Strain Hardening From Elastic-Perfectly Plastic to Perfectly Elastic Indentation Single Asperity Contact

Hamid Ghaednia; Gregory Mifflin; Priyansh Lunia; Eoghan O. O'Neill; Matthew R. W. Brake

doi:10.3389/fmech.2020.00060

Ghaednia, Hamid;Mifflin, Gregory;Lunia, Priyansh;O'Neill, Eoghan O.;Brake, Matthew R. W.;

2020-08-05 00:00:00

ORIGINAL RESEARCH published: 05 August 2020 doi: 10.3389/fmech.2020.00060 Strain Hardening From Elastic-Perfectly Plastic to Perfectly Elastic Indentation Single Asperity Contact 1 2 3 4 Hamid Ghaednia , Gregory Mifﬂin , Priyansh Lunia , Eoghan O. O’Neill and Matthew R. W. Brake 1 2 Department of Orthopaedic Surgery, Harvard Medical School, Boston, MA, United States, Department of Mechanical Engineering, Auburn University, Auburn, AL, United States, Department of Physics, William Marsh Rice University, Houston, TX, United States, Department of Mechanical Engineering, William Marsh Rice University, Houston, TX, United States, Department of Mechanical Engineering, William Marsh Rice University, Houston, TX, United States Indentation measurements are a crucial technique for measuring mechanical properties. Although several contact models have been developed to relate force-displacement measurements with the mechanical properties, they all consider simplifying assumptions, such as no strain hardening, which signiﬁcantly affects the predictions. In this study, the effect of bilinear strain hardening on the contact parameters for indentations is investigated. Simulations show that even 1% strain hardening causes signiﬁcant changes in the contact parameters and contact proﬁle. Pile-up behavior is observed Edited by: for elastic-perfectly plastic materials, while for strain hardening values greater than 6%, Elena Torskaya, only sink-in (i.e., no pile-up) is seen. These results are used to derive a new, predictive Institute for Problems in Mechanics (RAS), Russia formulation to account for the bilinear strain hardening from elastic-perfectly plastic to Reviewed by: purely elastic materials. J. Jamari, Keywords: bilinear strain hardening, elastic plastic contact, indentation, constitutive modeling, pile-up and sink-in Diponegoro University, Indonesia Sergei Mikhailovich Aizikovich, Don State Technical University, Russia 1. INTRODUCTION *Correspondence: Matthew R. W. Brake Fundamental to all assembled systems, contact mechanics is integral to mechanical design. This [email protected] is evident in many applications, such as: jointed structures (Brake, 2016), electrical contacts (Ghaednia et al., 2014), thermal contacts (Jackson et al., 2012), collision mechanics (Brake, 2012, Specialty section: 2015; Ghaednia et al., 2015; Gheadnia et al., 2015; Ghaednia and Marghitu, 2016; Brake et al., 2017), This article was submitted to Tribology, continum mechanics (Golgoon et al., 2016; Golgoon and Yavari, 2017, 2018), biomechanics (Zhao a section of the journal et al., 2007; Borjali et al., 2017, 2018, 2019; Langhorn et al., 2018; Mollaeian et al., 2018), turbines Frontiers in Mechanical Engineering (Firrone and Zucca, 2011), additive manufacturing (Kardel et al., 2017; Pawlowski et al., 2017), Received: 21 March 2020 bearings (Sadeghi et al., 2009), particle and powder interactions (Christoforou et al., 2013; Rathbone Accepted: 26 June 2020 et al., 2015), and seals (Green and Etsion, 1985; Miller and Green, 2001) amongst other applications. Published: 05 August 2020 Contact mechanics can be categorized into both single asperity and rough surface contact, where Citation: single asperity models are usually used in rough surface models. A third category, macroscale Ghaednia H, Mifﬂin G, Lunia P, applications of contact mechanics, tends to use similar models as for the contact of single asperities O’Neill EO and Brake MRW (2020) (for spherical contact at least; other geometries have solutions that are similar to Hertz’s original Strain Hardening From model, Johnson, 1987; Flicek, 2015). There is a multitude of analytical, numerical, and experimental Elastic-Perfectly Plastic to Perfectly studies on all types of single asperity contact: spherical, cylindrical, elliptical, conical, and ﬂat Elastic Indentation Single Asperity contact, all with the goals of developing predictive formulations of contact parameters, damage Contact. Front. Mech. Eng. 6:60. doi: 10.3389/fmech.2020.00060 predictions, or calculating design constraints and standards. Frontiers in Mechanical Engineering | www.frontiersin.org 1 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening Contact mechanics of engineering materials, such as metals, characterized as a deformable sphere pressed against a rigid can be divided into three regimes: purely elastic, elastic-plastic ﬂat, and indentation models were characterized as a rigid and fully plastic. For a majority of metallic contacts, elastic regime sphere pressed into a deformable ﬂat. It has to be considered ends at very small deformations, which are often impossible that the term single asperity contact has been applied to to avoid. The elastic-plastic regime initiates with the inception both of these contact groups, while indentation and ﬂattening of yield (at some depth below the contact surface), and then terms are specifying the contact types. Before Jackson and transitions into the fully plastic regime. As the fully plastic regime Kogut (2006), both ﬂattening and indentation contacts were often leads to mechanical failure, the majority of contacts exist assumed to be the same. Therefore, in almost all of the in the elastic-plastic regime, which is also the most complicated previous works, one of the objects in contact is assumed phase of contact for analysis. rigid. However, Ghaednia et al. (2016) showed that there Approximate analytical solutions are well-known for the is a transition from ﬂattening to indentation contact with elastic regime. Hertzian contact theory (Hertz, 1882) solves respect to the yield strength ratio of the contacting objects and the elastic spherical contact problem by ﬁtting a polynomial that contact of similar materials yields smaller contact areas on the interface and assuming a second order polynomial and larger contact pressures. Further, Ghaednia et al. (2016) for the pressure distribution. The theory is well-validated concluded that the change in the radius of curvature during for small deformations; however, for large deformations the contact determines whether the contact will lead to indentation approximations of Hertzian theory break down. or ﬂattening. For the perfectly plastic regime, there are a few analytical Another simplifying assumption in contact mechanics is solutions; however, all of the existing solutions include limiting related to the mechanics of strain hardening. The two primary assumptions, such as uniform contact pressure, or constant models for material strain hardening are power law and bilinear normal contact pressure or hardness. The majority of the strain hardening, and only a few contact