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Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. (English) Zbl 1346.74128

Summary: A model for the mechanics of woven fabrics is developed in the framework of two-dimensional elastic surface theory. Thickness effects are modeled indirectly in terms of appropriate constitutive equations. The model accounts for the strain of the fabric and additional effects associated with the normal bending, geodesic bending, and twisting of the constituent fibers.

MSC:

74K35 Thin films

References:

[1] Warren, W.E.: The elastic properties of woven polymeric fabric. Polym. Eng. Sci. 30, 1309-1313 (1990) · doi:10.1002/pen.760302008
[2] Wang, W.-B., Pipkin, A.C.: Inextensible networks with bending stiffness. Q. J. Mech. Appl. Math. 39, 343-359 (1986) · Zbl 0594.73047 · doi:10.1093/qjmam/39.3.343
[3] Wang, W.-B., Pipkin, A.C.: Plane deformations of nets with bending stiffness. Acta Mech. 65, 263-279 (1986) · Zbl 0602.73034 · doi:10.1007/BF01176886
[4] Pipkin, A.C.: Some developments in the theory of inextensible networks. Q. Appl. Math. 38, 343 (1980) · Zbl 0484.73009
[5] Pipkin, A.C.: Plane traction problems for inextensible networks. Q. J. Mech. Appl. Math. 34, 415 (1981) · Zbl 0466.73037 · doi:10.1093/qjmam/34.4.415
[6] Pipkin, A.C.: Equilibrium of Tchebychev nets. Arch. Ration. Mech. Anal. 85, 81 (1984) · Zbl 0556.73046 · doi:10.1007/BF00250867
[7] Luo, C.; Steigmann, DJ; Durban, D. (ed.); etal., Bending and twisting effects in the three-dimensional finite deformations of an inextensible network, 213-228 (2001), London
[8] Luo, C.: Nonlinear three-dimensional mechanics of fabrics. Dissertation, UC Berkeley (2000) · Zbl 1215.74016
[9] Indelicato, G.: The influence of the twist of individual fibers in 2D fibered networks. Int. J. Solids Struct. 46, 912-922 (2009) · Zbl 1215.74016 · doi:10.1016/j.ijsolstr.2008.10.002
[10] Steigmann, D.J., Pipkin, A.C.: Equilibrium of elastic nets. Phil. Trans. R. Soc. Lond. A335, 419-454 (1991) · Zbl 0734.73098 · doi:10.1098/rsta.1991.0056
[11] Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1-23 (1992) · Zbl 0762.01012 · doi:10.1007/BF00379680
[12] Steigmann, D.J.: Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int. J. Non-linear Mech. 47, 734-742 (2012) · Zbl 0424.73063
[13] Steigmann, D.J.: Effects of fiber bending and twisting resistance on the mechanics of fiber-reinforced elastomers. In: Dorfmann, L., Ogden, R.W. (eds.) CISM Course on Nonlinear Mechanics of Soft Fibrous Tissues. Springer, Wien and New York (in press) · Zbl 0734.73098
[14] Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78-79, 3-201 (2005) · Zbl 1086.74001
[15] Lovelock, D., Rund, H.: Tensors, Differ. Forms Var. Princ. Dover, New York (1989)
[16] Coleman, B.D.: Necking and drawing in polymeric fibers under tension. Arch. Ration. Mech. Anal. 83, 115-137 (1983) · Zbl 0535.73016 · doi:10.1007/BF00282158
[17] Coleman, B.D., Newman, D.C.: On the rheology of cold drawing: I. elastic materials. J. Polym. Sci. B: Polym. Phys. 26, 1801-1822 (1988) · doi:10.1002/polb.1988.090260901
[18] dell’Isola, F., Steigmann, D.J.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118, 113-125 (2015) · Zbl 1305.74024
[19] Spencer, A.J.M., Soldatos, K.P.: Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int. J. Non-Linear Mech. 42, 355-368 (2007) · doi:10.1016/j.ijnonlinmec.2007.02.015
[20] Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Arch. Ration. Mech. Anal. 72, 61-98 (1979) · Zbl 0424.73063 · doi:10.1007/BF00250737
[21] Indelicato, G., Albano, A.: Symmetry properties of the elastic energy of a woven fabric with bending and twisting resistance. J. Elast. 94, 33-54 (2009) · Zbl 1159.74306 · doi:10.1007/s10659-008-9183-z
[22] Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135-174 (1981) · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9
[23] Hilgers, M.G., Pipkin, A.C.: The Graves condition for variational problems of arbitrary order. IMA. J. Appl. Math. 48, 265-269 (1992) · Zbl 0767.49004 · doi:10.1093/imamat/48.3.265
[24] Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. Lond. A455, 437-474 (1999) · Zbl 0926.74016 · doi:10.1098/rspa.1999.0320
[25] Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85-112 (1964) · Zbl 0131.22001 · doi:10.1007/BF00253050
[26] Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415-448 (1962) · Zbl 0112.38906 · doi:10.1007/BF00253946
[27] Koiter, W.T.: Couple-stresses in the theory of elasticity. Proc. Knononklijke Nederl. Akad. van Wetensch. B67, 17-44 (1964) · Zbl 0124.17405
[28] Germain, P.: The method of virtual power in continuum mechanics, part 2: Microstructure. SIAM J. Appl. Math. 25, 556-575 (1973) · Zbl 0273.73061 · doi:10.1137/0125053
[29] dell’Isola, F., Seppecher, P., Madeo, A.: Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: A generalization of the Cauchy tetrahedron argument. Variational Models and Methods in Solid and Fluid Mechanics CISM Courses and Lectures, vol. 535, 17-106 (2012) · Zbl 1451.74046
[30] dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: Approach “à la D’Alembert”. Z. Angew. Math. Phys. 63, 1119-1141 (2012) · Zbl 1330.76016 · doi:10.1007/s00033-012-0197-9
[31] Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Z. Angew. Math. Phys. 65, 587-612 (2013) · Zbl 1302.74008
[32] Cusick, G.E.: The response of fabric to shearing forces. J. Text. Inst. 52, T395-T406 (1961) · doi:10.1080/19447027.1961.10750513
[33] Cao, J., Akkerman, R., Boisse, P., et al. Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos. Part A: Appl. Sci. Manuf. 39, 1037-1053 (2008)
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