The indentation of flat surfaces deforming in the plastic regime by various geometries has been well studied. However, there is relatively little work investigating cylinders indenting plastically deforming surfaces. This work presents a simple solution to a cylindrical rigid frictionless punch indenting a half-space considering only perfectly plastic deformation. This is achieved using an adjusted slip line theory. In addition, volume conservation, pileup and sink-in are neglected, but the model can be corrected to account for it. The results agree very well with elastic-plastic finite element predictions for an example using typical steel properties. The agreement does diminish for very large deformations but is still within 5% at a contact radius to cylinder radius ratio of 0.78. A method to account for strain hardening is also proposed by using an effective yield strength.

## Introduction

Despite a large amount of work existing on the contact of spherical elastic-plastic surfaces, little exists on the contact of cylinders in elastic-plastic contact. Cylindrical contact and indentation are important for wheel contacts, indentation experiments, wire compression, the contact of filleted edges, and many other applications. Hertz first solved the case of elastic contact between two cylinders or curved surfaces that were modeled by parabolic profiles [1]. In the current work, we confine our investigation to cylinders in contact whose axis of symmetries are in parallel. This case is also often referred to as line contact where the cylinders touch is initially a single line. This line will grow in width as the load is increased, and therefore, become a rectangle. A cross section of a cylindrical indentation contact is shown in Fig. 1.

*R*is the radius of the cylinder,

*S*is the yield strength of the weaker material, and

_{y}*E** is the effective elastic modulus that is given by $1/E*=1\u2212\nu 12/E1+1\u2212\nu 22/E2$ for the contact of two dissimilar elastic materials in plane strain, denoted by subscripts 1 and 2.

*L*is the length of cylinder.

*C*depends on Poisson's ratio and the assumption used to obtain two-dimensionality (i.e., plane strain or plane stress). For plane stress,

*C*= 1, and $\nu 1$ and $\nu 2$ are effectively set to zero [3]. For plane strain,

*C*is given by

where *v* is the Poisson's ratio of the weaker material (i.e., the material with the lowest $CSy$). Note that the plane strain case is the main focus of the current work, but at deformations far beyond the point of initial yielding.

Unfortunately, very little work has been performed on cylindrical contact once the critical force has been surpassed and significant plastic deformation occurs. There have been a few studies implementing the finite element method [4–6], but there is still much room for additional investigations.

As the load continues to increase, the plastic deformation in the material will increase until the entire contact is deforming plastically. This is known as the fully plastic regime and is the focus of the current work. Although little work has been done to characterize it, based on spherical contacts, the fully plastic regime is expected to be reached at approximately *δ* = 100*⋅δ _{c}* [7,8]. However, elastic deformation will be in close proximity to the contact region until values much greater than this are reached. In the fully plastic regime, the average pressure acting on the surfaces is often referred to as the hardness. The fully plastic regime is of great interest to many contact situations, such as rough surface contact and indentation. The closest existing work to the current one is by Bower et al. [9]. Reference [9] focuses mostly on indentation incorporating creep. However, it does provide some limiting solutions, including that for a cylinder indenting a perfectly plastic solid. This solution from Ref. [9] will be compared to the results of the current work.

## Methodology

### Slip-Line Solution.

The current work uses the concept of slip-lines to find the relationship between the average pressure and the yield strength as a function of the size of the contact width, which is proportional to the magnitude of the deformation. The slip-line theory and derivation is not thoroughly described here, but additional details can be found in the book by Tabor [10] and Hencky [11]. A recent work aimed at spherical indentation also used a similar methodology [12]. The assumptions required for the employed slip-line theory are (1) quasi-static loading; (2) body forces are nil; and (3) the material is rigid-perfectly plastic and yields according to the von Mises criterion. A rigid-perfectly plastic material shows no elastic deformation and flows plastically once the yield criterion is met. Prior to the yield criterion being met, the material is rigid and without deformation. A uniaxial stress–strain curve of a rigid-perfectly plastic material would be as shown in Fig. 2.

The case of plane strain is also assumed, and therefore, strain is assumed to only occur in one plane, while all other strains are zero. For instance, if the *x*–*y* plane is employed, then the only strains that have nonzero values are the normal strains in the *x* and *y* directions (*ε _{x}* and

*ε*) and the shear strain in the

_{y}*x*–

*y*plane (

*γ*). No elasticity or friction are considered. The effects of pileup and sink-in due to elasticity and volume conservation are also neglected. Pileup is the raising of the material surrounding a contact due to displaced material. Note that volume conservation has been often neglected in contact mechanics solutions. However, this and the other assumptions are quantified by comparing the analytical predictions to a finite element prediction.

