`ISRN TribologyVolume 2013 (2013), Article ID 871634, 13 pageshttp://dx.doi.org/10.5402/2013/871634`
Research Article

Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India

Received 30 October 2012; Accepted 19 November 2012

Academic Editors: J. Antunes and F. Findik

Copyright © 2013 Biplab Chatterjee and Prasanta Sahoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 1. Introduction

The present investigation therefore aims at studying the effect of strain hardening and hardening models on contact parameters during multiple normal loading-unloading of a deformable sphere against a rigid flat under full stick contact condition using commercial finite element software ANSYS.

#### 2. Theoretical Background

Figure 1 shows the contact of a deformable sphere and a rigid flat. The dashed line presents the original contour of the sphere, having a radius of , and the rigid flat before the deformation. The solid line shows the loading phase with the interference (), corresponding to the contact radius (), and the contact load ().

Figure 1: A deformable sphere pressed by a rigid flat.

Figure 2: Three different profiles of the sphere.

Figure 3: (a) Isotropic and (b) kinematic hardening models for two-dimensional stress field.

#### 3. Finite Element Model

A commercial finite element package ANSYS 11.0 is used in the present study in order to get the accurate solution of a complicated subject like multiple loading-unloading under full stick contact condition. The sphere is represented by a quarter of a circle, due to its axisymmetry. A line models the rigid flat. The elements as shown in Figure 4 consists the mesh of maximum 18653 number of plane183 finite elements. Plane 183 is six-node triangular axisymmetric element. The mesh density at the bottom of the sphere is coarsest one and is made gradually finer towards the sphere summit. The finest mesh density near the contact region simultaneously allows the sphere’s curvature to be captured and accurately simulated during deformation with a reduction in computation time. Window 2 of Figure 4 presents the enlarged view of the finest mesh density at sphere summit. The detail description and boundary condition of FE model can be found in [25].

Figure 4: Finite element mesh of a sphere generated by ANSYS.

#### 4. Results and Discussion

Engineering stress-strain curves are used within elastic limit. The dimension of the specimen changes substantially in the region of plastic deformation. The increment of strain in conjunction with true stress can be termed as strain hardening. Strain hardening causes an increase in strength and hardness of the metal. Strain hardening is expressed in terms of tangent modulus (), which is the slope of the stress-strain curve. Below the proportional limit the tangent modulus is the same as the Young’s modulus (). Above the proportional limit the tangent modulus varies with the strain. The tangent modulus is useful in describing the behavior of materials that have been stressed beyond the elastic region. In elastic perfectly plastic cases, the tangent modulus becomes zero. Very few material exhibit elastic perfectly plastic behaviors, generally all the materials follow the multilinear behavior with some tangent modulus. This multi-linear behavior can be assumed as bilinear behavior for analysis purpose in elastic-plastic cases. In this analysis a bilinear material property, as shown in Figure 5, is used for the deformable sphere. Shankar and Mayuram [27] mentioned that the tangent modulus for the most practical materials is less than , whereas Kadin et al. [16] found the tangent modulus for most practical materials below . However both the authors used tangent modulus up to for analytical purpose. On the other hand, Ovcharenko et al. [28] used stainless steel specimen with tangent modulus of (Figure 6(b)) in their in situ investigation). It is also available in literature that structural steel, aluminum alloys have significant amount of strain hardening.

Figure 5: Stress-strain diagram for a material with bilinear properties.

##### 4.1. Interfacial Parameters with Low Tangent Modulus

Figure 7: Normalized contact load, , as a function of loading cycles with (a) bilinear isotropic hardening, (b) bilinear kinematic hardening.

Figure 8: Normalized residual interference, /, as a function of loading cycles with (a) bilinear isotropic hardening, (b) bilinear kinematic hardening.

Figure 8(b) shows the normalized residual displacement, /, as a function of loading cycles with bilinear kinematic hardening. It is revealed from the figure that after ten loading unloading, the normalized residual displacement of elastic perfectly plastic material is 3.61, 8.26, and 20.57 percent higher than that with the materials having tangent modulus of , , and , respectively. Comparing the results with isotropic and kinematic hardening, it is observed that using kinematic hardening yields less residual displacement than that with isotropic hardening. This is more pronounced for the materials with high tangent modulus. Zolotarevskiy et al. [29] studied the elastic plastic spherical contact under cyclic tangential loading in presliding with 2% hardening and found the tangential displacement at the completion of the first cycle is less in the case of kinematic hardening compared to that in the isotropic hardening. The agreement between Zolotarevskiy et al. and present results is excellent.

