A modified Coulomb's law for the tangential debonding of osseointegrated implants

Guillaume Haiat; Katharina Immel; Roger A. Sauer; Thang X. Duong; Vu-Hieu Nguyen

REVIEW 4 major objections 5 minor 60 references

A modified Coulomb's law for the tangential debonding of osseointegrated implants

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes a modified Coulomb friction law whose coefficient decays with sliding distance, and shows it reproduces measured torque-versus-angle curves for osseointegrated implant debonding within 2.25% error.

desk verdict The state-variable friction law for implant debonding is a useful idea and the FE implementation fits the data well, but the analytical model in Appendix B contains a clear kinematic error that invalidates the reported analytical torque curves. read the letter →

arxiv 1908.04739 v3 pith:MFBA744Q submitted 2019-08-12 physics.med-ph cs.CE

Katharina Immel , Thang X. Duong , Vu-Hieu Nguyen , Guillaume Haiat , Roger A. Sauer This is my paper · ORCID

classification physics.med-phcs.CE

keywords osseointegrationbone-implantinterfacefrictionlawstatevariabledebondingfiniteelementmethodmodeIIIcleavageadhesionenergy

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

The paper aims to predict debonding of osseointegrated implants by replacing the constant friction coefficient of Coulomb's law with one that depends on the accumulated sliding distance. The friction coefficient starts at a high, unbroken value and smoothly decays to a lower, broken value, capturing the transition from a bonded to a sliding interface. This modified law is implemented in a 3D finite element framework and tested against a mode III cleavage experiment of a coin-shaped titanium implant in bone. The authors show that the torque-versus-angle curve is reproduced within 2.25% relative error, and that the model yields better estimates of bone shear modulus and adhesion energy than the reference analytical model. If correct, the model offers a simple macro-scale description of interfacial debonding that can handle partial osseointegration, which is relevant to implant stability prediction.

What carries the argument

The central object is the modified Coulomb law with a state-dependent friction coefficient, defined by $\mu(\varphi) = \varphi \mu_{\mathrm{ub}} + (1-\varphi)\mu_{\mathrm{b}}$ and the smooth state function $\varphi(g_s) = \varphi_0$ times a sinusoidal plateau–transition–zero profile in the normalized sliding distance $g_s/a_s$. This law converts Coulomb's friction into a local, history-dependent model that can represent a moving crack front and nonuniform bonding. It is implemented in a 3D nonlinear finite element framework with NURBS-enriched contact surfaces and also solved analytically for a radially symmetric mode III debonding, giving the traction distribution $\sigma_{\theta z} = t_{\max}^t r/c(\theta)$ in the sticking region and $\sigma_{\theta z} = \mu p$ in the sliding region.

What would settle it

Measure the normal contact pressure independently during a torsion debonding test, or run the test under controlled zero normal load, and check whether the fully debonded torque equals $T_\infty = \frac{2}{3}\pi R^3 \mu_b p$ with the calibrated friction coefficient; a mismatch, or a nonzero peak under zero pressure, would indicate that the interface resistance is not purely Coulomb-type friction with the proposed sliding-distance decay.

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Extended reading notes

Core claim

The central claim is that a state-variable friction law of the form $\mu(\varphi) = \varphi \mu_{\mathrm{ub}} + (1-\varphi)\mu_{\mathrm{b}}$, with $\varphi$ a smooth function of the accumulated sliding distance $g_s$, reproduces the full torque-angle response of an osseointegrated implant undergoing torsional debonding. The state function $\varphi(g_s)$ stays at 1 up to a sliding threshold $a_s$, then decays sinusoidally to 0 over a transition zone set by $b_s$. This turns the friction coefficient into a local, history-dependent quantity that describes the progressive loss of bonding. Calibrated against two experimental data sets, the model matches the initial stiffness, peak torque, softening branch, and residual torque, with mean relative errors below 2.25%, and yields bone shear moduli of 7–8 GPa and Young's moduli of 18–21 GPa, which are consistent with literature values, unlike the reference analytical model. The model also shows that partial osseointegration changes the extracted parameters but that the torque curve alone cannot uniquely determine the degree or distribution of osseointegration without knowing the friction coefficients.

