2 (2005) 153–165
A nonlocal model for size eﬀect and localization in plasticity
C. Comia,∗ and L. Driemeierb
Dept. Structural Engineering, Politecnico di Milano, Piazza L. da Vinci 32 - 20133 Milano Dept. Mechatronics and Mechanical Systems Engineering, University of Sao Paulo, Av. Prof. Mello Moraes, 2231, 05508-900, SaoPaulo, Brazil Abstract A simple nonlocal plasticity model is proposed to account for the size dependence of plastic deformation at the micro-scale and at the same time to regularize the response in the presence of localization phenomena. Nonlocality is introduced in the yield function through the deﬁnition of a nonlocal strain, which is the weighted average of the local strain over a suitableneighborhood, depending on the material length. We apply the model to a strain localization problem of a softening bar and we compare model predictions with experiments on microbending and microtorsion.
Classical local plasticity theory, in which no length scale enters, disregards the inﬂuence of the microscopic material structure on the macroscopic material behaviour.Although local theories are able to interpret the material behavior in a large number of applications, they become inadequate to model phenomena such as the experimentally observed size-dependence of the plastic response of micro-sized solids or the appearance of localization bands of ﬁnite width in the presence of softening or very large strains. In particular, tests performed at the micro- ornano-scale such as nano-indentation [3,27,28, 32], bending of thin metallic beams [21, 31] or micro-torsion of thin copper wires  have provided experimental evidence of strain gradient hardening, which makes the response dependent on the scale of the structure. Hardness and strength increase as the specimen size is decreased; this size eﬀect, which is negligible for macro-specimen, becomes importantat very small scales and cannot be captured by local models. Another noteworthy example is the localization phenomenon. Strain localization is characterized by pronounced displacement gradients – induced by geometry, boundary conditions, material heterogeneity or local defects – in restrict zones of the medium, called strain localization zones. With local models, the width of the localizationzones tend to zero, with the nowadays well known numerical consequences in terms of pathological mesh dependence. As
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Received 15 June 2005; In revised form 22 June 2005
C. Comi and L. Driemeier
pointed out in [7, 24], mathematically, the boundary value problem becomes ill-posed and the model no longer represents the physicalreality. The ill-posedness of the problem can be overcame by using regularization techniques, providing accurate numerical solutions. Nonlocal gradient models [8,10,13], nonlocal integral models [5, 9, 29] and micropolar models  which include a material internal length have been formulated and eﬀectively used. Enhanced gradient models have also been used to interpret size eﬀect at the microscale,see [1,2,4,16,18,20,23,25]. These models allow to account for the presence of geometrically necessary dislocations whose accumulation increases the ﬂow stress; this eﬀect becomes important when the scale of the specimen approaches the scale of the lattice. Gradient plasticity formulations lead to the deﬁnition of one or more internal length parameters, or even internal length tensors, whose valuescan be identiﬁed on the basis of sophisticated tests and/or atomistic considerations [15,17]. Gao and Huang in  explore the possibility of modeling size-dependent plasticity within the framework of nonlocal continuum theories. Considering the Taylor expansion of the strain, they represent strain gradient as a nonlocal integral of strain In this work we propose an alternative nonlocal plastic...