A Linear Strain Energy Function for Hyperelastic Transformation Method

A Linear Strain Energy Function for Hyperelastic Transformation Method Linli Chen, Chao Ma, Zheng Chang a) College of Science, China Agricultural University, Beijing 100083 changzh@cau.edu.cn Abstract: Hyperelastic transformation method provides a promising approach to manipulate elastic waves by utilizing soft materials. However, no existing constitutive model can rigorously achieve the requirement of such method. In this Letter, a linear strain energy function (SEF) is proposed, which can be implemented to control the longitudinal and shear elastic waves simultaneously. In comparison with the neo-hookean and the semilinear SEFs, the wave propagation and the impedance of pre-deformed linear hyperelastic material are exploited. Numerical simulations are performed to validate the theoretical results. The investigation may pave the ways for the design and realization of soft transformation devices. Keyword: Linear strain energy function; Hyperelasticity; Elastic wave; redeformation. a) Author to whom correspondence should be addressed. Electronic mail: changzh@cau.edu.cn. 1

1. Introduction In the past two decades, the transformation method 1, as a means of designing complex media that can bring about unprecedented control of wave field, has drawn considerable attention in different realms of physics 2,3. Recently, by analogy with such method, a hyperelastic transformation method (HTM) has been proposed 4-7, providing the elastic waves can be precisely controlled by a pre-deformed soft material with specified strain energy function (SEF). Compare to the traditional approach, HTM eliminates the requirement of microstructures, and therefore, exhibits remarkable potential for nondispersion and broadband wave manipulation. The elastodynamic behavior of a hyperelastic material is highly dependent on its constitutive model 8. In HTM, to bridge the small-on-large theory and the asymmetric transformation relation 9, the SEF W is required to satisfy 5 2 W F F ji lk A 0ijkl, (1) in which F is the deformation gradient tensor, and A 0 is a constant tensor equals to the stiffness of the un-deformed material. While the theory has been well established, the corresponding theory on material implementation is less well developed. For most of the existing hyperelastic models, Eq.(1) cannot rigorously hold. It has been reported 7 that neo-hookean SEF partially satisfies Eq.(1), so that only shear wave (S-wave) can be effectively manipulated. For semi-linear SEF, on the other hand, Eq.(1) holds exactly when the pre-deformation is symmetric 5, i.e. F T F. Additionally, a more recent research 10 reveals that without the out-of-plane extension constraint, only longitudinal wave (-wave) can be accurately controlled by semi-linear material. In this context, there is strong motivation to explore the existence of the SEF for simultaneous control of - and S-waves, which is essential for applications of HTM. In this Letter, we report a linear SEF that possesses as an ideal model for HTM. Comparing with the neo-hookean and semi-linear SEFs, we investigate the wave propagation and impedance of the linear hyperelastic material which are subjected to 2

pre-deformation. Both theoretical and numerical analyses demonstrate the effective wave manipulation of the linear SEF. The Letter is organized as follows. The proposed linear SEF and the other SEFs considered throughout this Letter are introduced in Section 2. Elastic wave propagation behavior in hyperelastic materials under simple-shear and uniaxial-tension is investigated in Section 3. Furthermore, transmission and reflection of elastic waves at the interface between un-deformed and pre-deformed hyperelastic materials are illustrated in Section 4. Finally, a discussion on our results and avenues for future work are provided in Section 5. 2. Strain energy functions for hyperelastic transformation method For the sake of simplicity, we consider all the SEFs to be in their two-dimensional forms. Therefore, the linear SEF can be expressed as 11 W 2 + dev tr 2, 2 (2) 1 dev2εε tr 2 ε I is the deviatoric L 2 in which εsym FI is the strain tensor, part of ε, is the Frobenius matric norm, and are the first and second Lamé constants, respectively. With reference to a previous study 12, W L is a reduced form of quadratic Hencky SEF with the higher-order terms being eliminated. By expressing W L in terms of F, it can be easily derived that the SEF satisfies the criterion of Eq.(1). For comparison, the neo-hookean 7 and semi-linear 5 SEFs are also considered. The neo-hookean model is given by 2 WNH ( J 1) ln( J) ( I1 2), (3) 2 2 in which 1 I is the first invariant of the right Cauchy-Green tensor, J det F is the volumetric ratio. Meanwhile, the semi-linear model can be written as 2 2 WSL tr tr, 2 E E (4) in which EUI with U T F F the right stretch tensor, and I is the unit 3

