Finite Element Simulation of Thermal Sliding Friction and Wear in an FGPM-Coated Half-Plane
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper can be accepted after the following corrections:
- The meshing concept has to be explained in detail.
- What were the meshing criteria? What exactly is the “meshing seed”?
- Meshing granularity is quickly reduced. Does it have an influence on the results?
- Figure 3 suggests that there is still no saturation. What is the influence on the final results?
- The quality of figure 1 is not acceptable.
- The conclusions are sound. However, please state the conclusions in a more quantitative way.
Author Response
Comment 1: The meshing concept has to be explained in detail.
Response: Thank you for pointing this out. The overall meshing concept was based on a balance between computational accuracy and efficiency. The core idea was to employ a gradient meshing strategy, where regions with high stress/strain gradients (contact surfaces) were discretized with a finer mesh, while other regions with relatively uniform stress distribution were meshed with a coarser mesh to reduce the total number of elements and computational cost. We have carefully revised the manuscript and incorporated a dedicated subsection titled "3.2 Mesh generation and independence study" on page 6 (line 179), which provides a comprehensive description of our meshing concept.
Comment 2: What were the meshing criteria? What exactly is the “meshing seed”?
Response: Thank you for this important question. And we apologize for the lack of clarity in our original manuscript. The following details have been added to Section 3.2 (Mesh generation and independence study) to address this comment directly. The additions include:
Meshing Criteria: We adoptedc3-node/4-node planar temperature-displacement coupled elements (CPE3T/CPE4T) and 3-node/4-node bilinear plane stress piezoelectric elements (CPS4E and CPS3E) as primary element types. A based seed size of 4 mm was applied to the entire model. In regions of high-stress concentration (the contact area), a much finer local seed size of 0.01 mm was applied. Given the significant impact of meshing granularity (element size) on the simulation results, a comprehensive mesh independence study was conducted.
Convergence Criterion: A mesh sensitivity study was performed by progressively reducing the local seed size from 1mm to 0.005mm (Fig. 2). The results showed that the maximum von Mises stress converged when the seed size reached 0.01mm, with variations of less than 2%. Consequently, a local seed size of 0.01mm was selected for local refinements for all subsequent simulations to balance accuracy and computational cost.
Comment 3: Meshing granularity is quickly reduced. Does it have an influence on the results?
Response: Thank you for your insightful comment regarding the influence of the meshing strategy on our results. The reviewer correctly observes a rapid reduction in mesh granularity in certain regions of the model. Please allow us to clarify the rationale behind this approach and demonstrate that it does not adversely affect the reliability of our key findings.
Rationale for Local Mesh Refinement: The rapid mesh transition was intentionally designed to achieve a balance between computational efficiency and accuracy. The critical region of interest, specifically the contact area under sliding wear, experiences severe stress and thermal gradients. Capturing these steep gradients accurately requires a very fine mesh. In contrast, regions far from the contact area experience much milder field variations. A coarse mesh in these areas is sufficient, and this strategy significantly reduces the total number of elements and computational cost.
Evidence from Mesh Independence Study: As detailed in our revised manuscript (Section 3.2, "Mesh generation and independence study"), we conducted a comprehensive mesh sensitivity analysis. We systematically refined the local seed size and monitored the convergence of key output parameters, namely the maximum von Mises stress. The results confirmed that the outputs converged (with variations of less than 2%) when the local seed size was reduced to 0.01 mm, which is the configuration used in our final model. This demonstrates that our results are insensitive to further local mesh refinement.
Analysis of the Transition Zone: To directly address the reviewer's concern, we have performed an additional post-processing analysis on the transition zone between the coarse and fine mesh. A contour plot of the von Mises stress distribution in transition zone has been added to the revised manuscript (Fig. 4). As Fig. 4 shows, the stress distribution is stable and continuous, without any abrupt, non-physical jumps or oscillations. This demonstrates that the solution is not adversely affected by the change in mesh density.
Therefore, we are confident that while the mesh granularity changes rapidly, our chosen meshing strategy, validated by a rigorous independence study and post-processing checks, provides reliable and accurate results for our study results.
Comment 4: Figure 3 suggests that there is still no saturation. What is the influence on the final results?
Response: Thank you for your valuable comment regarding the saturation behavior in Fig. 2. We verified smaller element sizes 0.005mm and updated Figure 2 on revised manuscript. As evident from the Fig. 2, the maximum von Mises stress exhibit a convergent trend as seed size decreases from 0.02mm to 0.0005mm. And refining the seed size from 0.02 mm to 0.01 mm yield only a marginal 1.89% (<2%) increase in the maximum von Mises stress. This indicates that the solution had essentially converged at the 0.01 mm level. Therefore, we determined that an element size of 0.01 mm represents the optimal balance, providing excellent numerical accuracy without incurring unnecessary computational cost.