_{xy}*k*), so that the maximum shear stress is always tangent to the lines along their length. These are known as slip-lines (see Fig. 3). Two curvilinear and orthogonal lines (

*α*and

*β*) make up the slip-lines in a material and are described by the following equations in plane strain (these were originally derived by Hencky [11], but an English derivation can be found here [13]):

*h*is the hydrostatic stress and

*k*is the shear yield strength of the material. Along a slip line, Eqs. (5) and (6) must hold such that

*k*,

*C*

_{1}, and

*C*

_{2}are constant and only

*h*and the change in the angle of the line, $\Delta \varphi $, vary. Hydrostatic stress does not cause yielding or plastic deformation according to most conventional plastic deformation theories and to the von Mises yield criterion. For the analysis, only the

*α*line requires consideration. According to the von Mises yield criterion,

*k*is also related to the yield strength,

*S*, by

_{y}The current work considers the indentation of a cylinder into a perfectly plastic half-space in plane strain as shown schematically in Fig. 2. Note that the slip-lines shown in Fig. 2 are approximate. In the past, slip-line theory was often solved graphically and the drawing of the lines was very important. However, in the current work, a simplified mathematical approach is used, and therefore, the graphical depiction is not extremely important.

*p*, is applied to the indenter surface, but the interaction is frictionless so no shear stress along the surfaces occurs. Note that the pressure is not constant and can vary across the surface. It is very difficult to obtain the slip lines for a general problem, especially if the surfaces are curved. However, in this work, it is assumed that the slip lines follow the same results as shown in Tabor [10] for a wedge indenter. For a wedge indenter, the pressure at a location only depends on the change in the angle of a slip-line. If one considers that the curvature of a cylinder changes relatively smoothly, and without any sharp edges, it might be reasonable to assume that this result for wedges also holds for cylinders. Similar to that obtained in Tabor [10] on page 102, these defined conditions will result in the following equation:

where $\Delta \varphi $ is the change in the angle of the slip line from the free surface to the pressurized indenter surface. $\Delta \varphi $ and *θ* are equivalent as shown in Fig. 4, and *θ* is now used in the remainder of the derivation to condense the equations. *θ* will decrease from *π*/2 rad at the point of maximum indentation depth at the tip of the cylinder to an angle defined by where the free surface meets the cylinder. This will be 0 rad if the cylinder is pressed into the half-space until the free surface is perpendicular to the cylinder (and the indentation depth is *R*).

*F*, Eq. (8) is then integrated over the indenter contacting surface. Since the contact pressure is always normal to the surface and could be in a nonvertical or horizontal direction, only the vertical portion,

*p*, contributes to

_{v}*F*

*θ*

_{1}is the angle,

*θ*, at

*x*=

*b*. Next, Eq. (8) is substituted in for

*p*

*b*/

*R*as

Equation (18) is a closed form prediction of the average pressure occurring in the cylindrical indentation in the fully plastic regime (there is no significant elastic deformation). It also does not consider volume conservation, the effect of curvature on slip-lines, and the occurrence of pileup or sink-in. It also captures the trend of decreasing average pressure with deformation often observed in other curved contacts, such as in spheres [8,12,14,15]. Clearly, this pressure is not constant as has often been assumed when describing the hardness between contact surfaces. As *b*/*R* approaches 0, $p\xaf/Sy$ approaches an upper limit of approximately 2.968. Alternatively, as *b*/*R* approaches unity, $p\xaf/Sy$ approaches a lower limit of 1.908, which in reality may be difficult to realize due to the large deformations and likely pileup that will have occurred. Interestingly, the solution to Eq. (8) shown in Ref. [10] for a flat punch problem aligns closely with the current work. Interestingly, this solution is $p\xaf/Sy=2.968$, which coincides to the limiting case of the cylindrical contact when the contact area is vanishing in the current work.

Therefore, Ref. [9] also predicts that the average pressure does not change with the deformed geometry of the contact. However, Eq. (19) is exactly the limit of Eq. (18) derived for $b/R\u21920.$ The authors believe that the equation derived in the current work is a more complete model of cylindrical indentation, although it is limited by the assumptions noted previously.