matrix. In the following analyses, we choose 6 6 4.32 10 a, 1.08 10 a, and 3 1080 kg m as the initial material parameters for all the aforementioned SEFs. The parameter combination refers to a compressible variant of the rubbery material SM-4. 3. Elastic wave propagation in homogeneously deformed hyperelastic materials The main feature of the HTM is to steer waves along with pre-deformation. Therefore, to testify about the suitability of a specified SEF for HTM, we first investigate the wave propagation in pre-deformed material. For the ease of analysis, the deformation is considered to be homogeneous, so that the elastic waves travel in straight paths. 3.1. Theoretical analyses Theoretically, for a hyperelastic material with homogeneous pre-deformation, the wave manipulation capability can be evaluated by the phase slowness S,S and the polarization direction x i of the elastic waves, both of which can be obtained by solving the eigenvalue problem governed by the Christoffel equation 13. In the first situation, we consider the hyperelastic materials which are subjected to simple-shear. The deformation can be specified by applying a body displacement U x 3m, x[0,1] on a 1m 1m material domain, as a schematic diagram shown y in Fig.1 (a). Correspondingly, the deformation gradient can be described as F 11 F22 1, F12 0 and F21 13, with the shear angle arctan1 3. The slowness curves for the SEFs have been illustrated in Fig.1 (b). For -mode, the curve of the linear SEF turns out to have a same configuration with that of the neo-hookean SEF. Conversely, for S-mode, the consistency appears between the linear and semilinear SEFs. The results reveal that the linear SEF acquires the advantages of both the neo-hookean and semi-linear SEFs, and can be utilized to manipulate - and S-waves simultaneously. More specifically, we consider the waves that originally horizontally propagate in an un-deformed hyperelastic material, and inquire into the shift angles p q of the wave 4

paths result from the simple-shear. The angles p q, with the superscript p, S denotes the incident wave mode, and the subscript q L, SL, NH denotes the SEF, can be obtained from the normal of the slowness curves at the right intersections of the slowness curves and the horizontal axis, as demonstrated in the insets of Fig. 1(b). It is evident that all the shift angles except for NH, equal to the shear angle, i.e. =. In addition to further demonstrating the wave control S S S L SL L SL NH capability of linear SEF, the result also indicates that - and S-waves propagate along the same path in semi-linear material with simple-shear deformation, although the slowness of the two wave modes slightly differ from each other. It is also worth noting that the coincidence of the wave paths is independent of the material parameters and the degree of the deformation. Fig.1. (Color online) Elastic wave propagation in simple-sheared and uniaxial-tensioned hyperelastic materials with different SEFs. (a), (c) The sketch maps of the deformation states. (b), (d) The slowness curves of the hyperelastic models. The shift angles of horizontally propagated elastic waves are illustrated in the insets of (b). In the second situation, we consider a symmetric deformation, e.g. uniaxialtension, which is applied on the hyperelastic materials. The deformation can be accomplished by applying a displacement U 37m on the right boundary of a 1m 1m material domain, with the rest boundaries set to be rollers, as shown in Fig.1 x 5

