Steady state cyclic behaviour of a half-plane contact in partial slip subject to varying normal load, moment, shear load, and moderate differential bulk tension

D.A. Hills; H.N. Andresen; J.R. Barber; J. V\'azquez

REVIEW 2 major objections 3 minor 13 references

Steady state cyclic behaviour of a half-plane contact in partial slip subject to varying normal load, moment, shear load, and moderate differential bulk tension

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A half-plane contact's permanent stick zone is obtained from its normal-load solution through a simple mapping.

desk verdict Genuine extension to moment-loaded half-plane contact with a useful mapping, but the bulk-tension branch has a sign error that flips the predicted stick-zone shift; correctable, but not as printed. read the letter →

arxiv 1908.07939 v1 pith:5FPAUZZR submitted 2019-08-13 cond-mat.soft

H.N. Andresen , D.A. Hills , J.R. Barber , J. Vázquez This is my paper · ORCID

classification cond-mat.soft

keywords ContactmechanicsHalf-planetheoryPartialslipVaryingnormalandshearloadsMomentModeratebulktensionMappingSteadystate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that a general half-plane contact in steady-state partial slip—loaded by oscillatory normal force, moment, shear force, and moderate differential bulk tension—has a permanent stick zone that can be read directly from the normal contact solution. The key is a mapping that replaces the normal load by $P_0-\Delta Q/(2f)$ and the tilt angle by $\alpha_0-A\Delta\sigma/(8f)$, with the half-plane compliance $A$ and friction coefficient $f$. If the mapping is right, engineers analysing fretting fatigue in components such as turbine dovetail roots can find the stick-slip boundaries and the maximum slip extents without solving the tangential problem afresh. The claim is worked out explicitly for a tilted wedge, where the permanent stick zone $[-m,n]$ follows from closed-form expressions.

What carries the argument

The load-bearing object is the mapping between the normal contact problem and the tangential steady-state problem, displayed as equation (25). It is derived by comparing the locked-in surface strain over the permanent stick zone at the two load-reversal points; the inversion of the resulting integral equation is identical in form to the normal-contact inversion, which is exactly why the mapping works. Under the assumption that the bulk tension never reverses slip at a contact edge, this machinery converts a tangential problem into a normal problem with adjusted load and tilt, requiring no further algebra.

What would settle it

Compute the steady-state slip-stick boundary for a tilted wedge contact with $\Delta\sigma$ large enough that one contact edge reverses its slip direction; if the observed permanent stick zone still matches the mapped normal solution (31)-(32), the moderate-$\Delta\sigma$ restriction is unnecessary. More directly, a direct numerical simulation or experiment measuring $[-m,n]$ for a known $P_0$, $\Delta Q$, $\Delta\sigma$, and $\alpha_0$ would settle whether the mapping is correct.

Watch

Extended reading notes

Core claim

The central discovery is a formal correspondence between the normal contact problem and the steady-state tangential problem. When the oscillatory loads are synchronous, the corrective shear traction $q^*_2-q^*_1$ over the permanent stick zone is a scaled copy of the pressure distribution of a normal contact whose load and tilt are adjusted. Explicitly, the mapping sends $[-a,c]$ to $[-m,n]$, $p(x)$ to $-\frac{1}{2f}[q^*_2-q^*_1](x)$, $P$ to $P_0-\Delta Q/(2f)$, and $\alpha$ to $\alpha_0-A\Delta\sigma/(8f)$. Once this mapping is accepted, the permanent stick-zone boundaries $[-m,n]$ are found by solving the normal contact problem at the adjusted load and tilt, and the maximum slip extents follow by subtracting these boundaries from the contact coordinates at the two load extremes.

Load-bearing premise

The entire construction assumes the oscillatory bulk tension is moderate enough that, during each half-cycle, slip at both contact edges runs in the same direction and no edge reverses its slip sense; if that fails, the permanent stick zone as defined here no longer exists.

