Finite element modeling of fretting wear in anisotropic composite coatings: Application to HVOF Cr3C2–NiCr coating

Journal Pre-proof Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating Akshat Sharma, Ph.D, Akhil Vijay, Ph.D, Farshid Sadeghi PII:

S0301-679X(20)30590-9

DOI:

https://doi.org/10.1016/j.triboint.2020.106765

Reference:

JTRI 106765

To appear in:

Tribology International

Received Date: 19 August 2020 Revised Date:

21 October 2020

Accepted Date: 4 November 2020

Please cite this article as: Sharma A, Vijay A, Sadeghi F, Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating, Tribology International, https://doi.org/10.1016/j.triboint.2020.106765. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Credit Author Statement:

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Akshat Sharma: Conceptualization, Methodology, Software, Writing - Original Draft, Visualization Akhil Vijay: Formal analysis, Investigation, Validation, Writing - Review & Editing Farshid Sadeghi: Resources, Writing - Review & Editing, Supervision, Project administration

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating

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Akshat Sharma1 Ph.D. Graduate Research Assistant Email: [email protected]

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Akhil Vijay1 Ph.D. Graduate Research Assistant Email: [email protected]

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Farshid Sadeghi1* Cummins Distinguished Professor of Mechanical Engineering Fellow ASME, STLE, Email: [email protected]

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School of Mechanical Engineering Purdue University West Lafayette, IN 47907, USA

*

Corresponding Author

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 1 of 33 Corresponding author: Farshid Sadeghi

Abstract This paper presents a two-dimensional (2D) plane strain finite element model to simulate fretting wear in composite cermet coating. The coating considered in this investigation is High Velocity Oxy-Fuel (HVOF) sprayed Cr3C2 – NiCr with 55% volume fraction of Cr3C2. The material microstructure is modelled using Voronoi tessellations with a log-normal variation of grain size. Moreover, the individual phases of the material in the coating were assigned randomly to resemble the microstructure from an actual SEM micrograph. The ceramic carbide phase is orthorhombic and the cubic matrix possesses a high anisotropy index. As a result, each grain

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was modeled with random orientation to account for material anisotropy. The RVE dimensions were chosen such that its elastic response represented the overall response of a poly-aggregate.

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In order to simulate debonding of the ceramic carbide phase from the matrix, cohesive elements were used at the grain boundaries. Damage mechanics was used to model degradation of A grain deletion algorithm was

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cohesive elements resulting from repeated fretting cycles.

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developed to simulate removal of material from fretting wear. The crack patterns predicted from the model match closely with the patterns observed in experimental studies on wear of HVOF

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Cr3C2 – NiCr coating. The model also predicts carbide pullout, a major damage mechanism in HVOF Cr3C2 – NiCr coating subjected to wear. Experiments were also conducted to evaluate

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and corroborate the wear rate of HVOF Cr3C2 – NiCr coating. The wear rate from the model matches closely with experiments at a constant load and displacement amplitude. The results

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from the model were then extended to obtain a fretting wear map under a combination of various loads and displacement amplitudes.

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Keywords: HVOF Cr3C2 – NiCr coating; Cohesive elements; Damage mechanics; Fretting wear

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 2 of 33 Corresponding author: Farshid Sadeghi

1. Introduction Fretting is a phenomenon observed when small vibratory motion occurs between the surfaces in contact. Damage due to fretting typically involves competing mechanisms of corrosion, wear and fatigue. The term ‘fretting corrosion’ was first used by Tomlinson et al. [1] for closely fitted surfaces undergoing surface degradation. They presented conclusive evidences that the damage was caused by vibrations and was mechanical in nature rather than chemical. Suh [2] presented the delamination theory of wear, where he hypothesized that subsurface dislocation pile ups are responsible for creating voids. When multiple voids coalesce, they form a crack which eventually grows towards the surface leading to delamination.

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His theory agreed with the findings of Waterhouse [3], who described fretting corrosion as a form of mild wear, where early surface damage occurs by adhesion followed by removal of material through

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delamination. Using his theory, Suh also provided an explanation of the dependence of fretting wear rates on displacement amplitude, which were later classified by Vingsbo and Soderberg [4] as regimes of wear

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starting from pure sticking condition to sliding wear.

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Although the conventional approach of investigating fretting phenomenon has been experimental in nature, there have been significant advancements in modeling the problem from analytical and numerical

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perspectives. Mindlin [5] studied the effects of tangential forces for elastic bodies in Hertzian contact. He predicted that slip would initiate at the edge of the contact and subsequently put forth an analytical

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solution of the stick region in terms of the contact width, coefficient of friction, normal and tangential force. Nowell and Hills [6] presented closed-form analytical solutions for shear traction distribution at

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different positions in a fretting cycle. Johansson [7] used finite element (FE) to simulate evolution of contact pressure in fretting. Szolwinski and Farris [8] showed that crack initiation and fatigue life in

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fretting fatigue followed Smith-Watson-Topper (SWT) multiaxial fatigue criterion. Goryacheva et al. [9] provided analytical solution of wear profile in partial slip for a 2D contact problem. McColl et al. [10] simulated fretting wear using an incremental wear approach based on Archard’s wear equation [11]. Paulin et al. [12] also used a progressive wear approach to simulate fretting wear in Ti-6Al-4V. However, their approach was based on energy dissipated during fretting. Leonard et al. [13], [14] developed a 2D model of fretting wear based on a combined discrete-finite element approach. Yue and Wahab [15] developed a 2D FE model to show that variable coefficient of friction has a considerable effect in running-in stage of gross-slip. Since wear debris plays a critical role in fretting, different authors [16], [17], [18], [19] have considered the influence of wear debris in their models. Semi-analytical methods have also been used to study fretting wear in dovetail joints at the turbine blade-disk interface [20], [21]. Surface coatings such as chromium carbide (Cr3C2-NiCr) and tungsten carbide (WC-CoCr) are extensively used to mitigate surface damage due to fretting wear in machine components and increase their service life. These coatings are deposited using thermal spraying techniques such as HVOF process

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 3 of 33 Corresponding author: Farshid Sadeghi

and offer excellent wear resistance [22]. The use of FE has also made it possible to investigate the influence of coating microstructure. Holmberg et al. [23], [24] presented a Scanning Electron Microscope (SEM) image-based computational modeling technique for modeling of thermally sprayed multiphase WC-CoCr coating subjected to wear.

