Review of "On the Moreau-Jean scheme with the Frémond impact law: energy conservation and dissipation properties for elastodynamics with contact, impact and friction"

Vincent Acary, Nicholas Anton Collins-Craft, Simon R Eugster, Remco Leine, Alexander Popp

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Vincent Acary, Nicholas Anton Collins-Craft, Simon R Eugster, Remco Leine, Alexander Popp. Review of "On the Moreau-Jean scheme with the Frémond impact law: energy conservation and dissipation properties for elastodynamics with contact, impact and friction". 2024. hal-04819313

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Submitted on 22 Jul 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
**Identifiers**

Open Review

OAI hal-04819313

Reviewed Article

DOI 10.46298/jtcam.13480

**Licence**

CC BY 4.0

©The Authors

# Review of “On the Moreau–Jean scheme with the Frémond impact law: energy conservation and dissipation properties for elastodynamics with contact, impact and friction”

© Vincent ACARY<fnref n="1" />*, © Nicholas Anton COLLINS-CRAFT<fnref n="1" />, © Simon R. EUGSTER<fnref n="2" />, R, © Remco LEINE<fnref n="3" />, R, and © Alexander POPP<fnref n="4" />, E

## Review of version 1

Permalink: hal-04230941v3

**Authors** In the first instance, we thank the reviewers for their kind words and insightful comments that have helped us substantially improve the manuscript. Their comments (in italics), and our responses to their comments are below, using their numbers where provided, and enumerated by us for clarity where not.

### 1 Reviewer 1 (Anonymous)

**Reviewer** This paper presents a solution to the well-known energetic inconsistency of classical frictional impact laws and their implementation in the Moreau-Jean scheme. Firstly, an adaptation of Frémond's frictional impact model is proposed and investigated in continuous time. Then, a discrete time version is established leading to a Moreau-Jean scheme with Frémond law. Both in continuous time and in discrete time, the energetic consistency is established based on theoretical arguments under the assumption of linear elastodynamics. The method is then tested on several numerical examples, which confirm these properties. The simulation of a system with finite rigid body motions also reveals improvements in the energetic behaviour, even though the rigid body dynamics is nonlinear. The paper brings an important and fundamental clarification regarding the formulation of frictional impact laws. In my opinion, it is of high value for the community.
I only have a few minor or typographical remarks and suggestions.

**Reviewer** In Eq. (9), the cone $K^\alpha$ is well-defined but a definition of the notation $N_{K^\alpha}(r^\alpha)$ seems missing.

**Authors** This is now added to the manuscript.

**Reviewer** In Eq. (21), the = sign on the right hand side seems erroneous.

**Authors** This is now corrected.

**Reviewer** In Eq. (46), the operator $\Theta^\alpha$ involves $\bar{u}^{\alpha, -}$. I believe that there is a typo and this variable should be $u^{\alpha, -}$. Please check.

**Authors** This is now corrected.

<footnote n="1">¹ Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, Institute of Engineering, LJK, 38000, Grenoble</footnote>

<footnote n="2">² Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands</footnote>

<footnote n="3">³ Institute for Nonlinear Mechanics, University of Stuttgart, Germany</footnote>

<footnote n="4">⁴ Institute for Mathematics and Computer-Based Simulation, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany</footnote>

<footnote>R Reviewer</footnote>

<footnote>E Editor</footnote>
**Reviewer** In Section 2.4, the operator $\Theta$ is defined as a function of a single argument $\bar{u}$. Strictly speaking, this is mathematically inconsistent with the definition in Eq. (46), which not only involves $\bar{u}^{\alpha}$ but also $u^{\alpha, -}$ (see previous remark). Therefore, both $\Theta$ and $\Theta^{\alpha}$ should be operators with 2 independent arguments.

**Authors** This is not actually inconsistent, as the value at the previous time step is not a variable of the problem, but is in fact data. In the one-step problem we solve, it is only $\bar{u}^{\alpha}$ (or to be even more precise, $u^{\alpha,+}$) that needs to be varied by the solver to obtain the solution. Hence, we leave the expression as is.

**Reviewer** In Eq. (48), it seems that two $\alpha$ exponents are missing on the right-hand-side of the inclusion. Don't we need to refer to $K^{\alpha}$ and $r^{\alpha}$?

**Authors** This is now corrected.

**Reviewer** In the last line before section 3, a closing parenthesis is missing in the expression of the average velocity.

