PhD Opening

Numerical modeling of adhesive wear in elastic-plastic contacts

We invite applications for a PhD position in numerical modelling of elastic-plastic contacts. The thesis will focuson the development of discrete and continuum models for the simulation of adhesive wear in asperity junctions.The thesis is embedded into the project SINFONIA, funded by the National Research Agency. The position isavailable starting from January 2022 and open until fulfilment.

Context Wear is ubiquitous in any mechanical system that experiences relative motion of its components.Wherever two solid surfaces slide or roll against each other, frictional resistance arises and results in the occurrenceof wear (Figure 1). The amount and rate of material loss is a key factor for determining the safety and lifespanof many systems and devices, ranging from machinery parts for engineering applications to medical implantsand prostheses. The unwanted occurrence of wear also comes with a high ecological impact, as the non-disposablechemicals used to lubricate are often discarded into the environment. This thesis will contribute to understand howthose phenomena originate, with the aim to promote both scientific advancement and technological development.

Job description As a successful candidate, you will develop numerical models to investigate elastic-plasticadhesive wear. This includes both discrete (e.g., Molecular Dynamics [1, 2]) and continuum (variational phasefield [3, 4]) approaches. You will implement your models in high performance computing software and assessperformance, accuracy and robustness in systems involving single and multi-asperity contacts. The resultingsoftware will be used to conduct high-fidelity numerical simulations. Our purpose will be two-fold:• Understanding the relationship among material microstructure, surface morphology and wear.• Using this understanding to design rough contacts with targeted wear behavior.Your work will benefit of the experimental activities of the group (3D-printing, laser texturing).

This is a shared position between the Laboratoire de Mécanique des Solides (LMS, École Polytechnique) andthe Institut Jean le Rond d’Alembert (IJLRA, Sorbonne Université). You will be under the joint supervision ofStella Brach, CR CNRS at LMS, and Prof. Djimedo Kondo (IJLRA).

Your profile• MSc in mechanical or computational engineering, mechanics of materials and structures, applied mathematics.• Programming experience (Python, Fenics, Fortran). HPC experience is an advantage.• Sound knowledge of the English language is expected.

Application Interested candidates should contact both Stella Brach (stella.brach@polytechnique.edu) and DjimedoKondo (djimedo.kondo@sorbonne-universite.fr) with the following documents: (i) cover letter, (ii) curriculum vitaewith academic record for the past three years, (iii) contact details of an academic reference.

The review of applications will start immediately, therefore early submissions are encouraged.

Non-interactingasperityjunctions

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

A reduced spacing λ between the junctions, though,results in a different mechanism [Figs. 2(e) and 2(g)]: Thejunction pair jointly forms a single, big debris particle.While the junctions are loaded individually at the veryonset of contact [Fig. 2(f)], their subsurface stress fieldsmerge with increasing sliding distance S. In this situation,the proportionality between the junction and debris sizes isviolated, as the debris size is associated with the apparentcontact area between the two asperity junctions. Thisobservation rationalizes the breakdown of the linear linearrelation between the wear volume and the normal load (i.e.,Archard’s wear law) in the severe wear regime.This merging can be understood by considering that the

formation of a particle is nothing but the development of a

crack that leads to its detachment [21]. Figure 3 presents adetailed analysis of the evolution of the stress componentσ45°, which is perpendicular to the crack tips and thereforedrives their nucleation and propagation. At the onset ofsliding, identical stress concentrations develop at the baseof both the leading and the trailing asperities [Fig. 3(a)],where subsurface cracks, marked as “inner” and “outer,”are nucleated [Fig. 3(b)]. With further sliding, however, theouter crack goes much deeper into the bulk as a conse-quence of the modified subsurface stress state. It can beseen that there is an asymmetry in that only the inner crackis unloaded. This is an effect of the somewhat peculiarloading conditions: The stress in the leading asperity (onthe right in our setup) cannot be relieved by detaching the

FIG. 2. Asperity-level wear mechanisms. Snapshots of debris formation in 2D (a), (e) and 3D (c), (g). The insets show the initial setupwith D ¼ 60r0 (2D) and D ¼ 70r0 (3D). The total box length is L ¼ 7D. The final detached volume (i.e., the debris atoms) ishighlighted in red. Panels (b) and (f) show the subsurface shear stress vertically averaged over a subsurface layer with thickness D.Panels (d) and (h) depict the total dislocation length per bin size of L=30 for the 3D simulations.

(a) (b) (c) (d)

FIG. 3. Crack shielding drives joint debris formation. (a)–(c) Map of the stress component σ45° perpendicular to the crack tips, whichdrives the nucleation and propagation of the subsurface cracks. Simulation with D ¼ 100r0 and λ ¼ 0.7D. The gray arrow indicates thesliding direction. At the onset of sliding (a), (b), the stress is concentrated identically at the corner of both asperities where subsurfacecracks, marked as “inner” and “outer,” are nucleated. After further sliding, however, the inner crack is shielded by the outer one andclosed (c), eventually resulting in joint debris formation. Panel (d) quantitatively supports this observation by showing a quick drop inthe stress intensity factor of the inner crack. For comparison, data of a simulation with λ ¼ 2D are shown with dotted lines. No shieldingoccurs, both cracks grow, and two individual debris particles form. To minimize the thermal noise, both simulations were performed at alower temperature T ¼ 0.045ϵ=kB.

PHYSICAL REVIEW LETTERS 120, 186105 (2018)

186105-3

Interactingasperityjunctions

Figure 1: Adhesive wear in asperity junctions. Phase-field [4] and Molecular Dynamics simulations [1].

[1] R. Aghababaei, T. Brink, and J.-F. Molinari. In: Physical review letters 120.18 (2018), p. 186105.[2] R. Aghababaei, D. H. Warner, and J.-F. Molinari. In: Nature communications 7 (2016), p. 11816.[3] S. Brach et al. In: Computer Methods in Applied Mechanics and Engineering 353 (2019), pp. 44–65.[4] S. Collet, J.-F. Molinari, and S. Brach. In: Journal of the Mechanics and Physics of Solids (2020), p. 104130.