Load-bearing premise

The model assumes that the resistance of the osseointegrated interface is governed by Coulomb friction with a decaying friction coefficient, and that the normal pressure $p$ entering that law is real and identifiable from the torque data—but the experiments could not measure or eliminate $p$, and the intact state is treated as high-friction rather than as an adhesive or cohesive interface.

Editorial extensions

If this is right

The same set of four calibrated parameters ($\mu_{\mathrm{ub}}$, $p$, $a_s$, $b_s$) together with a literature value of $\mu_{\mathrm{b}}$ reproduces the entire torque curve, so the model can serve as a predictive tool for torsional debonding of planar implants.
The model extracts bone shear modulus and adhesion energy more accurately than the previous analytical model, giving values consistent with independent measurements of cortical bone.
Because the state variable is local, the model naturally handles partial and inhomogeneous osseointegration, which is common after short healing times.
The torque-angle curve alone does not uniquely determine the distribution of osseointegration when friction coefficients are unknown, implying that additional measurements are needed for full characterization.
Under perfect twisting, the sliding distance at the center of the implant remains zero, so that point never debonds; this is a prediction about the crack-front arrest that could be checked experimentally.

Reading between the lines

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

The sliding threshold $a_s$ (about 22–26 µm here) may correspond to the clinically reported micro-motion tolerance of the bone–implant interface, suggesting a direct link between the friction-state parameter and biological tolerance limits.
The same state-variable philosophy could be extended to other loading modes such as pull-out or push-in, where the accumulated sliding distance would again drive the transition from bonded to frictional response.
Since the pressure $p$ is inferred but not measured, a natural testable extension is to design an experiment with independently measured normal force; such a test would either validate the identified pressure or reveal that the unbroken resistance is dominated by adhesion rather than friction.
The success of a single scalar state variable suggests that complex micro-mechanical adhesion and interlocking processes can be effectively coarse-grained into a history-dependent friction coefficient, which could simplify patient-specific implant stability simulations.

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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 core idea here is worth taking seriously: a sliding-distance-dependent friction coefficient that smoothly degrades from an unbroken to a broken state, implemented in a NURBS-based contact FE framework. That gives computational implant mechanics something it lacks — a pointwise, partial-osseointegration-capable debonding law. The FE results genuinely match the two experimental torque curves after calibration, the mesh and load-step sensitivity study is competent, and the partial-osseointegration examples are a nice demonstration of the model's reach. Credit is also due for the Discussion section, which candidly lists the unmeasured contact pressure, the lack of a priori parameter determination, and the limited experimental basis.

The soft spots are real, though. The analytical derivation in Appendix B has a load-bearing error. Eq. (49) gives the sliding distance in the sliding region as g_s = r(θ − θ_max + θ_lin). For a point at radius r, sliding actually begins when c(θ) = θ_lin R/θ reaches r, i.e. at θ_start = θ_lin R/r, so the accumulated sliding distance is rθ − θ_lin R, not the paper's expression. The two coincide only in special cases. For data set 1, the paper's formula gives g_s = 0 at θ = 1° and r = R, and negative values for θ < 1°, which is unphysical because sliding at that radius starts at θ_lin = 0.13°. So the analytical torque curves labeled "ana. Eq. (18)" in Fig. 4 are not valid consequences of the proposed friction law, and the reported analytical errors of 2.18% and 2.83% should not be trusted. The FE model accumulates sliding directly and may be unaffected, but the paper's central claims explicitly include the analytical model.

Beyond that, the calibration is entirely on the same two curves used to report error: all four friction parameters plus the bone shear modulus and the contact pressure are fitted from those curves. There is no independent validation, so the 2.25% error is a measure of curve fit, not prediction. The unmeasured contact pressure also weakens physical identifiability of the extracted parameters. The authors acknowledge most of this, which is fair, but it limits what the paper establishes.