**Authors** This is now corrected.

**Reviewer** Remark 2: "...it to cancel..." might be corrected as "... is to cancel...".

**Authors** This is now corrected.

**Reviewer** By curiosity, did the authors consider to define a different $\theta$ variable in Eq. (65c) and in Eqs. (65a-b)? We could have a $\theta$ and a $\theta^*$ for example, to give more flexibility in the choice of the algorithmic parameters (see also my remark 9 below). Would we then lose the dissipation property?

**Authors** We did consider it as in, but we eventually preferred the simpler presentation of not having an additional variable acting on the smooth terms, given the focus of our work is on the nonsmooth part of the problem. And the reviewer is correct, we would indeed need to set the two parameters equal for the proof.

**Reviewer** In Eq. (80), we see that if $e = 1$, $\theta$ can only be $1/2$, which is a rather strong restriction. It is not possible to impose numerical dissipation anymore. This observation could deserve a comment in the paper.

**Authors** We agree that this deserves a remark, which we add directly below the original equation, reading as follows: "In the case of $e = 1$, we have the very restrictive condition that $\theta = \frac{1}{2}$, which means that in the case of perfectly elastic contacts, we cannot impose artificial dissipation *via* the numerical scheme. In the case that additional dissipation is required to stabilise the integration, it must be imposed by adding viscous behaviour in the bulk, or by recourse to generalised nonsmooth integration schemes (Brüls et al. 2014)."

**Reviewer** After Eq. (88), the authors could mention that they also set $\Phi = \Gamma$, for the sake of readability.

**Authors** We have added this information.

**Reviewer** In the last sentence before Section 5, I think that the term "mid-step local relative velocities" is inappropriate as theta may differ from $1/2$. It could be replaced by "average local relative velocities".

**Authors** We have replaced mid-step with weighted average.

**Reviewer** In the numerical results, I find it difficult to see the fluctuations of the total energy curves on the plots. A vertical zoom on these curves might be helpful (this is only a suggestion).

**Authors** It is true that it is a bit difficult (indeed, impossible in the case of the rod) to distinguish the fluctuations in energy in figures 2 and 5. However, given the amount of space the existing figures take up, we found it hard to add an additional subplot without compromising the clarity of the others. Similarly, a zoomed-in inset plot in the last subplot also makes the legibility difficult. Ultimately, we decided to remain with our current figures as the best compromise, and trust that
the difference in subfigures (e) between the Moreau-Jean and Moreau-Jean-Frémond schemes convey the message adequately.

**Reviewer** Looking at Eq. (95), I expect that a similar formula was also needed in the preceding test case to represent the work produced by the external excitation, is it correct? Is a similar explanation also needed in the previous test case?

**Authors** No, equation 65 is adequate for this purpose, as the sliding block test case remains comfortably within the linear Lagrangian framework (really we just jiggle it back and forth, its overall displacement remains small, and rotations are negligible). We have added a small phrase recalling this equation, reading "In this case, as an external force is applied to the system, we use (65) to calculate the work input into the system."

**Reviewer** In the paragraph above Fig. 14. Firstly, the authors mention that the positive tangential work originates from the large time step and large solver tolerance. I fully understand the importance of the tolerance and would even suggest to include a study of the influence of this parameter in an additional figure. In contrast, I am less convinced by the influence of the time step because, in the linear case, this influence is shown to be zero in the first part of the paper. Maybe, this question should be further elaborated.

**Authors** We include below Figure 1 that studies the influence of the tolerance on the system. Given that the results are not surprising (a reader would always expect that a tighter tolerance results in less error) and our aim in the paper is not to study the quality of solvers for nonsmooth problems, but to compare two models and the associated integration schemes, we don't think it would add anything to include it in the paper.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Influence of the solver tolerance on the positive part of the tangent work, taking the maximum value over the simulation with respect to the time step</figcaption></figure>

It is also not quite right to say that we have demonstrated that the time step size has no influence on the system. Firstly, $p_{k+1}$ is calculated to balance the equation $H^T(q_k)p_{k+1} = M(v_{k+1} - v_k) + hKu_{k+\theta} + hCv_{k+\theta} - hF_{k+\theta}$, in which $h$ appears in some (but not all) terms on the right hand side (and will also appear as $h^2$ on expansion of the $u_{k+\theta}$ term). The particular value taken by $p_{k+1}$ thus depends on $h$ in a nonlinear way. Then, when we consider the (potentially
positive) work of friction we have $u_{T,k+\theta}^{\alpha,\top}p_{k+1}^{\alpha}$. Any positive energy we create due to numerical error should scale with $p_{k+1}^{\alpha}$ and $u_{T,k+\theta}^{\alpha,\top}$, which in turn are dependent on the time step since it is a numerical approximation of the trajectory.