## Finite Element Analysis

In order to verify the analytical results given by Eq. (18), a two-dimensional finite element model (FEM) has been developed in the commercial software ANSYS. The model simulates the indentation of a half space with a rigid cylinder. The materials were initially considered to deform elastic-perfectly plastically and in plane strain (later strain hardening will be considered). Typical material properties have been assumed (*E* = 200 GPa, *v* = 0.3) and the yield strength (*S _{y}*) is varied between 200 MPa and 1000 MPa. This differs from the assumptions of Eq. (18) as elastic deformation and volume conservation are included. Contact elements are used in between the rigid cylinder and the deformable flat. The contact elements use the augmented Lagrange method for enforcing contact and limiting penetration between the surfaces. The augmented Lagrange method is very similar to the pure penalty method, but it adjusts the contact force with a constant that is independent of the penetration stiffness. The contacting surfaces are also considered to be frictionless.

A fine mesh around the contact region has been applied on the flat surface. The cylinder is only considered by a rigid line and does not require meshing. Mesh convergence for both the small and the large deformations has been verified, and a mesh with a total of 67,288 elements has been chosen (see Fig. 5). Since the mesh is so fine in the contact region, it is difficult to discern. Therefore, an enlarged view is provided in Fig. 6. To apply the boundary conditions, the following constraints have been employed: the top edge of the flat has been fixed for displacements in the *y* direction (i.e., vertical) and the left edge of the flat have been fixed for displacements in the *x* or horizontal direction (this results in a symmetrical boundary). A uniform displacement downward on the top edge of the flat has been applied to load the flat against the rigid cylinder. This displacement is equivalent to the indentation depth, *δ*.

Different displacements have been applied, and the contact width and the resulting reaction forces have been recorded. The current results have focused on cases mainly in the fully plastic regime (approximately *δ* > 100*⋅δ _{c}*). Due to the contact and plasticity in the model, it is nonlinear and must be solved iteratively. In some cases of large deformations, the model has difficulty in converging. However, a large enough sampling of results has been achieved for the scope of this work.

## Results

After calculating the average contact pressure ($p\xaf$) and the contact width (*b*) for each case, they were normalized by the yield strength (*S _{y}*) and cylinder radius (

*R*), respectively. These results are plotted for several sets of material properties and alongside the predictions of Eq. (18) as shown in Fig. 7. Considering that the FEM model included elastic deformation and volume conservation, the agreement is surprisingly good. The agreement does diminish for very large deformations but is still within 5% up to a contact radius to cylinder radius ratio (

*b*/

*R*) of 0.78. At higher values of

*b*/

*R*, Eq. (18) underpredicts the FEM results, probably due to the effect of pileup, but it may also be due to other mechanisms. Later, a method for correcting this will be offered. Nonetheless, the differences between the FEM and slip-line theory in the pileup regions are no more than 7%. In much of the considered range, the differences between Eq. (18) and the FEM results for yield strengths of 200, 400, and 600 MPa are less than 1%. The results for the yield strengths of 800 and 1000 MPa also appear to lie about 2% above the predictions of Eq. (18). This may be due to elasticity becoming more influential since the yield strength has increased.

At some lower values of *b*/*R*, there is some disagreement between Eq. (18) and the FEM predictions, and this is due to the influence of elasticity (i.e., plastic deformation not completely dominating). This threshold can be further characterized by plotting the absolute error between each FEM case and the predictions of Eq. (18) as a function of the applied indentation depth normalized by the critical indentation depth (*δ*/*δ _{c}*). This is shown in Fig. 8. Note that

*δ*is calculated from Eq. (1). Although the initiation of the fully plastic region (i.e., when the entire contact region is plastically deforming) has not been characterized for cylindrical contact, for spherical contact, it is known to begin at approximately

_{c}*δ*/

*δ*= 100 [7,8]. Although the high errors can be noted at the low values of

_{c}*δ*/

*δ*, their point of initiation is difficult to recognize. Therefore, an enlarged view of the low

_{c}*δ*/

*δ*values region is included in Fig. 9. In Fig. 9, the influence of plastic deformation appears to cause the error to be above 5% at approximately

_{c}*δ*/

*δ*≈ 400.

