International Journal of Solids and Structures 126–127 (2017) 1–7 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr Elastic wave propagation in simple-sheared hyperelastic materials with different constitutive models ∗ Linli Chen, Zheng Chang , Taiyan Qin College of Science, China Agricultural University, Beijing 10 0 083, China a r t i c l e i n f o a b s t r a c t Article history: We investigate the elastic wave propagation in various hyperelastic materials which are subjected to Received 21 March 2017 simple-shear deformation. Two compressible types of three conventional hyperelastic models are con- Revised 17 July 2017 sidered. We found pure elastic wave modes that can be obtained in compressible neo-Hookean materials Available online 27 July 2017 constructed by adding a bulk strain energy term to the incompressible strain energy function. Mean- Keyword: while, for the compressible hyperelastic models which are reformulated into deviatoric and hydrostatic Elastic wave parts, only quasi modes can propagate, with abnormal ray directions that can be observed for longitudinal Hyperelastic waves. Moreover, the influences of material constants, material compressibility and external deformations Stain energy function on the elastic wave propagation and refraction in these hyperelastic models are systematically studied. Compressible Numerical simulations are carried out to validate the theoretical results. This investigation may open a promising route for the realization of next generation metamaterials and novel wave manipulation de- vices. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction even subversive, distinctions in the performance of the soft de- vices. Therefore, a comprehensive understanding of the mechanical Soft materials such as elastomers, gels and many biological tis- behaviors for different SEFs is essential, especially for applications sues usually exhibit rich and complex static and dynamic behav- where high-precision is demanded. iors when subjected to finite deformation. To describe the nonlin- Hyperelastic materials are usually considered to be incompress- ear mechanical behavior of such materials, hyperelasticity is usu- ible. However, compressible hyperelastic models are indispens- ally employed. Many hyperelastic models (or Strain Energy Func- able in considering the longitudinal wave motion in elastodynamic tions, SEFs) ( Simo and Pister, 1984; Arruda and Boyce, 1993; Gent, problems. There are a couple of approaches to extend an incom- 1996; Hartmann and Neff, 2003 ) are proposed based on experi- pressible hyperelastic model to a compressible form ( Boyce and Ar- mental data fitting ( Ogden, 1972; Yeoh, 1993; Boyce and Arruda, ruda, 2012 ). It has been demonstrated ( Ehlers and Eipper, 1998 ) 2012 ), and hereafter applied for theoretical and numerical anal- that these two versions of the SEFs have substantially differ- yses. Recently, soft materials with hyperelastic SEFs have drawn ent static behaviors, especially when the volume of the material considerable attention in the field of elastodynamics. In particular, changes significantly. Nevertheless, the way in which these SEFs by virtue of the high sensitivity to deformations and the remark- relate to their dynamic properties is still elusive. able capability of reversible structural instability of soft materials, Moreover, a previous study ( Chang et al., 2015 ) has proposed soft metamaterials or soft phononic crystals ( Bertoldi and Boyce, a feasible method to separate longitudinal and shear waves with a 2008; Auriault and Boutin, 2012; Wang and Bertoldi, 2012; Shim simple-sheared neo-Hookean solid. Considering the diversity of hy- et al., 2015 ) with tunable or adaptive properties have been demon- perelastic models, there remains an unmet need for understanding strated for wave applications. Moreover, a Hyperelastic Transforma- the capability of wave-mode separation for different hyperelastic tion theory ( Norris and Parnell, 2012; Parnell, 2012; Chang et al., models. 2015; Liu et al., 2016 ) has been reported, providing homogeneous To address the aforementioned issues, in this paper, we fo- soft materials with certain SEFs that can be utilized to manipulate cus on the propagation and refraction of elastic waves in simple- elastic wave paths. In all these works, it is generally revealed that sheared hyperelastic materials. In the framework of Small-on-Large subtle differences in hyperelastic models may cause significant, or theory ( Ogden, 2007 ), the dynamic behaviors of two different com- pressible types of the three conventional incompressible hyperelas- tic models are considered. We show that pure elastic wave modes ∗ Corresponding author. may propagate in a compressible neo-Hookean model which is E-mail address: [email protected] (Z. Chang). http://dx.doi.org/10.1016/j.ijsolstr.2017.07.027 0020-7683/© 2017 Elsevier Ltd. All rights reserved. 