Unloading Behavior of Elastic-power-law Strain Hardening Materials Indented by Elastic Indenter

Both strain hardening and indenter elastic deformation usually cannot be neglected in engineering contacts. By the finite element (FE) method, this paper investigates the unloading behavior of elastic-power-law strain-hardening half-space frictionlessly indented by elastic sphere for systematic materials. The effects of strain hardening and indenter elasticity on the unloading curve, cavity profile during unloading and residual indentation are analyzed. The unloading curve is observed to follow a power-law relationship, whose exponent is sensitive to strain hardening but independent upon indenter elastic deformation. Based on the power-law relationship of the unloading curve and the expression of the residual indentation fitted from the FE data, an explicit theoretical unloading law is developed. Its suitability is validated numerically and experimentally by strain hardening materials contacted by elastic indenter or rigid flat.


Introduction
Fundamental to all assembled systems, contact mechanics is integral to mechanical design. This is evident in various material processes and engineering applications, such as hardness measurement [1,2], particle and powder interactions [3,4], particle erosion [5], thermal spray [6][7][8], electrical contact [9][10][11], biomechanics [12][13][14][15] and additive manufacturing [16,17]. Surface curvature or roughness often causes the extremely small contact area, resulting highly local deformation and stress concentrations. Most local contact events are characterized by plastic deformation [18,19] that reduces the contact force, produces a permanent indentation and makes the unloading path to be different from the loading path. Due to the simplicity and motive by the hardness tests, most studies focus on the deformation analysis of the contact of elastic-plastic material by rigid indenter or rigid flat.
A complete elastic-plastic contact cycle consists of both the loading and unloading phases. For elastic contact problem, Hertz contact model [20] is widely applied. Using the methodology originated from Hertz's analysis of the contact between two nonconforming elastic solids [20], two main types of elastic-plastic contact models consisting of both the loading and unloading laws were developed, including the indenting models [18,[21][22][23][24][25][26][27][28] and the flattening models [29][30][31][32][33][34][35]. These contact models are derived from the theoretical analyses or by curve-fitting approach from the numerical results. Recently, both the deformations of the sphere and the half-space are considered by Dong, et al. [36], Ghaednia and Jackson [37], Ghaednia, Dan [38] and Cermik, Ghaednia [39]. Although the loading law is comprehensively studied, the unloading law is investigated insufficiently.
To predict the elastic-plastic contact behavior accurately, the unloading phase is as important as the loading phase [40][41][42][43][44]. The measurement of material properties, like yield strength, and elastic modulus is always relied on the analysis of the unloading phase [45]. However, the analysis of unloading phase is cumbersome and difficult due to the complicated deformation behavior, such as shallowing [46], pile-up [47][48][49][50][51], reyielding in the surface region just outside the contact area [47,51] and high residual stress [51]. Few studies have been devoted to the unloading behavior of elastic-plastic contact. The previous studies mainly aimed at the unloading behavior of the elasticperfectly plastic materials.
For the elastic-perfectly plastic materials, the unloading is generally assumed to be elastic [24-26, 36, 43, 52, 53]. The unloading law is usually characterized by the Hertzian-type and non-Hertzian-type. For Hertzian-type unloading laws, one of r  and *e R should be determined in advance and then another one can be calculated according to continuity condition between loading phase and unloading phase.
