A phenomenological theory for strain gradient effects in plasticity. (English) Zbl 0791.73029

Summary: A strain gradient theory of plasticity is introduced, based on the notion of statistically stored and geometrically necessary dislocations. The strain gradient theory fits within the general framework of couple stress theory and involves a single material length scale \(l\). Minimum principles are developed for both deformation and flow theory versions of the theory which in the limit of vanishing \(l\), reduce to their conventional counterparts: \(J_ 2\) deformation and \(J_ 2\) flow theory. The strain gradient theory is used to calculate the size effect associated with macroscopic strengthening due to a dilute concentration of bonded rigid particles; similarly, predictions are given for the effect of void size upon the macroscopic softening due to a dilute concentration of voids. Constitutive potentials are derived for this purpose.


74C99 Plastic materials, materials of stress-rate and internal-variable type
Full Text: DOI


[1] Ashby, M.F., The deformation of plastically non-homogeneous alloys, Phil. mag., 21, 399-424, (1970)
[2] Ashby, M.F., The deformation of plastically non-homogeneous alloys, (), 137-192
[3] {\scBrown L.M.},(1993) Private communication.
[4] Budiansky, B.; Hutchinson, J.W.; Slutsky, S., Void growth and collapse in viscous solids, (), 13-15 · Zbl 0477.73091
[5] Cottrell, A.H., The mechanical properties of matter, (), 277
[6] Drucker, D.C., A more fundamental approach to plastic stress-strain relations, (), 487-491
[7] Duva, J.M., A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material, J. engng mater. technol., 106, 317-321, (1984)
[8] Duva, J.M.; Hutchinson, J.W., Constitutive potentials for dilutely voided nonlinear materials, Mech. mater., 3, 41-54, (1984)
[9] Ebeling, R.; Ashby, M.F., Dispersion hardening of copper single crystals, Phil. mag., 13, 805-834, (1966)
[10] Fleck, N.A.; Muller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta metallurgica et materialia, (1993), To appear in
[11] Hill, R., New horizons in the mechanics of solids, J. mech. phys. solids, 13, 66-74, (1956) · Zbl 0073.23703
[12] Hill, R., Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. mech. phys. solids, 14, 95-102, (1966)
[13] Hill, R., The essential structure of constitutive laws for metal composites and poly-crystals, J. mech. phys. solids, 15, 79-95, (1967)
[14] Hull, D.; Bacon, D.J., Introduction to dislocations, (1984), Pergamon Press Oxford
[15] Kelly, A.; Nicholson, R.B., Precipitation hardening, Prog. mater. sci., 10, 3, 149-391, (1963)
[16] Koiter, W.T., General theorems for elastic-plastic solids, (), 167-221 · Zbl 0098.37603
[17] Koiter, W.T., Couple stresses in the theory of elasticity, I and II, Proc. ned. akad. wet., 67, 1, 17-44, (1964), (B) · Zbl 0124.17405
[18] Nye, J.F., Some geometrical relations in dislocated crystals, Acta metallurgica, 1, 153-162, (1953)
[19] Rice, J.R., On the structure of stress—strain relations for time-dependent plastic deformation in metals, J. appl. mech., 37, 728-737, (1970)
[20] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0202.14303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.