Secondly, the actual implementation of the scheme in this case relies on contact detection, discretising the rotations of the blocks etc. When we have coarser time steps, the performance of these aspects of the code also degrades, and this can cause energetic problems. As one example of how this can affect the scheme, we can have two blocks interpenetrate significantly, but the velocity returned at the end of the time step by the Moreau-Jean-Frémond solver not be sufficient for the interpenetration to be fully resolved. Then, initializing the new time step, the solver again sees interpenetration and solves the impact and friction system again, but with $u_{N,k}$ positive the resulting impact work at the end of this second time step will also be positive. When we have coarse time steps, we have this sort of "temporally nonconverged" system, and so expect the size of $h$ to play a role in whether we observe energy creation.

**Reviewer** Conclusion: the sentence "... we have demonstrated that the Frémond model..." appears as a too strong statement as the dissipation property is only demonstrated in the case of linear elastodynamics.

**Authors** We have added the caveat “for linear visco-elasto-dynamics,” in the phrase. We have also added an additional sentence before discussing the future prospects, namely “Even in cases that are outside the regime for which we have formally proved the dissipativity of the model, we still observe substantially better performance using the new scheme.”

**Reviewer** The authors may double-check the correctness of the information in the acknowledgements.

**Authors** The Acknowledgements now reads: “The authors acknowledge the support of the Marie Sklodowska-Curie Actions program under the Horizon Europe research and innovation framework program (Grant agreement ID 101064805 LEMMA). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or Horizon Europe. Neither the European Union nor the granting authority can be held responsible for them. The authors also thank Franck Bourrier and Ritesh Gupta for their helpful comments on the manuscript.” However, we’re not completely certain of what the reviewer wished us to check, and hence whether this satisfies their comment.

## 2 Reviewer 2 (Simon EUGSTER)

**Reviewer** Since the famous double pendulum example of Kane in “A dynamics puzzle”, we know that the Newton type impact law for frictional contacts can lead to an increase in energy over the impact. This energetic inconsistency of the Newton type impact law has been intensively studied in the article “Energetic consistency conditions for standard impacts” by Glocker 2013. One of the main results of this paper was: “The completely inelastic impact [when all restitution coefficients are zero] is the only case for which energetic consistency can be guaranteed for any and all combinations of impact elements in a system”.

The present manuscript introduces an impact law, inspired by the work of Frémond, which relates local pre- and post-impact velocities in normal and tangential directions with the impulsive forces in the normal and tangential directions of a frictional contact. The combination of normal and tangential directions is obtained by using the de Saxcé formulation, in which Coulomb friction together with the Signorini condition on velocity level can be formulated as a single cone complementarity condition as given in Eq. (16). Furthermore, it is shown that for a single frictional contact (or for isolated frictional contacts, i.e., the Delassus matrix is block diagonal), the Newton type impact law is energetically inconsistent but the Frémond impact law is energetically consistent. Finally, the new impact law is used for an event-capturing time-stepping scheme, which can be seen as a Moreau-Jean scheme with a different impact law. The energy inconsistency (consistency) of the Netwon (Frémond) impact law is also shown for the discrete contact/impact law. Finally, several numerical examples underline the proven results.

The manuscript is well written with many explanations and rigorous proofs. It is relevant to the whole community dealing with frictional contacts, and definitely deserves to be published.
However, I have a few comments and questions that should be taken into consideration before a
possible publication.

**Reviewer** Energetic consistency for a fully coupled multibody system is if the overall work done by all impact elements (all contact points) is non-positive. Only for isolated impact elements each impact element can be studied independently. If I am not mistaken, in Sect. 2.4 energetic consistency is only shown for one single contact, i.e., the entity of frictional and normal impacts. According to (Glocker 2013), a coupling of several impact elements could still lead to an energetic inconsistency. Only in the case of restitution coefficients $e=0$, no problems should appear. If this is the case, the limitation of the statement should be clearly mentioned within the manuscript.

