On worm-like chain models within the three-dimensional continuum mechanics framework. (English) Zbl 1149.74312
Summary: In this paper, we review critically some basic models derived from the statistics of long-chain molecules and then discuss the status of such models within the three-dimensional nonlinear theory of elasticity. We draw attention to some deficiencies of certain worm-like chain (WLC) models when viewed within the three-dimensional continuum mechanics framework. Modifications of the WLC models motivated by such considerations and avoiding these deficiencies are then discussed and shown to correspond well with data generated by the exact WLC model.
MSC:
| 74B20 | Nonlinear elasticity |
| 74A25 | Molecular, statistical, and kinetic theories in solid mechanics |
| 74L15 | Biomechanical solid mechanics |
| 92C40 | Biochemistry, molecular biology |
| 92D10 | Genetics and epigenetics |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
References:
| [1] | Arruda, E.M. & Boyce, M.C. 1993 A three dimensional constitutive model for the large deformation stretch behavior of rubber elastic materials. <i>J. Mech. Phys. Solids</i> <b>41</b>, 389–412, (doi:10.1016/0022-5096(93)90013-6). |
| [2] | Bao, G. & Suresh, S. 2003 Cell and molecular mechanics of biological materials. <i>Nat. Mater.</i> <b>2</b>, 715–725, (doi:10.1038/nmat1001). |
| [3] | Baumann, C.G., Bloomfield, V.A., Smith, S.B., Bustamante, C., Wang, M.D. & Block, S.M. 2000 Stretching of single collapsed DNA molecules. <i>Biophys. J.</i> <b>78</b>, 1965–1978. |
| [4] | Beatty, M.F. 2003 An average-stretch full-network model for rubber elasticity. <i>J. Elasticity</i> <b>70</b>, 65–86, (doi:10.1023/B:ELAS.0000005553.38563.91). · Zbl 1073.74013 |
| [5] | Bischoff, J.E., Arruda, E.M. & Grosh, K. 2002 Orthotropic hyperelasticity in terms of an arbitrary molecular chain model. <i>J. Appl. Mech.</i> <b>69</b>, 198–201, (doi:10.1115/1.1432664). · Zbl 1110.74345 |
| [6] | Bischoff, J.E., Arruda, E.M. & Grosh, K. 2002 A microstructurally based orthotropic hyperelastic constitutive law. <i>J. Appl. Mech.</i> <b>69</b>, 570–579, (doi:10.1115/1.1485754). · Zbl 1110.74344 |
| [7] | Bouchiat, C., Wang, M.D., Allemand, J.-F., Strick, T., Block, S.M. & Croquette, V. 1999 Estimating the persistence length of a worm-like chain molecule from force-extension measurement. <i>Biophys. J.</i> <b>76</b>, 409–413. |
| [8] | Boyce, M.C. 1996 Direct comparison of the Gent and Arruda–Boyce constitutive models. <i>Rubber Chem. Technol.</i> <b>69</b>, 781–785. |
| [9] | Boyce, M.C. & Arruda, E.M. 2000 Constitutive models of rubber elasticity: a review. <i>Rubber Chem. Technol.</i> <b>73</b>, 504–523. |
| [10] | de Gennes, P.G. 1979 Scaling concepts in polymer physics. Ithaca, NY: Cornell University Press. |
| [11] | Demiray, H. 1972 A note on the elasticity of soft biological tissues. <i>J. Biomech.</i> <b>5</b>, 309–311, (doi:10.1016/0021-9290(72)90047-4). |
| [12] | Erman, B. 2004 Molecular aspects of rubber elasticity. <i>CISM Courses and Lectures</i><i>Mechanics and thermomechanics of rubberlike solids</i> (eds. Saccomandi, G. & Ogden, R.W.), vol. 452. pp. 63–89, Wien: Springer |
| [13] | Flory, P.J. 1988 Statistical mechanics of chain molecules. Munich: Hanser Publishers. |
| [14] | Fung, Y.C. 1967 Elasticity of soft tissue in simple elongation. <i>Am. J. Physiol.</i> <b>213</b>, 1532–1544. |
| [15] | Gent, A.N. 1996 A new constitutive relation for rubber. <i>Rubber Chem. Technol.</i> <b>69</b>, 59–61. |
| [16] | Gent, A.N. & Thomas, A.G. 1958 Forms for the stored, strain energy function for vulcanized rubber. <i>J. Polym. Sci.</i> <b>28</b>, 625–628, (doi:10.1002/pol.1958.1202811814). |
| [17] | Gosline, J., Lillie, M., Carrington, E., Gurette, P., Ortlepp, C. & Savage, K. 2002 Elastic proteins: biological roles and mechanical properties. <i>Phil. Trans. R. Soc. B</i> <b>357</b>, 121–132, (doi:10.1098/rstb.2001.1022). |
| [18] | 2003 <i>CISM Courses and Lectures</i><i>Biomechanics of soft tissues in cardiovascular systems</i>, vol. 441. Wien: Springer |
| [19] | Horgan, C.O. & Saccomandi, G. 1999 Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. <i>J. Elasticity</i> <b>56</b>, 159–170, (doi:10.1023/A:1007606909163). |