models (Mesarovic numerical studies (Hardy et al., 1971; Lin and Lin, 2006) justify and Fleck, 2000; Brizmer et al., 2006; Brake, 2015; Zhao et al., the assumption of a uniform pressure distribution; however, 2015) incorporate strain hardening. Several researchers have it has been shown that the assumption of constant hardness considered power law strain hardening: by FEA (Zhao et al., is not accurate (Jackson and Green, 2005). Tabor suggested 2015), qualitatively (Mesarovic and Fleck, 2000), and analytically a constant H = 2.8S in his book (Tabor, 2000), this (Brake, 2015). In particular, Zhao et al. (2015) demonstrated that value was later analytically veriﬁed by Ishlinsky (Ishlinsky, strain hardening for a ﬂattening elastic-plastic sphere leads to a 1944) and has been used in the majority of the engineering decrease in the contact area while the contact force increases for applications and measurements. In contrast to this, Kogut and higher levels of strain hardening. Komvopoulos (2004) suggested that hardness depends on the Brake (2012, 2015) provided an analytical transition between material properties, and it was shown by Jackson and Green the purely elastic and fully plastic regimes by applying nine (2005) that hardness is a function of contact radius as well as the governing conditions on the transition functions. Brake’s material properties. This was further explored in Jackson et al. formulations is based on the Meyer’s hardness test (Meyer, 1908; (2015), which presented an analytical solution of hardness as a Biwa and Storåkers, 1995; Tabor, 2000), which leads to the function of the contact radius by using Ishlisky’s slip line theory. functional form for contact force in the fully plastic regime, They show that as the contact radius increases, the average normal pressure decreases. All of these works, however, were F = p π , (1) p 0 n−2 developed for elastic-perfectly plastic materials. a Due to the complexities of the stress distribution in elastic- plastic contact and the integral (path-dependent) nature of the where a is the contact radius, a is the contact radius at the material behavior, there is no closed form solution for the onset of the fully plastic regime, p is the average normal pressure elastic-plastic regime. Most studies use ﬁnite element analysis at perfectly plastic regime or hardness, and n is the Meyers (FEA) to develop predictive empirical formulations for contact strain hardening exponent, which comes from a power law parameters. Empirical formulations of FEA results started with relation between stress and strain (see Nomenclature). The main the works of Sinclair and Follansbee (Follansbee and Sinclair, complexities in this equation arise from the calculation of a and 1984; Sinclair et al., 1985). Later, this method was used and p , which aﬀect the ﬁnal predictions signiﬁcantly. improved by several researchers (Chang, 1986; Chang et al., Studies show that the bilinear strain hardening model is closer 1987; Komvopoulos, 1989; Kogut and Etsion, 2002; Jackson to the true stress true strain curve than power law hardening and Green, 2003, 2005; Ye and Komvopoulos, 2003; Kogut and (Sharma and Jackson, 2017). Kogut and Etsion (2002) stated that Komvopoulos, 2004; Green, 2005; Ghaednia et al., 2016), and the eﬀect of bilinear strain hardening with tangential modulus detailed discussions of the contact models can be found in the E < 0.05E (for a given elastic modulus E) on the contact force reviews of (Bhushan, 1996, 1998; Adams and Nosonovsky, 2000; and contact area is less than 4.5%; however, this comparison Barber and Ciavarella, 2000; Ghaednia et al., 2017). was based on very small deformations (δ < 20δ , where One important point that is often overlooked in contact δ is the critical interface at which yield initiates). For larger mechanics modeling is that the mechanicsm of indentation and deformations, this does not hold true as will be discussed in ﬂattening are fundamentally diﬀerent. This was demonstrated this work. An extension of Kogut and Etsion (2002) to include in Jackson and Kogut (2006), where ﬂattening models were bilinear strain hardening showed stable predictions for a large Frontiers in Mechanical Engineering | www.frontiersin.org 2 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening range of interfaces and material properties for a limited set of strain hardening coeﬃcients (Shankar and Mayuram, 2008). In Ghaednia et al. (2019), a new contact model for an elastic-plastic ﬂattening contact was developed that included both bilinear strain hardening on the indenter and elastic deformations of the ﬂat (which many previous ﬂattening model considered rigid). The present analysis seeks to advance this research by investigating the role of hardening behavior for indentation. In the previous work Ghaednia et al. (2019), the eﬀect of bilinear strain hardening for a ﬂattening contact was studied. Even though the present work addresses a similar issue for indentation contact of bilinear materials, it is paramount that these two regimes be considered separately due to fundamental diﬀerences in their mechanics. The eﬀect of pile-up and sink- in for an indentation contact signiﬁcantly aﬀects the stress distribution at the interface and the contact parameters (Jackson and Kogut, 2006; Ghaednia et al., 2017). To the best of the authors’ knowledge, no previous study has developed an FIGURE 1 | Finite element mesh for contact between a purely elastic sphere empirical formulation for the eﬀect of bilinear hardening on the and a bilinear plastic ﬂat. indentation of a surface. The present work uses a series of FEA simulations that are detailed in section 2. In section 3, the new empirical formulation for indentation contact with a strain hardening material model 1. The nodes on the bottom surface of the ﬂat were ﬁxed in both is derived, and is subsequently veriﬁed in section 4. The new x and y directions. formulation is derived to be general enough to span the full 2. The nodes on the vertical axis of symmetry were constrained range of strain hardening behavior, from elastic-perfectly plastic in the x-direction, but not in the y-direction. contact (i.e., the most compliant extreme) to purely elastic 3. Total of 29166 nodes were used in the mesh. contact (i.e., the most stiﬀ extreme). Case studies are then 4. The mesh was biased near the contact tip. presented to assess the eﬀect of strain hardening on the indenter 5. A uniform total displacement of 50 μm in the y-direction was pile-up and sink-in behavior. Finally, conclusions are presented applied to the top surface of the sphere in 40 sub-steps in section 5. 6. The sphere was modeled as a purely elastic material, while the ﬂat was modeled as a bilinear strain-hardened material with variable tangential modulus of Elasticity 0 ≤ E /E ≤ 1. 