**Authors** The work of contact reactions is an extensive quantity. If the work of all the contact points individually is negative, their sum is also negative. Demonstrating that the contacts are separately dissipative is a sufficient condition for total dissipative work. Of course, this is not a necessary condition. On this point, some explanations are in order. We have something of a philosophical disagreement with the reviewer. It is our firm belief that for a system to be energetically consistent, not only should the global work of impact and friction over all elements be nonpositive, but it should also be nonpositive for all individual elements. Given it is quite standard in thermodynamic treatments of bulk behaviour (plasticity, viscosity etc.) to "localise" the Clausius-Duhem inequality and insist on the thermodynamic admissibility of the behaviour at all material points, we don't see any reason not to insist similarly when we are considering a surface element. We have put an extensive response to the point of view of the reviewer (which is certainly widely shared in the community) as Section 2.5.

**Reviewer** In the section on Frémond's impact law, it is recognized that for the case of stick an inversion of the tangential velocities is obtained. When following the paper of Glocker 2013, a frictional contact can also be formulated by using two normal cone inclusions; one inclusion for normal and a second inclusion for the tangential directions with a force reservoir depending on the normal force/impulse. Using the same normal cone inclusions, Newtonian impact laws can be formulated with independent restitution coefficients both in normal (e) and tangential direction ($e_T$). An example in which a tangential restitution coefficient is required to model a certain behavior is given in Fig. 5 of the above mentioned paper. How is Frémond's impact law different from the case in which the restitution coefficients are $e$ in [0,1] for normal direction and $e_T = 1$ in tangential direction?

**Authors** It is the same, as is now explained in Section 2.5.

**Reviewer** In (Breuling et al. 2024), "A nonsmooth RATTLE algorithm for mechanical systems with frictional unilateral constraints", the "Rotating bouncing ball" was used as a benchmark system. I would like to see the outcome of this example, for the case of stick when using Frémond's impact law. I expect a somewhat different outcome than in (Breuling et al. 2024).

**Authors** We decided against adding this example, as our main interest is in showing situations in which the standard Moreau model fails, while the Frémond model can correctly resolve it. As in this case, there is no coupling between tangential and normal part in the Delassus matrix, the Moreau model should successfully resolve it, which means that we don't think it is relevant for the argument that we are trying to demonstrate. Without any assumption on the material or without experimental evidence, it is difficult to conclude which is the better solution, the Frémond model or the Moreau model.

**Reviewer** I have the following question, in case Frémond's impact law is indeed a particular case of the Newton type impact law described by Glocker 2013 for which restitution coefficients in normal and tangential directions are chosen independently (in contrast to the manuscript, where the Netwon impact law choses the tangential restitution coefficient to be zero). For the case of restitution coefficients in normal direction $e = 0$ and $e_T=1$, there is no condition that would proof energetic consistency for a fully coupled system. Hence, for a fully coupled system with many Frémond's impact elements, the energetic consistency of would still need to be proven;
or a counter example has to be found. This is definitively something for another paper, but a comment would be nice.

**Authors** We include this explanation in Section 2.5. To prove the results for $e = 0$ and $e_T = 1$, the cited approach is not adequate.

**Reviewer** The example “A sliding block” puzzles me. For the Newton impact law, restitution coefficients $e = 0$ were chosen. Hence all restitution coefficients, also in tangential direction, are zero. This is the case for which the Newton impact law should be energetically consistent. Why do we see positive work contributions in the results given in Figure 9? Do these work contributions also vanish with smaller solver tolerances as it is the case for example “impact on masonry structure”? Or does the discretization of the impact law destroy the energetic consistency?

**Authors** In fact this is quite a subtle point, which your comment has made us realize we should highlight in the paper. Essentially, numerical schemes do not have any guarantee that they will preserve invariants that are valid in the time-continuous system. This is exactly the case for the positiveness of dissipation by the impact law. Leine and Glocker’s results are only valid for the time-continuous system and therefore for the problem at instants with impacts. The advantage of Frémond’s law is twofold: it ensures dissipation in continuous time and for the Moreau-Jean scheme in discrete time. In Remark 4, we claim that the scheme will dissipate energy if all restitution vanishes *and also* $\theta = 1$ i.e. we have a fully implicit integration scheme. In the example, we use $\theta = 1/2$, both because it is the value that is energy-preserving for the smooth terms associated with the bulk, and in choosing it we more clearly draw the distinction between the Moreau-Jean-Frémond scheme and the original Moreau-Jean scheme.

