Peer review of: An active mechanical Willis meta-layer with asymmetric polarizabilities (2020)

Reviewers' comments: 
Reviewer #1 (Remarks to the Author): 

This manuscript presents an active elastic wave metamaterial, based around a sensor which 
measures the symmetric component of the incident field, and is then able to generate controlled 
levels of both symmetric and anti-symmetric waves. This is presented as an example of non-
reciprocal Willis coupling in a metamaterial, and this description is quite sound. 

However, the authors model this system with what they call a "polarizability" model, attributing 
the symmetric radiation to excitation of a monopole mode, and anti-symmetric radiation to a 
dipole mode. This terminology does not seem well justified, for example the bending moment M is 
a higher order moment of the stress tensor, so describing it as a monopolar term is quite 
confusing. We should expect the degree of a multipole to correspond to some order of near-field 
spatial variation, or to a far-field radiation pattern. 

Furthermore, the authors first introduce a 4x4 matrix description, with 2 "dipole" and 2 
"monopole" quantities in both source and induced quantities, but quickly reduce to only 
considering a 2x2 matrix, without any real justification. It is quite possible that the other 
quantities do play a role in the experiment, but given the number of observable 
transmison/reflection coefficients, it is not possible to resolve all 16 coefficients. It would be much 
to simplify the model to describe symmetric and anti-symmetric components, without making 
claims about multipole orders, or unnecessaril trying to differentiate indistinguishable scattering 
mechanisms. 

Overall this theoretical approach is quite difficult to follow, and seems unnecessary to describe the 
experiment. For example, the required voltage transfer functions given by Eq (24) in the 
supplementary material do not require the polarizability model, but just depend on simple 
quantities such as required transmission, separation between transducers etc. In particular, the 
source driven homogenization does not seem to relate to the experimental part, or to make any 
new predictions, and homogenization is not meaningful for the considered cases of a single 
scatterer ad an inhomogeneous line array of scatterers. 

On the other hand, the experimental aspects of this paper seem novel, well conducted, and quite 
interesting. In my view, this part of the paper clearly meets the level of novelty I would expect of 
Nature Communications, and could readily be understood by abroad audience. If the experimental 
results were given more prominence, and the theoretical part was reduced to the minimum 
necessary to interpret the experiments, this would be quite a solid paper. 

I note a few minor issues: 

- In several places numerical simulations are performed, however it is never described how these 
are done. It appears that some form of full wave elastic and piezoelectric coupled model is used. 

- On line 38 the statement "furnish unsurpassed control" seems like hype, and should be replaced 
with something more substantive 

- On lines 49-50, vague mention is made of the constraints of passive media for certain 
applications. This claim should be made more concrete and backed up by examples. 

- On lines 193-194, it is stated that the scattering in the passive state is negligible. This statement 
should be quantified 

- On line 266, reference titles suggest that reference [21] should have been cited instead of [20] 


Reviewer #2 (Remarks to the Author): 


The broad context of the work is Willis materials—a class of mechanical metamaterials with 
unusual dynamic constitutive relations, namely, velocity-dependent stress and strain-dependent 
linear momentum. The authors propose and experimentally realize a structural counterpart of 
Willis materials, by considering beams undergoing flexural motions with analogous cross-coupling. 
To my knowledge, the novelty in this work is the implementation of active elements in the settings 
of Willis beams. As a result of the integration of these elements, the authors demonstrate 
(theoretically, numerically, and experimentally) that the Willis beam exhibits exotic properties, 
such as nonreciprocal behavior, perfect absorption and beam steering over a wide range of 
frequencies. The final results are new in the topical field of Willis metamaterials, and generally 
interesting. From this perspective, I find them suitable for publication in Nat. Commun. However, I 
have several issues with the manuscript and some of the statements therein, which should be 
addressed prior to its acceptance. 

(A) The manuscript is hard to follow (at least for me), partially since necessary information is 
either assumed as common knowledge (an important example is given in the next point) or 
relegated to the supplementary information (SI). To my understanding, the manuscript should be 
able to stand on its own, where the SI should not be essential for the understanding of the main 
results. However, some of the main results appear only in the SI. For example, in the summary of 
the results appearing in the conclusion section, the authors list their development of a source-
driven homogenization theory—this development appears only in the SI. Similarly, beam steering 
is described in the body of the text, however the readers are directed to SI for corresponding 
figures. 