2. FINITE ELEMENT ANALYSIS 7. Friction between the sphere and ﬂat was ignored. A two-dimensional axisymmetric model similar to Ghaednia The displacement (compressive load) was applied in 15 load et al. (2019) was developed in ANSYS Mechanical APDL 18.0 to steps that each contained 100 sub-steps, during which none of simulate normal contact between a perfectly elastic sphere and the elements experienced large deformations. The von Mises an elastic-plastic ﬂat. The sphere has a radius of 1 mm, and the stress criterion was used to indicate the transition from elastic ﬂat is 3 mm thick and 5 mm wide. The reduction from a three- to elastic-plastic phase. The proﬁles of the ﬂat’s top surface were dimensional problem to a two-dimensional problem using the collected for 13 values of the tangential modulus and at 15 symmetry of the modeled system simpliﬁes the ﬁnite element load steps (load steps were required for convergence). This data calculations and signiﬁcantly reduces the error and computation was used to describe the relation between the pile-ups and the time of the analysis. The 2-D simpliﬁcation is applicable in this strain hardening. The real contact radius, a, was determined situation because the variables of interest, the real contact radius in each case by ﬁnding the last surface node in contact with (a) and the Hardness (H), are independent of the surface angle the sphere, taking its initial x-position, and subtracting the x- in the case of normal contact. There is no friction modeled deformation at each sub-step. The sphere deformation ratio, between the nodes, and nine node elements were used in the δ (see Nomenclature), was calculated by collecting the sphere model. Mesh convergence was performed for the extreme cases deformation, δ , at the contact tip and dividing it by the total of purely elastic deformations (i.e., small deformations), and the displacement, 1. The total reaction force, F, of the ﬂat was largest deformations studies (i.e., the limit of the plastic regime). measured by summing the individual reaction forces of each node The ﬁnal mesh was selected to ensure that the results across both in contact with the sphere. Finally, the normal average pressure, regimes were within 0.1% of the ﬁnest mesh considered. The ﬁnal H, was calculated for each sub-step and for each E . mesh is similar to the mesh used in Ghaednia et al. (2019) except In this work, unloading is not included in the modeling that a ﬁner mesh was required on the ﬂat’s surface due to large eﬀort as the available models in the literature are both mature pile-ups, Figure 1. The following conditions have been applied and applicable. The interested reader is referred to Ghaednia on the FE model: et al. (2015), Kogut and Komvopoulos (2004), and Wang et al. Frontiers in Mechanical Engineering | www.frontiersin.org 3 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening (2020) for more information regarding how the unloading phase boundary condition behaves in a purely elastic manner. As a sample result, Figure 2 shows the von-Mises stress for E = 0 (i.e., an elastic-perfectly δ E f f δ = = . (2) plastic material) and 1 = 50 μm. f 2 (1 − ν )E 2.1. Finite Element Observations: Contact For 1/R < 0.01, the ratio δ exhibits a high sensitivity with Parameters respect to the deformation ratio 1/R. As 1/R approaches 0.02, Two sets of materials were modeled using FE to analyze the δ transitions to being weakly dependent on the ratio of 1/R and, contact parameters. For both sets, the sphere is purely elastic in fact, is independent of 1/R for E /E > 0.1. With respect t f with modulus of elasticity E = 200 GPa and Poisson’s ratio to the tangential modulus E /E there is a continuous decrease ν = 0.3. In set one, the ﬂat is modeled as a bilinear material in the deformation ratio value from the elastic-perfectly plastic with modulus of elasticity E = 200 GPa, Poisson’s ratio ν = f f material model (E /E = 0) to purely elastic model (E /E = 1). t t f f 0.3 and yield strength S = 300 MPa, and for set two, the yf The results do not reach the Hertzian theory and shows around ﬂat is modeled as a bilinear material with modulus of elasticity 2% larger values. E = 71 GPa, Poisson’s ratio ν = 0.29 and yield strength f f A similar trend was observed in Ghaednia et al. (2019), where S = 200 MPa. Thirteen diﬀerent tangential moduli (E /E = 0, yf f the strain hardening eﬀect on a ﬂattening contact was studied, 0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.2, 0.3, 0.4, 0.6, 0.9, and 1) have and the the same diﬀerence between the Hertzian theory and the been used for each of the material sets. For each of the sets, four FEM results was found. This result is likely due to neglecting the contact parameters were analyzed: deformations, real contact diﬀerence between indentation and ﬂattening contact in Hertzian radius, contact force, and average normal pressure. For small theory, where for both the cases the same eﬀective radius of loads approaching zero, the system behaves in a purely elastic curvature is used. manner. Thus, this regime approaches the results of Hertz’s The FEA results for the real contact radius is shown in model (Hertz, 1882; Johnson, 1987). The numerical results shown Figure 4. The results show a large gradient with respect to the in Figures 3–6 at the lowest load step are linearly connected to tangential modulus E /E at the elastic-perfectly plastic limit t f the Hertzian solution for displacements smaller than considered (E /E = 0) and converges to the purely elastic (E /E = 1) t t f f in the empirical model formulation. solution at around E /E = 0.2. t f Figure 3 shows the numerical results for deformation ratio of The average normal pressure is shown in Figure 5. At the set one as a function of the tangential modulus E /E and the t f elastic-perfectly plastic limit (E /E = 0) the results match the t f applied displacement 1/R. The deformation ratio here is deﬁned hardness values from the Jackson et al. (2015) model. For as δ /1, where, δ and 1 are the deformation on the ﬂat at the f f E /E > 0, the average normal pressure should not be called tip of the indenter and the applied displacement, respectively. At hardness since the material never reaches the plastic ﬂow regime. 