Their model showed that stress concentration arises from a

nonhomogeneous multiphase microstructure. Bolelli et al. [25] developed a microstructure sensitive FE model of thermally sprayed coatings. They used their model to evaluate the elastic modulus of HVOF sprayed WC-CoCr and WC-FeCrAl coatings and verified it experimentally with three-point bend tests. Further, their model also reproduced plastic flow and extrusion of the matrix at the edge of the contact,

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which is characteristic of the surface profile observed in sliding wear experiments. The influence of crystallographic orientation of the material microstructure is also key area of focus in

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fretting analyses. Goh et al. [26], [27] used a 2D crystal plasticity FE model to study the influence of plasticity at the grain level in fretting fatigue of Ti-6Al-4V. They found that a random distribution of

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crystallographic orientation in the microstructure is able to better capture the deformation response as

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compared to isotropic J2 plasticity. Zhang et al. [28] used a three-dimensional (3D) Voronoi tessellation based fretting fatigue model to demonstrate the significant effect of grain size and crystallographic

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orientation on plastic deformation. More recently, Paulson et al. [29] and Vijay et al. [30], [31] have shown that a random distribution of crystallographic orientation provides a higher fatigue life scatter

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when modeling rolling contact fatigue failure in bearings. Repeated fretting cycles cause surface fatigue, due to which void formation takes place. This leads to

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progressive degradation of the material. Damage mechanics introduced by Chaboche [32] has been used to model material degradation in rolling contact fatigue [33], [34], fretting [35], [36], [37] and axial

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fatigue [38], where damage accumulates on grain boundaries, which are treated as weak planes for crack initiation and propagation. Ghosh et al. [39] used damage mechanics and FE to simulate fretting wear in partial slip. They were able to demonstrate material removal from the edge of the contact with the use of a stress-based damage model for fretting wear. Leonard et al. [40] have demonstrated that damage accumulation follows a linear trend with fretting cycles and wear volume has also been shown to vary linearly with number of cycles/frictional energy in fretting [41], [42]. More recently, Pereira and Wahab [43] used a damage mechanics based cohesive zone model for life estimation in fretting fatigue. In this investigation, a fretting wear model of HVOF Cr3C2 – NiCr coating is presented.

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microstructure of multiphase Cr3C2 – NiCr coating has been replicated in commercially available FE software; Abaqus using Voronoi tessellations. SEM micrograph of HVOF Cr3C2 – NiCr coating shows that Cr3C2 grain size follows a log-normal distribution and which was also incorporated in the model. These Voronoi polygons are randomly assigned Cr3C2 phase until the volume fraction of Cr3C2 in the RVE reached 55%. Material anisotropy was also considered in the model with random crystallographic

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 4 of 33 Corresponding author: Farshid Sadeghi

orientation of the grains. The elastic modulus of the microstructure is shown to converge as the size of RVE increases. Moreover, cohesive elements were used at the grain boundaries and damage mechanics was used to accumulate damage due to repeated fretting cycles in these cohesive elements. Stress concentration due to a nonhomogeneous microstructure leads to surface as well as sub-surface cracks during fretting. This crack pattern follows similar trend as shown in experimental wear studies on HVOF Cr3C2 – NiCr coating. Carbide pullout, which is a major failure mechanism in wear of HVOF Cr3C2 – NiCr coating, is also predicted by the FE model. Further, experiments were conducted to validate the wear rate obtained from the model. A fretting wear map of HVOF Cr3C2 – NiCr coating is also provided

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for a combination of loads and displacement amplitudes.

2. Modeling Approach

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2.1. Microstructure of the coating

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In Cr3C2 – NiCr coating, the spray powder typically consists of 75 % wt. of Cr3C2 phase and the remaining 25 % wt. is accounted by the NiCr matrix. However, during deposition, some Cr3C2 particles

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rebound from the substrate surface. As a result, the final microstructure of the coating contains lower

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proportion of carbide phase as compared to the original powder [44], [45]. Cr7C3 is an additional compound of chromium carbide formed during the deposition process [46]. However, it is formed in extremely small proportion as compared to Cr3C2, and hence, the entire chromium carbide phase has been

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approximated as Cr3C2 in this study. Figure 1 (a) depicts a representative backscattered SEM micrograph of HVOF Cr3C2 – NiCr coating microstructure. An image analysis software, (ImageJ), was used to

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distinguish between the color intensity of the carbide and matrix phase of the SEM image. Figure 1 (b)

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represents the threshold analysis of the SEM image using ImageJ. From this analysis, the volume fraction of the carbide phase was found to approximately 55%. A volume fraction of close to 55% has also been observed by some authors [46], [47], in HVOF sprayed Cr3C2 – NiCr coating. The backscattered SEM image was also used to obtain the grain size of nearly 400 grains. The distribution of grain size follows a log-normal distribution with log-normal mean and standard deviation of approximately 3 μm and 1 μm respectively. An average grain size close to 3 μm was also found in [47]. Figure 1 (c) illustrates the grain size distribution from the SEM image.

2.2. Representation of coating microstructure An FE model of a coating microstructure requires the geometric discretization of SEM micrographs into the FE mesh. One possible approach is to use image-based techniques such as OOF2 [23], [24], which use a pixel-based meshing technique. Although this method can construct the real microstructure of a material, a clear and detailed SEM image is required as an input.

Also, SEM images of high

magnifications [48] only reveal the local microstructure. Hence, to obtain microstructural variation at

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 5 of 33 Corresponding author: Farshid Sadeghi

different locations, multiple SEM images are needed. As a result, this method is not very efficient when modeling microstructural variations.

Moreover, because the mesh obtained near microstructural

inhomogeneities is extremely dense, modeling of fretting over many cycles becomes computationally intensive. An alternate method is Voronoi tessellations, which have been previously used to model granular topology variations in polycrystalline materials [49], [50]. In Voronoi tessellations, seed points are distributed randomly across the entire Euclidian space. Voronoi cells are generated using the seed points, such that the distance of any point in a cell to the seed point of that particular cell is less than the distance of that point to any other seed point. The Voronoi cells correspond to materials grains, and their boundaries represent grain boundaries [51].