(c). In this fashion, the corresponding deformation gradient can be written as F11 10 7, F22 1 and F12 F21 0, thus the elongation ratio in the horizontal direction is 10 7. The slowness curves have been plotted in Fig.1 (d). Similar qualitative information as those shown in Fig. 1 (b) can be observed, providing the wave control capability of the SEFs is independent of material deformation mode. Before and after the uniaxial-tension, the path of horizontally propagated waves will not be affected. Whereas to varying degrees, the wavelengths will change due to the material elongation. By utilizing the slowness at the right intersections of the slowness curves and the horizontal axis, the elongation ratio of the wavelengths can be easily obtained by some simple algebraic calculation. As shown in Fig.1 (d), for -mode, the phase slowness of the linear and semi-linear SEFs are both 0.0089 s m 1, which corresponds to the wavelength elongation equals to 10 7, the material elongation. The corresponding slowness of the neo-hookean SEF is 0.0093 s m 1, showing a small deviation between the wavelength elongation and material deformation. Similarly, for S-mode, both the linear and neo-hookean SEFs present the slowness of 0.022 s m 1, also verifying the consistency between the wavelength elongation and material deformation. 3.2. Numerical simulations Numerical simulations have been performed to validate the theoretical results by a two-step model 14 using the software COMSOL Multiphysics. In steady-state analyses, - and S-wave beams are imported at the left boundary of a 1m 1m pre-deformed material domain with the maximum amplitude as A 0.01 m and the angular frequencies being S 2kHz and 4kHz, respectively. In general, the results of the numerical simulations are in excellent agreement with those obtained from theoretical analyses. The divergence and curl of the displacement fields in the simple-sheared hyperelastic materials are plotted in Fig.2. For linear SEF, as depicted in Fig.2 (a) and (b), both - and S-waves are shifted along with the material deformation, indicating a full control of the elastic waves. For the neo-hookean SEF, as can be expected, only S-wave can be effectively controlled, as illustrated in Fig.2 (c) 6

and (d). For the semi-linear SEF, although the paths of - and S-waves coincide as shown in Fig.2 (e) and (f), the wave modes are converted into the quasi-ones, as illustrated in Fig.2 (g) and (h). Such mode conversion may lead to energy loss or failure of transformation devices, and therefore, should be avoided in HTM. Fig.2. (Color online) Divergence and curl fields in simple-sheared hyperelastic materials with different SEFs. (a), (b) linear SEF; (c), (d) neo-hookean SEF; (e)-(h) semi-linear SEF. (a), (c), (e), (h) -wave incidence; (b), (d), (f), (h) S-wave incidence. A steady-state analysis of the elastic waves propagate in uniaxial-tensioned hyperelastic material is also performed 14. Both qualitative result and quantitative information are shown to be in consistent with the theoretical predictions. 4. Elastic wave impedance of homogeneously deformed hyperelastic materials In transformation method, apart from wave path control, impedance matching between transformation media and background is also of great significance. In this sense, we consider the elastic waves incident on a plane interface between un-deformed and predeformed hyperelastic materials. For simplicity, normal incidence is considered to avoid mode conversion at the interface. 4.1. Theoretical analyses In the framework of small-on-large theory, pre-deformed hyperelastic material 7

usually exhibits effectively anisotropy. Therefore, the classical theory 13 of transmission and reflection of elastic waves has been modified and applied to our study, and the transmission and reflection coefficients m ns can be yielded 14. In m ns, the superscript m, S denotes the incident wave mode, the subscript n T, R distinguishes the transmission and reflection, and the subscript s, S denotes the transmission or reflection wave modes. Consider the transmission and reflection of elastic waves from an un-deformed domain to a simple-sheared domain with the deformation gradient F11 F22 1, F12 0 and F21 13. erfect impedance matching, i.e. and S TS 1, can be T 1 obtained for both linear and neo-hookean SEFs, regardless of any incident wave mode. For semi-linear SEF, total transmission, i.e., only occurs for -wave incidence. T 1 Slight impedance mismatch can be expected for S-wave counterpart, with the transmission and reflection coefficients are S TS 0.984 and S RS 0.016, respectively. The negative reflection coefficient means the waves receive a 180 phase shift. In the case of uniaxial-tension with the deformation gradient F11 10 7, F22 1 and F12 F21 0, the linear SEF and the neo-hookean SEF exhibit diverse properties. The perfect matching still holds for linear SEF, whereas a mismatch can be perceived for neo-hookean SEF with -wave incidence, i.e. T 1.022 and R 0.022. For the semi-linear SEF, although the -wave can be totally transmitted, i.e. T 1, significant reflection can be observed for S-wave incidence, as indicated by the coefficients S TS 0.843 and S RS 0.157. 4.2. Numerical simulations The theoretical results have been confirmed by numerical simulations. For a clear illustration, transient analyses 14 are adopted to distinguish the transmitted and reflected waves. As shown in Fig. 3, two 0.6 m 0.1 m rectangular domains are set as the undeformed and pre-deformed hyperelastic material. The harmonic wave sources, with the excitation lasts for three wavelengths, are set at the left boundary with maximum amplitude as A 0.01 m, the angular frequency of the - and S-waves being 8

