Contact Around a Sharp Corner with Small Scale Plasticity

Owing to elastic singularity, the contact stress around a sharp corner is highly sensitive to the boundary conditions and local geometrical details. Determination of such stress is critical in predicting failures such as wear, fretting fatigue and crack initiation. In this paper, the stress around such corner is analyzed based on linear elasticity and small scale plasticity. The stress on the contact interface is generalized in a way that the results can be easily converted to represent another corner with different dimensions or boundary conditions. An example is presented to show the determination of the stress scale and the formulation of a generalized solution. It is shown that the generalized macro stress field away from the corner dominates the contact behaviors around the corner.


Introduction
The stress field around a contact interface with a sharp corner is closely related to various failures such as wear, fatigue and plastic yielding, which significantly limit the life of various engineering components [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The analysis of contact stress around these area is important in understanding the onset and propagation of those failures: it will provide guidelines for the design of contact components so that the damage can be limited [11,12,[15][16][17][18][19][20][21]. Based on experimental observations, the sliding behavior between two elastic bodies in contact can be described by three regimes on a map of tangential and normal loads: full-stick, partial-slip and gross-sliding regimes [22][23][24]. The damage is determined by the energy dissipation process in the slip regions [22,25]. Such identification is crucial for predicting the life of many engineering components. For instance, studies have shown that crack growth is accelerated greatly at the boundary between the slip and stick zones, reducing the fatigue life significantly [25][26][27][28][29][30][31][32].
If the two contacting components are assumed to be elastic bodies with a constant coefficient of friction, the state of stick or slip at the corner of the contact interface, where partial slip regimes are often located [22,[33][34][35][36][37][38][39][40], is determined by the coefficient of friction and the stress-intensity factors for the singular stresses at the corner [41]. The stress state at the corner of a solid attached to a half-plane is equivalent to that of a notch with the same corner angle [42,43]. An in-depth analysis has shown that the contact stress between an elastic block and a half-plane can be treated as an eigenvalue problem [44]. The stress singularity at the corner has an exponent that is the same value for both the shear and normal compressive contract stresses, and is a function of the contact angle. For instance, a 90° corner has an exponent of -0.4555 [45]. The ratio between the interfacial shear and normal stresses pressure is independent of the applied forces, and only depends on the geometry. This leads to a critical friction coefficient of 0.543 (for the 90° corner) above which the interface sticks. Below this value of friction coefficient, slip will always occur at the corner even in the absence of an applied shear force [46].
In addition to the geometrical details, plasticity can affect frictional behavior in other ways. For example, yield can occur away from the interface. For example, below a critical normal load, yield occurs at the interface of a sphere on a flat surface, but it occurs below the contact region at higher loads [41], [47]. The effect of yielding can potentially significantly affect the stress field at the contact surface by relaxing the contact stresses. One methodology to study the plasticity of metals in microscale is to develop a statistical volume element (SVE) which takes into account the variability in the microstructure. Different from representative volume element (RVE) in which average value is assumed for all realizations, the SVE model can have a different value from one realization to another. SVE models have been developed to study polycrystalline microstructures. Zhang et al. developed a novel SVE model to improve computational efficiency compared to traditional RVE models [48]. The key grain features, including orientation, misorientation, grain size, grain shape, grain aspect ratio are considered in their SVE model. A slip-based damage criterion is developed to study the plasticity of aluminum alloy at the microscale [49]. Their study shows that the multiscale damage criterion can successfully characterize the microscale plasticity of FCC structure alloys and predict plasticity based damage evolution.
The goal of this paper is to investigate the effects of slip and small scale plasticity on the stress field on contacting surfaces, with a particular emphasis on the behavior at corners. By incorporating those contributions into the frictional analysis, it is expected that the stick and slip zones can be predicted more reliably, which will be important in subsequent investigation to model provide the much needed fundamentals for related studies such as fretting fatigue and wear. The investigations are done using a model configuration of a block on a flat surface, and comparing the slip behavior along the interface when Coulomb's law is assumed to the slip behavior when the effects of plasticity is included.

