. Thus, the contact pair has a face-to-face contact, as shown in Fig. 3(b). In this case, the normal stress distribution of the proposed model (the solid black line in Fig. 4) is found to be same as in the S&Z model (the red square in Fig. 4). Even for other scratch tips, of which  is not equal to zero, the difference between the two models may be ignored if the attack angle is small. For instance, for  = 20°, the maximum difference between the proposed (the dashed green line in Fig. 4) and S&Z models (the red square in Fig. 4) is less than 10%.
However, for a larger  , the difference has to be considered. For an extreme case with  = 90°, the shape of the conical tip emerges into a "needle-like" shape (as shown in Fig. 3(c)). The maximum normal   Fig. 4, this phenomenon can be described through the proposed model but not by the S&Z model.

Effects of substrate recovery on normal contact stress
As revealed in the literature [32−37], the significant influence of the substrate's elastic recovery has to be included. The resultant contact force existing on the rear half of the scratch tip actually pushes forward. As shown in Fig. 2(d), the recovery angle [0, / 2]    on the rear half of the contact surface is adopted to characterize the extent of the elastic recovery of the substrate. The included angle  changes to where  indicates the recovery angle. If  = 0, then we go back to case (c) or (b) of Fig. 2.
From the extreme case in which the normal stress at the edge of the contact surface is zero when  = 90°, as shown in Fig. 3(c), we obtain the following: The SCOF can then be formulated under consideration of the effect of the substrate recovery, contact geometry, and interfacial friction, namely, Here, A and C are functions of  and  , and B and D are functions of  ,  , and  . The detailed derivation of Eq. (7) of the proposed model can be found in the Appendix, where the computational details of the corresponding A, B, C, and D are also provided.

Results and discussion
The SCOFs calculated using the proposed and S&Z models, as well as obtained from the experimental data in the literature, are compared. In Sections 3.1 and 3.2, we discuss the effects of the contact geometry and interfacial friction, respectively, on the SCOF under a zero-recovery angle,  = 0°. In addition, the influence of the substrate recovery on the SCOF is discussed in Section 3.3.

Effects of contact geometry on the SCOF
The tangential scratch friction force consists of two components, i.e., the interfacial friction between the contact pair and the resistance from the deformed substrate material [25−28]. In an extreme case of  = 0°, i.e., when the conical tip has a 180° apex angle, as shown in Fig. 3(b), a face-to-face contact is found between the contact pair. The material flow on the contact surface is expected to be parallel to the scratch direction. Thus, the flow line parameter can be taken Eq. (8) converges to  2 tan π while  → 0°. This implies that the S&Z model is a good approximation for a small attack angle. However, for a large attack angle, the proposed model should be utilized to describe the SCOF. Ducret et al. [39] conducted scratch tests on UHMWPE under different constant normal loads with four attack angles, i.e., 15°, 30°, 45°, and 60°. The SCOFs from their experimental work, as well as from both analytical models, are plotted in Fig. 5. It is clear that, whereas the results from both analytical models agree well with the experimental data for a small attack angle  , the proposed analytical model provides a better prediction for cases with a large  . It should be noted that the value of s  used in both analytical models here is 0.03 [41−43]. Figure 6 shows the contours of the SCOF as the function of  and s  at two attack angles, i.e.,  = 10° and 60°, respectively. It is not surprising that, for a small attack angle, little difference is found between the two analytical models (as illustrated in Fig. 6(a)). However, for a large attack angle ( Fig. 6(b)), the SCOF predicted by the proposed model is larger than that by the S&Z model, and the difference (14%-36% for various values of  and s  ) cannot be ignored.

Effects of interfacial friction on the SCOF
It can also be seen from Fig. 6 that, for a given  , the SCOF increases with an increase in s  , and it increases more rapidly for a larger attack angle than a smaller one. The reason for this phenomenon is that,  whereas the friction-induced resistance dominates the scratch friction for a small attack angle  , the substrate deformation has a greater contribution to the scratch resistance for a large  . Felder et al. [21] numerically investigated a conical tip scratch for an elastic-perfect-plastic material with a small attack angle (  = 19.7°) when ignoring the recovery angle  . The authors took the interfacial friction coefficient to be 0, 0.05, 0.1, 0.15, and 0.2, respectively. The SCOFs of different values of s  from their numerical work and the two analytical models are shown in Fig. 7(a). It is clear that both analytical models show good agreement with Felder's work [21] when the attack angle is small.
Meanwhile, for large attack angles (  = 45° and 60°), the predicted SCOFs of both analytical models are compared with the numerical results [25], as plotted in Fig. 7(b). For a comparison with the results of the S&Z model [25],  is chosen as 3/2 and 4/3 for  = 45° and 60°, respectively. This shows that the proposed model provides a more accurate SCOF than the S&Z model.

