# The Static Frictional Behaviors of Rubber for Pipe-Laying Operation
## 4 Discussion
### Tangential Load Condition
#### 4.2.2 Increasing Rate of the Tangential Load
The schematic cutaway views of the contact are shown in Figure 11 when the increasing rate of the \(F_{T}\) is 0.062 N/s. Even when the fold forms at the front edge of the contact area (Figure 11b 1), its scale is smaller than that of which is shown in Figure 10b 1. It indicates that the motion of the pipe affects the formation of the fold. The pipe cannot move forward fast enough to supply sufficient space to initiate a wider fold. In addition, the fold lasts until the central adhesive zone shrinks, and the fold rarely appeared after the first fold is vanishing. The subsequent deformation of the contact area behaves gentler than that of high increasing rate of \(F_{T}\). Furthermore, the expansion of the apparent contact area (from Figure 11b 2 to Figure 11c 4) indicates that compression stress is concentrated at the rear edge of the contact by a rotation of the rubber sample which is caused by the slow increasing tangential load, whereas the front edge is lifted up from the pipe surface (Figure 11c 3). The shear stress which is concentrated at the central adhesive zone increases gradually with increase of the tangential load. Meanwhile, it is released when the partial slip zone replaces the central adhesive zone. Finally, the shear stress is beyond the limitation of the adhesive junction between the interfaces, the macroscopic slip is emerging. This stage lasts a long time due to the low increasing rate of the tangential load, and the viscoelastic property of rubber plays important role in affecting the separation of the interfaces. The creep deformation is a viscoelastic characteristic of rubber materials and it much depends on time [28; 29]. The strain of the rubber surface which sustains a constant load will exponentially increase to the limiting value [30]. The fact that viscoelastic deformation can affect the level of static friction of rubber had been verified by [PERSON] [31]. In his paper, when a rigid sphere is pressed against a smooth rubber substrate, the tangential load started to increase and the partial slip zone enlarged step by step, then the tangential load stops increasing and was holding for two minutes. During the holding time, when the partial slip zone occupied the whole apparent contact area gradually, the macroscopic slip was emerging. This phenomenon means that the viscoelastic deformation of the rubber surface decreases the adhesive junctions between interfaces. Even the maximum value of the tangential load which initiates full slip is not attached, and the contact area can also change its appearance. Furthermore, when there is an increasing load instead of the constant one, the increment of strain can be superposition. Figure 11: Schematic cutaway views of the rubber A-PMMA contact during a whole test run. Normal load: 30 N; Increasing rate of \(F_{T}\): 0.062 N/s; (**a**) Stable contact area without tangential load; (**b**) Few folds appear as the tangential load increases; (**c**) Deformation of the contact area due to a rotation of the rubber sample; (**d**) The instant of the macroscopic slip. Based on this principle, we have deduced the creep deformation on the rubber surface at the rear edge of the contact area. As is shown in Figure 12a, there are many adhesive junctions which can be seen as Standard Linear Solid models (SLS Model) between the interfaces. These adhesive junctions will be stretched and the rubber surface is dragged towards the limitation of elastic deformation (Figure 12b). The creep deformation is becoming active as the tangential loading time is growing. However, the creep deformation cannot hold more shear stress. On the contrary, as the tangential load is continuously increasing, some of the adhesive junctions (Figure 12c 1,2,7) take the lead in enlarging their strains by creep deformation, because they are reaching the limitation of elastic deformation ahead of the other adhesive junctions due to their locations. The other adhesive junctions do not have enough time to take creep deformation. As a consequence, they are broken by displacement between the interfaces.