Shear shock evolution in incompressible soft solids. (English) Zbl 1481.74411
Summary: Nonlinear evolution of shear waves into shocks in incompressible elastic materials is investigated using the framework of large deformation elastodynamics, for a family of loadings and commonly used hyperelastic material models. Closed form expressions for the shock formation distance are derived and used to construct non-dimensional phase maps that determine regimes in which a shock can be realized. These maps reveal the sensitivity of shock evolution to the amplitude, shape, and ramp time of the loading, and to the elastic material parameters. In light of a recent study, which hypothesizes that shear shock formation could play a significant role in Traumatic Brain Injury (TBI), application to brain tissue is considered and it is shown that the size matters in TBI research. Namely, for realistic loadings, smaller brains are less susceptible to formation of shear shocks. Furthermore, given the observed sensitivity to the imparted waveform and the constitutive properties, it is suggested that the non-dimensional maps can guide the design of protective structures by determining the combination of loading parameters, material dimensions, and elastic properties that can avoid shock formation.
MSC:
| 74J40 | Shocks and related discontinuities in solid mechanics |
References:
| [1] | Aboudi, J.; Benveniste, Y., One-dimensional finite amplitude wave propagation in a compressible elastic half-space, Int. J. Solids Struct., 9, 3, 363-378 (1973) · Zbl 0266.73015 |
| [2] | Aboudi, J.; Benveniste, Y., Finite amplitude one-dimensional wave propagation in a thermoelastic half-space, Int. J. Solids Struct., 10, 3, 293-308 (1974) · Zbl 0274.73019 |
| [3] | Bland, D., On shock structure in a solid, IMA J. Appl. Math., 1, 1, 56-75 (1965) |
| [4] | Budday, S.; Sommer, G.; Birkl, C.; Langkammer, C.; Haybaeck, J.; Kohnert, J.; Bauer, M.; Paulsen, F.; Steinmann, P.; Kuhl, E., Mechanical characterization of human brain tissue, Acta Biomater., 48, 319-340 (2017) |
| [5] | Carroll, M., Plane circular shearing of incompressible fluids and solids, Q. J. Mech. Appl.Math., 30, 2, 223-234 (1977) · Zbl 0361.73005 |
| [6] | Carroll, M., Reflection and transmission of circularly polarized elastic waves of finite amplitude, J. Appl. Mech., 46, 4, 867-872 (1979) · Zbl 0419.73023 |
| [7] | Carroll, M. M., Some results on finite amplitude elastic waves, Acta Mech., 3, 2, 167-181 (1967) |
| [8] | Carroll, M. M., Oscillatory shearing of nonlinearly elastic solids, Z. Angew. Math.Phys. ZAMP, 25, 1, 83-88 (1974) · Zbl 0291.73027 |
| [9] | Carroll, M. M., Plane elastic standing waves of finite amplitude, J. Elast., 7, 4, 411-424 (1977) · Zbl 0362.73016 |
| [10] | Catheline, S.; Gennisson, J.-L.; Tanter, M.; Fink, M., Observation of shock transverse waves in elastic media, Phys. Rev. Lett., 91, 16, 164301 (2003) |
| [11] | Chu, B. T., Finite amplitude waves in incompressible perfectly elastic materials, J. Mech. Phys. Solids, 12, 1, 45-57 (1964) · Zbl 0123.41403 |
| [12] | Chu, B. T., Transverse shock waves in incompressible elastic solids, J. Mech. Phys. Solids, 15, 1, 1-14 (1967) |
| [13] | Collins, W., One-dimensional non-linear wave propagation in incompressible elastic materials, Q. J. Mech. Appl.Math., 19, 3, 259-328 (1966) · Zbl 0146.22301 |
| [14] | Collins, W., The propagation and interaction of one-dimensional non-linear waves in an incompressible isotropic elastic half-space, Q. J. Mech. Appl.Math., 20, 4, 429-452 (1967) · Zbl 0152.44205 |
| [15] | Davison, L., Propagation of plane waves of finite amplitude in elastic solids, J. Mech. Phys. Solids, 14, 5, 249-270 (1966) |
| [16] | Destrade, M.; Murphy, J. G.; Saccomandi, G., Simple shear is not so simple, Int. J. Non Linear Mech., 47, 2, 210-214 (2012) |
| [17] | Destrade, M.; Pucci, E.; Saccomandi, G., Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids. proceedings of the royal society a, Math. Phys. Eng. Sci., 475, 2227, 20190061 (2019) · Zbl 1472.74043 |
| [18] | Destrade, M.; Saccomandi, G., Finite amplitude elastic waves propagating in compressible solids, Phys. Rev. E, 72, 016620 (2005) |
| [19] | Destrade, M.; Saccomandi, G.; Sgura, I., Methodical fitting for mathematical models of rubber-like materials. proceedings of the royal society a: mathematical, Phys. Eng. Sci., 473, 2198, 20160811 (2017) · Zbl 1404.74016 |