Who should read this? People working on bone–implant interface modeling, especially those needing a smooth debonding law for computational studies. It deserves a serious referee, but only with a major revision that fixes Eq. (49), re-runs the analytical model, and either validates on unseen data or clearly separates calibration claims from prediction claims. As it stands, the numerical model is a promising tool, but the analytical part is not correct.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Modified Coulomb's law with a state-variable friction coefficient that decays as a function of accumulated sliding distance (Eqs. (11)-(13)) to describe the tangential debonding of osseointegrated implants. The law is implemented in a 3D nonlinear finite element framework and in a new analytical model (Sec. 3.3, Appendix B) for mode III cleavage of a coin-shaped titanium implant, and is calibrated against two experimental torque-angle datasets from Mathieu et al. [32]. The numerical results reproduce the experimental curves with a mean relative error below 2.25% (Table 4), and the paper reports improved estimates of the bone shear modulus and adhesion energy compared with the reference analytical model.

Significance. If the proposed law is accepted, it offers a simple and computationally efficient way to model the transition from intact osseointegration to frictional sliding, and the FE implementation naturally handles partial osseointegration patterns. The paper is thorough in its parameter studies and mesh convergence analysis, and it explicitly lists the limitations of the experimental data. However, the central claim is weakened by the fact that the reported errors are in-sample calibration errors and by an error in the analytical derivation that calls the analytical torque curves into question.

major comments (4)

[Appendix B, Eq. (49)] The accumulated sliding distance in the sliding region is incorrect. For a point at radius r, sliding begins at θ_start = θlin R/r, so the physical sliding distance is g_s = r(θ - θ_start) = rθ - θlin R. The paper's expression g_s = r(θ - θmax + θlin) is negative for a significant range of θ (for data set 1, for θ between 0.13° and 1.0° at r = R) and yields zero at θ = 1.0°, whereas the correct value is approximately 38 μm at θ = 1.0°. Since Eq. (18) places all points with r ≥ c(θ) in the sliding region, the analytical torque curves in Fig. 4 labeled 'ana. Eq. (18)' are not valid consequences of the proposed friction law. The analytical errors of 2.18% and 2.83% are therefore expressed relative to a model that is inconsistently derived.
[Sec. 4.1, Eq. (19), Table 4] The parameters μub, as, bs, d, and Gb are determined by minimizing the mean relative error defined in Eq. (19) on the very same experimental curves that are later used to report the error values. The statement that the model reproduces the experimental curves 'with a relative error of less than 2.25%' is thus an in-sample calibration error, not a measure of predictive accuracy. With approximately five free parameters fitted to a single torque curve per data set, the close agreement is to be expected. The paper should state this explicitly and, if possible, provide a validation on an independent data set (e.g., one of the two datasets held out during calibration) to support the claim of predictive power.
[Sec. 4.5, Table 4] Because the contact pressure p is not measured and is instead generated by an imposed displacement d that is itself calibrated from the same torque data, the friction coefficients μub and μb are not individually identifiable; only the products μub p and μb p are determined. The paper acknowledges that no statement can be made about the accuracy of p, but it still reports μub values with three significant digits and interprets them as physiological parameters. The authors should discuss the non-identifiability more explicitly and perhaps report the results in terms of μ p or provide bounds on p from independent measurements.
[Sec. 4.3 vs. Sec. 4.4] The average initial osseointegration fraction for the first data set is given as φ̄0 = 0.73 in Sec. 4.3 (and used in Eq. (21) and Table 5), while Sec. 4.4 states that 'the sample in Fig. 7a is considered to have average osseointegration φ̄0 = 0.55' for the reconstructed pattern. This discrepancy is not explained and changes the computed values of the area-specific work of adhesion substantially. The authors need to clarify which value is used and why the same photograph yields two different estimates.

minor comments (5)

[Sec. 4.5] 'Circular friction law' is a typo; it should likely read 'Coulomb friction law' or 'state-variable friction law'.
[Sec. 3.3] The statement 'c≥r≥R denotes the sliding region' contains a reversed inequality; it should read 'c ≤ r ≤ R' since the critical radius c separates the sticking region r < c from the sliding region r ≥ c.
[Abstract] The phrase 'relative error of less than 2.25%' should specify that this applies to the numerical solution; the analytical solution errors are 2.18% and 2.83%.
[Table 4] The rows labeled 'ana. [32]' report errors for the reference analytical model; clarifying this in the caption would avoid confusion with the new analytical model.
[Sec. 4.4] The sentence 'The model only depends on four physiological parameters (µub, p, as, and bs)' treats the contact pressure p as a parameter; p is a mechanical field rather than a physiological parameter, and µb is also fixed from literature.