The use of Voronoi tessellation also facilitates the

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incorporation of inherent material topology by introducing grain size and geometrical distributions, The Voronoi

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thereby making it a better choice as opposed to using a structured mesh approach.

microstructure used in this study was developed using the software package, Neper [52]. The grain

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morphology parameters were setup such that a log-normal grain size distribution as depicted in Figure 1 (c) can be replicated. Several domains were generated by choosing unique seed distribution to account

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for microstructural variability. Cr3C2 phase was randomly assigned to the grains until its volume fraction

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in the domain reached 55%. Figure 2 shows a domain generated using Neper with Cr3C2 in black and NiCr in grey.

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Thermal spray coatings are generally composed of a heterogeneous microstructure with various constituents. These individual constituents further possess unique crystallographic properties. As a

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result, the mechanical behavior of thermal spray coatings is highly anisotropic [53], [54]. Hence, it is important to consider the effects of a multiphase microstructure along with anisotropic crystallographic

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orientation. In HVOF sprayed Cr3C2-NiCr coating, the individual phases viz., Cr3C2 and NiCr possess orthorhombic and cubic crystal structure respectively [55], [56]. The corresponding stiffness matrices that define the constitutive stress-strain relationship for orthorhombic Cr3C2 and cubic NiCr are: =

=

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0

0

0

(1)

(2)

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 6 of 33 Corresponding author: Farshid Sadeghi

The values of the material constants for Cr3C2 and NiCr are obtained from Li et al. [55] and Faraoun et al. [56]. These material constants are given in Table 1 and they depict the stiffness in local coordinate system. Anisotropy was defined by assigning a random orientation to each Voronoi grain through a unique Euler angle. The local stiffness matrix was then rotated by the Euler angles to obtain the global stiffness matrix as: !"#$%"

= &' &( &)

+ + + "#*%" &) &( &'

(3)

where, Cglobal and Clocal are the global and local stiffness matrices and Ri is the rotation matrix associated with the corresponding axis, i. The global stiffness matrix was used to define material properties in

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Abaqus using UMAT subroutine. It is important to note that although the problem is modelled using a 2D plane strain assumption, the grain orientations were allowed to rotate in all three directions. The full

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3D stiffness matrix of each grain was formulated based on the grain orientations. It was only after this More details on the rotation

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matrix construction that the assumption of plane strain was applied.

procedure adopted in this study can be found in [29], [30]. The equivalent anisotropy index (Aeq) [57]

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was also evaluated for Cr3C2 and NiCr using the constants mentioned in Table 1. Cr3C2 was found to have Aeq value of 2.02, whereas the value for NiCr was found to be 8.67. Aeq values typically range from

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1 to 9, with values closer to 1 for isotropic materials. This further reveals that the material under consideration in this study, specially the matrix phase, is highly anisotropic.

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2.3. Micromechanical estimation of effective mechanical properties

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The presence of a heterogeneous anisotropic microstructure in HVOF Cr3C2-NiCr coating leads to a twostep process for evaluation of the effective mechanical properties.

The first step is related with

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homogenization of a multiphase microstructure, while the second step deals with random orientation of microstructure obtained from the first step. The homogenization process averages the effects of a heterogeneous microstructure to arrive at the overall macroscopic behavior of the material. This can be achieved using approaches like Voigt and Reuss rule of mixtures (ROMs), which represent the upper and lower bounds of stiffness of a multiphase material [58], [59]. The Voigt rule is based on equal strain field assumption in the representative domain whereas the Reuss rule is based on equal stress field assumption in the representative domain. The method of evaluation of Voigt and Reuss ROMs for orthotropic constituents is mentioned in [60]. These are given by Equations (4) and (5) respectively. ,

,-./0

=

,

,

0 0 0

,

,

,

0 0 0

,

,

,

0 0 0

0 0 0

,

0 0

0 0 0 0

,

0

0 0 0 0 0

(4)

,

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 7 of 33 Corresponding author: Farshid Sadeghi

where,

, 1

=2

and NiCr respectively. Cr3C2 and NiCr.

.-./0

4. 4. 4. = 0 0 0

+ 2

1

4. 4. 4. 0 0 0

1

4. 4. 4. 0 0 0

0 0 0 4. 0 0

and

1

1

0 0 0 0 4. 0

with 2

and 2

representing volume fraction of Cr3C2

are the elements of the corresponding stiffness matrices of

0 0 0 0 0 4.

-

(5)

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where, 4 .1 = 2 41 + 2 41 . 41 and 4 1 are the elements of the corresponding compliance matrices of Cr3C2 and NiCr, which are obtained by inverting their respective stiffness matrices. From the Voigt and Reuss ROMs, the arithmetic Hill average [61] was evaluated as: = (

,-./0

+

.-./0 )

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6

(6)

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Although, CH calculated by Equation (6) represents the homogenized stiffness, but it does not account for random orientation of the poly-aggregate microstructure.

In order to further account for material

anisotropy, the method outlined by Hearmon [62] was used to calculate the Voigt-Reuss-Hill (VRH) (6).

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average Young’s modulus (9,.6 ) and Shear modulus (:,.6 ) using the stiffness matrix from Equation This formed the second of the two-step process.

9,.6 and :,.6 were calculated to be

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approximately 264.64 GPa and 100.01 GPa respectively. This resulted in ;,.6 of 0.323. The FE

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simulations presented in Section 2.5 also corroborate with these values.

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2.4. Modeling of grains and interfaces in finite elements The discretization of Voronoi cell microstructure for FE analysis involves two aspects, viz. (i) bulk response of each grain and (ii) response of each grain boundary interface. In this study, each grain was meshed with linear triangular (CPE3) elements to capture the material response in the bulk, whereas the grain boundaries were modeled using cohesive elements. The interface between particles and matrix [63] as well as at grain boundaries [31], [64] has been extensively characterized with cohesive elements. Using these elements, damage can be localized at an interface such as a grain boundary, which are potential locations of crack initiation and propagation [65]. In this study, it has also been hypothesized that damage from fretting wear leads to debonding between the carbide and matrix phase and subsequently loss of material.