Editorial extensions

If this is right

The permanent stick zone depends on $P_0$, $\Delta Q$, $\Delta\sigma$, and $\alpha_0$ only; the mean shear $Q_0$ and mean bulk tension $\sigma_0$ drop out of the steady-state result.
For any incomplete half-plane contact with equal elastic constants, the permanent stick zone is obtained from the normal solution with no extra algebra.
The maximum slip extents, $(a_i-m)$ and $(c_i-n)$, follow by computing the contact coordinates at the two load extremes, which requires $\Delta P$ and $\Delta M$ as additional inputs.
The shear traction distribution in the permanent stick zone is a scaled copy of the mapped normal pressure, so quantities such as slip displacement can be evaluated after the mapping.
Application to a flat-and-rounded contact is straightforward although algebraically heavier than the wedge example.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical consequence the authors do not spell out: any existing normal-contact solver, analytical or numerical, can be reused as a fretting predictor by feeding it the adjusted load and tilt, with the only extra bookkeeping being the threshold at which the moderate-$\Delta\sigma$ assumption fails.
A testable extension would be to derive the explicit reversed-slip boundary, where $m=a$ or $n=c$, as a function of $P_0$, $\Delta Q$, $\Delta\sigma$, and $\alpha_0$; the paper stops at noting that the solution limit is reached there.
The mapping exploits synchronous oscillations; an extension to phase-shifted loads would have to revisit the locked-in strain equality that anchors the permanent stick zone.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The mapping idea is the real contribution here, and the wedge example is displayed carefully. But the bulk-tension branch has a sign error: using the paper's own Δσ = σ1 − σ2, Eq (16) together with the normal contact identity gives +Δσ/4 on the left of Eq (17), not the printed −Δσ/4. The subsequent formulas (23), (25), (32) are all consistent with the wrong sign, so the entire Δσ dependence is flipped. That is a load-bearing defect, not a typo.

What is genuinely new: [1] treated the no-moment steady state and [10] handled periodic P–Q loading, but neither included an offset normal load or the resulting asymmetric contact. Formalizing the mapping [−a,c] → [−m,n], with the pressure replaced by the corrective shear traction and the tilt shifted by the bulk term, is a useful recipe for fretting-fatigue analysis of dovetail roots. The derivation is systematic, the example is worked out in full, and no free parameter is fitted to the result. The reliance on the authors' own prior papers is fine because those are the exact predecessors.

The sign problem propagates. With α0 = 0 and Δσ > 0, the printed Eq (32) gives t < 0, so the permanent stick zone moves right, whereas Section 2.5 and Figure 5 describe a shift to the left as Δσ increases. The derivation from (13)–(16) says the shift should indeed be to the left, so the corrective mapping should read α → α0 + AΔσ/(8f), and Eq (23) and (32) should have the opposite sign. This needs a proper fix and a recomputation of the figures before the paper can be accepted. One secondary point that is handled honestly: the 'moderate' bulk-tension assumption, needed so slip does not reverse at an edge, is stated clearly and is a limitation of the method, not an internal inconsistency.

For the editorial decision: this deserves referee time, not a desk reject. The problem is practically important, the core derivation is mostly sound, and the error is localized. I would send it out with a clear request to fix the sign and check the example; with that corrected, it should be publishable. The audience is specialists in contact mechanics and fretting fatigue; for them the mapping is directly usable once fixed. I would not cite it in its current form.

Referee Report

2 major / 3 minor

Summary. The paper presents an analytical solution for the steady-state partial-slip response of a two-dimensional half-plane contact under synchronously oscillating normal load P, shear force Q, moment M, and differential bulk tension σ. The authors write the tangential problem in terms of sliding shear tractions plus a corrective traction over the permanent stick zone, and by equating surface strains at the two load extremes they derive an integral equation for the corrective traction difference. This equation is mapped onto the corresponding normal contact problem, yielding a recipe in which the permanent stick zone [−m, n] is obtained from the normal solution with modified load and tilt parameters (Eq. (25)). The mapping is applied to a tilted wedge, giving closed-form expressions for the stick-zone boundaries (Eqs. (31)-(32)) and for the corrective shear traction (Eq. (33)). The paper emphasizes the restriction to 'moderate' bulk tension, meaning that the slip direction is never reversed at a contact edge.

Significance. If correct, the proposed mapping would be a valuable extension of the Ciavarella-J"ager and Barber-Davies-Hills ideas, converting a nontrivial steady-state partial-slip problem with normal, shear, moment, and bulk-tension variation into a known normal contact calculation. The derivation is systematic, uses no fitted parameters, and the wedge example provides closed-form, independently checkable predictions. The paper is also clearly written and the intended application to fretting problems is plausible. However, the significance is conditional: the bulk-tension branch of the central mapping is internally inconsistent, as detailed below, so the claimed recipe cannot be accepted as printed.

major comments (2)