Effects of the substrate recovery on the SCOF
Gauthier and Schirrer [33,34] conducted in-situ observations of the polymer scratch process. They found that an instantaneous recovery of the substrate material, after the tip passes by, will create contact with the rear part of the scratch tip. Thus, an additional resultant forward pushing force is applied at the tip. The substrate elastic recovery shows an important influence as can be phenomenologically determined from Fig. 2(d). A small change in the recovery angle  which will converge to a perfectly elastic contact. For a similar shape, a larger contact area on the rear half of the tip will induce a larger forward pushing force.
For two attack angles, i.e.,  = 10° and 60°, the contour planes of SCOF with different substrate recovery angles (  = 0°, 30°, and 60°) are shown in Fig. 9. When the overall shape of the contour planes of the SCOF remains similar for various attack angles, the recovery angle can clearly reduce the SCOF regardless of the attack angle. For  = 10°, the maximum relative differences in the SCOF between  = 0° and  = 30°, 60° reach 35.75% and 69.78%, respectively. The maximum relative differences reach 35.16% and 57.84%, respectively for  = 60°.
From the illustrations of the two tips (see Figs. 9(a) and 9(b), respectively), it can be seen that, for a small attack angle, such as  = 10° in Fig. 9(a), in which the conical tip is very close to a flat punch, the SCOF is dominated by the interfacial friction. However, for a large attack angle, such as  = 60° in Fig. 9(b), the deformation resistance of the substrate also has a significant contribution to the SCOF. Thus, the SCOF becomes larger.
Bucaille et al. [35] conducted a numerical simulation of a frictionless scratching of an elastic-perfect-plastic substrate using a conical tip with an attack angle of  = 19.7°. They obtained the SCOFs corresponding to various elastic recovery angles. In the extreme case of s  = 0, the proposed model is reduced to SCOF which is a function only related to  and  , whereas the S&Z model is reduced to a function related only to  . It is found that  shows no influence on the SCOF for either model. The SCOFs, as a function of the recovery angle, obtained from the proposed analytical model and the numerical simulation results [35], along with a constant value from the S&Z model, are given in Fig. 10. It is clear that the influence of the recovery angle described by the proposed analytical model agree well with the literature [35], whereas the S&Z model is not capable of this because no elastic recovery is included. It should be noted that the material pile-up in front of the scratch tip, commonly observed during the scratch process, has yet to been considered for the proposed analytical model. It is also known that the substrate recovery is closely correlated with the material properties such as the elastic modulus, yield strength [21,23], and viscoelastic characteristic [44][45][46][47][48], as well as the possible influence of the crystallinity on these properties [49], which are inherently important for the scratch friction performance of polymeric materials. A comprehensive analytical investigation of polymer scratching should be further conducted when considering the above factors.

Summary
In this paper, an analytical model is proposed to investigate the effects of the contact geometry, the interfacial friction, and the substrate elastic recovery on the scratch performance for a conical tip-polymer substrate system. The following conclusions were drawn: (1) For a small attack angle, the proposed analytical model is somewhat close to the S&Z model. For a large attack angle, the effects of the contact geometry, i.e., the included angle θ and the attack angle  , can be described more accurately using the proposed model.
(2) Validated through experimental and numerical studies, the proposed analytical model can describe well the scratch behavior with a consideration of the contact geometry, interfacial friction, and substrate elastic recovery.
(3) The proposed analytical model provides meaningful guidance to understanding the scratch friction behavior when using a conical tip.
A three-dimensional diagram of conical tip scratching on a polymer substrate is schematically depicted in Fig. 1(a). We assume a perfect contact between the conical tip and the polymer substrate in the proposed model. The contact surface, with all necessary variables established in the Cartesian coordinates, is shown in Fig. 1(b). The conical apex angle is defined as  . Line OD is the axis of the conical tip, and its length, d, is the instantaneous indentation depth of the tip. Line DB is the radius of the circle on the top contact surface, and its direction is in accordance with the scratch direction. The angle  represents the included angle between planes OCD and OBD, and an arbitrary chosen point K located on line OC on the contact surface has a distance of r from the original point O. During the scratch process, at point K, the normal and tangential stresses are denoted by  and  , respectively. Here, a simple formulation is given as where s  is the local friction coefficient on the contact surface. The normal stress  is perpendicular to line OC toward the outside of the conical surface located on plane OCD. The direction of  coincides with the material flow direction at K. The angle between  and line OC is defined as    , as shown in Fig. 1 where parameter  is utilized to characterize the material flows around the scratch tip. For simplicity, the tangential stress  is decomposed into  r along the line OC and   perpendicular to the plane OCD, respectively (see Fig. 1(b)). Therefore, we have Finally, the total tangential force T F and normal force N F applied on the scratch tip can be calculated by integrating t  and n  over the entire contact surface, respectively. The contact surface is symmetric about the x-z plane, as shown in Fig. 1(b). In this work, only one-half of the contact surface (i.e., surface OA'E'B or OAEB in Fig. 1(b)) is utilized in the integration. Therefore,