In order to simulate this debonding, four-noded cohesive elements,

COH2D4, were inserted at the grain boundaries. Figure 3 shows cohesive elements inserted at the grain boundaries in the FE domain. HVOF Cr3C2 – NiCr coating investigated in this study is a type of particle-reinforced metal matrix composite coating. The bi-linear traction separation law has been used previously [63], [66], [67], [68] to Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 8 of 33 Corresponding author: Farshid Sadeghi

model various metal matrix composites. Also, it is important to consider that the primary aspects that govern the shape of the cohesive model are the peak traction and the area under the traction displacement curve. While the shape of the traction-separation curve can indeed play some role in the response of material, it is generally accepted that the bi-linear law represents the optimal choice between computational cost, numerical stability and approximation [69]. In the case of a Voronoi framework with discrete cohesive zone elements along a grain boundary [31], [64], [70], [71], the bi-linear relation is the most common choice. Hence, a bi-linear traction-separation curve was used in this study. The initial constitutive traction-separation relationship is given by Equation (7). => B < = @ = A >> 0 ?

0 E> CD F B?? E?

(7)

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where, => , B>> , E> and =? , B?? , E? are the traction, stiffness and separation in the normal and tangential

directions respectively. A maximum nominal stress criteria was used to initiate damage in these cohesive

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< => > =? max J , #N = 1 =># =?

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elements. This is given by Equation (8).

(8)

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where, =># , and =?# are the critical tractions. The Macaulay step function ‘< >’ represents that the material is not damaged in compression. Once the tractions on these cohesive elements reach the value of critical

traction, softening of the cohesive elements takes place. This softening behavior is implemented through

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a static damage variable, PQ . The damage variable PQ increases from a value of zero at critical separation

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E # to a value of unity when maximum separation E R%) is reached. The evolution of PQ in terms of the

separations and is given by Equation (9). U

TVW -S X ) ST (ST T U

TVW (S -S X ) ST T T

where, ER = Y< E> > + E?

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PQ =

(9)

where, E Z is the separation at a particular instance in the damaged configuration. Figure 4 depicts the

schematic representation of the bi-linear traction-separation response. A more detailed explanation on the constitutive model for cohesive elements can be found in [72]. The normal and tangential separations E>

and E? , are normalized with respect to the initial thickness [# , to represent the nominal normal and shear

strains. The initial thickness [# , was set to 1 μm in this model. The undamaged stiffness in the normal # direction, B>> was set to be equal to 109,.6 , while the undamaged stiffness in the tangential direction

was set to be equal to 10:,.6 . When these values of undamaged stiffnesses of the cohesive elements were used in the FE simulation discussed in Section 2.5, they resulted in the overall Young’s modulus and

Poisson’s ratio of the coating to be approximately equal to 9,.6 and ;,.6 respectively. The undamaged critical tractions were derived from the yield strength of the material [31], [64]. A general rule for

estimating the yield strength (σy) from hardness (H) is mentioned in [73], [74] where it is stated that σy ≈ H/2.8. This relation was used in the current study with H = 9785 MPa measured for HVOF Cr3C2 – NiCr Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 9 of 33 Corresponding author: Farshid Sadeghi

coating with conventional grains [75]. The area under the traction-separation response curve is equal to the cohesive fracture energy :* [72] and is represented by:

1 :* = =E Z 2

(10)

The fracture toughness (B] ) of HVOF Cr3C2 – NiCr coating has been mentioned to be around 4 MPa m1/2 [46]. This was used to calculate the fracture energy [26] as per Equation (11), which was subsequently used in the cohesive zone model. Table 2 lists the properties of the cohesive elements used in this study. : =

B] (1 − ; ) 9

(11)

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2.5. Determination of minimum size of the representative domain

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The representative domain of a multiphase material with random orientation of its constituents should be

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of sufficient size, such that, it characterizes the overall macroscopic response of the bulk material. The smallest volume element which can represent the effective response of the bulk material is called the With the use of RVE, simulation of large domains can be

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representative volume element (RVE).

eliminated. In this investigation, a study was conducted on a 2D plane strain model to determine the size

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of the RVE by evaluating the effective Young’s modulus and Poisson’s ratio for RVEs of different sizes. The approach for evaluating these properties follows the method used by Toonder et al. [76]. A uniaxial

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strain of 0.5% was applied to the right face in the X direction as shown in Figure 5. The top and bottom faces were constrained from moving in the Y direction and the left face was constrained from moving in

(12)

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= %__ (( = 0 ))

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the X direction. This results in the following strains on the right and bottom face respectively.

For 2D plane strain condition, the constitutive stress-strain relationship is given by Equations (13) and (14).

9abc [(1 − ;abc ) (1 + ;abc )(1 − 2;abc ) 9abc = [(1 − ;abc ) (1 + ;abc )(1 − 2;abc )

`)) = `((

))

+ ;abc

((

+ ;abc

(( ]

(13)

)) ]

(14)

where, 9abc and ;abc are the overall Young’s modulus and Poisson’s ratio evaluated using FE analysis.

Similarly, `)) and `(( are the average of

))

and

((

on the right and bottom faces respectively, which

can be computed from the FE simulation. Upon substitution of Equation (12) into Equations (13) and (14), a solution can be found for 9abc and ;abc in terms of `)) , `(( and

9abc =

f `)) + 2 `(( gf `)) − `(( g %__ ( `)) + `(( )

%__

as: (15)

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 10 of 33 Corresponding author: Farshid Sadeghi

;abc =

`(( `)) + `((

(16)

The FEM simulations were implemented in Abaqus and a MATLAB script was used to evaluate 9abc and

;abc . Figure 6 illustrates 9abc and ;abc evaluated for square RVE domains of edge length L with L

ranging from 25 μm to 150 μm in increments of 25 μm. For each RVE edge length, four different

domains were used. These values converge from RVE size of 75 μm onwards. Hence, the minimum dimension of the RVE to represent HVOF Cr3C2 – NiCr coating microstructure is 75 μm. The converged

values of 9abc and ;abc also match 9,.6 and ;,.6 respectively, calculated in Section 2.3. It is worth

mentioning that, the elastic modulus of HVOF Cr3C2 – NiCr coating measured using indentation

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technique [53] shows a significant amount of variation. This is due to the fact that indentation measures

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the local properties whereas, the FE model in this study considers the macroscopic response of the bulk material. The value of Young’s modulus obtained from the FE model falls inside the variation observed

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in [53]. However, it needs to be noted that the current model does not account for porosity in the material

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which is observed to some extent in HVOF Cr3C2 – NiCr coatings [46].

