Disentangling Longitudinal and Shear Elastic Waves by NeoHookean Soft Devices: Supplemental Material
Zheng Chang, Hao-Yuan Guo, Bo Li, and Xi-Qiao Feng *
Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua
University, Beijing 100084, China.
*
Author to whom correspondence should be addressed. Electronic mail: fengxq@mail.tsinghua.edu.cn
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I.
BRIEF INTRODUCTION OF HYPERELASTIC TRANSFORMATION THEORY
The theory of small-on-large is applied to investigate the problem of linear waves propagating in
finite-deformed materials. For a hyperelastic solid with the strain energy density function W , the
equilibrium equation of the finite deformation can be written as
( AijklU l , k ),i  0 ,
(S1)
where U i denotes the displacement, Aijkl  2W Fji Flk are the components of the fourth-order
elastic tensor expressed in the initial configuration, and Fij  xi  X j the deformation gradient.
Further, the incremental wave motion ui superimposed on the finite deformation U i is governed by
( A0i jk l ul , k  ),i   0 u j ,
(S2)
with a pushing forward operation on the elastic tensor A and the initial mass density  ,
A0ijk l  J 1 Fi ' i Fk ' k Aijkl , 0  J 1  ,
(S3)
where J  det(F) is the volumetric ratio.
The governing equation of linear elastic waves has the same form as Eq. (S2):
(Cijkl ul , k ),i   u j ,
(S4)
where Cijkl are linear elastic constants. In the theory of transformation elastodynamics, Eq. (S4)
preserves its form under asymmetric transformation relations1,2
Cijk l  J 1 Fi ' i Fk ' k Cijkl ,    J 1  ,
(S5)
where C ,  and C ,   are the material parameters expressed in the virtual space x and the
physical space X , respectively. These two spaces are connected by the mapping Fij  xi  X j . In Eq.
(S5), J  det(F) is the Jacobian of F . According to Eq. (S5), one can map the wave field from the
virtual space to the physical space by introducing the anisotropic and/or inhomogeneous material
parameters C and  ' . Moreover, the Cosserat form asymmetry elastic tensor is required in Eq. (S5)
.
Note that the pushing forward formulation of Eq. (S3) has the same form as the asymmetric
transformation relations, Eq. (S5). This consistency naturally bridges the theory of small-on-large and
the transformation theory under the condition expressed in Eq. (1). In other words, if Eq. (1) holds, a full
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analogy between the pushing forward formulation and the asymmetric transformation relations can be
obtained, i.e.
[C,  , F]  [A0 , 0 , F] .
(S6)
In this fashion, the deformation gradient F of the hyperelastic material automatically becomes the
mapping F required by the asymmetric transformation relations. Simultaneously, the asymmetric
elastic tensor C required by the asymmetric transformation relations can be maintained by A 0 ,
implying that “smart metamaterial” used for transformation devices can be implemented by such
deformed hyperelastic solids without introducing any microstructures.
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II. NUMERICAL IMPLEMENTATION
The numerical examples are accomplished by a two-step model using the software COMSOL
Multiphysics. In the first step, the finite deformation of the hyperelastic material is calculated with the
solid mechanics module. After which, the deformed geometry configuration, together with the
deformation gradient F , is imported into the second step to simulate the linear elastic wave motion
governed by Eq. (S2). In this step, the weak form PDE module is applied to deal with the asymmetry of
the elastic tensor A 0 . In the calculation of wave propagation, in order to save the computational
resources, only a portion of the deformed neo-Hookean material and the undeformed domains on its left
and right hand sides are included, as surrounded by the blue lines in Fig. 3(a) and (d). Additionally, the
perfect matched layers3 are applied to avoid the reflection on the left and right boundaries of the
computational domain.
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III. DETERMINATION OF REFRACTION ANGLES OF THE WAVE-MODE SEPARATION
DEVICE
For a homogenously deformed hyperelastic material, consider the incremental plane waves of the
form
ui  mi f (kl j  x j  t ) ,
(S7)
where m is a unit polarization vector, f is a twice continuously differentiable function and l is the
unit vector in the wave direction. Inserting Eq. (S7) into (S2), the propagation condition4 or the
Christoffel equation5 can be written as
A0ijk l lilk  ml  c2 0 m j ,
(S8)
where c   k is the wave speed. For a neo-Hookean material, inserting Eqs. (2) and (S3) into (S8),
the Christoffel matrix A0i jk l li lk  has the form of
A0ijk l lilk   (  1 )l j ll  2 jl ,
(S9)
where
   (2 J  1),
1   (1  J )  J 1 ,
(S10)
2  J  Fis Fk s lilk  .
1
In this fashion, the characteristic solutions of Eq. (S8) can be easily obtained as Eq. (5), which denotes
the speeds of the P- and S-waves, respectively.
With the definition of si  ci1 , according to Eq. (5), the phase-slowness of the P- and S-waves are
respectively
sP  0 (  1  2 ) ,
sS   0  2 .
(S12)
Thus, the phase-slowness curves can be plotted to show the wave propagation property in the certain
deformed hyperelastic materials. Moreover, the refraction angles of the elastic waves on a straight
interface between the undeformed and deformed neo-Hookean material can be obtained by the graphical
approach proposed by Auld5 and Rokhlin et al.6 . The key feature of the approach is “all projections of
the slowness vectors on the interface are equal to one another” 6 . With this criterion, we can easily find
the phase-slowness of the refraction wave (point rP and rS on the phase-slowness curve for P- and S-
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wave, respectively) and further determine the refraction angle, which is normal to the phase-slowness
curve on the corresponding point ( rP or rS ).
For the wave-mode separation device proposed in the main text, which is a simple-sheared neoHookean material with the deformation gradient of Eq. (4), the phase-slowness curves of such material
can be obtained by inserting Eq. (4) into Eq. (S12) as
 