[Section 2.2, Eq. (16)-(17)] The derivation of Eq. (17) from Eq. (16) contains a sign error. Substituting the normal-contact identity (5) into each of the two sliding-traction integrals in Eq. (16) gives 2f(dg/dx+α0) for those two terms. The remaining bulk term is −A(σ2−σ1)/4, which, with the paper's definition Δσ=σ1−σ2, equals +AΔσ/4. After dividing by A, the constant term on the left of Eq. (17) must therefore be +Δσ/4, not −Δσ/4. This is not a convention-dependent sign: the error propagates into Eq. (23), Eq. (25), and Eq. (32). With the corrected sign, Eq. (23) should read ∫g'/w = −πα0 − AπΔσ/(8f), the mapping in Eq. (25) should read α → α0 + AΔσ/(8f), and Eq. (32) should contain +πAΔσ/(16fφ) in the argument of the sine.
[Section 2.5, Figure 5 and Eq. (32)] The printed sign in Eq. (32) reverses the predicted direction of the permanent-stick-zone shift, contradicting the paper's own text and figure. For α0=0 and Δσ>0, Eq. (32) as printed gives t<0, so m<K and n>K, i.e. the stick zone moves toward the right-hand contact edge. Section 2.5 states, and Figure 5 depicts, that increasing Δσ shifts the permanent stick zone toward the left-hand edge. The internally consistent derivation gives t>0 and the leftward shift, so the displayed example results and the formula set disagree. Since this sign appears in the central mapping claim, the bulk-tension branch of the solution is not reliable as printed.

minor comments (3)

[Section 2, opening] The 'moderate' bulk-tension restriction is only described verbally as insufficient to reverse the slip direction at either edge; the paper would benefit from a quantitative criterion in terms of P0, ΔQ, Δσ, and geometry, so that the domain of validity of Eqs. (31)-(33) is explicit.
[Figures 5-6] Several axis labels and subscripts in the arXiv figures are poorly rendered, for example the fractions Δσa/ΔQ and the α0/φ labels, which makes independent checking of the plotted results unnecessarily difficult.
[Eq. (13)-(14)] The signs of the σ1 and σ2 terms are consistent with the rest of the paper, but the convention is easy to lose; a sentence explicitly fixing the positive directions of σ1, σ2 and Δσ in Figure 1 would help.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the mapping from the normal to the tangential problem is derived in-paper from the integral equations, and cited prior solutions are used only as non-fitted inputs.

full rationale

The derivation chain is self-contained. Equation (16) is obtained by writing the surface strains at the two load points as a sliding traction plus a corrective term; the normal-contact identity (Eq. (5)) converts the sliding terms into profile and tilt terms, giving Eq. (17). The bounded inversion (Eq. (22)) and the consistency condition (Eq. (23)) reproduce the normal-problem integral equation with mapped parameters, which is then formalized as the mapping (25). The wedge example evaluates this mapped normal problem using the known wedge solution [13]; no parameter is fitted to the target permanent-stick zone, and the claimed stick-zone boundaries are not fed back as inputs. The self-citations ([1], [10], [13]) are historical, provide standard inversion formulas, or supply an input normal solution; the key equations are re-derived in the paper, so none of these citations is load-bearing. The moderate-bulk-tension restriction is an explicitly stated modelling assumption, not an imported uniqueness theorem. Independently of circularity, the printed sign of the Delta-sigma term in Eqs. (17), (23), (25), and (32) appears inconsistent with Eq. (16) and the definition Delta-sigma = sigma1 - sigma2; this is a correctness risk, not a circularity, because the mapping would still be a derivation even if the transcription of that term is wrong.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are introduced. The mapping (25) is derived from equilibrium and integral equations; all quantities are loads, profile data, material constants, and the friction coefficient. The only non-standard inputs are the published wedge normal solution and the standard singular integral inversion.

assumptions (6)

domain assumption Contacting bodies are elastically similar with equal compliance parameter A (Dundurs' second constant vanishes).
Section 2.1 states the solution is mathematically exact only if EA, nuA = EB, nuB; the same A connects the pressure and shear-traction integral equations.
domain assumption The loading trajectory remains inside the partial-slip wedge, so the contact never slides completely.
Section 1 assumes mean and oscillatory loads keep the trajectory within the partial-slip wedge; the solution does not cover full sliding.
domain assumption Bulk tension range is moderate: slip direction at each contact edge is the same during each half-cycle and is never reversed.
Section 2 opening and Conclusions state the solution applies only when bulk stress is too small to reverse slip sense at an edge; this is the paper's explicit scope limit.
domain assumption A steady state exists in which the permanent stick zone [−m,n] has the same extent just before every load reversal.
Section 2.2 argues the stick zones before each reversal must be equal for material continuity over cycles; transient behavior is neglected.
standard math Known inversion formula for the Cauchy singular integral equation with weight w(x,a,c) is used.
Eqs. (8) and (22) rely on the standard inversion from [11]; no proof is repeated.
domain assumption The tilted wedge normal-contact pressure solution from [13] is correct.
The example in Section 2.4 uses Eq (26) from Sackfield et al. as input to the mapping; the paper does not re-derive it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Steady state cyclic behaviour of a half-plane contact in partial slip subject to varying normal load, moment, shear load, and moderate differential bulk tension." pith.science (2026). https://pith.science/paper/5FPAUZZR