(B) The concept of bianisotropic polarizability tensor has been recently introduced in [1] for 
acoustics, and used thereafter by others in this context (acoustics), e.g., [2]. To my knowledge, 
the introduction in this manuscript of a polarization tensor for flexural waves is new. If it is not, 
please add references for more clarification. If it is indeed new, there is a need to clarify more its 
physical meaning and connection to the constitutive equations in the manuscript. Specifically, to 
me it is not clear how the authors associate some of the quantities and monopole (in-phase) and 
some as dipole (out-of-phase) quantities, e.g., how is the curvature associated with out-of-phase 
propagation? Perhaps an illustration as in [1] will be useful. The connection between the 
polarizability tensor and the constitutive properties of beams in Willis form also needs clarification. 
(Again, there is some explanation in the SI, but I would expect it to appear in a clearer and more 
concise form in the manuscript itself.) 

(C) The authors refer to the symmetry property of the Willis coupling in the abstract and the body 
of the text. I find this statement as ambiguous: initially I understood it as a symmetry with respect 
to the components of the Willis coupling, but in the text it turns out it refers to the loss of 
reciprocity [3], or, equivalently, the loss of self-adjointness of effective constitutive operator [4]. 
This should be clarified. 

(D) The authors claim that “It is also important to mention that  the scattering properties of a 
Willis scatterer cannot be properly captured using simple the Euler beam theory because it lacks 
the degree of freedom associated shearing motion within the beam that is required to capture 
monopole-dipole coupling that is the finger-print of the Willis material response”. Why an Euler 
beam whose bending moment depends (not only on the curvature but also) on the transverse 
velocity, and whose linear momentum depends (not only on the transverse velocity but also) on 
the curvature [5] is not considered a beam with a Willis response? 

(E) Some of the statements made in the text should be toned down. For example, in the abstract 
the authors claim that “Willis coupling coefficients in Willis solids designed so far are symmetric”. 
However, there are several works who already broke reciprocity in Willis materials theoretically [6] 
and experimentally [7] (I think that the last author also has a relevant theoretical work on the 
topic [8]). 
Another statement that I find inexact is that “In summary, we have provided a theoretic and 
experimental study on an active Willis meta-layer in elastic plates…”. The theoretical analysis is 
completely a one-dimensional analysis, and therefore is not applicable to plates, which are 
governed by equations of two spatial coordinates. In the SI, the authors themselves say that “Note 
that the approach demonstrated below can be easily extended to a Willis plate through the use of 
Mindlin–Reissner plate theory”, i.e., they did not execute this extension. I suggest that the authors 
would either extend their approach as they say, or rephrase their statements. 
(F) Finally, the manuscript should undergo a thorough proofreading, as is contains several typos 
(e.g., line 68 is missing the word “to” after “ability”, in line 181 the words “is” and “leads” should 
be “are” and “lead” since they refer to the word “functions”). 

References 

[1] C. F. Sieck, A. Alù, and M. R. Haberman, Phys. Rev. B 96,  104303 (2017).  [2] X. Su and A. 
N. Norris, Phys. Rev. B 98, 174305 (2018) 
[3] M. B. Muhlestein, C. F. Sieck, A. Alù, and M. R. Haberman, Proc. R. Soc. A 472, 20160604 
(2016) 
[4] J. R. Willis, Proc. R. Soc. A 467, 1865 (2011) 
[5] R. P. Salomon, G. Shmuel, J. Mech. Phys. Solids 119 (2018) 
[6] L. Quan, D. L. Sounas, and A. Alu, Phys. Rev. Lett., 123, 064301(2019) 
[7] Y. Zhai, H.-S. Kwon, and B.-I. Popa, Phys. Rev. B 99, 220301(R) (2019) 
[8] H. Nassar, X. Xu, A. Norris, and G. Huang, J. Mech. Phys. Solids 101, 10 (2017)