zbMATH — the first resource for mathematics

Dominant negative Poynting effect in simple shearing of soft tissues. (English) Zbl 1360.74110
Summary: We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.

74L15 Biomechanical solid mechanics
Full Text: DOI
[1] British Standard BS ISO 8013:2006 Rubber, vulcanized—determination of creep in compression or shear · Zbl 1023.74033
[2] Poynting, JH, On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted, Proc R Soc Lond Ser A, 82, 546-559, (1909) · JFM 40.0875.02
[3] Rivlin, RS, Large elastic deformation of isotropic materials iv: further developments of the general theory, Philos Trans R Soc Lond Ser A, 241, 379-397, (1948) · Zbl 0031.42602
[4] Mihai, LA; Goriely, A, Positive or negative Poynting effect? the role of adscititious inequalities in hyperelastic materials, Proc R Soc Lond A, 467, 3633-3646, (2011) · Zbl 1243.74011
[5] Destrade, M; Murphy, JG; Saccomandi, G, Simple shear is not so simple, Int J Non-Linear Mech, 47, 210-214, (2012)
[6] Horgan, CO; Smayda, M, The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials, Mech Mater, 51, 43-52, (2012)
[7] Janmey, PA; McCormick, ME; Rammensee, S; Leight, JL; Georges, PC; MacKintosh, FC, Negative normal stress in semiflexible biopolymer gels, Nat Mater, 6, 48-51, (2007)
[8] Destrade, M; Gilchrist, MD; Motherway, J; Murphy, JG, Slight compressibility and sensitivity to changes in poisson’s ratio, Int J Numer Methods Eng, 90, 403-411, (2012) · Zbl 1242.74006
[9] Horgan, CO; Murphy, JG, On the normal stresses in simple shearing of fiber-reinforced nonlinearly elastic materials, J Elast, 104, 343-355, (2011) · Zbl 1269.74023
[10] Wu, MS; Kirchner, HOK, Nonlinear elasticity modeling of biogels, J Mech Phys Solids, 58, 300-310, (2010) · Zbl 1193.74095
[11] Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In Spencer AJM (ed) Continuum theory of the mechanics of fibre-reinforced composites. CISM Courses and Lectures Series No. 282. Springer-Verlag, Vienna
[12] Murphy, JG, Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants, Eur J Mech A/Solids, 42, 90-96, (2013) · Zbl 1406.74501
[13] Dokos, S; Smaill, BH; Young, AA; LeGrice, IJ, Shear properties of passive ventricular myocardium, Am J Physiol Heart Circ Physiol, 283, h2650-h2659, (2002)
[14] Saccomandi, G; Beatty, MF, Universal relations for fiber-reinforced elastic materials, Math Mech Solids, 7, 95-110, (2002) · Zbl 1032.74006
[15] Ogden RW (2003) Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue. In Biomechanics of soft tissue in cardiovascular systems. CISM Courses and Lectures Series No. 441. Springer, Vienna, pp 65-108 · Zbl 1151.74386
[16] Merodio, J; Ogden, RW, Mechanical response of fiber-reinforced incompressible non-linearly elastic solids, Int J Non-Linear Mech, 40, 213-227, (2005) · Zbl 1349.74057
[17] Vergori, L; Destrade, M; McGarry, P; Ogden, RW, On anisotropic elasticity and questions concerning its finite element implementation, Comput Mech, 52, 1185-1197, (2013) · Zbl 1388.74015
[18] Gennisson, J-L; Catheline, S; Chaffaõ, S; Fink, M, Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles, J Acoust Soc Am, 114, 536-541, (2003)
[19] Papazoglou, S; Rump, J; Braun, J; Sack, I, Shear wave group velocity inversion in MR elastography of human skeletal muscle, Magn Reson Med, 56, 489-497, (2006)
[20] Sinkus, R; Tanter, M; Catheline, S; Lorenzen, J; Kuhl, C; Sondermann, E; Fink, M, Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography, Magn Reson Med, 53, 372-387, (2005)
[21] Morrow, DA; Haut Donahue, TL; Odegard, GM; Kaufman, KR, Transversely isotropic tensile material properties of skeletal muscle tissue, J Mech Behav Biomed Mater, 3, 124-129, (2010)
[22] Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed) Encyclopedia of Physics, vol III/3, 3rd edn. Springer-Verlag, Berlin · Zbl 0779.73004
[23] Beatty, MF, Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissue, Appl Mech Rev, 40, 1699-1734, (1989)
[24] Feng, Y; Okamoto, RJ; Namani, R; Genin, GM; Bayly, PV, Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter, J Mech Behav Biomed Mater, 23, 117-132, (2013)
[25] Holzapfel, GA; Gasser, TC; Ogden, RW, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J Elast, 61, 1-48, (2000) · Zbl 1023.74033
[26] Le Tallec P (1994) Numerical methods for nonlinear three-dimensional elasticity. In: Ciarlet PG, Lions JL (eds) Handbook of Numerical Analysis, vol III. Elsevier, Amsterdam · Zbl 0875.73234
[27] ADINA R&D Inc (2005) ADINA theory and modeling guide. ADINA R&D Inc, Watertown · Zbl 0031.42602
[28] ARES Rheometer manual (2006) Rheometrics series user manual. Revision J, TA Instrument-Waters LLC, New Castle
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.