2 L. Chen et al. / International Journal of Solids and Structures 126–127 (2017) 1–7 constructed by adding a bulk strain energy term to an incompress- ( Ehlers and Eipper, 1998 ) ible SEF. For a compressible model reformulated from a conven- μ κ = − + 2, tional incompressible SEF into a deviatoric part and a hydrostatic W NH2 I¯1 2 ( ln J) (4) 2 2 part, only quasi wave modes exist. When considering the refraction κ = λ + μ of elastic waves which are normally incident on a plane interface where is the 2-D bulk modulus. from an un-deformed hyperelastic material to a pre-deformed one, Similarly, two typical compressible forms of the Arruda–Boyce significant differences can be observed in the refraction angle of model can also be found in previous contributions ( Boyce and Ar- ruda, 2012; Kaliske and Rothert, 1997 ), which can be written as the longitudinal wave between the two versions of SEFs. Both the- oretical analysis and numerical simulations are carried out to con- 1 1 2 11 3 firm each other. The paper is organized as follows: in Section 2 , the W = C ( I − 2) + (I − 4) + (I − 8) AB1 1 2 1 20 N 1 1050 N 2 1 approaches to extend the incompressible SEFs to the compressible ones are briefly reviewed; the Small-on-Large theory, which de- 19 4 519 5 + (I − 16) + (I − 32) scribes linear wave motion propagation in a finitely deformed hy- 70 0 0 N 3 1 673750 N 4 1 perelastic material, is reviewed in Section 3 . Moreover, the behav- λ ior of elastic wave propagation and refraction in a simple-sheared − μ ln (J) + (J − 1) 2, (5) hyperelastic material with different SEFs is shown in Section 4 , 2 with numerical validations illustrated in Section 5 . Finally, a dis- and cussion on our results and on the avenues for future work is pro- vided in Section 6 . 1 1 2 11 3 W = C ( I¯ − 2) + ( I¯ − 4) + ( I¯ − 8) AB2 1 2 1 20 N 1 1050 N 2 1 2. Strain energy functions for compressible hyperelastic 19 4 519 5 + ( I¯ − 16) + ( I¯ − 32) materials 70 0 0 N 3 1 673750 N 4 1 In this section, we briefly review the two approaches λ + μ J 2 − 1 + − ln J , (6) ( Boyce and Arruda, 2012 ) for expanding an incompressible SEF into 2 2 a compressible one. = μ/ ( + / + / 2 + / 3 + / 4) Approach I involves adding a bulk strain energy term WB to an where C1 1 2 5N 44 175N 152 875N 834 67375N existing incompressible isotropic SEF WC . Thus, the compressible and N is a material constant that denotes a measure of the limiting SEF can be expressed as network stretch. Note that the form of C1 used here is only for 2- D SEFs. When N → ∞ , the Arruda–Boyce model degenerates to the = ( , , ) + ( ), W1 WC I1 I2 J WB J (1) neo-Hookean model. Another commonly utilized hyperelastic model is the Gent where I1 and I2 are the first and second invariant of the right Cauchy–Green tensor, respectively. J = det (F ) is the volumetric ra- model (Gent, 1996). Here we only consider its compressible form = ∂ ∂ as constructed by Approach I, which can be referred from an ear- tio. Fij xi / Xj denotes the deformation gradient, in which Xj and x are the coordinates in the initial and the current configurations, lier work (Bertoldi and Boyce, 2008), namely i respectively. μ − λ μ Jm I1 2 2 Approach II involves applying a multiplicative decomposition on W G1 = − ln 1 − − μ ln J + − (J − 1) , (7) 2 J m 2 J m the Cauchy–Green deformation tensor, and reformulating the in- compressible SEF into a deviatoric one. A hydrostatic strain energy where J m is a material constant that is related to the strain satura- term WH is then added to extend the SEF into a compressible form, tion of the material. Similar to the Arruda–Boyce model, the Gent namely model also degenerates to the neo-Hookean model when J → ∞ . m W 2 = W D I¯ 1 , I¯ 2 + W H ( J ), (2) 3. Small-on-Large wave motion in hyperelastic materials −1 −2 where I¯ 1 = J I 1 and I¯ 2 = J I 2 are the invariants of the deviatoric stretch tensor. The Small-on-Large theory provides an ideal platform for in- In Eqs. (1) and (2), the bulk stain energy terms WB or WH vestigating linear wave propagation in finite-deformed hypere- = can be selected from some empirical formulas, such as WH1 (J) lastic materials. For a hyperelastic solid with a certain SEF, the 2 = 2 − − = − 2 B(ln J) /2, WH2 (J) B((J 1)/2 ln J)/2, WH3 (J) B(J 1) /2, and equilibrium equation of the finite deformation can be written as = α − − α2 WH4 (J) B{cosh [ (J 1)] 1}/ , which are provided in former (Ogden, 2007) literature ( Bischoff et al., 2001; Hartmann and Neff, 2003 ). In these ( A U , ) , = 0 , (8) formulas, B and α are material parameters, which can be deter- ijkl l k i mined from the conditions ( Simo and Pister, 1984 ) of W ( F = I ) = ∂ 2 ∂ ∂ where Ui denotes the displacement, Aijkl W/ Fji Flk are the = ∂ ( = )/ ∂ = ∂ 2 ( = )/ ∂ ∂ = λδ δ + μδ δ + 0, W F I F 0 and W F I F F ij kl ik jl components of the fourth-order elastic tensor expressed in the ini- μδ δ il jk .