**Reviewer** To further improve the manuscript, I have the following comments which could be taken into account by the authors.

**Reviewer** Text between Eq. (6) and (7): I suggest to add an article “the” in front of “set C”.

**Authors** This is now corrected

**Reviewer** In Eq. (9), the normal cone inclusion is used. For the sake of completeness, I would give here the definition of the normal cone; or at least introduce the notation.

**Authors** This is now added to the manuscript.

**Reviewer** I have had some difficulties to follow point 3 in the proof of Lemma 1.

- there is a typo in (21). The equality sign is wrong.

- What would have helped me: “Using $\tilde{u} \in \dots$ together with $0 \le a \le b$ and $0 \le c \le d$ implies $0 \le ac \le bd$.

- Two other hints which would be nice to write: $\tilde{u}_N r_N \ge 0$ as well as that the Cauchy-Schwarz inequality is used for $\tilde{u}_T r_T$

**Authors** The typo is now corrected, and further hinting has been added, although we're not quite sure where we should refer to the Cauchy-Schwarz inequality other than where we already have.

**Reviewer** In Eq. (36) what do you mean by $d(v^T M v)$?

**Authors** The measure of the kinetic energy, which is defined (in extremely extensive mathematical detail) in (Moreau 1988).

**Reviewer** In the text below Eq. (44), I believe that it should be a post impact velocity $u^+$

**Authors** This is now corrected.

**Reviewer** Text below (83) instead of “an function” → “a function”. Moreover, the clarification of the meaning of proj within the norm would be helpful?

**Authors** This is now corrected, and an explanation of proj has been added.

**Reviewer** Because Kane’s double pendulum is so famous, it would be interesting to see this example when using Frémond’s impact law.
**Authors** Kane's double pendulum is certainly a famous example. However, it is also a simple system that does not fall into the category of elastic linear systems. An extension of our results to nonlinear Lagrangian systems and to Hamiltonian systems for solids in large rotations seems quite feasible, but for the sake of brevity, this is not done in this article.

**Reviewer** In the examples, could you specify the vector of generalized position and velocity coordinates $q$ and $v$?

**Authors** We have added the information requested in Section 5.4.

**Reviewer** In example "impact on masonry structure" is it the angular velocity represented in the body fixed frame?

**Authors** We have added the information requested in Section 5.4.

### 3 Reviewer 3 (anonymous)

**Reviewer** The paper is well written and should be published with a minor change regarding the order of the plots in Figs 2,3 and 5. There the velocity and position are (a) and (b) respectively but caption reads displacement and velocity are (a) and (b) respectively.

**Authors** The figures have been appropriately re-ordered.

**Reviewer** The authors demonstrate that the method is strictly dissipative and the results appear to be very excellent. But from the proof in Sect. 3.3, is the reviewer correct in stating that the method doesn't exactly guarantee energy conservation in the frictionless case? A statement regarding that would be instructive. Nonetheless, for all practical purposes this probably not that important give the quality of the results.

**Authors** This is not correct. However, the nature of the conservative properties in this case is certainly interesting, and so we have inserted Corollary 1 into the paper to explore this question.

### 4 Reviewer 4 (Reimco LEINE)

**Reviewer** The paper under review addresses the possible energy inconsistency of the generalized Newtonian impact law with Coulomb friction. As remedy, the authors propose to use the Frémond impact law and present a Moreau-Jean type scheme with a discretization of this impact law. A large number of very interesting numerical examples is reported.

The paper is well written and is dealing with a relevant and timely topic. The scientific quality of the paper is excellent. I advocate the acceptance of the manuscript but I do have some points of criticism (or suggestions for improvement).

**Reviewer** The authors present the extended (or generalized) Newtonian impact law with Coulomb friction in (29) and (34), which involves a normal restitution coefficient for each contact. The more general version of this frictional impact law which also considers a tangential restitution coefficient for each contact is not considered. Apparently, without writing this explicitly, the authors consider in Section 2.2 the tangential restitution coefficient to be zero. At the end of Section 2.3, the authors correctly state that this frictional impact law may lead to an unphysical increase of energy, as noted by Kane and others. However, for what is to come, it may be helpful to realize that, if ALL restitution coefficients are equal (i.e. all normal AND all tangential restitution coefficients are the same), then this law is energetically consistent (see Leine and Wouw 2008, page 160). As important special case, if the contacts are completely inelastic (both in normal and tangential direction), then the frictional impact law is energetically consistent. I would like to urge the authors to introduce tangential restitution coefficients at this point in the paper.

