For homogeneous nonlinear isotropic micropolar solids a class of universal deformations is obtained. The kinematics of micropolar media is determined by two independent fields, which are the position vector of material particles and the microrotation tensor which describes the rotational degrees of freedom. We present two types of universal deformation. The first type is uniform deformations. Here the deformation gradient tensor and the microrotation tensor are both constant. In addition the microrotation tensor coincides with the macrorotation tensor from the polar decomposition of the deformation gradient or differs from the latter by a half-turn about one of the principal axes of the stretch tensor. The second type of universal deformation consists of six families of non-uniform deformations. It is obtained for incompressible materials. Each family includes a few subfamilies which differ from each other by the form of the microrotation tensor.
RivlinRS. Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos Trans R Soc London Ser A Math Phys Sci1948; 241(835): 379–397.
2.
RivlinRS. Large elastic deformations of isotropic materials. V. The problem of flexure. Proc Trans R Soc London Ser A Math Phys Sci1949; 195(1043): 463–473.
3.
RivlinRS. Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos Trans R Soc London Ser A Math Phys Sci1949; 242(845): 173–195.
4.
EricksenJL. Deformations possible in every isotropic, incompressible, perfectly elastic body. Z Angew Math Phys (ZAMP)1954; 5: 466–486.
5.
KlingbeilWWShieldRT. On a class of solutions in plane finite elasticity. Z Angew Math Phys (ZAMP)1966; 17(4): 489–511.
6.
TruesdellCNollW. The nonlinear field theories of mechanics. In: FlüggeS (ed) Handbuch der Physik, vol. 3. Berlin: Springer, 1965, 1–602.
7.
TruesdellC. A first course in rational continuum mechanics. New York, NY: Academic Press, 1977.
8.
LurieAI. Nonlinear Theory of Elasticity. Amsterdam: North Holland, 1990.
9.
ZubovLM. Universal quasistatic deformations for isotropic incompressible bodies with memory. Mech Solid1979; 3: 57–62. (In Russian.)
10.
BeattyMFZhouZ. Universal motions for a class of viscoelastic materials of differential typeContin Mech Thermodyn1991; 3: 169–191.
11.
BeattyMF. A class of universal relations for constrained, isotropic elastic materials. Acta Mech1989; 80(3-4): 299–312.
12.
PucciESaccomandiG. Universal relations in constrained elasticity. Math Mech Solid1996; 1(2): 207–217.
13.
PucciESaccomandiG. Universal motions for constrained simple materials. Int J Non-Lin Mech1998; 34(3): 469–484.
14.
SaccomandiGBeattyMF. Universal relations for fiber-reinforced elastic materials. Math Mech Solid2002; 7(1): 95–110.
15.
DorfmannAOgdenRWSaccomandiG. Universal relations for non-linear magnetoelastic solids. Int J Non-Lin Mech2004; 39(10): 1699–1708.
16.
PucciESaccomandiG. On universal relations in continuum mechanics. Contin Mech Thermodyn1997; 9(2): 61–72.
17.
SaccomandiG. Universal results in finite elasticity. Non-linear elasticity: Theory and applications. London Math Soc Lect Note2001; 283: 97–134.
18.
BustamanteROgdenRW. On nonlinear universal relations in nonlinear elasticity. Z Angew Math Phys (ZAMP)2006; 57(4): 708–721.
19.
AeroELKuvshinskiiEV. Fundamental equations of the theory of elastic media with rotationally interacted particles. Sov Phys Solid State1961; 2(7): 1272–1281.
20.
AeroELKuvshinskiiEV. Continuum theory of asymmetric elasticity. Equilibrium of an isotropic body. (In Russian.)Fis Tverd Tela1964; 6: 2689–2699.
21.
KoiterWT. Couple-stresses in the theory of elasticity. Pt. I–II. Proc Koninkl Neterl Akad Weten1964; 67: 17–44.
22.
Pal’movVA. Fundamental equations of the theory of asymmetric elasticity. J Appl Mech Math1964; 28(6): 1341–1345.
23.
EringenAC. Linear theory of micropolar elasticity. J Math Mech1966; 15(6): 909–923.
24.
NowackiW. Theory of Asymmetric Elasticity. Oxford: Pergamon Press, 1986.
25.
EringenAC. Microcontinuum Field Theories. I. Foundations and Solids. New York, NY: Springer, 1999.
26.
MauginG.A.On the structure of the theory of polar elasticity. Philos Trans R Soc Lond A1998; 356: 1367–1395.
27.
ToupinRA. Theories of elasticity with couple-stress. Arch Rat Mech Anal1964: 17(2): 85–112.
28.
ShkutinLI. Mechanics of Deformations of Flexible Bodies. (In Russian.) Novosibirsk: Nauka, 1985.
29.
ZubovLM. Nonlinear Theory of Dislocation and Disclinations in Elastic Bodies. Berlin: Springer, 1997.
30.
NikitinEZubovLM. Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. J Elast1998; 51: 1–22.
31.
PietraszkiewiczWEremeyevVA. On natural strain measures of the non-linear micro-polar continuum. Int J Solids Struct2009; 46(3-4): 774–787.
32.
ErofeevVI. Wave Process in Solids with Microstructure. Singapore: World Scientific, 2003.
33.
AltenbachJAltenbachHEremeyevVA. On generalised Cosserat-type theories of plates and shells: A short review and bibliography. Arch Appl Mech2010; 80: 73–92.
34.
MauginGA. A historical perspective of generalized continuum mechanics. In: AltenbachHErofeevVIMauginGA (eds) Mechanics of Generalized Continua. Berlin: Springer, 2011, pp. 3–19.
35.
MauginGA. Continuum Mechanics Through the Twentieth Century. Heidelberg: Springer, 2013.
36.
ForestS. Micromorphic media, in: AltenbachHEremeyevVA. (Eds.), Generalized Continua from the Theory to Engineering Applications. Springer Vienna. 2013. Volume 541 of CISM International Centre for Mechanical Sciences, pp. 249–300.
37.
AifantisE. Update on a class of gradient theories. Mech Mater2003; 35: 259–280.
38.
Dell’IsolaFSeppecherP. The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. CR Acad Sci Sér21995; 321: 303–308.
39.
Dell’IsolaFSciarraGVidoliS. Generalized Hooke’s law for isotropic second gradient materials. Proc Trans R Soc London Ser A Math Phys Sci2009; 465: 2177–2196.
40.
Dell’IsolaFSeppecherPMadeoA. 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 (ZAMP)2012; 63: 1119–1141.
41.
CaprizG. Continua with Microstructure. New York, NY: Springer, 1989.
42.
PietraszkiewiczWEremeyevVA. On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int J Solids Struct2009; 46(11): 2477–2480.
43.
KafadarCBEringenAC. Micropolar media - I. The classical theory. Int J Eng Sci1971; 9(3): 271–305.
44.
EremeyevVAPietraszkiewiczW. Material symmetry group of the non-linear polar-elastic continuum. Int J Solids Struct2012; 49(14): 1993–2005.
45.
AltenbachHEremeyevVA. Strain rate tensors and constitutive equations of inelastic micropolar materials. Int J Plast2014; 63: 3–17.
46.
GreenAEAdkinsJE. Large Elastic Deformations and Non-Linear Continuum Mechanics. Oxford: Clarendon Press, 1960.
