Plasticity at the micron scale

Over a scale which extends from about a fraction of a micron to tens of microns, metals display a strong sizedependence when deformed non uniformly into the plastic range: smaller is stronger. This e€ect has important implications for an increasing number of applications in electronics, structural materials and MEMS. Plastic behavior at this scale cannot be characterized by conventional plasticity theories because they incorporate no material length scale and predict no size e€ect. While micron sized solid objects are too small to be characterized by conventional theory, they are usually too large to be amenable to analysis using approaches presently available based on discrete dislocation mechanics. The relatively large numbers of dislocations governing plastic deformation at the micron scale motivate the development of a continuum theory of plasticity incorporating size-dependence. Strain gradient theories of plasticity have been developed for this purpose. The motivation and potential for such theories will be discussed. Important open issues surrounding the foundations of strain gradient plasticity will also be addressed and a few critical experiments identi®ed. # 1999 Elsevier Science Ltd. All rights reserved. 1. Size-dependence at the micron scale Applications of metals and polymers at the micron scale are multiplying rapidly. Plasticity as well as the elasticity of the materials are important in many of these applications, and e€orts are underway in the materials and mechanics communities to measure and characterize behavior at the micron scale. Indentation tests are a common means of assessing material yield strength. Instruments have been developed which permit measurement of indentation hardness at the micron and nano scales. A large size-dependence is observed in indentation tests on metals. Data for tungsten single crystals at three orientations relative to the indenter are shown in Fig. 1. The hardness is de®ned as the indentation load divided by the area of the indent after unloading. The Vickers indenter is relatively shallow so that the indentation depth is a fraction of the indentation diagonal plotted in Fig. 1. There is some dependence International Journal of Solids and Structures 37 (2000) 225±238 0020-7683/00/$ see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0020-7683(99 )00090-6 www.elsevier.com/locate/ijsolstr * Tel: +1-617-495-2848; Fax: +1-617-495-9837. E-mail address: [email protected] (J.W. Hutchinson) of hardness on crystal orientation, but the size-dependence is the predominant e€ect. Indents with diagonals longer than about 100mm cease to display any size-dependence. By simple dimensional arguments, it follows that any conventional plasticity theory (e.g. one that does not possess a constitutive length scale) necessarily implies indentation hardness would be size-independent. The strong size-dependence evident at the micron scale in Fig. 1 and in data on other metals (Atkinson, 1995, Ma and Clarke, 1995; De Guzman et al., 1993; McElhaney et al., 1998; Poole et al., 1996) constitutes one of the compelling pieces of experimental evidence for the need of an extension of plasticity theory to the micron scale. Here and throughout this paper, the term `micron scale' will be used to refer to the range which extends roughly from a fraction of a micron to tens of microns. At even smaller scales, dislocation mechanics is required to understand nucleation of dislocations at the indenter and the interaction between relatively small numbers of individual dislocations. Recent modeling e€orts of nano scale indentation of single crystals based on discrete dislocation mechanics appear promising (Tadmore et al., 1998). At the micron scale, the number of dislocations involved in the indentation zone is usually large Ð too large to be amenable at the present time to quantitative analysis using dislocation mechanics. A second set of experiments displaying strong size-dependence of plastic deformation in the micron range is displayed in Fig. 2. Annealed copper wires of diameter ranging from 170 down to 12mm are twisted well into the plastic range. For each wire, the torque, Q, versus twist per unit length, k, is plotted as Q/a 3 versus ka, where a is the radius of the wire (Fig. 2a). Dimensional arguments dictate that, had the wires been governed by a continuum theory with no constitutive length parameter, the curves of Fig. 2a should plot on top of one another. The factor of three strength advantage of the smaller wires over the largest wire re ̄ects the size e€ect. Tensile stress-strain data for these same wire are plotted in Fig. 2b. These do superimpose, to within experimental error. The strengthening e€ect at the micron scale is intrinsically associated with non-uniform deformation, as will be discussed shortly. Test data for plastic bending of a series of `micron' thick nickel ®lms is presented in Fig. 3. Strips of the ®lm were bent around a series of ®bers of di€erent diameters. Elastic springback upon release of the Fig. 1. Hardness data for tungsten single crystals at three orientations relative to the Vickers indenter (Stelmashenso et al., 1993). The hardness (load divided by the area of the indent) is plotted against the diagonal of the four-sided pyramidal indent displaying the increase in hardness with decrease in indent size. J.W. Hutchinson / International Journal of Solids and Structures 37 (2000) 225±238 226