Summary
In this paper attention is focussed on the derivation of higher-order isotropic tensors and their application in the formulation of enhanced continuum models. A mathematical theory will be discussed which relates formal orthogonal invariant polynomial functions to isotropic tensors. Using this theory, the second-order to the sixth-order isotropic tensor will be derived. When the tensor order increases, the derivation procedure clearly reveals a repeatable character. Thereafter, an example will be given of how the higher-order isotropic tensors can be used in the formulation of an enhanced continuum model. It will be demonstrated that symmetry conditions significantly reduce the number of material parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Noll, W.: A mathematical theory for the behavior of continuous media. Arch. Rat. Mech. Anal.2, 197–226 (1958).
Cosserat, E., Cosserat, F.: Théorie des corps deformables. Paris: Herman et fils, 1909.
Mindlin, R. D.: Micro-structure in linear elasticity. Arch Ration. Mech. Anal.,16, 51–78 (1964).
Eringen, A. C.: Theory of micro-polar elasticity. In Liebowitz, H. (ed.) Fracture—An Advanced Treatise, Vol. II, Ch. 7, pp. 621–693. New York London: Academic Press, 1968.
Mühlhaus, H.-B., Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique37, 271–283 (1987).
De Borst, R.: Simulation of strain localisation: A reappraisal of the Cosserat continuum. Eng. Comp.8, 317–332 (1991).
De Borst, R., Mühlhaus, H.-B.: Gradient-dependent plasticity: Formulation and algorithmic aspects. Int. J. Num. Meth. Eng.35, 521–539, (1992).
Chang, C. S., Ma, L.: Elastic material constants for isotropic granular solids with particle rotation. Int. J. Solids Struct.29, 1001–1018, (1992).
Sluys, L. J.: Wave Propagation. Localisation and Dispersion in Softening Solids. PhD thesis, Delft University of Technology 1992.
Fleck, N. A., Hutchinson, J. W.: A phenomenological theory for gradient effects in plasticity. J. Mech. Phys. Solids41, 1825–1857 (1993).
Pamin, J.: Gradient-dependent plasticity in numerical simulation of localization phenomena. PhD thesis, Delft University of Technology 1994.
Chang, C. S., Gao, J.: Second-gradient constitutive theory for granular material with random packing structure. Int. J. Solids Struct.16, 2279–2293 (1995).
Suiker, A. S. J., Chang, C. S., De Borst, R.: Micro-mechanical modelling of granular material-Part 1-Derivation of a second-gradient micro-polar constitutive theory. Acta Mech. (forthcoming).
Suiker, A. S. J., Chang, C. S., De Borst, R.: Micro-mechanical modelling of granular material-Part 2-Plane wave propagation in infinite media. Acta Mech. (forthcoming).
Mühlhaus, H.-B., Oka, F.: Dispersion and wave propagation in discrete and continuous models for granular materials. Int. J. Solids Struct.33, 2841–2858 (1996).
Weyl, H.: Classical Groups. Princeton: Princeton University Press 1946.
Sokolnikoff, I. S.: Tensor Analysis: New York: Wiley 1951.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Suiker, A.S.J., Chang, C.S. Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mechanica 142, 223–234 (2000). https://doi.org/10.1007/BF01190020
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01190020