| [20] | Espíndola, D.; Lee, S.; Pinton, G., Shear shock waves observed in the brain, Phys. Rev. Appl., 8, 4, 044024 (2017) |
| [21] | Fung, Y., Biomechanics: Mechanical Properties of Living Tissues. Biomechanics (1993), Springer New York |
| [22] | Gent, A., A new constitutive relation for rubber, Rubber Chem. Technol., 69, 1, 59-61 (1996) |
| [23] | Hamilton, M. F.; Blackstock, D. T., Nonlinear Acoustics, Vol. 237 (1998), Academic press San Diego |
| [24] | Hamilton, M. F.; Ilinskii, Y. A.; Zabolotskaya, E. A., Separation of compressibility and shear deformation in the elastic energy density (l), J. Acoust. Soc. Am., 116, 1, 41-44 (2004) |
| [25] | Horgan, C. O., The remarkable gent constitutive model for hyperelastic materials, Int. J. Non Linear Mech., 68, 9-16 (2015) |
| [26] | Horgan, C. O.; Murphy, J. G., Simple shearing of soft biological tissues, Proceedings of the Royal Society of London A: Mathematical Physical and Engineering Sciences. Vol. 467. The Royal Society, 760-777 (2011) · Zbl 1428.74156 |
| [27] | Horgan, C. O.; Murphy, J. G., A boundary-layer approach to stress analysis in the simple shearing of rubber blocks, Rubber Chem. Technol., 85, 1, 108-119 (2012) |
| [28] | Lee-Bapty, I.; Crighton, D., Nonlinear wave motion governed by the modified burgers equation, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 173-209 (1987) · Zbl 0643.35091 |
| [29] | Marchand, A.; Duffy, J., An experimental study of the formation process of adiabatic shear bands in a structural steel, J. Mech. Phys. Solids, 36, 3, 251-283 (1988) |
| [30] | Mercier, S.; Molinari, A., Steady-state shear band propagation under dynamic conditions, J. Mech. Phys. Solids, 46, 8, 1463-1495 (1998) · Zbl 0955.74011 |
| [31] | Mihai, L. A.; Budday, S.; Holzapfel, G. A.; Kuhl, E.; Goriely, A., A family of hyperelastic models for human brain tissue, J. Mech. Phys. Solids, 106, 60-79 (2017) |
| [32] | Mihai, L. A.; Chin, L.; Janmey, P. A.; Goriely, A., A comparison of hyperelastic constitutive models applicable to brain and fat tissues, J. R. Soc. Interface, 12, 110, 20150486 (2015) |
| [33] | Mooney, M., A theory of large elastic deformation, J. Appl. Phys., 11, 9, 582-592 (1940) · JFM 66.1021.04 |
| [34] | Ogden, R.; Saccomandi, G.; Sgura, I., Fitting hyperelastic models to experimental data, Comput. Mech., 34, 6, 484-502 (2004) · Zbl 1109.74320 |
| [35] | Ogden, R. W., Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond. A, 326, 1567, 565-584 (1972) · Zbl 0257.73034 |
| [36] | Pucci, E.; Saccomandi, G., A note on the gent model for rubber-like materials, Rubber Chem. Technol., 75, 5, 839-852 (2002) |
| [37] | Pucci, E.; Saccomandi, G.; Vergori, L., Linearly polarized waves of finite amplitude in pre-strained elastic materials, Proc. R. Soc. A, 475, 2226, 20180891 (2019) · Zbl 1472.74108 |
| [38] | Rivlin, R., Large elastic deformations of isotropic materials iv. further developments of the general theory, Phil. Trans. R. Soc. Lond. A, 241, 835, 379-397 (1948) · Zbl 0031.42602 |
| [39] | Saccomandi, G.; Vergori, L., Generalised Mooney-Rivlin models for brain tissue: a theoretical perspective, Int. J. Non Linear Mech., 109, 9-14 (2019) |
| [40] | Saccomandi, G.; Vitolo, R., On the mathematical and geometrical structure of the determining equations for shear waves in nonlinear isotropic incompressible elastodynamics, J. Math. Phys., 55, 8, 081502 (2014) · Zbl 1366.74031 |
| [41] | Storm, C.; Pastore, J. J.; MacKintosh, F. C.; Lubensky, T. C.; Janmey, P. A., Nonlinear elasticity in biological gels, Nature, 435, 7039, 191 (2005) |
| [42] | Wochner, M. S.; Hamilton, M. F.; Ilinskii, Y. A.; Zabolotskaya, E. A., Cubic nonlinearity in shear wave beams with different polarizations, J. Acoust. Soc. Am., 123, 5, 2488-2495 (2008) |
| [43] | Zabolotskaya, E., Sound beams in a nonlinear isotropic solid, Sov. Phys. Acoust., 32, 4, 296-299 (1986) |
| [44] | Zabolotskaya, E. A.; Hamilton, M. F.; Ilinskii, Y. A.; Meegan, G. D., Modeling of nonlinear shear waves in soft solids, J. Acoust. Soc. Am., 116, 5, 2807-2813 (2004) |
| [45] | Ziv, R.; Shmuel, G., Smooth waves and shocks of finite amplitude in soft materials, Mech. Mater., 135, 67-76 (2019) |
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