Circularity Check

4 steps flagged · score 6.0 of 10

The headline <2.25% agreement is an in-sample error on the same torque curves used to fit all friction parameters; the reported shear modulus and adhesion energy are derived from the same fitted curve.

fitted input called prediction [Sec. 4.1, Eq. (19); Sec. 4.2, Fig. 4 and Tab. 4]
"The remaining parameters Gb, d, µub, as, bs are determined by minimizing the mean relative error emp T = mean θ∈[0◦,10◦] (|Texp(θ)−T(θ)|/Texp(θ)) (19) ... Finally, the parameters as and bs are then determined by minimizing Eq. (19) for the whole torque-per-angle curve."

The reported peak/softening agreement (<2.25%, Tab. 4) is the value of the same objective function (19) that is minimized to select d, µub, as, and bs. The torque curves in Fig. 4 are therefore least-squares fits to the very experimental curves they are claimed to 'reproduce'. No hold-out data or cross-validation is used, so the residual is a training error, not an independent prediction.
self definitional [Sec. 2.2.3, Eq. (13)]
"This function was designed such that it captures the experimental behavior shown in Sec. 4.2 (see Fig. 4)."

The smooth state function φ(gs) in Eq. (13) controls the entire shape of the torque peak and softening. It is explicitly designed to match the experimental torque behavior that is later presented as the model's successful reproduction of the experiments. The target behavior is thereby built into the model definition rather than derived from independent first principles.

2 more flagged steps

fitted input called prediction [Sec. 4.3, Eqs. (20)-(21), Tab. 5]
"Edeb = ∫_{θ=0◦}^{10◦} T(θ) dθ, E fric = ∫_{θ=θmax}^{10◦} T∞ dθ, W adh = Edeb−Efric. (20) ... The area-specific average work of adhesion ¯wadh is then given by ¯wadh = Wadh/(πR² ¯φ0). (21)"

The 'very good agreement' for the adhesion energy (e.g. 98 N/m for data set 1) is obtained by integrating the torque curve T(θ) produced with parameters fitted to minimize Eq. (19) on the same experimental curve. Wadh is therefore a functional of the fitted curve, and ¯φ0 is taken from the same post-test implant images, so the agreement is inherited from the calibration rather than being an independent estimate.
fitted input called prediction [Sec. 4.1; Sec. 4.2, Tab. 3]
"The shear modulus Gb is calibrated using the initial slope of the linear part of the torque-per-angle curve (i.e. T (θ ≤ θlin)) ... The estimated shear moduli of 7 and 8 GPa are higher than the reported values of 2–6 GPa [48, 53]), while the corresponding Young’s moduli of 18 and 21 GPa are in good agreement with experimental data from the literature."

The bone shear modulus is read off from the slope of the same experimental torque curve that the model is said to reproduce. Comparing this fitted value with literature values and calling it a 'better estimate' than Ref. [32] is a statement about the calibration, not an independent validation of the friction model, since the initial stiffness is enforced by the fit.

full rationale

The central derivation is a parameter-calibration exercise rather than an independent prediction. Every input of the modified Coulomb model (d, µub, as, bs) is chosen by minimizing Eq. (19) on the full experimental torque curve, and the reported errors are the minimized values of that same objective; the shear modulus is taken from the initial slope of the same curve and the adhesion energy is the integral of the fitted curve. These are standard identifiability and calibration statements, but presenting them as 'predictions' or 'better estimates' reduces the claimed verification to in-sample fit quality. The paper itself concedes that 'no a-priori determination of the input parameters can be made yet' and that the contact pressure required by the Coulomb law could not be measured or eliminated (Sec. 4.5), which reinforces that the extracted parameters are not independently identifiable. The self-citations to Duong/Sauer and Corbett/Sauer concern the contact finite-element machinery and are not load-bearing for the debonding claim. A separate correctness risk is the apparent kinematic inconsistency in Appendix B Eq. (49) for the analytical sliding distance, but that is an error concern, not a circularity, so it is not scored here. Overall, the paper contains several fitted-inputs-called-predictions, giving partial circularity with score 6.

