Though targeting at different scales, the theory of objective structures and the theory of materially uniform bodies have some issues in common. We highlight those aspects of the two theories that share similar ideas, as well as delineate areas where they are inherently different. Prompt by the fact that materially uniform but inhomogeneous bodies ultimately describe dislocations into solids, we propose a way for defining continuous distribution of dislocations for objective structures. In the course of doing, so we draw upon combined theories of algebraic topology and discrete exterior calculus to model crystal elasticity. We also need a generalization of the theory of materially uniform bodies suitable for a class of micromorphic bodies.
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Ariza M.P., Ortiz M.: Discrete crystal elasticity and discrete dislocations in crystals. Arch. Ration. Mech. Anal. 178, 149–226 (2005)
Bhattacharya K., James R.D.: A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47, 531–576 (1999)
Choquet-Bruhat Y., DeWitte-Morette C., Billard-Bleick M.: Analysis, Manifolds and Physics. North Holland, Amsterdam (1977)
Dayal K., James R.D.: Nonequilibrium molecular dynamics for bulk materials and nanostructures. J. Mech. Phys. Solids 58, 145–163 (2010)
Dontsova E., Dumitrica T.: Nanomechanics of twisted mono-and few-layer graphene nanoribbons. J. Phys. Chem. Lett. 4, 2010–2014 (2013)
Dumitrica T., James R.D.: Objective molecular structures. J. Mech. Phys. Solids 55, 2206–2236 (2007)
deLeon M., Epstein M.: The geometry of uniformity in second-grade elasticity. Acta. Mech. 114, 217–224 (1996)
Edelen D.G.B., Lagoudas D.: Gauge Theory and Defects in Solids. North Holland, Amsterdam (1989)
Epstein M., deLeon M.: Geometrical theory of Cosserat media. J. Geom. Phys. 26, 127–170 (1998)
Epstein M., Elzanowski M.: Material Inhomogeneities and Their Evolution. A Geometric Approach. Springer, Berlin (2007)
Epstein M., Maugin G.A.: The energy momentum tensor and material uniformity in finite elasticity. Acta. Mechanica. 83, 127–133 (1990)
Epstein, M., Segev, R.: Geometric aspects of singular dislocations. Math. Mech. Sol., in press
Eringen A.C.: Theory of micropolar elasticity. In: Liebowits, H. (ed.) Fracture an Advanced Study, pp. 621–729. Academic Press, New York (1968)
Eringen A.C., Suhubi E.: Nonlinear theory of simple microelastic solids-I. Int. J. Eng. Sci. 2, 189–203 (1964)
Friesecke G., James R.D.: A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48, 1519–1540 (2000)
Friesecke G., James R.D., Muller S.: A hierarchy of plate models derived from nonlinear elasticity by gamma convergence. Arch. Ration. Mech. Anal. 180, 183–236 (2006)
Grammenoudis P., Tsakmakis Ch.: Micromorphic continuum part I: strain and stress tensors and their associated rates. Int. J. Non Linear Mech. 44, 943–956 (2009)
Hakobyan Ya., Tadmor E.B., James R.D.: Objective quasicontinuum approach for rod problems. Phys. Rev. B. 86, 245435 (2012)
Hirani, A.: Discrete Exterior Calculus. Ph.D. thesis, California Institute of Technology (2003)
James RD.: Objective structures. J. Mech. Phys. Solids 54, 2354–2390 (2006)
Kosevich A.M.: Crystal dislocations and the theory of elasticity. In: Nabarro, F.R.N. (ed.) Dislocationsin Solids, Vol. I: The Elastic Theory, pp. 33–142. North-Holland, Amsterdam (1979)
Kroner E. et al.: Continuum theory of defects. In: Balian, R. Physics of Defects, pp. 215–315. North-Holland, Amsterdam (1981)
Leok, M.: Foundations of Computational Geometric Mechanics. Ph.D. thesis, California Institute of Technology (2004)
Morgan A.J.A.: Inhomogeneous materially uniform higher order gross bodies. Arch. Ration. Mech. Anal. 57, 189–253 (1975)
Muncaster R.G.: Invariant manifolds in mechanics I: the general construction of coarse theories from fine theories. Arch. Ration. Mech. Anal. 84, 353–373 (1984)
Noll W.: Materially uniform simple bodies with inhomogeneities. Arch. Ration. Mech. Anal. 27, 1–32 (1967)
Sfyris D.: Propagation of a plane wave to a materially uniform but inhomogeneous body. ZAMP 62, 927–936 (2011)
Sfyris D.: Comparing the condition of strong ellipticity and the solvability for a purely elastic problem and the corresponding dislocated problem. Math. Mech. Solids 17, 254–265 (2012)
Sfyris D.: The role of the symmetry group in the non-uniqueness of the uniform reference. Case study: an isotropic solid body. Math. Mech. Solids 18, 738–744 (2013)
Sfyris D.: Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group. Mech. Res. Commun. 51, 56–60 (2013)
Sfyris D.: Autoparallel curves and Riemannian geodesics for materially uniform but inhomogeneous bodies. Math. Mech. Solids 19, 152–167 (2004)
Sfyris D., Charalambakis N., Kalpakides V.K.: Continuously dislocated elastic bodies with a neo-Hookean like expression for the energy subjected to antiplane shear. J. Elast. 93, 245–262 (2008)
Wang C.-C.: On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distribution of dislocations. Arch. Ration. Mech. Anal. 27, 33–94 (1967)
Zhang D.-B., James R.D., Dumitrica T.: Dislocation onset and nearly axial glide in carbon nanotubes under torsion. J. Chem. Phys. 130, 071101 (2009)
Communicated by Andreas Öchsner.
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Sfyris, D. A proposal for defining continuous distribution of dislocations for objective structures. Continuum Mech. Thermodyn. 27, 399–407 (2015). https://doi.org/10.1007/s00161-014-0362-9
- Objective structures
- Materially uniform bodies
- Inhomogeneous bodies