**Authors** This comment made us realise that we needed to do a better job of highlighting the difference between our work, and that of Glocker 2013 and Leine and Wouw 2008, which are undoubtedly the most important contributions in this area. In essence, we have a philosophical difference with these authors, in that for them energetically consistent means *globally dissipative*, whereas for us it means *locally dissipative at all points*. We have added Section 2.5 explaining our differences with these authors (among other things).
**Reviewer** The authors introduce the impact law (45)+(46), inspired by the work of Frémond. As a side remark, here it would be good to briefly state the differences with the law of Frémond. The authors derive that, in the stick case, the tangential relative velocity (or local contact velocity) inverts, i.e. $u_T^+ = -u_T^-$. This corresponds to a purely elastic behaviour in tangential direction, i.e. a tangential restitution coefficient equal to unity ($e_T = 1$). For one, the authors could better make this clear in their work and also discuss the differences with the generalized Newtonian impact law with tangential restitution coefficient equal to one (they should not be the same). But, more importantly, and this is my point of criticism, what is the practical use of this law with $e_T = 1$? A tangential restitution coefficient larger than zero can be used to describe the extremely jumpy dynamics of the Superball, see e.g. the PhD thesis of Glocker, Section 5.3.2. This behaviour, where the ball is jumping back and forth, is quite exotic. For practical applications, a tangential restitution coefficient of zero (or close to zero) is considered. A nice test for a ball to know if a nonzero tangential restitution coefficient is to be taken into account is to throw the ball under a table (first bounce on the ground, second below the table desk, and so on). If the ball comes back, then tangential restitution is prominent. For a tennis ball or table tennis ball, however, it does not. Interestingly, and I am somewhat surprised about that, the authors remark (Remark 1) that experiments of Frémond and Cholet would show that Coulomb's friction law for percussions holds with respect to the AVERAGED tangential velocity, not the post impact tangential velocity. This would indicate that tangential restitution does play a role in the experiments of Frémond and Cholet... To check this, I tried finding the original publications, but I only have access to Figure 8.11 in the book of Frémond. I was unable to check, but I assume that the "small triangles colliding on a marble table" were made of highly elastic rubber, which may explain the tangential restitution. To conclude, I urge the authors to much better discuss this point in their paper.

**Authors** In essence, Frémond presents some experimental results that indicate that the average of pre- and post-impact velocities should be used, and defines (very concisely!) a continuous time model of contact by specifying a dissipation pseudo-potential, and stops the development there (or rather, goes off in other directions, adding temperature *etc*. to the continuous-time model). Hence, at the continuous time level, our impact model is straightforwardly Frémond's, with no difference to our work. We then extend Frémond's work by demonstrating that his model is consistent, while other possible Moreau-type models are not, and show how to integrate this model in discrete time, which was not a subject considered by Frémond. As far as any differences to a generalised Newtonian impact law goes, there are none. The Frémond law can be thought of as just this sort of law, with $e_T = 1$ the only allowable value. But we don't think that the observed experimental behaviour that would require $e_T = 1$ is particularly exotic! While we are also unable to access the thesis of C. Chollet in which the experimental results were originally presented, we generally trust Frémond to report them accurately, and in (Frémond 2017) he reports that the impactors are steel, which seems like an ideal material to treat with a constitutive framework designed for rapid impact of quasi-rigid bodies. We have added that the Frémond model can be conceptualised as setting $e_T = 1$, as well as a small phrase to direct readers to the "best" reporting of Frémond's work, in addition to the extensive discussion in Section 2.5.

As far as concerns the behaviour of bouncing balls with respect to tangential restitution coefficients, we don't think throwing them under a table is a sufficiently well-controlled experiment to draw any conclusions about their contact behaviour (essentially any behaviour can arise as a result of ill-controlled top-spin or back-spin on the balls)

**Reviewer** In Remark 3, the authors write (and cite Glocker 2013) that $e = 0$ is the only value of the restitution coefficients which can assure energetic consistency. This is not completely true, see Point 1 and also (Leine and Wouw 2008): energetic consistency can be assured if all restitution coefficients are equal ($e_N = e_T$) OR if all restitution coefficients are small enough with respect to the condition number of the Delassus matrix OR if inertial decouplings exist.