Assumptions & free parameters 7 free parameters · 9 assumptions · 0 invented entities

The model's parameters are mostly calibrated from the very experimental curves it is compared against. The only quantities taken from outside are the friction coefficient mu_b (literature range), the titanium material properties, and the geometric dimensions. The state function Eq. (13) is an ad hoc construction tailored to the data shape.

free parameters (7)

mu_b (broken-state friction coefficient) = 0.4 (chosen from literature range 0.2-0.5)
Not fitted but assumed from literature for the interface between cortical bone and polished metal; all other fitted parameters are conditional on this choice (Sec. 4.1).
d (imposed vertical displacement generating contact pressure) = 4.9 um (data set 1), 5.1 um (data set 2) at mu_b=0.4
Calibrated from the plateau torque for theta in [9 deg, 10 deg] (Sec. 4.1).
mu_ub (unbroken-state friction coefficient) = 0.58 (data set 1), 0.55 (data set 2) at mu_b=0.4
Calibrated from Tmax via tmax_t = mu_ub p (Sec. 4.1).
a_s (sliding threshold) = 22 um (data set 1), 26 um (data set 2)
Calibrated by minimizing Eq. (19) over the full torque curve (Sec. 4.1).
b_s (transition zone factor) = 0.74 (data set 1), 1.86 (data set 2)
Same calibration as a_s (Sec. 4.1).
G_b (bone shear modulus) = 7 GPa (data set 1), 8 GPa (data set 2)
Calibrated from the initial slope T(theta <= theta_lin) (Sec. 4.1).
phi0_bar (average initial osseointegration fraction) = 0.73 (data set 1), 0.72 (data set 2), 0.55 in partial cases
Estimated by visual inspection of implant surfaces (Sec. 4.3, 4.4), used to convert total adhesion work to area-specific values.

assumptions (9)

domain assumption Bone and implant are hyperelastic Neo-Hookean with isotropic response (Eq. (1)).
Used for the FE model (Sec. 2.1); acknowledged as an approximation for anisotropic bone.
domain assumption In the analytical model, bone and implant are linear elastic.
Appendix B, around Eq. (39).
domain assumption Normal contact pressure is homogeneous in the analytical model.
Appendix B, before Eq. (48).
domain assumption In the numerical model, a uniform vertical displacement d creates a normal pressure p that is sufficient for friction.
Sec. 3.4; the experimental pressure was unmeasurable (Sec. 4.5).
ad hoc to paper The state variable phi depends only on accumulated sliding distance, not on slip rate or pressure.
Eqs. (12)-(13); the sine form is designed to capture the experimental torque curve (Sec. 2.2.3), not derived from a physical mechanism.
domain assumption Crack and debonding front propagates circularly from the outer radius toward the center.
Adopted from Mathieu et al. [32] and used in both analytical models (Sec. 3.2, 3.3).
domain assumption Main calibration assumes full initial bonding phi0=1.
Sec. 4.1 and Tab. 4; visual analysis indicates phi0_bar around 0.73, so this assumption is violated, and the paper only later tests partial bonding patterns.
domain assumption The fully debonded torque T_infinity is reached at 10 degrees.
Sec. 4.3, Eq. (20); used to split the energy into adhesion and friction parts.
domain assumption Poisson ratio of bone is fixed to nu=0.3.
Sec. 4.1; standard value.

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Cite this review

Pith. "Pith review of A modified Coulomb's law for the tangential debonding of osseointegrated implants." pith.science (2026). https://pith.science/paper/MFBA744Q

@misc{pith2026190804739,
  author       = {Pith},
  title        = {Pith review of: A modified Coulomb's law for the tangential debonding of osseointegrated implants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFBA744Q}},
  note         = {Machine review of arXiv:1908.04739}
}

read the original abstract

Cementless implants are widely used in orthopedic and oral surgery. However, debonding-related failure still occurs at the bone-implant interface. It remains difficult to predict such implant failure since the underlying osseointegration phenomena are still poorly understood. Especially in terms of friction and adhesion at the macro-scale, there is a lack of data and reliable models. The aim of this work is to present a new friction formulation that can model the tangential contact behavior between osseointegrated implants and bone tissue, with focus on debonding. The classical Coulomb's law is combined with a state variable friction law to model a displacement-dependent friction coefficient. A smooth state function, based on the sliding distance, is used to model implant debonding. The formulation is implemented in a 3D nonlinear finite element framework, and it is calibrated with experimental data and compared to an analytical model for mode III cleavage of a coin-shaped, titanium implant (Mathieu et al. 2012). Overall, the results show close agreement with the experimental data, especially the peak and the softening part of the torque curve with a relative error of less than 2.25 %. In addition, better estimates of the bone's shear modulus and the adhesion energy are obtained. The proposed model is particularly suitable to account for partial osseointegration, as is also shown.

