For a given class of materials, universal displacements are those displacements that can be maintained for any member of the class by applying only boundary tractions. In this paper, we study universal displacements in compressible anisotropic linear elastic solids reinforced by a family of inextensible fibers. For each symmetry class and for a uniform distribution of straight fibers respecting the corresponding symmetry, we characterize the respective universal displacements. A goal of this paper is to investigate how an internal constraint affects the set of universal displacements. We have observed that other than the triclinic and cubic solids in the other five classes (a fiber-reinforced solid with straight fibers cannot be isotropic), the presence of inextensible fibers enlarges the set of universal displacements.
RivlinRSSaundersDW.Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos Trans R Soc Lond A Math Phys Sci1951; 243(865): 251–288.
2.
TadmorEBMillerREElliottRS.Continuum mechanics and thermodynamics: from fundamental concepts to governing equations. Cambridge: Cambridge University Press, 2012.
3.
GorielyA.The mathematics and mechanics of biological growth. New York: Springer Verlag, 2017.
4.
EricksenJL.Deformations possible in every compressible, isotropic, perfectly elastic material. J Math Phys Camb1955; 34(1–4): 126–128.
5.
EricksenJL.Deformations possible in every isotropic, incompressible, perfectly elastic body. Z Angew Math Phys1954; 5(6): 466–489.
6.
RivlinRS. Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philos Trans R Soc Lond A1948; 241(835): 379–397.
7.
RivlinRS. Large elastic deformations of isotropic materials. V. The problem of flexure. Proc R Soc Math Phys Eng Sci1949; 195(1043): 463–473.
8.
RivlinRS.A note on the torsion of an incompressible highly elastic cylinder. Proc Cambridge Philos Soc1949; 45(3): 485–487.
9.
SaccomandiG.Universal solutions and relations in finite elasticity. In: HayesMSaccomandiG (eds) Topics in finite elasticity. New York: Springer, 2001, pp. 95–130.
10.
SinghMPipkinAC.Note on Ericksen’s problem. Z Angew Math Phys1965; 16(5): 706–709.
11.
KlingbeilWWShieldRT.On a class of solutions in plane finite elasticity. Z Angew Math Phys1966; 17(4): 489–511.
12.
FosdickRL. Remarks on compatibility. In: EskinaziS (ed.) Modern developments in the mechanics of continua. Cambridge, MA: Academic Press, 1966, pp. 109–127.
13.
YavariA.Universal deformations in inhomogeneous isotropic nonlinear elastic solids. Proc R Soc A2021; 477(2253): 20210547.
BeattyMF.General solutions in the equilibrium theory of inextensible elastic materials. Acta Mech1978; 29(1): 119–126.
28.
PipkinAC.Constraints in linearly elastic materials. J Elast1976; 6(2): 179–193.
29.
AdkinsJERivlinRS. Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords. Philos Trans A Math Phys Eng Sci R Soc Lond A Math Phys Sci1955; 248(944): 201–223.
30.
TruesdellCNollW.The non-linear field theories of mechanics. New York: Springer, 2004.
31.
SaccomandiGBeattyMF.Universal relations for fiber-reinforced elastic materials. Math Mech Solids2002; 7(1): 95–110.
32.
HayesMHorganCO.On mixed boundary-value problems for inextensible elastic materials. Z Angew Math Phys1975; 26: 261–272.
33.
CowinSCMehrabadiMM.Anisotropic symmetries of linear elasticity. Appl Mech Rev1995; 48(5): 247–285.
34.
ChadwickPVianelloMCowinSC.A new proof that the number of linear elastic symmetries is eight. J Mech Phys Solids2001; 49(11): 2471–2492.
35.
TingTCT.Anisotropic elasticity: theory and applications. Oxford: Oxford University Press on Demand, 1996.
36.
TingTCT.Generalized Cowin–Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight. Int J Solids Struct2003; 40(25): 7129–7142.
37.
CowinSCDotySB.Tissue mechanics. Berlin: Springer Science & Business Media, 2007.
