A study of microindentation hardness tests by mechanism-based strain gradient plasticity
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H. Gao Division of Mechanics and Computation, Stanford University, Stanford, California 94305
W.D. Nix Department of Materials Science and Engineering, Stanford University, Stanford, California 94305
Z.C. Xia Ford Research Laboratory, P.O. Box 2053, MD 3135/SRL, Dearborn, Michigan 48121 (Received 27 September 1999; accepted 24 May 2000)
We recently proposed a theory of mechanism-based strain gradient (MSG) plasticity to account for the size dependence of plastic deformation at micron- and submicronlength scales. The MSG plasticity theory connects micron-scale plasticity to dislocation theories via a multiscale, hierarchical framework linking Taylor’s dislocation hardening model to strain gradient plasticity. Here we show that the theory of MSG plasticity, when used to study micro-indentation, indeed reproduces the linear dependence observed in experiments, thus providing an important self-consistent check of the theory. The effects of pileup, sink-in, and the radius of indenter tip have been taken into account in the indentation model. In accomplishing this objective, we have generalized the MSG plasticity theory to include the elastic deformation in the hierarchical framework.
I. INTRODUCTION
Microindentation and nanoindentation experiments provide a useful means to determine the mechanical properties of materials at the micron or submicron level, and are therefore important to microscale applications such as thin films and microelectromechanical systems (MEMS). These experiments have repeatedly shown that the microscale mechanical properties of materials are significantly different from those of bulk materials. For example, the measured indentation hardness of metallic materials increases by a factor of two or even three as the depth of indentation decreases from 50 to 1 m.1–10 Classical plasticity theories fail to predict this size dependence of material behavior at the micron scale because their constitutive models possess no internal length scale. The predicted indentation hardness based on classical plasticity theories would not depend on the indentation depth, contrary to the experimental observations. Over this 1–10 m scale, however, there are still hundreds or even thousands of dislocations such that there should be a continuum theory of plasticity (but not classical plasticity!) that can describe the collective behavior of dislocations. The rational design and manufacturing issues at the microscale also warrant the development of 1786
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J. Mater. Res., Vol. 15, No. 8, Aug 2000 Downloaded: 01 Dec 2014
such a micro-level continuum theory because it is still not possible to perform quantum and atomistic simulations for micron-level structures over realistic time scale. Therefore, there is an impending need to develop a continuum theory in order to bridge the gap between classical plasticity and dislocation theory. The recent development of strain gradient plasticity theories represents such an effort to bridge this gap. Fleck, Hutchinson, and co-workers11,12 have develop
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