2.6. Finite element simulation of fretting wear

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The material and finite element framework was then extended to develop a 2D plane strain model for modeling of fretting wear. The fretting contact was simulated by moving a rigid cylindrical upper body

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of 8.4 mm diameter, over a representative coating microstructure embedded in a lower body as shown in Figure 7. The assumption of a rigid upper body was made to suppress wear in it as the purpose of this

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study was to characterize fretting wear of HVOF Cr3C2 – NiCr coating alone. The coating domain consisted of the multiphase microstructure of the coating with random orientation of its constituents. The

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size of the coating domain was 150 μm in width and 75 μm in depth, thus meeting the minimum size requirement discussed in Section 2.5. The coating domain was surrounded by an extended homogeneous domain. This extended domain was assigned isotropic material properties viz., 9,.6 and ;,.6 . Further,

infinite elements were used at the bottom and side faces of the extended domain to represent the elastic half-space and eliminate any edge effects in the model. The out-of-plane thickness for all fretting wear simulations was kept at 6 mm, similar to the experimental setup discussed in Section 2.6.3. Surface roughness has negligible effect in gross sliding fretting wear [77] and thus, it was ignored in this study. In fretting simulation, firstly a load balance step was used with a maximum load of 90 N applied on the upper body, which resulted in a half contact width of about 17 μm. Then, a coefficient of friction (μ) of

0.475 was used at the interface of the upper and lower bodies. This was measured experimentally as discussed in Section 2.6.3.

To impose friction at the interface, a penalty based surface-to-surface

interaction method was used. Lastly, sinusoidal displacements of small amplitudes (up to 25 μm) were applied to the rigid upper body to simulate fretting wear. Since the model is stress-based, the number of

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 11 of 33 Corresponding author: Farshid Sadeghi

steps were controlled to accurately capture the stress state of individual grains in a fretting cycle. As a result, 40 discrete steps were used with a displacement amplitude (δ) of 25 μm.

It is also worth

mentioning that the depth of the coating domain was more than four times the maximum half contact width. As a result, the sub-surface Hertzian stresses were mostly confined within the coating domain.

2.6.1. Mesh convergence Contact between the upper and lower bodies necessitates that the mesh to be sufficiently refined in the contact region in order to accurately determine the contact stresses. Hence, a mesh convergence study was conducted to determine the required element size. Different meshes with grain size to element size

of

ratios of 2.5-5.5 were considered. The contact pressure was integrated over the contact width to evaluate the reaction force. This force was compared with the applied load on the upper body and percentage error

ro

was calculated. Figure 8 (a) illustrates that the error percent decreases as the element size decreases.

-p

Figure 8 (b) and (c) show the coarsest and finest mesh used in this analysis. From this analysis, a mesh with grain to element size ratio of 5.5 was used at the contact surface to achieve an error of < 1%.

re

2.6.2. Cohesive zone damage from fretting wear

lP

The damage variable Ds, presented in Section 2.4, takes into account only the static degradation in cohesive elements. This would be useful when the material is subjected to static overloads. However,

na

when used in fatigue loading, this would lead to an infinite life estimation. Hence, modeling of fatigue failure takes into consideration an additional irreversible damage parameter [78]. Since fretting wear is a

ur

surface fatigue process, this variable needs to be accounted to represent damage from repeated fretting cycles. Thus, the total accumulated damage in cohesive zone models typically comprises of damage due to quasi-static overloads and from cyclic loading [64], [70]. This can be written as:

Jo

P = PQ + PZ

(17)

where, P is the total damage variable and PZ is the fatigue damage parameter. When, P is zero, it

represents an undamaged material and when P is unity, it represents a completely damaged material.

Based on the total damage, the stiffness of the cohesive element decreases as: # B>> = B>> (1 − P) # B?? = B?? (1 − P)

(18)

# where, B>> and B??# are the initial undamaged stiffness of the cohesive element in normal and tangential

directions. It is assumed that this degradation only applies when the normal separation opens the crack in tension. In compression, crack closure takes place and the stiffness remains unchanged. The total damage variable, P, also leads to reduction in the critical tractions as follows: # =># = =>_R%) (1 − P) # # =? = =? R%) (1 − P)

(19)

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 12 of 33 Corresponding author: Farshid Sadeghi

where, =># and =?# are the undamaged critical tractions in the cohesive element in normal and tangential directions.

The principles of damage mechanics [79], [80], which are constituted within the framework of

thermodynamics can be used to obtain the evolution of fatigue damage parameter, PZ . This evolution can

be written as:

R iPZ k =A C (1 − P) ij

(20)

where, k is the critical stress reversal quantity causing damage,

is the resistive stress; it quantifies the

of

resistance of a material from fatigue damage and m is a material parameter. Equation (20) has previously been used in damage evolution of fretting fatigue [35], [36], axial fatigue [38], [81] and rolling contact

ro

fatigue [33], [34], [82].

-p

Fretting wear is a surface phenomenon which involves removal of material under the application of load and sliding. The amount of material removed is governed by Archard’s wear equation: n 1 p. 4 = p. 4 o

re

lm =

(21)

lP

where, lm is wear volume, n is Archard’s wear coefficient, H is the hardness of the material, F is the

= H/k is dependent on the material pair

na

applied normal load and S is the sliding distance. The quantity

and measures their resistance towards wear. The higher this resisting wear term, the lower the wear volume and vice versa. Hence, a parallel can be drawn between this term and

in Equation (20). As

ur

wear volume is linear in terms of the applied load, the value of m can also be taken as unity [39], [40].

Jo

The choice of the critical stress reversal quantity causing failure can be accounted from the stress state in the fretting process. Friction and hence shear stress changes direction, when the direction of motion is reversed in a fretting cycle. Shear stress reversal is maximum at the surface, where wear occurs. As a result, it is hypothesized similar to [39], that shear stress can be taken as a critical stress reversal quantity in fretting wear. Therefore, the damage evolution equation for wear can be written as:

iPZ k= =A C (1 − P) ij

(22)