For some Hertzian-type unloading laws, the expression of r  is fitted from the FE data or from the limited experimental data [22,38,54]. For other Hertzian-type unloading laws, *e R is solved by some theoretical assumptions [25, 26, 28, 30-32, 36, 43, 52, 53, 55-58]. Du and Wang [55] enforced *e R to be the effective contact radius * R . Thornton [30] assumed a continuity of a defined contact area to calculate *e R . Li, Wu [52] solved *e R from the assumed parabola relation between indentation and contact area. Vu-Quoc, Zhang [32] calculated *e R by a defined relationship of *e R with contact force. Stronge [25] solved *e R by assuming a geometry similarity. Dong, Yin [36] and Chen, Yin [43] revised Strong's geometry similarity to calculate *e R by introducing a scaling factor. Different solving equations for *e R were suggested by Brake [53] and Big-Alabo, Harrison [28] for unloading during elastic-plastic and fully plastic loading phases, respectively.
For non-Hertzian-type unloading laws, Li and Gu [44] has presented the empirical relation of contact force to indentation by curve-fitting approach from the FE and experimental results. Christoforou, Yigit [59], Christoforou, Yigit [60] and Ibrahim and Yigit [61] simplified the unloading law by the linearizing approach.
In fact, most engineering materials exhibit strain hardening during inelastic deformation, which causes the yield strength to increase with the evolution of plasticity [62]. The investigations of unloading behavior are mainly devoted to two types of strain  , the exponents are found to depend on the power-law strain-hardening exponent for the frictionless contact by Zhao, Nagao [34], or depend on both the power-law strain-hardening exponent and Passion's ratio for the stick contact by Zhao, Zhang [35], but be independent upon the yield strain.
However, Song and Komvopoulos [62] found that the residual indentation is dependent on both the power-law strain-hardening exponent and yield strain.
Till now, most studies are focused on the unloading behavior of the elasticperfectly plastic materials or the strain hardening elastic-plastic materials contacted by a rigid sphere or a rigid flat. Taljat, et al. [45,69] found that there are discernible or large differences between the unloading force-indentation curves obtained by the experimental test and calculated by the FE method when the indenter is modeled as rigid. They pointed out that even though the ratio of elastic moduli between the indenter and indented materials is at least three times, the indenter compliance cannot be neglected. Rodriguez, Alcala [70] showed that there are 26% and 17% permanent indentation errors for the spherical indenter and for the conical and Berkovich indenters, respectively, even though the reduce modulus are used in the conventional model to take into account the indenter elastic deformation. For the unloading behavior of elasticperfectly plastic materials, the indenter elastic deformation has been considered and found to have significant influence recently by Dong, Yin [36] and Chen, Yin [43]. A comprehensive study of unloading behavior for power-law strain hardening materials accounting for the indenter elasticity is still lacked.
The objective of this paper is to study comprehensively the unloading behavior of the frictionless contacts for a wide range of material combinations between elastic sphere and elastic-plastic half-space by considering material strain hardening effect.
Abundant information on unloading is provided by intensive FE simulations. Based on the FE results, the effects of both strain hardening and indenter elasticity on the unloading curve, cavity profile during unloading and residual indentation are analyzed.
Analytical expression of the residual indentation is fitted from the FE data. Then the unloading law is developed based on the analysis of the unloading curve and the expression of the residual indentation. The unloading law is sufficiently validated numerically and experimentally. It demonstrates that the present unloading law is accuracy and can be suitable for elastic-plastic materials with or without strain hardening contacted by elastic indenter or rigid flat ranging from small to large deformations.