@misc{pith2026190807939,
  author       = {Pith},
  title        = {Pith review of: Steady state cyclic behaviour of a half-plane contact in partial slip subject to varying normal load, moment, shear load, and moderate differential bulk tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FPAUZZR}},
  note         = {Machine review of arXiv:1908.07939}
}

read the original abstract

A new solution for a general half-plane contact in the steady state is presented. The contacting bodies are subject to a set of constant loads - normal force, shear force and bulk tension parallel with the interface - together with an oscillatory set of the same quantities. Partial slip conditions are expected to ensue for a range of these quantities. In addition, the line of action of the normal load component does not necessarily need to pass the centre-line of the contact, thereby introducing a moment and asymmetry in the contact extent. This advancement enables a mapping to be formalised between the normal and tangential problem. An exact and easy to apply recipe is defined.

Figures

Figures reproduced from arXiv: 1908.07939 by the authors.

**Figure 1.** Generic half-plane contact subject to normal load, moment, shear load and bulk tension. so that the trajectory in (P, Q, σ) space consists of a line from the origin to some point in the steady state followed by reciprocating behaviour along a straight line between two points whose separation from the mid-point (P0, Q0, σ0) is (±∆P/2, ∆Q/2, ∆σ/2). This procedure is very useful for analysing a range of practical probl… view at source ↗

**Figure 2.** Two (a) and three-dimensional (b) illustration of a load space for a P-Q-M problem. tonically increasing shear force, was found by Cattaneo [3], and, apparently unaware of this solution, Mindlin [4] developed the same solution and went on to look at unloading and reloading problems [5], [6]. These were the only significant solutions for some time, and then Nowell and Hills [7] looked at what happened when a bulk ten… view at source ↗

**Figure 3.** Contact as the ends of loading cycle are approached, including a permanent stick zone. For load point 1 (P1, Q1, M1, σ1) we write the relative surface strain parallel with the surface, ∆εxx,1, as the sum of the effect of the sliding shear traction −fp1(x) over the whole 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: shows a shallow wedge of apex angle (π − 2φ), φ ≪ 1, pressed into an elastic half plane by a force P acting through the vertex, and also subject to a moment M. The moment causes the wedge to tilt through an angle α as shown. The normal contact problem was solved by Sac…

**Figure 5.** Figure 5: Steady state solutions of (a) slip and stick zone extents and (b) shear tractions for different values of a change in bulk stress, ∆σ. In [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Steady state solutions of (a) slip and stick zone extents and (b) shear tractions for different values of a change in average angle of tilt, α0. From [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Andresen, D

H. Andresen, D. Hills, J. Barber, J. V´ azquez, Frictional half-p lane contact problems subject to alternating normal and shear loads and tension in the st eady state, Int. Jnl. Solids Struct. (in press, DOI: j.ijsolstr.2019.03.025) (2019)

work page 2019
[10]

Barber, M

J. Barber, M. Davies, D. Hills, Frictional elastic contact with per iodic loading, Int. Jnl. Solids Struct. 48 (2011) 2041–2047 (2011)

work page 2011
[2]

Hertz, ¨ uber die ber¨ uhrung fester elastischer k¨ orper, Journal f¨ ur die reine und ange- wandte Mathematik 92 (1881) 156–171 (1881)

H. Hertz, ¨ uber die ber¨ uhrung fester elastischer k¨ orper, Journal f¨ ur die reine und ange- wandte Mathematik 92 (1881) 156–171 (1881)

work page
[3]

Cattaneo, Sul contato di due corpo elastici, Atti Accad

C. Cattaneo, Sul contato di due corpo elastici, Atti Accad. Na z. Lincei, Cl. Sci. Fis., Mat. Nat., Rend. 27 (1938) 342–348, 434–436, 474–478 (1938)

work page 1938
[4]