1/R = 0, Hertzian theory is applied on the results to create the Therefore the term average normal pressure represents the physical meaning more precisely. Figure 5 shows a continuous increase from the elastic-perfectly plastic material model at E /E = 0 to the purely elastic material model at E /E = 1. t f t f FIGURE 2 | Von-Mises stress distribution for a contact between a purely FIGURE 3 | Variation of ﬂat deformation ratio with respect to the strain hardening and applied displacement for material set 1 (numerical results). elastic sphere and a bilinear elastic-plastic ﬂat. Frontiers in Mechanical Engineering | www.frontiersin.org 4 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening FIGURE 4 | Variation of real contact radius with respect to strain hardening FIGURE 6 | Variation of contact force with respect to the strain hardening and and applied displacement for material set 1 (numerical results). applied displacement for material set 1 (numerical results). E /E = 0 to 1. The position of the highest point at the proﬁle t f of the ﬂat top surface is used to measure the pile-up height. If the highest point is bellow the initial surface of the ﬂat then the contact resulted in sink-in, and if this point is higher than the ﬂat initial surface then the contact resulted into pile-up. It can be seen in Figure 7A that the pile-up happens for E /E ≤ t f 6%. For E /E > 6% the contact proﬁle exhibits sink-in. At t f E /E = 0 the proﬁle shows an extremely sharp pile-up, as t f large as 30% of the indentation depth, that drops signiﬁcantly to 3% of the indentation depth at E /E = 0.01. The maximum t f indentation depth decreases from the elastic-perfectly plastic to the purely elastic material model. In Figure 7B, the volumetric ratio between the piled-up and sinked-in region y(x)xdx p y = , (3) FIGURE 5 | Changes in the normal average pressure with respect to the strain V y(x)xdx hardening and applied displacement for material set 1 (numerical results). is shown, where, V and V are the pile-up and sink-in volumes, p s respectively, and y and y are the domains at which y(x) ≥ 0 + − The variation of contact force with respect to deformations and y(x) < 0, respectively. The volumetric ratio exponentially and strain hardening is depicted in Figure 6. The results show decays from E /E = 0 and converges to 0 at around 5%, which t f a continuous increase from the elastic-perfectly plastic material again shows the signiﬁcant eﬀect of strain hardening on pile-up model (E /E = 0) to the purely elastic material (E /E = 1). at very small tangential moduli. t f t f The gradient with respect to E /E is large at the elastic-perfectly One of the parameters that can aﬀect the pile-up/sink-in t f plastic limit (E /E = 0) and at large deformations, 1/R. The deformations is the elastic deformation on the indenter. Figure 8 t f Hertzian solution shows smaller values for E /E = 1 that is likely shows the contact proﬁle for six diﬀerent sphere moduli E = t f s due to the ﬁrst order approximation of the contact radius, which 10, 50, 100, 150, 200, and 250 GPa, and elastic-perfectly plastic ﬁts a parabola onto on the sphere. ﬂat, E /E = 0, the rest of the material properties are the same t f as set one. It can be seen that the pile-up is directly related to the 2.2. Finite Element Observations: Pile-Up stiﬀness of the sphere; however, this does not transform the pile- vs. Sink-In ups into sink-in and the proﬁle piles-up for all values of E . There An Important characteristic of indentation is the proclivity of is a signiﬁcant change from E = 10 GPa to E = 50 GPa, which s s the material to exhibit pile-up, or sink-in. Figure 7A shows the is due to the ﬂat not deforming as much as the sphere (recall, results of the FEM simulations’ contact proﬁle on the ﬂat for E = 200 GPa), as seen in Figure 8. diﬀerent tangential modulus E /E values using the properties Another important factor for the contact proﬁle is the contact of material set one with 21 diﬀerent tangential moduli from force or the total displacement of the indenter. In the FE Frontiers in Mechanical Engineering | www.frontiersin.org 5 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening FIGURE 9 | Effect of loading (in 15 steps) on surface proﬁle for elastic-perfectly plastic material. presented in Figure 9 are larger than the critical load needed to initiate yield in the ﬂat. It can be seen that proﬁle piles-up independent of the loading. FIGURE 7 | (A) Evolution of pile-up with respect to strain hardening. The contact parameters for set two show a similar trend as (B) Volumetric pile-up-sink-in ratio. set one’s. Thus, the graphs are not shown here; however, the data from set two have been used in the model development, presented in the next section. 3. MODEL DEVELOPMENT In the proposed formulation, the contact has been divided into two phases: elastic and elastic-plastic regimes. The contact starts with the elastic regime for very small deformations and continues to the elastic-plastic regime. Eventually the elastic-plastic regime converges to the fully plastic regime for large deformations. 3.1. Elastic Regime The elastic regime follows the Hertzian theory. In this phase, the deformation on each object is calculated as E(1 − ν ) δ = 1, δ = 1 − δ , (4) f f s where E is the eﬀective modulus of elasticity FIGURE 8 | Effect of the indenter modulus of elasticity on surface proﬁle for a elastic-perfectly plastic material. (1 − ν ) 1 (1 − ν ) = + . (5) E E E s f In the Hertzian theory, the contact curvature is approximated model, (depending on whether the application is displacement with a polynomial and the contact radius is calculated up to the controlled or force controlled) the load is applied as the ﬁrst order accuracy as penetration of the indenter to the ﬂat. Figure 9 shows the contact proﬁle for 15 load steps from 1/R = 0.0042 to 1/R = 0.05 for set 1 used in the initial observations. All of the load steps a = R1, (6) Frontiers in Mechanical Engineering | www.frontiersin.org 6 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening in terms of the equivalent radius, which is given as 3.2.1. Deformation The deformation ratio of the ﬂat is deﬁned as 1 1 1 = + . (7) R R R s f δ = . (13) The contact force for the elastic regime becomes Several assumption governing the form of this ratio are made. First, it is explicitly a function of the ratio of the elastic moduli 1/2 3/2 E /E and the non-dimensionalized deformation 1/R. Second, F = R E1 . (8) e t f it must satisfy a set of governing conditions that are deﬁned in what follows. Other variables, such as the eﬀective modulus of elasticity and Poisson ratio, are implicitly incorporated into 3.2. Elastic-Plastic Regime the elastic limit and the large deformation limit, deﬁned below. For large deformations, plastic strains are present in Under these assumptions, the functional form of the deformation the softer object (i.e., the object with the smaller yield ratio, in the most general case, is expressed as strength). By deﬁnition, the elastic-plastic regime begins at the initiation of yield, which occures at some point 1 E δ = f , . (14) beneath the contact surface. Following Johnson (1987), f R E this critical point is calculated using the von Mises stress criterion, which gives the critical deformation for the onset of Here, both ratios (1/R and E /E ) only exist on the intervals of t f yield as: [0,1]. The deformation ratio (Equation 14) must also satisfy the governing conditions (GCs) that are based on the results shown πS yf in Figures 3–6: δ = , (9) 3(ν ) 2E For small deformation ratios, the elastic solution must be recovered where the material properties of the ﬂat (as the more compliant E(1−ν ) material) are used, and 3(ν) is solved from Johnson (1987): f E ∗ ∗ t (I) lim δ = δ = , 0 ≤ ≤ 1. f fe E E f f →0 " # R Likewise, for a purely elastic material, the elastic solution must z a 3 1 −1 3(ν) = max −(1 + ν) 1 − tan + . hold for all deformations z≥0 a z 2 1 + E(1−ν ) (10) f 1 ∗ ∗ (II) lim δ = δ = , 0 < ≤ 1. E R f fe E f →1 Jackson and Green (2005) proposed an approximation for the In GCs (I,II), δ is deﬁned as the elastic limit of the deformation fe solution of Eq. (10): ratio, which is calculated via Equation (4). 