Comment 5: The quality of figure 1 is not acceptable.
Response: Thank you for pointing out the issue with the quality of Fig. 1. We have updated Figure 1 on page 3 of the revised manuscript to accurately represent the simulation model.
Comment 6: The conclusions are sound. However, please state the conclusions in a more quantitative way.
Response: Thank you for your valuable comments. We fully agree that a quantitative presentation strengthens our conclusions. And we have revised the “Conclusions” section of the manuscript to supplement specific numerical results and quantitative comparisons.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsYou are using rigid punch to slide over the coating.
how did you quantify the mechanical wear?
is it visible in the simulation, can you add some pictures?
how exactly wear is modelled? if the failed elements are being removed, it should be visible in the simulation, right? what I am trying to say, is shown in the following papers.
The reciprocating motion path, is the motion happening only tangential direction or normal/depth direction too? Explain the motion path.
Azmeera, A.K.; Jadhav, P.; Lande, C. Effect of Change in Material Properties of the Abradable Coating on the Wear Behavior of It—Microstructure Model-Based Analysis Approach. Lubricants 2025, 13, 22. https://doi.org/10.3390/lubricants13010022
Is it possible to visually show and also quantify the wear as shown in the above papers? and then link the amount of wear to the other parameters that are talked about here.
Figure 12b - cosine plot looks outlier, what could be the reason?
Also why 2D model is used?
Why not 3D model by giving some out of plane thickness to the 2d model?
3D model will give more information, surface variation plots can be seen if its 3D.
Seems like you are measuring contact stress, pressure, temperature etc but not measuring wear.
It will be interesting to relate amount of wear to the other parameters like gradient index.
For 2D wear simulation, how can you justify that its giving correct result. You are validating with literature, which may be wrong too.
3D wear results may be more reliable.
Author Response
Comment 1: You are using rigid punch to slide over the coating.how did you quantify the mechanical wear?How exactly wear is modelled?
Response: Thank you for this insightful question regarding the core of our wear modeling methodology. The following details have been added to Section 3.3 (the wear modeling) in the revised manuscript, providing a detailed description of the establishment of the wear model. The additions include:
Since the punch is assumed to be rigid with no self-wear, all wear quantification focuses exclusively on the surface of coating material. This approach avoids the complexity of calculating “two-body wear,” ensuring that the quantified results directly reflect the coating’s intrinsic performance.
Theoretical Fundamentals of wear: A modified Archard wear model was employed to characterize the surface geometric evolution induced by material wear, based on the contact characteristics of the “rigid punch- piezoelectric coating structure”. The Archard wear model was implemented via the well-established UMESHMOTION subroutine. When coupled with the Arbitrary Lagrange-Euler (ALE) adaptive meshing technique, this approach enables the mesh to remain independent of the model’s motion while preserving its original topology.
The Archard wear model correlates wear severity with contact pressure and relative slip. Given that the pressure varies with the sliding position, the integral form of the model was applied. The wear depth at time t is expressed as follows:
where denotes the contact pressure at time t, represents the slip increment, and is the wear coefficient with a value range of .
The implementation approach in the FEM: The Archard wear formula is implemented via the user-defined subroutine UMESHMOTION. The wear accumulation rule is defined as follows: the total sliding process is divided into multiple incremental steps; at each step, the wear induced in that step is calculated based on the instantaneous contact pressure and sliding distance, and the mesh geometry of the piezoelectric layer (e.g., nodal coordinate offsets in the worn region) is updated in real time. This approach enable simulate of the gradual surface material loss of the piezoelectric coating during sliding process.
All supplementary content related to wear has been added to Lines 205-227 (pages 6-7) of the revised manuscript, highlighted with yellow background.
Comment 2: The reciprocating motion path, is the motion happening only tangential direction or normal/depth direction too? Explain the motion path.
Response: Thank you for this important question regarding the motion path in our simulation. The reciprocating motion implemented in our model is purely tangential. There is no prescribed motion in the normal (depth) direction. The motion path of the punch consists of simple linear reciprocating frictional sliding along a single axis parallel to the contact surface. This is trajectory is illustrated in Fig.1 and detailed in Section 2 of the manuscript. The punch is constrained to move only in this tangential direction. The normal direction, by contrast, is controlled by the applied normal load-no displacement or oscillation occurs in the normal direction throughout the simulation. As wear progresses and the coating’s surface geometry is updated (via the UMESHMOTION subroutine), the contact stress distribution evolves dynamically; however, the prescribed kinematic motion of the punch remains exclusively tangential.