 (tan 2  sin 2   2 tan  sin  cos   1)]1/ 2 ,



sS  [ (tan 2  sin 2   2 tan  sin  cos   1)]1/ 2 ,

sP  [
(S13)
with the definition of lx  cos  and l y  sin  . An example of the phase-slowness curves of the wavemode separation device with the normalized parameters of   4 ,   1 ,   1 and tan   1 3 is
illustrated in Fig. S1. Based on Eq. (S13), the normal of the phase-slowness curves has the slope
tan  P,S 
sP,S sin   (dsP,S d ) cos 
sP,S cos   (dsP,S d )sin 
.
(S14)
For the case of horizontal incidence (   0 ), the refraction angle of the P- and the S-waves can be
obtained as
 P  arctan(
 tan 
),
  2
(S15)
S   ,
respectively.
FIG. S1. The slowness curves of P-wave (blue) and S-wave (red) propagating in a wave-mode separation device. The refraction
angles  P and  S of an entangled wave beam horizontally incident on the interface of the wave-mode separation device are also
denoted.
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IV. OTHER NUMERICAL EXAMPLES
The following two examples are the variants of the wave-mode separation device to show the
robustness of the soft device under different loading situations. To implement the simple shear
deformation state, two neo-Hookean materials with the nearly incompressible material parameters of
  2 GPa ,
  1.08 MPa ,
0  1050 Kg/m3 , and compressible ones
  4.32 MPa ,
  1.08 MPa ,   1050 Kg / m3 are fixed on the left boundary and slide upward for a certain distance
of U y  0.4 m on the right boundary. In these cases, the deformation state is no longer rigorously
simple shear, as illustrated in the material coordinates in Fig.S2 (a) and (d). Nevertheless, it is shown
through numerical examples in Fig. S2 that the inhomogeneity will not affect the wave control ability
and the effect of wave-mode-separation.
FIG. S2. The same as Fig. 3, but the simple-shear deformation is produced by a displacement on the right boundary. (a) Normalized
wave parameters NP and NS in a nearly incompressible neo-Hookean material under simple shear. (b) The distributions of
NP and NS on the right boundary (green line) of the computational domain. For comparison, NP and NS of the elastic
waves propagating through an undeformed neo-Hookean material are given in (c). Figures (d), (e), and (f) are the calculation results
for a compressible neo-Hookean material.
In addition, a transient analysis of entangled elastic waves propagate in such wave-mode-separation
device is also performed. Consider a neo-Hookean material with the material parameters of
  4.32 MPa ,   1.08 MPa , and   1050 Kg / m3 in a simple-shear deformation state with the
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shear angle of   arctan(1/ 3) . A transient signal of u x  a  exp(  r 2 / w2 )  sin(7500t ) m and
u y  a  exp(  r 2 / w2 )  sin(1500t ) m with a  1 103 m , r  0.2 π (y  0.2) m , and w  0.025 m
are applied on the left boundary. The remaining three boundaries are applied to be low-reflecting
boundaries. As illustrated in Fig. S3, both P- and S-waves propagate in the same manner as our theoretical
prediction, and the results are consistent with the frequency domain analysis.
t
(a)
(b)
(c)
abs(u)
div(u)
rot(u)
[s]
(1)
0.005
(2)
0.010
(3)
0.015
(4)
0.020
FIG. S3. Transient analysis of entangled elastic waves propagating in a simple-sheared neo-Hookean material: Row (a)–(c)
represent the total displacement field, the P-wave field (divergence of the displacement field), and S-wave field (curl of the
displacement field), respectively, while line (1)–(4) represent the snapshots at t  0.005 s , 0.01s , 0.015 s and 0.02 s ,
respectively. The blue and red dashed lines represent the P- and S-wave paths predicted by the theoretical analysis, respectively.
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p. 1.
5
B. A. Auld, Acoustic Fields and Waves in Solids (R.E. Krieger, Malabar, Fl., 1990).
6
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