2 For the large deformation limit, since the ﬂat is assumed πCS yf 1 = R, (11) to have a bilinear material model, the solution for large 2E deformations is assumed to be the same as an elastic material with the same elastic modulus as the system (E = E ) t f such that from Jackson and Green (2005) C is a ﬁt to the numerical solution of Eq. (10) ∗ ∗ (III) lim δ = δ , 0 ≤ E /E ≤ 1. t f f fp →1 0.736ν Here, δ is deﬁned as the deformation ratio at the large C = 1.295e . (12) fp deformation limit, which can be estimated as Therefore, the elastic-plastic phase starts and continues for ∗ 2 2 E (1 − ν ) 1 − ν deformations 1 > 1 . t 1 1 − ν f f δ = , = + . (15) To develop a predictive formulation for the elastic-plastic fp ∗ E E E E t s t regime, several governing conditions have been applied to the model development to ensure that continuity, boundary Equation (15) is deﬁned in terms of E , which is the reduced conditions, continuity of the ﬁrst derivatives, and the physical modulus of elasticity for a purely elastic ﬂat having modulus meaning are preserved in the formulation. In the following, E = E . This simpliﬁcation is veriﬁed against the ﬁnite element f t the model development for the ﬂat’s deformation ratio, δ , the results, and is shown for material sets one and two in Figure 10. real contact radius, a, and contact force, F are discussed. The In this comparison, the FEM results are shown for an applied formulations focus on two of the main eﬀects that are neglected displacement of 1/R = 0.05, which is well below the maximum in almost all previous models: ﬁrst, the eﬀect of the bilinear strain displacement ratio considered of 1/R = 1. It is observed from hardening from elastic-perfectly plastic E = 0 to fully elastic this comparison that the deformation ratio quickly converges E = E , and second, the elastic deformations on the indenter to the large deformation limit, which justiﬁes the assumption during contact. implicit in GC (III). At E = E , even though FEM results are t f Frontiers in Mechanical Engineering | www.frontiersin.org 7 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening expected to reach the Hertzian solution, Figure 10 shows 2% rewritten in terms of φ and ψ as: smaller values compared to Hertzian theory. The reason is most ! 1 E likely the deﬁnition of the equivalent radius of curvature in the ∗ ∗ δ (1/R = 0) = φ = 0 ψ + D = δ ; (17) f fe Hertzian theory, which does not consider any diﬀerence between R E the ﬂat and the sphere. Since this diﬀerence is not signiﬁcant, the upper limit, E = E , is left to be the Hertzian theory. This diﬀerence has been considered in the ﬁnal formulation to φ = 0 = 0, D = δ , (18) fe decrease the errors in the predictions. For large deformations, δ asymptotically converges to the ! large deformation limit. This is modeled by assuming that the ψ = 1 = 0. (19) gradient of δ with respect to the applied displacement is 0 for f large 1/R. Substitution of these two constraints on φ and ψ into GC (III) ∂f E yields a new constraint equation on the product of φ, ψ at one of (IV) lim = 0, 0 ≤ ≤ 1, 1 E the boundaries: →1 ∗ ! From Figure 10, no convergence for δ with respect to E /E can t f 1 E ∗ ∗ ∗ δ = φ = 1 ψ = 0 + δ = δ . (20) be seen, which can be expressed mathematically in GC (V) as f fe fp R E ∂f (V) lim 6= 0, 0 < ≤ 1. E E t As ψ is not constant, the application of GC (IV) to the functional →1 ∂ E E form of Equation (16) establishes that The ﬁnal two GCs deﬁne constraint conditions for the derivatives " !# of the deformation ratio along the boundaries (with respect to ∂f 1 E lim = lim φ ψ = 0, (21) both 1/R and E /E ): 1 1 R E →1 ∂ →1 R R R ∂f (VI) lim 6= 0, 0 < ≤ 1, 1 R E which yields t ∂ →0 R ∂f E lim φ = 0. (22) (VII) lim 6= 0, 0 ≤ ≤ 1. E →1 E f 1 R →0 ∂ R E A ﬁnal set of constraints on φ and ψ come from GCs A seperation of variables solution is proposed for f via (V) and (VI): E 1 ! ! ′ ′ lim ψ 6= 0, lim φ 6= 0. (23) δ E 1 1 E 1 E t E R f t t →0 f →1 = f , = φ ψ + D. (16) 1 R E R E f f To satisfy the constraints established in the GCs and detailed through Equations (16) to (23), the functional form From GC (I) and (II), it is apperant that neither ψ not φ are s 1−1 1 −2E E λ f 1 s constants. Further, in applying these constraints to the functional φ = 1 − e , λ = , λ = 0.55 1 2 form of Equation (16), the constraints of GCs (I) and (II) can be R 40S E yf f FIGURE 10 | Comparison of the FEA results (x) and the assumption of GC III at the large deformation limit (1/R = 0.05). (A) Material set 1. (B) Material set 2. Frontiers in Mechanical Engineering | www.frontiersin.org 8 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening   ! " # where P is the average normal pressure is Hertzian theory and E E t t ∗ ∗   ψ = (δ − δ ) β − β , β = 0.9 (24) 1 2 1 is calculated as fp fe E E f f h i 4 1 a P = E 1 + (29) β = , β = 0.1 H 2 3 3π R 3R and H is the hardness calculated from the analytical solution by   " # ! Jackson et al. (2015) for elastic-perfectly plastic materials as: 0.1 0.55 1−1c 1 E t λ ∗ ∗ ∗ ∗ R   δ = δ + δ − δ 0.9 − 1 − e , f ef fp fe −2 4S 10 E yf H = √ 3 3 is proposed, where the coeﬃcients, β and λ have been calculated 3/2 h i h i i i 3 2 1 a a a a π −1 − 1 + cos 1 − − + + 1 . from ﬁtting to FEM results for sets one and two, with E /E = t s 3 R R R R 2 0, 0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.2, 0.3, 0.4, 0.6, 0.9, 1 and for 40 (30) increments of 1/R from 1/R = 0 to 1/R = 0.05. Note that Equation (25) automatically satisﬁes GC (VII). Equation (28) presents a linear transition from elastic-perfectly plastic to purely elastic materials with respect to the tangent 3.2.2. Real Contact Radius modulus. The limit of the contact force at large deformations is Using the model of Equation 25 and the FEA results presented proposed as: in Section 2, it is proposed that the real contact radius during elastic-plastic contact follows the form of " # ! E γ +γ 2 3 t E ! f ∗ F = πa P 1 + e , γ = 0.1551, (31) LD LD 1 ep f E a = a , (25) γ = −2, γ = −1. 