| [20] | Horgan, C.O. & Saccomandi, G. 2002 A molecular-statistical basis for the Gent model of rubber elasticity. <i>J. Elasticity</i> <b>68</b>, 167–176, (doi:10.1023/A:1026029111723). |
| [21] | Horgan, C.O. & Saccomandi, G. 2002 Constitutive modelling of rubber-like and biological materials with limiting chain extensibility. <i>Math. Mech. Solids</i> <b>7</b>, 353–371. · Zbl 1063.74012 |
| [22] | Horgan, C.O. & Saccomandi, G. 2003 Finite thermoelasticity with limiting chain extensibility. <i>J. Mech. Phys. Solids</i> <b>51</b>, 1127–1146, (doi:10.1016/S0022-5096(02)00144-8). |
| [23] | Humphrey, J.D. 2002 Cardiovascular solid mechanics. New York: Springer. |
| [24] | Kratky, O. & Porod, G. 1949 Röntgenuntersushung gelöster Fadenmoleküle. <i>Recueil des Travaux Chimiques des Pays Bas–J. R. Neth. Chem. Soc.</i> <b>68</b>, 1106–1123. |
| [25] | Kuhl, E., Garikipati, K., Arruda, E.M. & Grosh, K. 2004 Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. <i>J. Mech. Phys. Solids</i> <b>53</b>, 1552–1573, (doi:10.1016/j.jmps.2005.03.002). · Zbl 1120.74635 |
| [26] | Kuhl, E. Menzel, A. & Garikipati, K. In press. On the convexity of transversely isotropic chain network models. <i>Phil. Mag.</i> |
| [27] | Marko, J.F. & Siggia, E.D. 1995 Stretching DNA. <i>Macromolecules</i> <b>28</b>, 8759–8770, (doi:10.1021/ma00130a008). |
| [28] | Miehe, C., Göktepe, S. & Lulei, F. 2004 A micro-macro approach to rubber-like materials–Part I: the non-affine micro-sphere model. <i>J. Mech. Phys. Solids</i> <b>52</b>, 2617–2660, (doi:10.1016/j.jmps.2004.03.011). · Zbl 1091.74008 |
| [29] | Ogden, R.W. 1984 Non-linear elastic deformations. Chichester: Ellis Horwood. · Zbl 0541.73044 |
| [30] | Ogden, R.W., Saccomandi, G. & Sgura, I. 2004 Fitting hyperelastic models to experimental data. <i>Comput. Mech.</i> <b>34</b>, 484–502, (doi:10.1007/s00466-004-0593-y). · Zbl 1109.74320 |
| [31] | Ogden, R. W. Saccomandi, G. & Sgura, I. Submitted. Computational aspects of the worm-like-chain interpolation formulas. · Zbl 1190.74018 |
| [32] | Pucci, E. & Saccomandi, G. 2002 A note on the Gent model for rubber-like materials. <i>Rubber Chem. Technol.</i> <b>75</b>, 839–851. |
| [33] | Rivlin, R.S. 1960 Some topics in finite elasticity. <i>Proc. First Naval Symp. on Structural Mechanics</i> (ed. Perrone, N.), pp. 169–198, Oxford: Pergamon Press |
| [34] | Rosa, A., Hoang, T.X., Marenduzzo, D. & Maritan, A. 2005 A new interpolation formula for semiflexible polymers. <i>Biophys. Chem.</i> <b>115</b>, 251–254, (doi:10.1016/j.bpc.2004.12.030). |
| [35] | Sun, Y.-L., Luo, Z.-P., Fertala, A. & An, K.-N. 2004 Stretching type II collagen with optical tweezers. <i>J. Biomech.</i> <b>37</b>, 1665–1669, (doi:10.1016/j.jbiomech.2004.02.028). |
| [36] | Urry, D.W., Hugel, T., Seitz, M., Gaub, H.E., Sheiba, L., Dea, J., Xu, J. & Parker, T. 2002 Elastin: a representative ideal protein elastomer. <i>Phil. Trans. R. Soc. B</i> <b>357</b>, 169–184, (doi:10.1098/rstb.2001.1023). |
| [37] | White, J.H. & Bauer, W.R. 2004 Finite-element analysis of the displacement of closed DNA loops under torsional stress. <i>Phil. Trans. R. Soc. A</i> <b>362</b>, 1335–1353, (doi:10.1098/rsta.2004.1379). · Zbl 1091.92009 |
| [38] | Wineman, A. 2005 Some results for generalized neo-Hookean elastic materials. <i>Int. J. Non-Linear Mech.</i> <b>40</b>, 271–279, (doi:10.1016/j.ijnonlinmec.2004.05.007). · Zbl 1349.74064 |
| [39] | Saccomandi, G. 2004 Phenomenology of rubber-like materials. <i>CISM Courses and Lectures</i><i>Mechanics and thermomechanics of rubberlike solids</i> (eds. Saccomandi, G. & Ogden, R.W.), vol. 452. pp. 91–134, Wien: Springer |
| [40] | 2004 <i>CISM Courses and Lectures</i><i>Mechanics and thermomechanics of rubberlike solids</i>, vol. 452. Wien: Springer |
| [41] | Shaqfeh, E.S. G. 2005 The dynamics of single-molecule DNA in flow. <i>J. Non-Newton. Fluid Mech.</i> <b>130</b>, 1–28, (doi:10.1016/j.jnnfm.2005.05.011). |
| [42] | Strick, T.R., Dessinger, M.-N., Charvin, G., Dekker, N.H., Allemand, J.-F., Bensimon, D. & Croquette, V. 2003 Stretching of macromolecules and proteins. <i>Rep. Progr. Phys.</i> <b>66</b>, 1–45, (doi:10.1088/0034-4885/66/1/201). |
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