1kHz and S 0.4 khz, respectively. The top and lower boundaries are set to be the Floquet periodic, and the right boundary is left to be free. Normalized parameters inc 1, x x discussion. D u A and Dy uy A are introduced for the ease of comparison and Fig.3. (Color online) Transmission and reflection of elastic waves at the interface between undeformed and uniaxial-tension hyperelastic materials with different SEFs. (a), (b) linear SEF; (c), (d) neo-hookean SEF; (e), (f) semi-linear SEF. (a), (c), (e) -wave incidence; (b), (d), (f) S-wave incidence. The transmission and reflection of elastic waves at the interface between an undeformed and a uniaxial-tensioned hyperelastic material are plotted, as illustrated in Fig.3. For each SEF, to display the status before and after the wave impinging, two typical snapshots of the normalized displacement field D x or D y are displayed in color maps. In the two moments, the distribution of D x and D y along the horizontal direction is illustrated in the line graphs, with the theoretical results marked by the dash lines. With a good agreement with the theoretical predictions, total transmission is apparent in the cases of linear SEF with both - and S-wave incidence, neo-hookean SEF with S-wave incidence, and semi-linear SEF with -wave incidence, as shown in Fig. 3 (a), (b), (d) and (e), respectively. Minor reflection is noticeable for neo-hookean SEF with -wave incidence, as shown in the line graph of Fig. 3 (c), whereas a 9

comparatively large reflection is noticeable for semi-linear SEF with S-wave incidence, as shown in Fig. 3 (f). Moreover, we also investigate the transmission and reflection of elastic wave at the interface between the un-deformed and the simple-sheared hyperelastic materials 14. Once again, the results confirm the wave manipulation capability of the linear SEF. 5. Conclusion In this Letter, we have proposed a linear SEF which can satisfy the condition required by HTM. By conducting the theoretical and numerical analyses, it has demonstrated that the hyperelastic material characterized by linear SEF can fully steer the elastic wave by pre-deformation. Meanwhile, the wave impedances of the pre-deformed material perfectly match with those of the deformation-free one. For comparison, the wave propagation behavior and impedance of pre-deformed neo-hookean and semi-linear materials have also been studied. The results on wave propagation are in consistent with previous findings 7,8,10. Besides, it has been observed that even for asymmetrical pre-deformation, the wave path can still be accurately controlled by semi-linear material. However, the wave modes will transform into quasiones. At the interface between un-deformed and pre-deformed hyperelastic material, both neo-hookean and semi-linear SEFs do not acquire the excellent feature of linear SEF in impedance matching. The effect of deformation mode on the impedance of such materials deserves a systematical study. This work may provide a necessary guideline for the material design. With the development of the modern chemical synthesis processes, soft material with proposed linear SEF may be synthesized in near future. Besides, as demonstrated in a recent work 10, microstructure design may be another route to conceive the novel hyperelastic metamaterials. Acknowledgement Support from the National Natural Science Foundation of China (Grant No. 11602294) is acknowledged. 10

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A Linear Strain Energy Function for Hyperelastic Transformation Method:Supplemental Material Linli Chen, Chao Ma, Zheng Chang a) College of Science, China Agricultural University, Beijing 100083 changzh@cau.edu.cn a) Author to whom correspondence should be addressed. Electronic mail: changzh@cau.edu.cn