2.6.3. Determination of material pair constant from wear experiments Fretting wear experiments were conducted on the Bruker UMT TriboLab to determine the material pair constant, . For this purpose, an upper cylindrical body made of alumina was used against a lower flat HVOF Cr3C2 – NiCr coated specimen. Figure 9 (a) shows the experimental setup for fretting wear tests. Alumina has been used in previous studies as a rigid body and it has been shown to undergo negligible wear and is chemically inert [46], [83]. The diameter and length of alumina cylinder used were 8.4 mm and 6 mm respectively. A frequency of 5 Hz was used to minimize the effect of vibration during the Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 13 of 33 Corresponding author: Farshid Sadeghi

experiments. A load of 45N was applied which resulted in a half contact width of about 12 μm. The thickness of the coating used in the experiments was approximately twenty five times this half contact width. Hence, all stresses are confined within the coating. This allows for only the coating to be modelled using FE and the substrate can be neglected for fretting contact simulations. The number of fretting cycles was fixed to 20,000. It should be noted that the hysteresis loop at the ends of the fretting stroke is influenced by compliance of the experimental setup [84]. As a result, an effective hysteresis loop can be represented with the same amount of dissipated energy and steady state friction force, but reduced displacement amplitude. Figure 9 (b) illustrates the hysteresis loop obtained from the experiments and the effective hysteresis loop. As depicted in Figure 9 (b), the applied δ was 300 μm, whereas δ of the The compliance effect of the experimental setup imposed

of

effective hysteresis loop was 150 μm.

ro

limitation on the displacement amplitude, due to which an effective δ of 150 μm was chosen to determine the constant, γ. The coefficient of friction (μ) was measured to be 0.475 by dissipated energy approach

-p

[85]. The wear scar was analyzed using the Bruker NPFLEX optical surface profilometer. The image of the wear scar as obtained from the profilometer and its cross-sectional wear profile are shown in Figures 9

re

(c) and (d) respectively. The worn out area from cross sectional wear profile was multiplied by the

lP

contact length to evaluate the wear volume, as performed by the authors in [12]. The wear volume data was linearly approximated to evaluate 7

2

using Equation (21), which was found to have a value of

na

1.639x10 N/mm for alumina - HVOF Cr3C2 – NiCr material pair.

2.6.4. Overview of fretting wear simulation

ur

The numerical modeling the fretting phenomenon is a computationally intensive non-linear contact analysis. As a result, the evolution of damage on a per cycle basis is not computationally feasible.

Jo

Therefore, most fretting wear and fatigue analyses use either a damage indication parameter like the Smith-Watson-Topper criteria for fretting fatigue [86] or Archard’s law formulations [40] for fretting wear. Alternately, for simulations involving a large number of cycles, an accelerated simulation approach called ‘jump-in-cycles’, developed by Chaboche [80] is extensively used. This approach has been used in fatigue simulations [33], [34], [35], [36], [38], [81] and also in wear [39], [87].

In this approach the

evolution of damage is considered constant for some block of cycles. This accelerates the computational procedure. In this study, this approach was adopted with ΔD = 0.2, which has been previously used in fretting [39], [40]. A brief overview of this procedure is mentioned here: (i) The initial damage was set to zero for all cohesive elements Pq = 0 where, r = 1,2,3….n cohesive elements

(23)

(ii) Fretting wear simulation over one cycle was conducted in Abaqus and a C++ code was used to record

the stress history over discrete number of steps. Moreover, the static degradation parameter, PQ was also recorded.

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 14 of 33 Corresponding author: Farshid Sadeghi

(iii) A post-processor code written in MATLAB, reads the stress history and averages the stresses along the cohesive elements at a grain boundary in each load step. The damage evolution rate given by Equation (19) was evaluated for all grain boundaries. (iv) The grain boundary with highest damage evolution rate was selected to determine the current block of cycles as per jump-in cycle approach. kj =

st

vwU z

u

y

(24)

vx TVW

|tU

where, kj is the finite block of cycles and { } |

(v) The number of cycles was updated as: ~

|

along the critical

= j + kj

ro

j

|tU

of

grain boundary after ith simulation.

R%)

is the maximum value of

(25)

1

(26)

re

|tU

P1 ~ = P1 + PQ 1 + kj { } |

-p

(vi) Damage for all grain boundaries was updated at the start of the next simulation as:

lP

where, j is the grain boundary number. Theoretically, the value of the critical damage P* should be unity,

for a fully damaged material. However, this causes instabilities in the simulation and hence, the critical The damage field variable was defined in Abaqus using USDFLD

na

damage P* is kept at 0.999.

subroutine. A crack is assumed to be initiated, if damage along a grain boundary reaches the value of

ur

critical damage. Once, all the grain boundaries of a particular grain reach the critical damage value, the grain is removed to simulate fretting wear. Wear volume per unit length is computed from the area of the

Jo

removed grain.

3. Results and Discussion

3.1. Subsurface Stresses in the Coating The load balance step in fretting simulation of Hertzian line contact leads to generation of subsurface stresses in the coating. A theoretical solution of these stresses for homogeneous isotropic materials in available in literature [60]. A comparison was made between the stresses obtained from a homogeneous isotropic domain with Young’s modulus and Poisson’s ratio as 9,.6 and ;,.6 respectively and a

heterogeneous anisotropic domain. No friction was considered for this comparison. The heterogeneous anisotropic domain of the coating was represented with 55% volume fraction of Cr3C2 and random orientation of the grains.

Figure 10 shows the centerline stress distribution of

),

(,

'

and

corresponding absolute maximum shear stress =R%) from these domains compared with the theoretical

solution. The centerline stress distribution for the homogeneous isotropic domain is smooth and closely matches with the theoretical solution. This trend is also shown by the Von Mises stress contours for the Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 15 of 33 Corresponding author: Farshid Sadeghi

isotropic domain in Figure 11 (a). The heterogeneous anisotropic domain shows localized deviation at discrete depths below the surface. However, the general trend resembles that of the homogeneous domain and theoretical results. The local jumps in the stress distribution in the heterogeneous anisotropic domain is clearly demonstrated by its Von Mises stress contour shown in Figure 11 (b). The heterogeneous anisotropic coating domain creates mismatch of elastic stiffness between neighboring grains and this leads to local stress concentration. Moreover, the cohesive element at the interface of the grains acts as a junction of this stress concentration. Prior research [30] has shown that maximum stress along a grain boundary has a mesh dependency, however, the average stress along the grain boundary is bounded [30], [88].