##### 4.2. Interfacial Parameters with High Tangent Modulus

It is necessary to investigate the effect of strain hardening and hardening rule on interfacial parameters with higher tangent modulus to understand the response of the materials with significant amount of strain hardening such as stainless steel, structural steel and aluminum alloys in repeated loading unloading. In our forth-coming simulations, the tangent modulus () is varied according to a hardening parameter (). The hardening parameter is defined as . We have considered four different values of , covering wide range of tangent modulus to depict the effect of strain hardening in single asperity multiple loading unloading contact analysis with other material properties being constant. The values of used in this analysis are within range as most of the practical materials falls in this range [30]. The value of equals to zero indicates elastic perfectly plastic material behavior, which is an idealized material behavior. The hardening parameters used for this analysis and their corresponding values are shown in Table 1.

Table 1: Different and values used for the study of strain hardening effect.
###### 4.2.1. Mean Contact Pressure Distribution

Figure 10(c) represents the evolution of dimensionless mean contact pressure at the end of each seven loading cycles for the maximum dimensionless interference of . The evolution of mean contact pressure clearly indicates the same qualitative trend of Figures 10(a) and 10(b). It can be seen from the figure that the evolution of mean contact pressure with isotropic hardening is higher than that with kinematic hardening. The dimensionless mean contact pressures with isotropic hardening are 6.4, 8.9, and 8.8 percent higher after first loading than that with kinematic hardening for the materials having tangent modulus of , , and , respectively.

Kral et al. [17] simulated repeated indentation of a half space by a rigid sphere with isotropic hardening. They used three strain hardening exponent as 0, 0.3, and 0.5 and indentation load up to 300 times the load necessary for initial yield. Kral et al. inferred that the distribution of contact pressure over the contact area increases with increasing load and strain hardening characteristics (Figures 4(b) and 8). There is no scope of comparison as we have simulated with a deformable sphere against a rigid flat but the qualitative natures of present results (Figures 10(a)10(c)) are in good agreement with the results of Kral et al.

###### 4.2.3. Residual Displacements

Figure 14(a) presents the increment of residual interference with the increase of unloading cycles compared with the residual interference after first unloading. is the residual interference after first unloading. It reveals from the figure that the increases in residual interferences with isotropic hardening are significantly more than that with kinematic hardening. The figure also demonstrates that the increment of residual interference with the increase in unloading cycles depends on tangent modulus. Higher tangent modulus yields slightly less increment of residual interference with the increase in unloading cycles.

Figure 15: Normalized contact area, , as a function of the number of loading cycles; (a) comparison of different studies; (b) comparison of hardening models for maximum loading, .

Figure 15(b) shows the growth of normalized contact area after each loading cycle with four different tangent modules using isotropic and kinematic hardening. It is evident from the plot that the contact area increases slightly after each loading cycles indicating marginal effect of strain hardening in isotropic hardening. The materials with kinematic hardening exhibit very less increase of contact area during ten loading cycles with maximum dimensionless interference of loading . Small discrepancy was observed for the results of higher tangent modulus ( and ) in kinematic hardening, probably due to the discretization of contact elements as discussed earlier.

#### 5. Conclusion

The residual interference varies with both tangent modulus and hardening model. The residual interference decreases with the increase in tangent modulus and it is significantly higher in isotropic hardening compared to that with kinematic hardening. The increase in residual interference with the increasing number of unloading cycles was severely affected by hardening model at high tangent modulus. The contact area increases slightly after each loading cycles in isotropic hardening whereas the materials with kinematic hardening exhibit very less increase of contact area. The growth of contact area was found to be independent of strain hardening but increases with the increase in intensity of loading. The effect of hardening model on contact parameters at high tangent modulus clearly indicates different shakedown behavior for isotropic and kinematic hardening for repeated normal loading unloading under full stick contact condition. The analysis of hysteretic loops at high tangent modulus is an interesting field for future study.

#### Nomenclature

 Contact area radius : Modulus of elasticity of the sphere : Yield Strength of the sphere material : Real contact area : Radius of the sphere : Contact load : Interference : Poisson’s ratio of sphere : Mean contact pressure : Tangent modulus of the sphere : Dimensionless contact load, , in stick contact : Dimensionless contact area, , in stick contact : Dimensionless interference, /, in stick contact.
Subscripts
Superscripts
 : Dimensionless.

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