1. Small-on-large theory and hyperelastic transformation method The theory of small-on-large is applied to investigate the problem of linear waves propagation in finitely deformed material. For a hyperelastic solid with the strain energy function W, the equilibrium equation of the finite deformation can be written as ( A U ) 0, (S1) ijkl l, k, i where U denotes the finite displacement, A W F F is the component of 2 ijkl ji lk the fourth-order elastic tensor expressed in the initial configuration, W denotes the strain energy function (SEF), and Fij xi X j the deformation gradient. Further, the incremental wave motion u superimposed on the finite deformation U is governed by 1 ( A u ) u, (S2) 0 ijkl l, k, i 0 j with a pushing forward operation on the elastic tensor A and the initial mass density 1 where J det( F ) is the volumetric ratio. A J F F A, J, (S3) 1 1 0 ijkl ii ' k' k ijkl 0 The governing equation of linear elastic waves has the same form as Eq.(S2), i.e. ( C u ) u, (S4) ijkl l, k, i j where C is linear elastic tensor. In the theory of transformation elastodynamics, Eq.(S4) preserves its form under asymmetric transformation relations 2 C J F ' F ' C, J, (S5) 1 1 ijkl ii k k ijkl 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 F ij x i 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 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) of the Letter. In other words, if Eq. (1) holds, a full analogy between the pushing forward formulation and the asymmetric transformation relations can be obtained, i.e. [ C,, F ] [ A,, F]. (S6) 0 0 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 the materials used for transformation devices can be implemented by such deformed hyperelastic solids without introducing any microstructures.

2. Elastic wave propagation in homogenously deformed hyperelastic material For a homogenously deformed hyperelastic material, incremental plane waves can be expressed in the form of 1 u i ikl jx jt Ae, (S7) i where A i is the wave amplitude, i denotes the imaginary unit, k is the wave number, l is the unit vector in the wave direction and is the angular frequency of the elastic waves. By inserting Eq.(S3) and Eq.(S7) into Eq.(S2), the Christoffel equation can be obtained as where m is a unit polarization vector, c C l l m c m (S8) 2 0ijkl i k l 0 j, k denotes the velocity of the wave. By solving the eigenvalue problem of Eq.(S8), we can obtain the phase velocities ( V and V S ) and polarization direction ( x i ) of the - and S-waves which propagate in the hyperelastic material.

3. Transmission and reflection of elastic waves at the interface between undeformed and deformed hyperelastic material At the interface between un-deformed and deformed hyperelastic material, the particle velocity and the stress which are caused by elastic waves must be continuous, i.e. 3 v I II i vi, (S9) (S10) I II ij ij, in which vi ui t is the particle velocity, and ij C0 ijklul, k is the stress. The superscripts I and II denote the un-deformed and deformed domain, respectively. In two-dimensional (in-plane) problems, consider an S-wave propagates alone the x1 -direction and thus polarizes in x2 -direction from the un-deformed domain to the deformed one. By inserting Eq.(S7) into Eq.(S8), we can obtain For a plane interface at x1 12 V. S v2 (S11) a, the particle velocity of the incident S-wave, and the stress caused by particle vibration at the interface of the un-deformed domain can be expressed as v I i 1 i 1 Ae kl a t I kl a t 2 12 A V e S inc i, i, (S12) in which the subscript inc indicates the incident wave, the subscript i 1, 2 denotes the component of the spatial coordinate x, and A is the amplitude of incident S-wave. Similarly, the particle velocity and the stress of the - and S-waves emerged during reflection and transmission at the interface between the un-deformed and the deformed areas can be expressed as I ikl1 a t I I ikl1 a t v1 Be 1 BVe R inc i, i, (S13) R