Problem description and elastic unloading law
As a discriminating feature for the contact problem, the profile of the indented surface is characterized by pile-up or sink-in depending on the hardening properties of material [71,72]. By considering the hardening strain effect, Fig. 1

Sink-in
Pile-up 1 1 During the loading phase, the sphere penetrates the half-space to a total indentation depth  with an indentation of the half-space 1  , an indentation of the sphere where * 1  (6) In this paper, to investigate more general unloading law, the normalized contact

FE model
In this section, the FE model of an elastic-plastic half-space indented by an elastic sphere is built to consider the effects of both strain hardening and indenter elasticity on unloading behavior. The half-space material is assumed to be elastic-plastic, whose plastic behavior obeys the 2 J flow theory and satisfies a power-law strain hardening law reconstructed by the classical Ramberg-Osgood curve [75]. The deformation gradient is decomposed into elastic and plastic components, and the equivalent plastic strain eq  is defined as where the integration is carried out over the strain path  , and The relation of the strain  to the stress  for an elastic-power-law strainhardening material is given as where n is the strain hardening exponent. 0 n  indicates the elastic-perfectly plastic case, which can be consider as a special strain hardening case.
To ensure the computational accuracy, the contact area is meshed regularly and densely as shown in Fig. 2 (b). A coarse mesh is selected in the low stress zone far away from the contact area, so as to improve the computation efficiency. In order to avoid numerical errors caused by a sudden change in the mesh size, the element size gradually increases in a radial gradient for outside high stress zone as shown in Fig. 2 (c). The value of stiffness for the contact element and the tolerance of the current work are set according to the mesh convergence tests. The contact area is controlled to be meshed at least 80 contact elements in contact for each applied indentation according to the convergence tests. As the indentation increases, the contact zone evolves from the small deformation in early stage to large deformation. To achieve the balance at computations in the full indentation range, the sequential finite element sub-models are designed in order to resolve indentations ranging from small to large deformation. In the present study, / aR is designed to range from 0.002 to 0.42. For the fast modeling, the finite element sub-models are designed parametrically utilizing the APDL language and created automatically by ANSYS software. For different sub-models, the half-space consists of 10434~13074 elements with a total of 10552~13342 nodes, and the sphere consists of 7200~10800 elements with a total of 7411~11101 nodes.
To verify the accuracy of the FE model, the half-space ( 800 L  mm) indented by the sphere ( 35 R  mm) is simulated for both loading and unloading phases, where The loading and unloading curves are almost the same. The simulation errors of the maximum contact force and maximum contact radius are 1.2% and 2.2%, respectively.
The close agreements between the FE simulation results and the Hertz solutions validate the modeling assumptions and the suitability of the FE mesh in Fig. 2.

Results and Discussion
By means of the FE models developed above, the unloading behavior of elasticplastic contact is investigated for the elastic-power-law strain-hardening materials

Strain hardening effect
Most engineering metals and alloys exhibit strain hardening. Strain hardening increases the capacity of the material to accumulate plastic deformation at stress levels above the initial yield strength. In this section, the effect of strain hardening on the unloading behavior is investigated.

Unloading curve
To examine the effect of strain hardening on the unloading curves, Fig. 4   from the total indentation  , in order to make the normalized curves to pass through a common origin. The careful examinations observe that none of the curves is linear. Each unloading curve is slightly concave up over its entire span.
As mentioned by Oliver and Pharr [76], unloading curves can be well described by a power-law relation like F    (9) Therefore, in Fig. 4, the power-law fitting curves are plotted and values of  are calculated as listed in the plots.  is found to vary with the strain hardening exponents n , illustrating that the unloading curve is strongly dependent on the strain hardening.
As can be seen from Fig. 4, for Mat. 11 and Mat. 14 with n = 0 (elastic-perfectly plastic material),  =1.501 and 1.502. They are nearly as the same as  =1.5 for the Hertzian-type unloading law [43]. However, when n > 0,  is distinctly smaller than 1.5. It varies from 1.327 to 1.402. It demonstrates that for strain hardening materials the unloading curve no longer meets the Hertzian-type unloading law, but it still meets a power-law relationship.

Pile-up deformation and cavity profile
For unloading, the spherical surface assumption is generally applied to the indentation of elastic-perfectly plastic materials by rigid sphere [25,26,53]. The shape of the residual cavity is assumed to be a spherical surface with a uniform radius res R after the indenter is unloaded and the material elastically recovers. This assumption was validated by Chen, Yin [43] for the elastic-perfectly plastic materials indented by elastic sphere. The importance of this assumption is that the elastic contact solution exists during the unloading of spherical indentation. For the indentation of elastic-power-law strain-hardening materials by elastic spherical indenter, the spherical assumption should be validated. For a rigid sphere indenting into elastic-power-law strain-hardening materials, Kral, Komvopoulos [47] obtained the same result. The residual pile-up is caused by the residual compressive stress, which is the greatest for the nonhardening materials and becomes smaller with increasing strain hardening [47].
To analyze the effect of strain hardening on the cavity profile during unloading process, the cavity profiles at four unloading indentations, As can be seen from Fig. 8 (a), for the elastic-perfectly plastic material ( 0 n  ), the fitting spherical surface can always be in perfect agreement with the whole cavity profile including both the contact area and outside region of the cavity, even though the contact area decreases to zero. It illustrates that the whole cavity profile maintains a spherical surface during the entire unloading process.
As can be seen from Fig. 8 (b), for the strain hardening material ( 0.6 n  ), the fitting surface can only be in coincident with the part of the contact area. The outside cavity profile cannot maintain the same spherical surface. After the fully unloading, the whole cavity profile becomes entirely aspheric.