Hence, the averaging approach along a grain boundary has been adopted in this study to

of

characterize the material response. Figures 12 (a) and (b) depict the quarter cycle position of sinusoidal

ro

fretting displacement and Von Mises stress contour at that instance respectively. Here, μ of 0.475 was used at the contact. The Von Mises contour plot shows that stresses during fretting are also locally

-p

affected by the heterogeneous anisotropic microstructure of the coating.

re

3.2. Crack Pattern from Simulation of Fretting Wear

Fretting wear simulations were performed with F = 45 N, μ = 0.475 and δ = 25 μm. The coefficient of

lP

friction was found from the performed experiments. The evolution of damage with a damage increment of ΔD = 0.2, resulted in crack formation when damage along a grain boundary reached the critical value The coating domain with a heterogeneous microstructure and anisotropic material

na

of damage, P* .

definition was used. Figure 13 (a) illustrates the crack pattern obtained from the FE simulation. The

ur

average crack length from nearly 100 cracks generated by the FE models was found to be 2.8 μm. The heterogeneous microstructure and anisotropic material definition lead to both surface as well as sub-

Jo

surface cracks in random direction. This can again be attributed to stress concentration that arises in the cohesive elements due to elastic stiffness mismatch from a heterogeneous microstructure and random orientation of the grains. Figure 13 (b) depicts the crack pattern obtained from sliding wear experiments conducted by Matikainen et al [47].

The SEM image in Figure 13 (b) was used to estimate the

experimentally observed average crack length, which was observed to be approximately 3.6 μm. The experimental crack pattern confirms that surface as well as sub-surface cracks oriented in random direction are observed in wear of HVOF Cr3C2 - NiCr coating. Although, experimental studies in [47] are for sliding wear regime, the distinct crack pattern arises due to a heterogeneous and anisotropic coating microstructure and hence, the same results can be expected in fretting wear regime too. Previous damage mechanics based FE study on fretting wear [39] conducted with a homogeneous microstructure and isotropic material definition resulted in only surface cracks.

This reveals that a heterogeneous

microstructure and anisotropic material definition of the coating is essential to replicate its experimental characteristics.

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 16 of 33 Corresponding author: Farshid Sadeghi

3.3. Wear Pattern from Simulation of Fretting Wear The grain removal technique adopted in the post-processor MATLAB code removes a grain, when damage in all cohesive elements at a grain boundary reaches the critical value of damage, P* . As shown

previously, this can happen on the surface as well as sub-surface of a heterogeneous anisotropic domain of the coating. However, material removal by wear can occur only from the surface. Thus, the model was restricted to the condition that only the grains on the surface can be removed in the simulation. After removal of the grains, the node list, element list as well as surface element list were updated in the next simulation. Since the carbide phase has a higher stiffness, it leads to higher stresses in the carbide grains.

of

As a result, the reversal of shear stress is also higher in the cohesive elements at its grain boundary and hence, carbide grains at the surface are typically removed first from the simulation as depicted in Figure

ro

14 (a-b). Carbide pullout is one of the major damage mechanism observed in wear of HVOF Cr3C2 NiCr coating [46] and is shown in Figure 14 (c). The FE model developed in this study can accurately

-p

predict this phenomenon as it is a stress-based failure mechanism. The model was further used to

re

measure the ratio of wear volume lost through carbide pullout to the total wear volume. This ratio was defined by the parameter, β. Figure 14 (d) illustrates the variation of β with number of fretting cycles for

lP

three different coating domains. It is noticed that β is typically higher than the 55% volume fraction of carbide in the coating domain. Hence, the model predicts more carbide removal during wear process.

na

This again agrees well with the findings in literature [46], [47], which show that carbide pullout is a dominant damage mechanism in wear of HVOF Cr3C2 – NiCr coating.

ur

Fretting wear profiles were obtained when sufficient grains were removed and considerable wear had taken place. In fretting wear, a partial slip wear profile is one in which material is removed from the edge

Jo

of the contact [40]. This is due to the fact that shear stress is highest at the ends of the stick portion [89]. A gross slip wear profile occurs when shear stress is maximum at the center of the contact. Figure 15 illustrates both partial and gross slip fretting wear profiles obtained from the simulation of different coating domains. As described earlier, stresses in the carbide phase are higher than in the matrix phase. As a result, when the carbide phase is present near the edge of the contact and the matrix phase is present near the center of contact, there is increased likelihood of obtaining a partial slip wear profile. This is shown in Figure 15 (a), where the matrix phase is present near the center of the contact and does not wear out. Hence, the FE model is also sensitive to the stress-based nature of distinct fretting wear profiles.

3.4. Comparison with Experimental Results The evolution of damage in the FE model and the subsequent removal of material can be used to obtain the total wear volume per unit length with respect to fretting cycles. The total wear volume per unit length and number of cycles were stored after each simulation. Three distinct domains were used for this purpose and F = 45N with δ = 25 μm were applied. The results from these domains were compared with Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 17 of 33 Corresponding author: Farshid Sadeghi

experimentally determined wear rates from two experiments. Figure 16 provides comparison of the results obtained from these domains with experimental results. The FE model slightly over predicts the number of fretting cycles which leads to a reduced slope of the wear rate but nonetheless, the wear rate obtained from the FE model is in close approximation with the experimental results. One plausible reason for the lower wear rate from the FE model is the ignorance of porosity in the domains. Porosity in the coating microstructure would create more stress concentration [24], which can make the wear rate curves from the FE model more aligned with the experimental results. Fatigue crack behavior is highly dependent on the inherent material microstructure as mentioned in literature [90], [91] and these three domains present the scatter in wear rate from microstructural variability. The inherent influence of the

of

material microstructure lies more in the initial period of running-in wear and reduces with more cycles. It

ro

should be noted that the incubation period until the removal of first grain leads to no wear volume [39], [40]. However, this is not consistent with Archard’s wear equation (Equation (21)), which states that

-p

wear volume is zero only at the beginning when number of cycles are zero.

Hence, to maintain

re

consistency with Archard’s wear equation, the incubation period was neglected in this analysis.

3.5. Fretting Wear Map

lP

After validating the FE simulation with experimental results, the model was used to generate a fretting wear map over a range of applied load and displacement amplitudes. A numerical generation of fretting

na

wear map eliminates the need to perform multiple fretting wear experiments. F was varied from 30 to 90 N, while δ was varied from 6 to 25 μm. Total of 25 simulations were conducted to create a fretting wear

ur

map. Figure 17 shows the fretting wear map of HVOF sprayed Cr3C2 – NiCr coating with wear volume per unit length per cycle generated from the FE simulation. The fretting wear map represents wear rates

Jo

at different loading conditions. Figure 17 demonstrates that wear rate increases with increase in normal load and displacement amplitude and this trend is consistent with Archard’s wear equation. It should be noted that wear volume per unit length per cycle can be written as: lm np = .4E •. j o •

(26)

Thus, wear volume per unit length per cycle is proportional to the product of applied load per unit length and displacement amplitude. Hence, for a given length, the curves of constant wear volume per unit length per cycle can be represented by:

p. E = •

(27)

These curves represent a rectangular hyperbola and were plotted on top of the fretting wear map in Figure 17. As seen in the figure, the curves of constant F.δ are close to the contour lines of the fretting wear map. This again proves that the fretting wear map generated from the FE model follows a similar trend to Archard’s wear equation. Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 18 of 33 Corresponding author: Farshid Sadeghi

4. Summary & Conclusions This paper presents a microstructure sensitive FE model for fretting wear of HVOF Cr3C2 – NiCr coating. To represent the microstructure of the coating, Voronoi tessellations were used with random distribution of Cr3C2 phase.

