Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T07:46:29.896Z Has data issue: false hasContentIssue false

On macroscopic effects of heterogeneity in elastoplastic media at finite strain

Published online by Cambridge University Press:  24 October 2008

R. Hill
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge

Abstract

For elastic or plastic media the linkages between basic properties at two levels of description are rigorously analysed. At one level the material is structurally heterogeneous, while at the other it behaves macroscopically as if homogeneous. The deformation is unrestricted as to magnitude, but is treated incrementally. Full account is taken of coupling between stresses and rotations within a representative volume. Theorems on spatial averages of tensor variables and their products are reviewed and extended. Relations between constitutively significant quantities at both levels are developed by means of tensor influence functions associated with auxiliary elastic fields. The analysis is conducted primarily in terms of non-objective variables and a Lagrangian reference configuration. A final transformation to intrinsic variables is effected with the help of some novel operators.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Havner, K. S.. Aspects of theoretical plasticity at finite deformation and large pressure. Z. Angew. Math. Phys. 25 (1974), 765781.CrossRefGoogle Scholar
[2]Havner, K. S.. The theory of finite plastic deformation of crystalline solids. Mechanics of Solids: The Rodney Hill 60th Anniversary Volume (ed. Hopkins, H. G. and Sewell, M. J.), 265302 (Pergamon Press, 1982).CrossRefGoogle Scholar
[3]Hill, R.. Discontinuity relations in the mechanics of solids. Progress in Solid Mechanics (ed. Sneddon, I. N. and Hill, R.), 2, 247276 (North-Holland, 1961).Google Scholar
[4]Hill, R.. Acceleration waves in solids. J. Mech. Phys. Solids 10 (1962), 116.CrossRefGoogle Scholar
[5]Hill, R.. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11 (1963), 357372.CrossRefGoogle Scholar
[6]Hill, R.. The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15 (1967), 7995.CrossRefGoogle Scholar
[7]Hill, R.. Eigenmodal deformations in elastic/plastic continua. J. Mech. Phys. Solids 15 (1967), 3711–386.CrossRefGoogle Scholar
[8]Hill, R.. On macroscopic measures of plastic work and deformation in micro-heterogeneous media. Prikl. Mat. Mekh. 35 (1971), 3139. English translation: J. Appl. Math. Mech. 35 (1971), 11–17.Google Scholar
[9]Hill, R.. On constitutive macro-variables for heterogeneous solids at finite strain. Proc. Roy. Soc. London A 326 (1972), 131147.Google Scholar
[10]Hill, R.. Aspects of invariance in solid mechanics. Advances in Applied Mechanics 18 (1978), 175.Google Scholar
[11]Mandel, J.. Généralisation dans R9 de la règle du potential plastique pour un élément polycrystallin. C. R. Acad. Sc. Paris 290 (1980), 481484.Google Scholar
[12]Ogden, R. W.. Inequalities associated with the inversion of elastic stress-deformation relations and their implications. Math. Proc. Cambridge Philos. Soc. 81 (1977), 313324.CrossRefGoogle Scholar