2 3 where a = R1 is the Hertzian contact radius and χ is The coeﬃcients γ , j = 1, 2, 3, are ﬁtted to the FEM results. The e j elastic-plastic contact force is thus calculated via:   ! " # 0.8376 1/3 E E s t   χ = 0.63 1 − 0.8 , (26) F = W F + W F , (32) e e LD LD E E f f where F and F are from the Hertzian theory (Johnson, 1987) e LD with χ found from ﬁtting to the FEM results (with mean absolute and Equation (31), respectively, and W and W are ﬁtted to e LD ∗ ∗ error less than 2%) and δ and δ can be found from Hertzian FEM results as theory and Equation (25), respectively. The elastic-perfectly 1 − 1 3 α c 4 α 1− E /E (1/1 −1) 2 t c plastic, E = 0, limit of Equation (25) reduces to W = 1 + α e , (33) e 1 0.8376 0.63 E /E s f 3 3 5 a = a . (27) ep e α = , α = −0.25, α = , α = (34) 1 2 3 4 2 4 12 The elastic-perfectly plastic limit, a , at large deformations is the ep E 1/1 −1 t c h i − 1− real contact radius that is used in the majority of the hardness α E 5 E /E W = 1 − e , (35) measurements using spherical indenters, such as nano and micro LD indentation tests. Even though the eﬀect of pile-up is not directly mentioned in development of the contact radius formulation, the yf α = , (36) eﬀect of pile-up and sink-in is considered as the eﬀect of strain 5 2 E 1 f c hardening and is thus embedded in the equations. and the coeﬃcients α , j = 1, 2, 3, 4, are ﬁtted to the FEM results 3.2.3. Contact Force with absolute mean error less than 2%. To formulate the contact force, the average normal pressure at very large deformations, conventionally called Hardness, 4. RESULTS AND DISCUSSION is assumed to increase linearly from hardness for elastic- perfectly plastic materials to average normal pressure from 4.1. Model Veriﬁcation Hertzian theory To verify the predictions of the new model, the new formulation is compared with the contact of six diﬀerent material combinations with properties listed in Table 1. Here, Mat1 to P = H + P − H , (28) LD H Mat6 are indentation contacts of: Frontiers in Mechanical Engineering | www.frontiersin.org 9 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening TABLE 1 | Material sets used in the comparisons. Mat. sets Mat1 Mat2 Mat3 Mat4 Mat5 Mat6 E (GPa) 193 205 630 630 630 193 E (GPa) 69 69 201 205 193 103 E /E (%) 7 5 2 14 18 4 t f S (MPa) 276 186 385 436 760 241 yf ν 0.26 0.26 0.31 0.31 0.31 0.265 ν 0.33 0.33 0.29 0.29 0.26 0.31 Mat1 Aluminum (Al) 6061 ﬂat and Stainless Steel (SS) 304 indenter, Mat2 Al 5005 ﬂat and Alloy Steel (AS) 4130 indenter, Mat3 Carbon Steel (CS) 1070 ﬂat and Tungsten Carbide (WC) indenter, Mat4 AS 4130 ﬂat and WC Indenter, Mat5 SS 304 ﬂat and WC indenter, Mat6 Titanium-G1 ﬂat and SS 304 indenter. These material sets (in particular, sets 3, 4, and 5) were chosen FIGURE 11 | Mat 1 comparisons between FEM results, new formulation and to validate the model outside of the material range that was Hertzian theory for deformation ratio, contact area and contact force. used for model development. For each of the comparison sets, three parameters are studied: the deformation ratio, real contact radius, and contact force. Unfortunately, comparison TABLE 2 | Maximum normalized deformation and mean absolute errors. with previous models (such as Ye and Komvopoulos, 2003; Mat1 Mat2 Mat3 Mat4 Mat5 Mat6 Jackson and Green, 2005; Ghaednia et al., 2016; Wang et al., 2020) was not possible because those models do not consider the eﬀect 1 /δ 336.8 768.8 1430.0 1142.0 333.7 751.6 m c of strain hardening. Thus, comparison between them and the e 0.5 0.3 0.7 0.7 0.3 0.7 present model would be dominated by the signiﬁcant diﬀerences e 0.9 1.2 1.6 1.5 1.3 1.1 in material models and would be unfair. e 4.7 1.3 3.5 3.5 5.6 3.8 As an exmaple of using the presented equations, the contact parameters for Mat1 has been calculated. A Matlab script was written to calculate the deformations, contact area, and contact force for diﬀerent applied displacements. In each iteration of on each of the material combinations. All of the maximum the loop, the critical deformation from Equation (11) is ﬁrst normalized deformations that have been analyzed are well over calculated. The deformation was then determined to be either 110, which is the limit for fully plastic ﬂow in Kogut and Etsion in the elastic regime (if 1 ≤ 1 ), in which case Equations (4– (2002). Thus, the analysis shows the transition from purely elastic 8) would be employed, or in the plastic regime (1 > 1 ). For to elastic-plastic to purely plastic regimes. The deformation deformations within the plastic regime, the deformation ratio, ratio results show a maximum mean absolute error of 0.7% δ , was calculated from Equations (13–25), contact radius from among all of the compared materials. The maximum error for Equation (25), and for the contact force Equation (32) is used. the deformation ratio is 2.4%. The contact radius shows the Figure 11 shows the comparison for Mat1, the contact of Al maximum mean absolute error 1.6% with maximum error being 6061 ﬂat and a SS 304 sphere with R = 1 mm. It can be seen that 6.7%. For the contact force the maximum mean absolute error for the deformation ratio, the predictions match the FE results and maximum error are 5.6 and 8.3%, respectively. very well with mean absolute error less than e = 0.5% and f Figure 12 shows the loglog comparison of real contact maximum error, e = 1.2%. For the real contact radius, the mδ radius vs. normalized deformation between the FEM results model shows a very good match with maximum error e = and formulation predictions for all of he six material properties ma 2.6% and mean absolute error e = 0.8%. The predictions for the shown in Table 1. For each set, the closest line to the FEM results contact force also shows a reasonable match with maximum and is the prediction for that material set. Material sets Mat3 and mean absolute error e = 5.9% and e = 4.7%, respectively. Mat4 have the smallest contact radii as functions of deformation; mF F The same quantities are compared for each of the other ﬁve Mat3 represents a highly plastic case with modulus E /S = f yf material combinations, and are summarized in Table 2, which 522 and very small strain hardening E /E = 0.02, and Mat4 t f shows the mean absolute error values for all of the cases, where represents a highly plastic contact with E /S = 470 and large f yf 1 /δ shows the maximum normalized deformation applied tangential modulus E /E = 0.14. Mat1 and Mat5 both show the m c t f Frontiers in Mechanical Engineering | www.frontiersin.org 10 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening largest contact radii for a given displacement, and they represent (Jackson and Green, 2005), Ye-Komvopoulos (Ye and more compliant materials compared to Mat3 and Mat4. Komvopoulos, 2003), and Kogut-Etsion (Kogut and Etsion, Figure 13 shows the comparison of contact force with respect 2002) models are compared to the experimental data too as the to the normalized deformation between the FE simulations and available data in the literature is similar to elastic-perfectly plastic the predictions for material properties presented in Table 1. contact (i.e., the experimental data available in the literature Overall, the predictions show a very good match with the for the indentation contact of metallic materials does not diﬀerences presented in Table 2. adequately span the strain hardening regime for validating the new model). In Brake et al. (2017), amongst other experiments, the indentation contact of common aerospace materials by 4.2. Comparison With Experimental Data a sapphire sphere for peak loads of 25 