This was performed until the volume fraction of Cr3C2 reached 55%, which is

characteristic of the HVOF Cr3C2 – NiCr coating. To account for anisotropy of the cubic NiCr matrix and orthorhombic Cr3C2 phase, a procedure that allowed for random crystallographic orientation was used. Cohesive elements were used at the grain boundary to simulate debonding of Cr3C2 from the matrix phase. A micromechanical estimation of the effective properties of the coating was corroborated with FE Moreover, the size of the RVE required to represent the coating

of

simulations of uniaxial strain.

microstructure was found to be 75 μm from this simulation.

A stress-based damage mechanics

ro

formulation was used to account for degradation of cohesive elements at the grain boundary. Once damage on all grain boundaries of a grain at the surface was reached, a grain removal technique was

-p

incorporated to simulate fretting wear in HVOF Cr3C2 – NiCr coating. The crack pattern obtained from

re

the FE model matches the experimental crack pattern observation from literature. The model also predicts carbide pullout from the matrix, which is a major damage mechanism in wear of HVOF Cr3C2 –

lP

NiCr coating. The results from the FE model were compared with experimental results and a close match was found. It was also observed that variability in the coating microstructure has a noticeable effect on

na

the wear rate at least in the initial running-in period. Lastly, a fretting wear map of HVOF Cr3C2 – NiCr coating was generated with a combination of various loads and displacement amplitudes. The fretting

5. Acknowledgements

Jo

ur

wear map of the coating was found to be consistent with Archard’s wear equation.

The authors would like to express their deepest appreciation to the sponsors of METL for their support to this project.

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 19 of 33 Corresponding author: Farshid Sadeghi

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Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 24 of 33 Corresponding author: Farshid Sadeghi

Constant in GPa Cr3C2 [48] NiCr [49]

Table 1: Elasticity Constants for Cr3C2 and NiCr C11 C12 C13 C22 C23 C33 C44 C55 C66 447.1 217.5 243.3 545.3 217.9 471.2 237.7 116.6 241.3 221.4 193.8 119.8 Table 2: Properties of Cohesive Elements Value Unit 1 μm 2646.4 GPa 1000.15 GPa 3494.64 MPa 2016.4 MPa 0.054 N/mm

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Property [# # B>> B??# # => R%) ( ( ) # =? R%) (0.577 ( ) :*

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 25 of 33 Corresponding author: Farshid Sadeghi

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(c) Figure 1: (a) Backscattered SEM micrograph of HVOF Cr3C2 – NiCr coating (b) Threshold analysis in ImageJ to evaluate carbide volume fraction (c) Grain size distribution from SEM micrograph

Figure 2: Domain of HVOF sprayed Cr3C2 – NiCr coating generated through Neper (Cr3C2 in black and NiCr in grey)

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 26 of 33 Corresponding author: Farshid Sadeghi

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Figure 3: Cohesive elements (in red) at grain boundaries between CPE3 triangle elements

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Figure 4: Schematic representation of bi-linear cohesive traction – separation law

… †

Figure 5: Boundary conditions with the application of uniaxial strain on the FE model

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 27 of 33 Corresponding author: Farshid Sadeghi

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(a) (b) Figure 6: (a) Young’s modulus and (b) Poisson’s ratio computed from FE simulation for different RVE sizes

Figure 7: Contact setup for fretting wear simulation

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 28 of 33 Corresponding author: Farshid Sadeghi

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(b) (c) Figure 8: Mesh convergence study showing (a) % error of the contact force with (b) coarsest mesh (c) finest mesh

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Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 29 of 33 Corresponding author: Farshid Sadeghi

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Figure 10: Sub-surface centerline stress distribution of ‡ˆ , ‡‰ , ‡Š and ‹Œ•ˆ max

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(a) (b) Figure 11: Von Mises stress contour plot in the (a) Homogeneous isotropic half domain (b) Heterogeneous anisotropic half domain

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 30 of 33 Corresponding author: Farshid Sadeghi

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(a) (b) Figure 12: Representation of (a) Quarter cycle fretting position, A (b) Von Mises stress in the heterogeneous anisotropic half domain at quarter cycle fretting position

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(b) Figure 13: Crack pattern obtained from (a) FE model (b) wear experiments on HVOF Cr3C2 – NiCr coating [47]

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 31 of 33 Corresponding author: Farshid Sadeghi

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(c) (d) Figure 14: Carbide pullout from (a) & (b) FE model and (c) experimental observation in HVOF Cr3C2 – NiCr coating [46] (d) Variation of β against fretting cycles

(a) (b) Figure 15: Surface wear profiles of HVOF Cr3C2 – NiCr coating obtained from the FE model for (a) partial slip and (b) gross slip condition

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 32 of 33 Corresponding author: Farshid Sadeghi

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Figure 16: Comparison of the results obtained from the FE model and experiments

Figure 17: Fretting wear map of HVOF Cr3C2 – NiCr coating generated through the FE model

Finite Element Modeling of Fretting Wear in Anisotropic Composite Coatings: Application to HVOF Cr3C2 – NiCr Coating - Page 33 of 33 Corresponding author: Farshid Sadeghi

Highlights

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Damage mechanics based cohesive zone finite element model for fretting wear in HVOF Cr3C2 – NiCr multiphase coating Individual phases of the anisotropic coating material were randomly assigned to resemble the microstructure from an SEM micrograph Model predicts stress-based failure mechanisms in wear of HVOF Cr3C2 – NiCr coating Model also simulates partial and gross slip fretting wear profiles Wear rate from the model is in close approximation to experimentally measured wear rate

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Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.