Figures

Figures reproduced from arXiv: 1908.04739 by the authors.

**Figure 1.** Modified Coulomb’s law: Tangential contact traction kttk /tmax t as a function of the sliding distance gs for varying sliding threshold as and transition factor bs for φ0 = 1. ∗ In principle Eq. (13) can be brought into the form of Eq. (10) if a dot is applied to Eq. (13) and then gs is eliminated by the inverse function of (13). † In principle, φ0 follows from an evolution law that describes the healing/osseointegr… view at source ↗

**Figure 2.** Illustrations for the analytical and numerical setup. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** New analytical model: Normalized tangential traction σθz/tmax t as a function of the radius r/R of the contact surface (see Eq. (18)) at different rotation angles. 3.4 Numerical setup As in the experiments, we consider a coin-shaped cylindrical implant with dimensions R = 2.5 mm and H = 3 mm. The bone sample is modeled as a rectangular cuboid with dimensions 12.5 × 12.5 × 5 mm. The implant is positioned at the cente… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Variation of the Torque T as a function of the imposed rotation angle θ. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Evolution of the debonding of the BII: Value of the friction coefficient µ (top) and the sliding distance gs (bottom) on the contact area of the implant for different angles of rotation. 0 0.5 1 1.5 2 2.5 0.4 0.5 0.6 10◦ 5 ◦ 3 ◦ 2 ◦ θ ≤ θmax radius r [mm] friction coef…

**Figure 6.** Figure 6: Evolution of the debonding of the BII: Behavior of the friction coefficient and the transition zone for the first data set. 4.3 Work of adhesion Due to the poor agreement between the experimental results and the analytical model developed in Mathieu et al. [32], an ene…

**Figure 7.** Figure 7: Original and constructed osseointegration patterns with average osseointegration φ¯ 0 = 0.55. Light areas represent full osseointegration (φ0 = 1), while dark areas represent no osseointegration (φ0 = 0). For the corresponding elements in the reconstructed and artifici…

**Figure 8.** Figure 8: Variation of the friction coefficient µ at the BII for different angles of rotation for different patterns of partial osseointegration. 0 1 2 3 4 5 6 7 8 9 10 0 0.01 0.02 0.03 0.04 0.05 0.06 rotation angle θ [deg] torque T [Nm] (a) experiment [32] homogeneous bonding (…

**Figure 9.** Figure 9: Partial osseointegration: Variation of the Torque T as a function of the imposed rotation angle θ for different cases of the initial bonding state of the interface with the material parameters shown in Tab. 6. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: Partial osseointegration: Variation of the Torque T as a function of the imposed rotation angle θ for different patterns of the initial bonding state of the interface with φ¯ 0 = 0.55 and the same material parameters for all patterns. T∞ = 0.0368 N m according to data…

**Figure 11.** Figure 11: Frictional contact kinematics at current time tn+1: current slave point xk, current position of the previous interacting point x`( ˆξ n ), current interacting point x`( ˆξ), the current elastic gap vector ge and its components, previous elastic gap g n e , and the sli…

**Figure 12.** Figure 12: New analytical model: Critical radius for stick/ slip transition c as a function of the imposed rotation angle θ from Eq. (17). We also know that for linear elasticity dT(θ) dθ = C, θ ≤ θlin, (42) where C is the effective shear stiffness of the system, that is approxi…

**Figure 13.** Figure 13: shows the convergence behavior of the different meshes. It can be seen that e mp T reaches its limit for all meshes after 1,000 load steps to 0.0175, 0.0171, and 0.017, respectively. This is also the case for the mean percentage error shown in Tab. 7, with its lowest …

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