Mindlin, Compliance of elastic bodies in contact, AASME Trans

R. Mindlin, Compliance of elastic bodies in contact, AASME Trans. Jn l. Appl. Mech. 16 (1949) 259–268 (1949)

work page 1949
[5]

Mindlin, W

R. Mindlin, W. Mason, T. Osmer, H. Dereiewicz, Eﬀect of an oscillatin g tangential force on the contact surfaces of elastic spheres, Proc. First Na t. Congress of Applied Mechanics (1951) 203–208 (1951)

work page 1951
[6]

Mindlin, H

R. Mindlin, H. Dereiewicz, Elastic spheres in contact under varying oblique forces, J. of Applied Mechanics 20 (1953) 327–344 (1953). 12

work page 1953
[7]

Hills, D

D. Hills, D. Nowell, Mechanics of fretting fatigue tests, Int. Jnl. M ech. Sci. 29 (1987) 355–365 (1987)

work page 1987

Show all 13 references

[8]

J¨ ager, A new principle in contact mechanics, Jnl

J. J¨ ager, A new principle in contact mechanics, Jnl. Tribology 120 (1998) 677–684 (1998)

1998
[9]

Ciavarella, The generalised cattaneo partial slip plane contact problem, part i theory, part ii examples, Int

M. Ciavarella, The generalised cattaneo partial slip plane contact problem, part i theory, part ii examples, Int. Jnl. Solids Struct. 35 (1998) 2349–2378 (19 98)

1998
[11]

Hills, D

D. Hills, D. Nowell, A. Sackﬁeld, Mechanics of Elastic Contacts, Bu tterworth- Heinemann, 1993 (1993)

1993
[12]

Sackﬁeld, C

A. Sackﬁeld, C. Truman, D. Hills, The tilted punch under normal a nd shear load (with application to fretting tests), Int. Jnl. Mech. Sci. 43 (2001) 1881 –1892 (2001)

2001
[13]

Sackﬁeld, D

A. Sackﬁeld, D. Dini, D. Hills, The tilted shallow wedge problem, Euro pean Journal of Mechanics A/Solids 24 (2005) 919–928 (2005). 13

2005

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Andresen, D

H. Andresen, D. Hills, J. Barber, J. V´ azquez, Frictional half-p lane contact problems subject to alternating normal and shear loads and tension in the st eady state, Int. Jnl. Solids Struct. (in press, DOI: j.ijsolstr.2019.03.025) (2019)

work page 2019

[2] [10]

Barber, M

J. Barber, M. Davies, D. Hills, Frictional elastic contact with per iodic loading, Int. Jnl. Solids Struct. 48 (2011) 2041–2047 (2011)

work page 2011

[3] [2]

Hertz, ¨ uber die ber¨ uhrung fester elastischer k¨ orper, Journal f¨ ur die reine und ange- wandte Mathematik 92 (1881) 156–171 (1881)

H. Hertz, ¨ uber die ber¨ uhrung fester elastischer k¨ orper, Journal f¨ ur die reine und ange- wandte Mathematik 92 (1881) 156–171 (1881)

work page

[4] [3]

Cattaneo, Sul contato di due corpo elastici, Atti Accad

C. Cattaneo, Sul contato di due corpo elastici, Atti Accad. Na z. Lincei, Cl. Sci. Fis., Mat. Nat., Rend. 27 (1938) 342–348, 434–436, 474–478 (1938)

work page 1938

[5] [4]

Mindlin, Compliance of elastic bodies in contact, AASME Trans

R. Mindlin, Compliance of elastic bodies in contact, AASME Trans. Jn l. Appl. Mech. 16 (1949) 259–268 (1949)

work page 1949

[6] [5]

Mindlin, W

R. Mindlin, W. Mason, T. Osmer, H. Dereiewicz, Eﬀect of an oscillatin g tangential force on the contact surfaces of elastic spheres, Proc. First Na t. Congress of Applied Mechanics (1951) 203–208 (1951)

work page 1951

[7] [6]

Mindlin, H

R. Mindlin, H. Dereiewicz, Elastic spheres in contact under varying oblique forces, J. of Applied Mechanics 20 (1953) 327–344 (1953). 12

work page 1953

[8] [7]

Hills, D

D. Hills, D. Nowell, Mechanics of fretting fatigue tests, Int. Jnl. M ech. Sci. 29 (1987) 355–365 (1987)

work page 1987