I ikl1 a t I I ikl1 a t v2 Ce 12 CV R R Se II ikl1a t II II ikl1 a t v1 De 1 D T T 0V e II ikl1a t II II ikl1 a t v2 Ee 12 E0VS e T i, i, (S14) i, i, (S15) i, i, (S16) where the subscripts R and T indicate the reflection transmission waves, the subscript i 1,2 denotes the polarization coordinate x, while B, C, D and E are the amplitudes of - and S-waves during reflection and transmission. As quasi-mode elastic waves may exist in the pre-deformed hyperelastic materials, it is necessary to introduce a polarization angle as the angle between the T polarization direction x i and the spatial direction x i. In this fashion, Eq.(S15) and Eq. (S16) can be expressed in the spatial coordinate x. Together with Eq. (S13) and (S14), Eq.(S9) can be written as B Dcos Esin, (S17) AC Dsin Ecos. (S18) Meanwhile, Eq.(S10) can be expressed as BV D V cos E V sin, (S19) I II II 0 0 S AV CV D V sin E V cos. (S20) I I II II S S 0 0 S By solving Eq.(S17) to Eq.(S20), the transmission and reflection coefficients m ns, can be obtained as S D 2V sin V V T A I II I S 0 S S E 2V cos V V TS A I I II S 0 S B 2 V sin cos V V R A I II II 0 S S,,, (S21) (S22) (S23)

cos S C VV V V sin VV VV VV V V RS A 2 I I 2 II II 2 I II I II 2 I II II I S 0 S 0 S S 0 S S, (S24) where = 2 VV I I 2 II II 2 I II I II 2 I II II I S 0VSV 0sin VV S S VV 0cos VV S VSV. For m, the superscript m, S denotes wave mode of the incident wave, the ns subscript n T, R denotes the transmission and reflection waves, while the subscript s, S denotes the wave-mode of the transmission or reflection waves. Similarly, for the -wave propagates alone the x1 -direction, the transmission and reflection coefficients can be obtained as R T TS 2V cos V V I II I 0 S S 2V sin V V I I II S 0,. cos VV V V sin VV VV V V VV 2 I I 2 II II 2 I II I II 2 II I I II S 0 S 0 S S 0 S S (S25) (S26), (S27) RS 2 V sin cos V V I II II 0 S, (S28)

4. 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. 4.1 Steady state analysis For steady state analysis, the second step is to simulate the linear elastic wave motion governed by ( A u ) u, (S29) 2 0 ijkl l, k, i 0 j in which u is independent of time. In this step, the weak form DE module is applied to deal with the asymmetry of the elastic tensor A 0. Additionally, the perfect matched layers 4 are applied to avoid the reflection on the boundaries of the computational domain. The divergence and curl of the displacement fields travel in the uniaxial tensioned hyperelastic domain are plotted in Fig.S2. The deformation gradient is set to be F11 10 7, F44 1 and F12 F21 0, corresponding to the elongation ratio 10 7. The sources of elastic waves are set at the left side of the model with the maximum amplitude as A 0.01 m, the angular frequencies being S 2kHz and 4kHz, respectively. The -wave lengths in the three deformed SEFs are almost identical. Moreover it is worthy to notice that the S-wave length in deformed semilinear material is different from that of the other two SEFs.

Fig. S1 Divergence and curl field of elastic waves in hyperelastic materials under uniaxial tension. (a), (b) linear SEF; (c), (d) neo-hookean SEF; (e)-(h) semi-linear SEF. (a), (c), (e), (h) -wave incidence; (b), (d), (f), (h) S-wave incidence. 4.2 Transient Analysis For transient analysis, the second step is to simulate the linear elastic wave propagation governed by Eq.(S2). Again, the weak form DE module is applied to deal with the asymmetry of the elastic tensor A 0. The transmission and reflection of elastic wave at the interface between the undeformed and the simple-sheared hyperelastic materials are shown in Fig. S2. With a good agreement with the theoretical predictions, total transmission can be observed in the case of linear SEF and neo-hookean SEF with both - and S-wave incidence, and semi-linear SEF with -wave incidence, as shown in Fig. S2 (a), (b), (c), (d) and (e), respectively. Minor reflection can be observed in the line art of Fig. S2 (f) in the case of semi-linear SEF with S-wave incidence, where S RS 0.016.

Fig. S2 The transmission and reflection of elastic wave at the interface between the stress-free and the simple-sheared deformed areas with different hyperelastic models. (a), (b) linear SEF. (c), (d) neo-hookean SEF. (e), (f) semi-linear SEF.

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