Parameters of the Model
This study will focus on a commonly used contact setup: the plane stress model of a rectangular block in contact with a large substrate of the same material. As shown in Fig. 1, the bottom of the substrate is fixed in both the horizontal and vertical directions. It was confirmed that the substrate is large enough for boundary effects to be neglected for small scale plasticity (Slip zone or plastic zone size is less than 10% of the contact interface). The indenter is subjected to a uniformly distributed normal pressure load, p and a uniformly distributed tangential load, q. The vertical load is applied first, as a preload, and then the tangential load is applied to investigate the slip behavior. The response at the contact interface depends on factors such as the magnitude of p and q, the young's modulus E, and the dimensions of the system. The nomenclature used in this paper is given in Table  1, and values of the non-dimensional parameters used in the simulations are given in Table 2.

Friction Law
Coulomb's friction law describes the relationship between the frictional force and the normal load by means of a constant friction coefficient. Based on this law, slip will occur when the condition as follows is satisfied, where µ is the coefficient of friction, τ i is the shear stress on the interface, σ i is the normal stress on the interface.

Plastic Deformation
We consider plasticity by assuming the material to be elastic / perfectly-plastic during the loading step. A von Mises yielding criteria is used, i.e. yielding will occur if the von Mises stress as shown below reaches the yielding strength.
Here, the x σ and y σ are the stress components in x and y direction.

Scaling of the System
A contact with little slip is elastically similar to the notch problem [50]. We can normalize the length and stress field of the system with the scales defined based on the solution of an infinite large wedge [51,52] to make the results general. In such a case the detailed loading and geometry of the model are not important; what matters are the stress-intensity factors. We will also use the asymptotic stress solution as benchmark to validate our numerical simulations. When the slip is large, such a normalization approach does not offer a benefit anymore. Instead, we will use the physical dimension and applied shear load to normalize the length and stress.
The asymptotic stress solution of the notch problem can be written as [53] where K I and K II are the stress-intensity factors corresponding to the eigenvalues I λ and II λ respectively, r is the distance from the wedge corner, and θ is the angle shown in Fig. 1 We define a length scale 0 x and a stress scale 0 τ by Note that the normalized stresses do not depend on the dimensions of the system or on the loading conditions, which makes the analysis more general. Along the interface, these equations reduce to 1 1 Note that Eqs. 13  depend on the loads and geometry, can be obtained by analyzing the stress field around the corner. We have built finite-element models (FEM) using ABAQUS. Figure 2 shows an example with 25881 elements. At the corner of the contact at the interface, where a singularity occurs, the element size is refined to be about 10 11 smaller than that of the regular elements in order to capture the stress gradient.
for all elements on a circle close to and around the corner, as shown in the bottom of Fig. 2.

An Example to Determine Stress and Length Scales
The loading conditions and dimensions in Figure 3 (a) and Table 2 are used to illustrate the procedure of determining the stress and length scales. When x is small, the second term of Eq. (2) is negligible, and this equation can be rewritten as, The second order intensity factor K II becomes significant when the radius is relatively large, so it can be calculated by fitting the stress field of a circle region as is shown in the magnified view in Figure 2. Now Eq. (2) is rewritten as, Based on the K I and K II obtained above and Eqs. (9) and (10) can obtained which can represent any loading conditions or dimensions that has the same corner angle. This generality will be illustrated in the next section.   Figure 5. The second stress intensity factor KII can also be determined by curve fitting.

Generalized Stress Around a Sharp Corner
In order to illustrate the generality of the normalization approach using 0 x and 0 τ , the stress fields under 3 different loading conditions and system dimensions, as shown in Fig.  3, were computed. By repeating the calculation procedures in section 3.2, the length term scale 0 x and stress term 0 τ can be obtained. Figure 6 shows that with elastic analysis the three cases shown in Fig. 3 give the same normalized normal and shear stresses along the interface. The contact shear stress and contact pressure are also modeled when there is a normalized yield stress of 0 y σ τ =25, showing that all the plastic and elastic curves collapse outside the plastic zone. So the dimensions and loading conditions do not influence the normalized result. In other words, the normalized slip results will be representative of various combinations of loading conditions and dimensions as long as they have the same K field.

Conclusions
The slip behaviors around a sharp corner is highly sensitive to the specific loading condition, corner angle, dimensions and local yielding. In many applications, only small portion of the contact interface has slip or yielding. In this case, the contact behaviors generally follows the elastic solution. In this paper, a generalized form of the elastic stress filed is formulated. It is found this generalized form can well describe the stress field even if a small slip or plastic yielding zone is present around the corner. This generalized solution is independent on the loading condition or dimensions and therefore can be easily scaled to represent various boundary conditions or geometries.