**Authors** The remark has been corrected, and now reads "A possible way to dissipate energy is to cancel the second term in (70) using $\theta = 1$. In this case, the condition (69) will imply that $e^\alpha = 0$ for all $\alpha$. This result coincides with the results obtained by Leine and Wouw 2008; Glocker 2013, who demonstrated that if all restitution coefficients are equal, the system will be globally dissipative.
Hence, in this special case, the system is also locally dissipative at all points."

**Reviewer** The authors write the frictional contact/impact laws using the normal cone formulation of de Saxcé. Indeed, this is very elegant and only involves a single normal cone inclusion. However, it would be good to write at one point in the paper that this formulation is equivalent to the somewhat more usual formulation using two normal cone inclusions (one in normal direction and one in tangential direction involving a set which depends on the normal force/percussion). Not all readers may be that familiar with de Saxcé's formulation.

**Authors** After introducing the De Saxcé formulation, we have added a remark that demonstrates the equivalence of the two formulations, and justifies why we prefer the De Saxcé formulation.

**Reviewer** Notation: "The value of a real function $t$) ... is approximated by $x_k$". Perhaps it makes sense to insert the word "right-continuous" if $t$) is not a continuous function in time.

**Authors** This has been inserted.

**Reviewer** "non positive" → "non-positive".

**Authors** This is now corrected.

**Reviewer** "little work", perhaps another formulation? "not much literature'

**Authors** This is now corrected.

**Reviewer** Eq. (9): in the adopted notation of the authors, the capital "N" of the normal cone should be upright

**Authors** This is now corrected.

**Reviewer** Lemma 1: the sliding case should explicitly state $u_T \neq 0$. Otherwise, it would include the sticking case, which is not wanted here.

**Authors** This is now included in the definition of the sliding case.

**Reviewer** Eq. 26 "$-\tilde{u} = \Phi(u)$": this is a definition of $\Phi(u)$. However, from (27e) it becomes clear that it should read "$\tilde{u} = \Phi(u)$". Compare with (32).

**Authors** This is now corrected.

**Reviewer** Section 2.3: The authors restrict their analysis to systems which are linear with constant system matrices. This is not essential. In (Leine and Wouw 2008), the energy balance is given for general multibody systems with scleronomic unilateral constraints.

**Authors** Given the different philosophical outlook between us and Leine and Wouw 2008, we will keep our restriction on the type of systems that we claim our proof is valid for, as we're not certain that our point-by-point energy balance necessarily holds outside this class of systems. We plan to attempt a more generalised proof in future work. We list some extensions at hand in the conclusion.

**Reviewer** Perhaps mention somewhere that isotropic Coulomb friction is considered.

**Authors** We have inserted isotropic in front of the first mention of Coulomb friction, but we generally expect that the results should also hold with anisotropic friction as well.

**Reviewer** End of Section 2.3: ".. it would appear to be difficult to propose a consistent numerical scheme that dissipates energy when the underlying law ... ". Can you make this more precise? I would say, that an energetically inconsistent law cannot be used to come to an energetically consistent numerical scheme.

**Authors** This was a bit of humorous understatement on our part, but we have replaced it with "There does not seem to be any prospect of developing a consistent numerical scheme that dissipates energy when the underlying law does not possess this property." in favour of clarity of meaning.

**Reviewer** Remark 1: check the closing brackets on the last line.

**Authors** This is now corrected.
5 Editor's assessments (Alexander POPP)

**Editor** The manuscript presents a novel numerical time-integration scheme for nonsmooth elastodynamics with contact, impact, and friction, incorporating the Frémond impact law into the Moreau-Jean scheme. This approach ensures energy consistency and positive dissipation, both analytically and in practical numerical simulations, across a range of examples. The initial submission underwent a thorough peer review process, with four expert reviewers providing detailed assessments. All reviews were highly positive, commending the originality and rigor of the work, and requested only minor revisions to improve clarity and presentation. The authors responded diligently, addressing all comments and suggestions in a carefully prepared minor revision. The revised manuscript was directly assessed by the handling editor and found to have fully incorporated the requested changes. Given that all reviewer concerns were satisfactorily resolved, a second round of external review was deemed unnecessary, and the manuscript was accepted for publication accordingly.

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