Indenter elasticity effect
The indenter deformation cannot be neglected for the contact by a softer indenter.
The level of indenter elasticity can be represented by ** 12 / EE . In this section, the effect of indenter elasticity on the unloading behavior is investigated. Fig. 9 shows the effect of elastic deformation of indenter on the unloading curve.

Unloading curve
Under / mY  =200, the simulated unloading curves are plotted for Mats. 6-10 with 0 n  and 0.6 n  . As can be seen from Fig. 9 (a), for the elastic-perfectly plastic materials ( 0 n  ), all the normalized unloading curves for different values of ** 12 / EE coalesce to a single master power-law curve with the exponent  =1.5. The Hertziantype unloading law can be applied. The elastic deformation of indenter has almost no influence on the unloading curve.
As can be seen from Fig. 9

Cavity profile
To consider the effect of indenter elasticity on the residual cavity, Fig. 12  ).
As shown in Fig. 12 (a), if the residual cavity is fitted by a sphere surface, the sphere surface can be in perfect agreement with the residual cavity profile for the elastic-perfectly plastic materials ( 0 n  ). All the residual cavity profiles maintain perfectly spherical surface. The spherical shape of the residual cavity is not influenced by elastic deformation of indenter. As shown in Fig. 12 (b), the fitting circles deviate from the simulated profiles for the strain hardening materials. The residual cavity profiles are aspheric surface. The aspheric surface of the residual cavity profile becomes more significant for higher ** 12 / EE (softer spherical indenter). It means that the elastic indenter deformation significantly affects the residual cavity profile for strain hardening materials, while has no influence on the spherical shape of the residual cavity for elastic-perfectly plastic materials.   Fig. 8. Fig. 13 shows the cavity profiles at the four unloading indentations when / mY  =200. The contact area is marked by the green color. Compared with the spherical indenter with ** 12 / EE = 4.038 used in Fig. 8, the spherical indenter with ** 12 / EE =0.252 used in Fig. 13 is harder. To compare the results for hard spherical indenter in Fig. 13 with those for soft spherical indenter in Fig. 8 can examine the effect deformation of indenter on the cavity profile.
By comparing the cavity profiles in Fig. 13 (a) with those in Fig. 8 (a), all the cavity profiles are spherical surfaces for the elastic-perfectly plastic materials ( 0 n  ). The spherical shape of the cavity profile has not changed for the hard and soft spherical indenters. By comparing of the cavity profiles in Fig. 13 (b) with those in Fig. 8 (b), all the cavity profiles are aspheric surfaces for the stain hardening materials ( 0.6 n  ), the aspheric cavity profile has not changed for the hard and soft spherical indenters. It can be concluded that the indenter elastic deformation does not change the spherical or aspheric feature of the residual cavity during unloading for both elastic-perfectly plastic and strain hardening materials.

Unloading law
In this section, according to the analyses of unloading behaviors as mentioned in sections 3.1 and 3.2, a new unloading model is presented by fitting the FE data to consider the effect of both strain hardening and indenter elasticity. both n and ** 12 / EE . As can be seen from Fig. 6 and Fig. 11, for a given n or