This setup aligns with standard sliding wear test configurations, enabling us to isolate the effects of tangential fretting while avoiding the complexity induced by superimposed normal oscillations.
Comment 3: Is it possible to visually show and also quantify the wear as shown in the above papers? and then link the amount of wear to the other parameters that are talked about here.
Response: Thank you for your valuable comments. We fully agree that visually presenting and quantifying wear significantly strengthens the rigor of our analysis.
In response to your comment, we have incorporated a new figure (now Fig.8) in the revised manuscript, which clearly depicts the coating’s worn profile post-sliding. Furthermore, as recommended, we have directly linked the quantified wear amount to the key parameters discussed in our paper. This analysis-establishing a quantitative correlation between wear and these governing factors-is now presented and discussed in detail in Section 5.1 the effect of wear.
We believe these additions greatly enhance the depth and impact of our work, and we sincerely appreciate your valuable guidance.
Comment 4: Figure 12b - cosine plot looks outlier, what could be the reason?
Response: Thank you for highlighting the anomalous behavior observed in Figure 12b. We have conducted a thorough re-examination of this specific data point and concur that the cosine plot is an outlier. Our investigation suggests that the anomaly likely originated from a suboptimal configuration in the initial model parameters for this particular simulation, which led to an unphysical response. The figure and the corresponding discussion in the main text have been updated accordingly. We sincerely appreciate your insightful observation, which has undoubtedly enhanced the clarity and accuracy of our work.
Comment 5: Also why 2D model is used?Why not 3D model by giving some out of plane thickness to the 2d model?3D model will give more information, surface variation plots can be seen if its 3D. For 2D wear simulation, how can you justify that its giving correct result. You are validating with literature, which may be wrong too.
3D wear results may be more reliable.
Response: Thank you for this insightful comment and for raising the important question regarding the choice of a 2D model versus a 3D model for our wear simulation. We agree that a 3D model can provide additional information, such as detailed surface variation plots. In the following, we will justify our decision to use a 2D approach and address your concerns regarding the validation of our results.
Rationale for Employing a 2D Mode:
The primary aim of this study is to investigate the fundamental relationship between the sliding friction and wear behavior of FGPM coatings and mechanical responses (such as contact stress and wear rate) when the rigid punch slides in a specific direction, rather than to explore the effects of complex geometries in three-dimensional space (such as edge fillets and asymmetric loads) on wear. The 2D plane strain model can accurately capture key information in the two-dimensional plane composed of the sliding direction and the normal direction (such as the stress distribution in the contact area and the variation of wear depth along the sliding path), fully covering the core data required for the research objectives.
The friction and wear simulation needs to consider material nonlinearity, contact nonlinearity (such as frictional sliding), and wear accumulation (requires iterative incremental steps). If a 3D model is used, the number of meshes will be dozens of times that of a 2D model. The computational time for a single simulation would increase from a few hours in 2D to several days in 3D. Consequently, the computational burden of performing a multi-parameter study (such as on the effects of sliding friction coefficient and sliding speed) would become prohibitively high. This would fundamentally undermine our primary research aim of systematically quantifying parametric relationships.
Justification and Validation of the 2D Wear Model:
We recognize the urgent need to verify any computational model and have taken several steps to ensure the reliability of 2D wear simulation:
Model Verification: We performed a mesh independence analysis to ensure that our wear results were independent of the element size.
Validation with Literature: In Section 4 of the manuscript (page 8-9), the results of this study were validated against two types of literature: one focusing on theoretical analysis (Ke et al.) and the other on finite element calculations (Zhou et al.). The close agreement was observed between our results and these references, which strongly confirms the predictive capability of our proposed model. Notably, the cited literature represents widely recognized and highly cited work in this research field—this further reinforces the credibility of our validation process.
Outlook on 3D Modeling:
We fully concur with your insightful perspective that 3D wear simulations hold significant value, as they can provide more detailed information—such as surface variation plots—for comprehensive analysis. Developing a 3D model to investigate more complex geometries and surface effects is indeed a logical and necessary extension of the present work. Notably, the current study, featuring a thoroughly validated 2D framework, calibrated material parameters, and established wear law, serves as a critical foundational step for our future research. This subsequent work will explicitly focus on advancing toward 3D wear simulation, with the findings of this 2D study laying the groundwork for its reliability.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper was corrected and can be accepted in the present state.
Reviewer 2 Report
Comments and Suggestions for AuthorsRevised version looks ok
Relevant questions are answered
It could have been better but I think this is ok