mN, 100, mN, 5 N, As a ﬁnal comparison, the experimental data recorded by and 10 N was analyzed. From Brake et al. (2017) the data for Brake et al. (2017) was used to validate the proposed model indentation of Aluminum 6160 (Al 6160) and Stainless Steel against experimental data. Additionally, the Jackson-Green 304 (SS 304) with peak load of 10 N is used here. The material properties reported in Brake et al. (2017) for these experiments are summarized in Table 3. Both of the materials are considered to be elastic-perfectly plastic. Figure 14 shows the comparison between the new model, the previous models, and the data from Brake et al. (2017). The results show relatively small elastic-plastic deformations with normalized deformations up to δ ≃ 6. This is due to the applied force of 10 N during the experiments. Therefore, the contact has just entered the elastic-plastic regime. Hertzian (elastic) contact is also shown on the graph as a baseline for comparison. The new model, Kogut-Etsion Kogut and Etsion (2002), and Jackson-Green (Jackson and Green, 2005) models are all acceptable compared to the experimental data. The previous ﬂattening model (Ghaednia et al., 2019) and the Jackson-Green model (Jackson and Green, 2005) are coincident for the case of elastic-perfectly plastic materials, as modeled here, and is thus not shown. One interesting observation is that from 1 < 1 < 4 experimental data shows larger results than Hertzian theory, which is considered the upper limit of the contact. FIGURE 12 | Comparison between all materials for real contact radius in In Figure 15, the same comparison as in Figure 14 is shown loglog scale. The markers show the results from the FE simulations, and the between the experimental data (Brake et al., 2017), the proposed continuous lines show the predictions. model, and the previous models. The experimental results show normalized deformation of up to δ = 17, which is still in the lower ranges of elastic-plastic regime. The new model compares better with the experimental data than the other models; however, at larger deformations (1 > 15) the experiments show a slight decrease in the slope and a negative second derivation. 4.3. Inﬂuence on Frictional Sliding Together with Ghaednia et al. (2019), four diﬀerent conditions of contact can be considered: 1. Rigid on rigid, 2. Flattening (an elastic-plastic sphere against an elastic or rigid substrate), TABLE 3 | Material properties used from Brake et al. (2017). Mat. sets Al 6160 SS 304 Sapphire E (GPa) 71.47 187.02 370 FIGURE 13 | Comparison between all materials for contact force in loglog S (MPa) 353.70 331.72 - yf scale. The markers show the FEM results and the continuous lines show the Brinell Hardness 99.36 206.91 1740 predictions. ν 0.29 0.29 0.22 Frontiers in Mechanical Engineering | www.frontiersin.org 11 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening FIGURE 16 | Surface proﬁles for (A) rigid on rigid contact, (B) ﬂattening contact, (C) pile-up contact, and (D) indentation contact. TABLE 4 | Effective coefﬁcients of friction for different tangential displacements. Contact condition Effective μ Rigid on rigid 0.6 FIGURE 14 | Comparison between the new model and experimental data Flattening 0.599 from Brake et al. (2017) for Al 6160. Pile-up 0.838 Sink-in 0.577 μ = 0.6, the frictional force deﬁned for rigid on rigid contact is f = μN for a given normal load N. From preliminary simulations of tangential loads applied after normal loads to a ﬂattening case (Al 6061 sphere, WC ﬂat), a pile-up case (WC sphere, Al 6061 ﬂat with E /E = 0), and a sink-in case (WC t f sphere, Al 6061 ﬂat with E /E = 1), the frictional behavior is summarized in Table 4. For all cases, the material properties of Table 1 are used unless otherwise noted; a normal displacement of 0.05 mm is ﬁrst applied, then tangential displacements of 0.1 mm are applied across 100 load steps. Coulomb friction is modeled with μ = 0.6 for all cases. As is evidenced by the table, the contact condition can result in eﬀective coeﬃcients of friction that are up to 50% greater than the rigid on rigid case. These eﬀective μ are, of course, dependent on a number of parameters: normal indentation, tangential displacement, and FIGURE 15 | Comparison between the new model and experimental data bilinear stiﬀness amongst others. As these parameters are varied, from Brake et al. (2017) for Stainless Steel 304. the pile-up contact condition is found to have an eﬀective μ close to 1, while the ﬂattening and skin-in conditions can exhibit eﬀective μ close to 0.5. This makes sense as signiﬁcantly more material is displaced by tangential motion in the pile-up case 3. Indentation (pile-up; an elastic sphere against an elastic- than in the sink-in or ﬂattening cases. It is therefore clear that perfectly plastic ﬂat), the conditions of ﬂattening, pile-up, and sink in must be treated 4. Indentation (sink-in; an elastic sphere against a strain diﬀerently and that the historical approach of using one contact hardening ﬂat). model to describe all three cases is insuﬃcient. As these results are The condition of rigid on rigid could be more broadly preliminary, they merit further investigation in subsequent work. contextualized as elastic on elastic for small deformations; once the deformations become large, the contact evolves into one of the other three conditions. Additionally, ﬂattening could also 5. CONCLUSION include large deformations of an elastic sphere against a rigid (or very stiﬀ) substrate. For these four diﬀerent conditions of contact, In this work, a new formulation for a frictionless elastic-plastic the surface deformations are substantially diﬀerent, as shown single asperity indentation contact of a sphere and a ﬂat has in Figure 16. Once tangential loads are applied to these four been presented. The work focuses on two aspects of elastic- diﬀerent conditions of contact, signiﬁcantly diﬀerent frictional plastic contact, the eﬀect of bilinear strain hardening and the forces are to be expected. Given a coeﬃcient of friction of elastic deformations on the indenter. The presented formulations Frontiers in Mechanical Engineering | www.frontiersin.org 12 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening considers the bilinear strain hardening from elastic-perfectly Finally it has been shown in this work that even 1% tangential plastic to perfectly elastic contact. modulus signiﬁcantly aﬀects the contact parameters. A new The formulation presented in this work provides an empirical predictive formulation based on an empirical formulation of the ﬁt to the FEM results for a wide range of engineering FEM results has been provided for deformations on the objects, metals. Deformations on both of the objects, real contact contact radius, and contact force. The current work, along with radius, and contact force have been considered in the model. previous work on the eﬀect of strain hardening in ﬂattening Several diﬀerent governing conditions