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



RR2
 REVIEWER COMMENTS
Reviewer #1 (Remarks to the Author):
  The revised manuscript is greatly improved, and gives more appropriate emphasis on the
 important experimental aspects. Also, the multipole model is now much better justified. The
 manuscript should be suitable for publication, once the following issues are clarified:
 On line 237, the reader is referred to the Supplementary Information for interpretation of beta_34'
 and beta_44'. While it makes sense that the full definition should be kept in the Supplementary
 Information, these coefficients should be given some name and interpretation to help the reader
 understand their significance. To my understanding these represent "bending to shear" and
 "bending to bending" terms respectively.
 Now that it has been clarified that p_0 and M_loc are quadrupole terms, and that only bending and
 shear terms are significant in the structure, it seems that the Willis coupling is in the quadrupole-
 dipole terms. As far as I know, all Willis coupling reported to date in acoustics is for monopole-
 dipole coupling, so this difference should be mentioned in the manuscript.
 It is not clear that the claims (including in the abstract) about being broadband and overcoming
 strong frequency-dependence are fully justified. My understanding of Figure 3 (e) is that results
 for each frequency were obtained by reconfiguring the transfer function each time. If this
 understanding is correct, then the structure is not truly broadband, in the sense of being able to
 fully manipulate a short pulse, so claims of being broadband should be toned down (e.g. state only
 that the structure is reconfigurable over a broad bandwidth). The key theoretical point, which the
 authors may wish to discuss, is whether the functions H_1 and H_2 given in Eq (1) correspond to
 causal functions in the time-domain.
 I am still not convinced that the section on Homogenization really adds any value to the paper. On
 the other hand, it takes up minimal space in the revised manuscript.
 Minor points:
Reviewer #2 (Remarks to the Author):
 - The order of supplementary Figure S5 (c) and (d) should be swapped, so that it corresponds to
 the order these terms appear in Q
  The revision has significantly improved the manuscript, and most of the points made were
 addressed. I disagree, however, with the reply to point (D). In the first part of the reply, the
 authors claim that Eq. (R6) implies that Willis couplings cannot exist between the bending moment
 and the velocity, nor between the linear momentum and the curvature; this claim is inconsistent
 with Ref. [5], which shows the converse. The issue with Eq. (R6) is that it is obtained by assuming
 a single constituent whose constitutive response is described by a Willis material, and then
 applying the standard procedure from beam theory to obtain equations in terms of in terms of
 forces, bending moments etc. In a sense, it carries out an homogenization from the equations of
 (Willis) elastodynamics to the equations of the mechanics of structures. However, Eq. (R6) does
 not model the homogenized constitutive relation of a heterogeneous beam whose structural
 properties vary in space, and the arguments that are based on it in the second part of the reply do
 not apply in this case. In accordance, Eq. (R10) is an assumption, and not the end of a
 homogenization process at the structural level. The inconsistency with the analysis based on the
 source-driven homogenization used by the authors may be caused by the fact that the body force
 density is missing; the transverse body force they consider (f in the notation of the manuscript) is
 a higher order moment of it. It is this density that appears in the beam governing equation. Its

 analogue in elastodynamics is the one that is prescribed as the source in the literature, see, e.g.,
 [4,8], in the same way that the analogue of the bending curvature source appears in [4,8],
 referred to as an “eigen-strain”. In summary, the authors should either disprove the arguments
 above, or remove the statement that an Euler beam cannot exhibit Willis coupling.