Unloading model
The unloading curve is a key unloading behavior and represents the relationship of contact force with indentation. Recently, Zhao, Nagao [34] studied the unloading curve of the contact between a power-law hardening material with a rigid flat. The equations governing the unloading curve of frictionless indentation of an elastic-power-law strain-hardening material by an elastic sphere will be presented in this section.
The unloading behavior is strongly affected by the maximum loading indentation m  [43,63]. To analyze the unloading behavior of an indentation of a material combination, the unloading curves at different indentations needs to be simulated.
Hence, a total of 1525 unloading curves are simulated for the twenty material combinations listed in Table. 1, to reveal the effect of strain hardening and indenter elasticity on unloading. For example, Fig. 14   For strain-hardening elastic-plastic materials, as discussed in the sections 3.1 and 3.2, the unloading curves can be described by a power-law relation, but the power-law exponents are not 1.5 as the same for the Hertz contact law. For the 2% linear strain hardening elastic plastic sphere compressed by rigid flat, Etsion, Kligerman [63] proposed a unloading law, 1.5( ) / e mY m   where the coefficient e was fitted from the FE data. This unloading law was applied by Song and Komvopoulos [62], Zhao, Nagao [34] and Zhao, Zhang [35] for powerlaw strain hardening materials indented by rigid sphere or compressed by rigid flat.
As analyzed above, under the same / mY  , the exponents of the power-law relation is merely dependent upon the strain hardening exponent n . By detailed observations, the exponent m is also related to the maximum indentation as reported by Etsion, Kligerman [63], Zhao, Nagao [34] and Zhao, Zhang [35]. In the present study, the fundamental form of the unloading law (12) is adopted to express the unloading relation for elastic-power-law strain-hardening materials. By curve-fitting approach, the coefficient e is fitted from the 1525 unloading curves. e is merely dependent upon the strain hardening exponent and can be expressed as:

Numerical verifications
To validate the present unloading law numerically, the existing FE results obtained by Zhao, Nagao [34] and some new contact simulations based on the real engineering materials are used for comparisons.
By using a FE model, Zhao, Nagao [34] studied the power-law hardening elastic-  Table 2. The three elastic-perfectly plastic materials with high yield stress of materials 1-3 are selected as the sphere materials. The two power-law strain hardening materials with low yield stress of materials 4-5 are selected as the half-space materials.
To ensure the purely elastic deformations of the spheres, the ratio of the sphere yield stress to the half-space yield stress should be larger than 2.5 as suggested by Tabor [46] and 2.0 as suggested by Larsson and Olsson [74]. Four contact cases as listed in Table   3 are simulated. For all contact cases, the ratio of the sphere yield stress to the halfspace yield stress is larger than 3.5.

Experimental validation
Six compliance curves including loading and unloading phases shown in Fig. 18 were measured by Bartier, Hernot [77]. The experiment used a tungsten carbide sphere to indent an AISI 1065 steel plate. The material properties are listed in Table 4. By the use of the measured force and indentation at the inception of unloading, the present unloading law predicts the residual indentations and unloading curves as shown in Fig.   18. It can be seen that the predictions agree perfectly with the experimental data without any calibration parameters. The maximum predicting error of the residual indentation is 7.0% and the mean error is only 3.79%. Even for the large contact deformation with m  up to 3969.9 Y  , the prediction is still accurate.

Conclusions
In the present study, the unloading behavior of elastic-power-law strain-hardening half-space indented without friction by elastic spherical indenter is investigated comprehensively in the light of FE method. All 1525 FE simulations are performed for twenty material combinations covering a wide range of strain hardening exponents, yield strengths and indenter elasticity. The intensive FE results show that both the strain hardening and the indenter elastic deformation have significant influences on unloading behaviors. The main contributions of the present study are concluded as the followings: (1) Unloading curves can be well described by a power-law relation. The power-law exponent of unloading curve is strongly dependent on strain hardening while independent on ** 12 / EE . For elastic-perfectly plastic materials, the exponents are always 1.5 and the unloading curve follows Hertzian-type unloading law. For strain hardening materials, the exponents are distinctly smaller than 1.5, implying the inadequate of the Hertzian-type unloading law.
(2) Based on the FE analyses, both strain hardening and indenter elasticity have strong (3) During unloading, there are two types of cavity profiles, spherical and aspheric.
For the elastic-perfectly plastic materials, the cavity profiles always maintain spherical during the entire unloading process and after fully unloading. For the strain hardening materials, the cavity profile merely remains spherical within the contact area and is aspheric outside the contact area during an unloading process.
The cavity profile is sensitive to strain hardening. For high strain hardening materials, the residual cavity profile becomes aspheric more significantly. However, the elastic deformation is found to not affect the type of the cavity profile during unloading and the residual cavity after fully unloading. It only affects the aspheric degree of cavity profile.
(4) By curve-fitting approach, a new unloading law is presented to consider the effects of both strain hardening and indenter elastic deformation. The new unloading law is suitable for both the elastic-perfectly plastic and strain hardening materials.