have been applied on contact (Ghaednia et al., 2019) are providing a comprehensive the formulation to ensure that the continuity, boundary predictive formulation for a majority of engineering applications. conditions, and common physics limits are satisﬁed. The There is signiﬁcant work to be done for a better understanding formulation was compared with FE simulations for six diﬀerent of single asperity contact. One of the areas that is lacking in material combinations, and the accuracy of the predictions the literature is lack of experimental data for pile-up during was veriﬁed. the loading phase. The challenge is that the measurements need In addition to the contact parameters, the occurrence of pile- to be conducted during compression of the ﬂat. During the up and sink-in on the contact surface have been analyzed. For unloading phase, the pile-ups change signiﬁcantly. There are two elastic-perfectly plastic materials the contact surface shows very main parameters that should be considered in future studies: the large pile-ups. Further, pile-up transforms to sink-in rapidly with eﬀect of friction on pile-up and sink-in, and the eﬀect of strain respect to the strain hardening. From E /E = 0 to E = E = hardening on both of the objects in contact. t t f f 0.01 the maximum peak height of the surface proﬁle decreases by an order of magnitude, and at E /E = 0.06, the pile-up has been DATA AVAILABILITY STATEMENT completely transformed into sink-in. Moreover, the dependency The raw data supporting the conclusions of this article will be of the pile-up on the elastic deformations of the indenter and the made available by the authors, without undue reservation. loading has been analyzed. It was shown that the strain hardening has the dominant eﬀect compared to the loading and indenter’s elastic deformations. AUTHOR CONTRIBUTIONS In this work the indenter was considered to be perfectly HG designed the project, conducted the analytical modeling, elastic. Due to the strain hardening of the ﬂat, for very oversaw all of the work, and wrote the paper. GM and PL large deformations, the contact stresses reach very large states conducted all of the normal indentation FEA modeling. EO’N that, in reality, would cause the indenter to yield; however, it must be considered that for a high strength indenter, the conducted the frictional FEA modeling. MB oversaw the project and edited and revised the paper. ﬂat will fail before the indenter yields. This scenario becomes problematic when the yield strength of the contact materials are close, in which case both of the objects reach the elastic- FUNDING plastic regime at similar deformations. This phenomena is not This research was partially supported by a grant from the Taiho within the scope of the present study, and is relegated to the future work. Kogyo Tribology Research Foundation. REFERENCES Brake, M. (2012). An analytical elastic-perfectly plastic contact model. Int. J. Solids Struct. 49, 3129–3141. doi: 10.1016/j.ijsolstr.2012.06.013 Adams, G., and Nosonovsky, M. (2000). 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Trans. ASME F J. Tribol. 127, 343–354. doi: 10.1115/1.1866166 Conﬂict of Interest: The authors declare that the research was conducted in the Jackson, R. L., and Kogut, L. (2006). A comparison of ﬂattening and indentation absence of any commercial or ﬁnancial relationships that could be construed as a approaches for contact mechanics modeling of single asperity contacts. J. potential conﬂict of interest. Tribol. 128, 209–212. doi: 10.1115/1.2114948 Johnson, K. L. (1987). Contact Mechanics. Cambridge, UK: Cambridge University Copyright © 2020 Ghaednia, Miﬄin, Lunia, O’Neill and Brake. This is an open- Press. access article distributed under the terms of the Creative Commons Attribution Kardel, K., Ghaednia, H., Carrano, A. L., and Marghitu, D. B. (2017). Experimental License (CC BY). The use, distribution or reproduction in other forums is permitted, and theoretical modeling of behavior of 3d-printed polymers under collision provided the original author(s) and the copyright owner(s) are credited and that the with a rigid rod. Addit. Manufact. 14, 87–94. doi: 10.1016/j.addma.2017.01.004 original publication in this journal is cited, in accordance with accepted academic Kogut, L., and Etsion, I. (2002). Elastic-plastic contact analysis of a sphere and a practice. No use, distribution or reproduction is permitted which does not comply rigid ﬂat. J. Appl. Mech. 69, 657–662. doi: 10.1115/1.1490373 with these terms. Frontiers in Mechanical Engineering | www.frontiersin.org 14 August 2020 | Volume 6 | Article 60 Ghaednia et al. Indentation Modeling With Strain Hardening 6. NOMENCLATURE a Real contact radius. a Real contact radius at which the contact reaches the fully plastic regiem. a Real contact radius for purly elastic materials. a Real contact radius for elastic-perfectly plastic materials ep R Equivalent radius of curvature. R Sphere’s radius of curvature. R Flat’s radius of curvature. 1 Total relative normal displacement of the objects during contact. 1 Critical deformation at which the elastic-plastic regime effectively starts. 1 Normalized applied normal displacement, 1/1 . n c 1 Maximum applied normal displacement. δ Deformation of one of the objects. δ Deformation of the sphere. δ Deformation of the ﬂat. δ Deformation at which yield initiates. δ Deformation ratio of the ﬂat. δ Flat deformation ratio limit for elastic contact. fe δ Flat deformation ratio limit for larger deformations. fp μ Coefﬁcient of friction. ν Poisson’s ratio. ν Poisson’s ratio of the sphere. ν Poisson’s ratio of the ﬂat. z Distance depth on the axis of the symmetry from the contact tip. S Yield strength. S Yield strength of the ﬂat. yf n Meyer’s hardness exponent. n Strain hardening exponent. E Effective modulus of elasticity. E Modulus of elasticity of the sphere. E Modulus of elasticity of the ﬂat. E Tangent modulus of elasticity of the softer material (ﬂat). E Effective modulus of elasticity for a ﬂat with E = E . C Coefﬁcient deﬁned by Green Green (2005) to account for the effect of poisson ratio in the initation of yield. F Contact force. F Purely plastic contact force for elastic-perfectly plastic materials. F Contact force for purely elastic materials. F Contact force at very large deformations. LD H Hardness. HP Hardening Parameter. N Normal load. P Average normal pressure in Hertzian theory. P Average normal pressure at very large deformations. LD V Piled-up volume. V Sinked-in volume. φ Function accounting for the effect of applied displacement on the sphere deformation ratio. ψ Function accounting for the effect of bilinear strain hardening on the sphere deformation ratio. χ Function accounting for the effect of bilinear strain hardening on the real contact radius. α,β,λ Fitting parameters. Frontiers in Mechanical Engineering | www.frontiersin.org 15 August 2020 | Volume 6 | Article 60

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