_{c}The slow increase with error due to pileup can also be seen for larger values of *δ*/*δ _{c}* in Fig. 8. However, this error can be better correlated to the nondimensional contact width (

*b*/

*R*), as shown in Fig. 10. The presence of pileup can also clearly be seen in the comparison of the FEM predicted deformation on the von Mises stress distribution of a relatively small indentation depth to a large one, as shown in Fig. 11. In Fig. 11(a), there is not much recognizable pileup, while in Fig. 11(b) there is clear pileup. Note that the difference might not all be due to pileup alone as other assumptions were also made in deriving Eq. (18). It also appears that all the cases converge to a single line in this region, and therefore, might be corrected by a single equation. The error also does not appear to become influential until approximately

*b*/

*R*= 0.5. By fitting to the error, the following simple corrective equation was found:

where $p\xaf/Sy$ is calculated from Eq. (18). Equation (20) is only used for *b*/*R* > 0.5. When *b*/*R* < 0.5, Eq. (18) should be used. The predictions of Eq. (20) are also shown alongside the FEM results and Eq. (18) in Fig. 7. This correction results in a model that is in extremely good agreement with all of the FEM results, excluding those closer to the elastic-plastic regime (*δ/δ _{c}* > 400). The average error between the adjusted model (Eqs. (18) and (20)) and the FEM results is only 0.8% with a maximum error of 4.6%. This model could also be used in other materials which show plastic deformation failure modes and ratios of (0.001 <

*E*/

*S*< 0.01). A wider range of usage may also be possible, but has not been explored.

_{y}It should be noted that although Eqs. (18) and (20) relate the average pressure to the contact width, it can also be used to relate the contact force to the contact area for a fully plastic cylindrical indentation. The process may be iterative because it is unknown what the contact width, *b*, or average contact pressure to yield strength ratio, $p\xaf/Sy$, will be initially. First, an initial guess of the ratio can be made of 2.97. Then, the contact area can be calculated by dividing the force by the average pressure. Using the calculated contact width, *b*, the average pressure can then be updated by Eqs. (18) and (20). This would then be repeated iteratively until the contact area and average pressure are found to the desired accuracy.

*E*. Therefore, to consider the effect of hardening, the tangent modulus (

_{t}*E*) is varied to values of 0.02⋅

_{t}*E*, 0.05⋅

*E*, and 0.1⋅

*E*, which covers the typical range of most metals [16]. The results of the finite element model considering these degrees of hardening along with an original elastic-perfectly plastic case are shown in Fig. 12 (E = 200 GPa,

*v*= 0.3, and

*S*= 200 MPa). In Fig. 12, the pressures are normalized by an effective yield strength,

_{y}*S*, that has taken into account the bilinear strain hardening. The following

_{e}*approximate*prediction of the effective yield strength,

*S*, is found by fitting to the finite element data:

_{e}Equation (21) can then be used to replace *S _{y}* with

*S*in all equations in order to account for the effect of strain hardening. Note that Eq. (21) follows a form similar to that in Ref. [3]. The form of Eq. (21) is based on the effective strain approximation by Tabor for indentation [10]. As

_{e}*E*approaches nil, the effective yield strength,

_{t}*S*, predicted by Eq. (21) approaches

_{e}*S*(the perfectly plastic case). When Eqs. (18), (20), and (21) are implemented together, the finite element data and the models differ on average by 1.0% and by no more than 3.9%. The predicted values of

_{y}*S*are up to tenfold greater than

_{e}*S*in the cases considered in this work due to hardening. This confirms that the effect of hardening is indeed very significant. Note that care should be taken if employing these equations outside of the range of material properties considered in this work.

_{y}## Conclusion

There has been a relatively scarce amount of work on the contact of elastic-plastic cylinders or the more simple case of indentation of a rigid cylinder into an elastic-plastic flat. First, this work derives analytically using slip-line theory, a prediction of the average pressure occurring in the indentation of a rigid frictionless cylindrical punch into a perfectly plastic flat surface. Theoretically, the upper limit to the pressure appears to be a factor of approximately 2.97 times the yield strength. However, as indentation is increased and the contact effectively changes shape, the pressure reduces. The lower limit is approximately 1.91. This is also confirmed by a FEM analysis of a rigid cylinder indenting an elastic-plastic flat. The FEM and slip-line predictions are in surprisingly good agreement. Above the region where elasticity influences the problem, the error is less than 1% for most cases, and less than 5% until *b*/*R* reaches approximately 0.78. An adjustment for pileup is also made to the model for *b*/*R* > 0.5. In addition, the effect of bilinear strain hardening has been analyzed using finite element results. An effective yield strength effectively brings all cases of perfectly plastic and bilinear hardening cases into good agreement. Therefore, the resulting model has very good agreement for all the fully plastic FEM cases, although care should still be taken when employing the equations outside of the range of material properties and deformations considered in this work.

## Funding Data

Directorate for Engineering (Grant No. 1362126).