A Dynamical Approach to Plastic Deformation of Nano-Scale Materials : Nano and Micro-Indentation
Recent studies demonstrate that mechanical deformation of small volume systems can be significantly different from those of the bulk. One such interesting length scale dependent property is the increase in the yield stress with decrease in diameter of micrometer rods, particularly when the diameter is below a micrometer. Intermittent ﬂow may also result when the diameter of the rods is decreased below a certain value. The second such property is the intermittent plastic deformation during nano-indentation experiments. Here again, the instability manifests due to smallness of the sample size, in the form of force ﬂuctuations or displacement bursts. The third such length scale dependent property manifests as ’smaller is stronger’ property in indentation experiments on thin ﬁlms, commonly called as the indentation size eﬀect (ISE). More specifically, the ISE refers to the increase in the hardness with decreasing indentation depth, particularly below a fraction of a micrometer depth of indentation. The purpose of this thesis is to extend nonlinear dynamical approach to plastic deformation originally introduced by Anantha krishna and coworkers in early 1980’s to nano and micro-indentation process. More specifically, we address three distinct problems : (a) intermittent force/load ﬂuctuations during displacement controlled mode of nano-indentation, (b) displacement bursts during load controlled mode of nano-indentation and (c) devising an alternate framework for the indentation size eﬀect. In this thesis, we demonstrate that our approach predicts not just all the generic features of nano-and micro-indentation and the ISE, the predicted numbers also match with experiments. Nano-indentation experiments are usually carried-out either in a displacement controlled (DC) mode or load controlled (LC) mode. The indenter tip radius typically ranges from few tens of nanometer to few hundreds of nanometers-meters. Therefore, the indented volume is so small that the probability of ﬁnding a dislocation is close to zero. This implies that dislocations must be nucleated for further plastic deformation to proceed. This is responsible for triggering intermittent ﬂow as indentation proceeds. While several load drops are seen beyond the elastic limit in the DC controlled experiments, several displacement jumps are seen in the LC experiments. In both cases, the stress corresponding to load maximum on the elastic branch is close to the theoretical yield stress of an ideal crystal, a feature attributed to the absence of dislocations in the indented volume. Hardness is deﬁned as the ratio of the load to the imprint area after unloading and is conventionally measured by unloading the indenter from desired loads to measure the residual plastic imprint area. Then, the hardness so calculated is found to increase with decreasing indentation depth. However, such size dependent eﬀects cannot be explained on the basis of conventional continuum plasticity theories since all mechanical properties are independent of length scales. Early theories suggest that strong strain gradients exist under the indenter that require geometrically necessary dislocations (GNDs) to relax the strain gradients. In an eﬀort to explain the the size eﬀect, these theories introduce a length scale corresponding to the strain gradients. One other feature predicted by subsequent models of the ISE is the linear relation between the square of the hardness and the inverse of the indentation depth. Early investigations on the ISE did recognize that GNDs were required to accommodate strain gradients and that the hardness H is determined by the sum of the statistically stored dislocation (SSD) and GND densities. Following these steps, Nix and Gao derived an expression for the hardness as a function of the indentation depth z. The relevant variables are the SSD and GND densities. An expression for the GND density was obtained by assuming that the GNDs are contained within a hemispherical volume of mean contact radius. The authors derive an expression for the hardness H as a function of indentation depth z given by [ HH 0 ]2 = 1+ zz ∗ . The intercept H0 represents the hardness arising only from SSDs and corresponds to the hardness in the limit of large sample size. The slope z ∗ can be identified as the length scale below which the ISE becomes significant. The authors showed that this linear relation was in excellent agreement with the published results of McElhaney et al for cold rolled polycrystalline copper and single crystals of copper, and single crystals of silver by Ma and Clarke. Subsequent investigations showed that the linear relationship between H2 verses 1/z breaks down at small indentation depths. Much insight into nano-indentation process has come from three distinct types of studies. First, early studies using bubble raft indentation and later studies using colloidal crystals (soft matter equivalent of the crystalline phase) allowed visualization of dislocation nucleation mechanism. Second, more recently, in-situ transmission electron microscope studies of nano-indentation experiments have been useful in understanding the dislocation nucleation mechanism in real materials. Third, considerable theoretical understanding has come largely from various types of simulation studies such as molecular dynamics (MD) simulations, dis¬location dynamics simulations and multiscale modeling simulations (using MD together with dislocation dynamics simulations). A major advantage of simulation methods is their ability to include a range of dislocation mechanisms participating in the evolution of dislocation microstructure starting from the nucleation of a dislocation, its multiplication, formation of locks, junctions etc. However, this advantage is oﬀset by the serious limitations set by short time scales inherent to the above mentioned simulations and the limited size of simulated volumes that can be implemented. Thus, simulation approaches cannot impose experimental parameters such as the indentation rates or radius of the indenter and thickness of the sample, for example in MD simulations. Indeed, the imposed deformation rates are often several orders of magnitude higher than the experimental rates. Consequently, the predicted values of force, indentation depth etc., diﬀer considerably from those reported by experiments. For these reasons, the relevance of these simulations to real materials has been questioned. While several simulations, particularly MD simulation predict several force drops, there are no simulations that predict displacement jumps seen in LC mode experiments. The inability of simulation methods to adopt experimental parameters and the mismatch of the predicted numbers with experiments is main motivation for devising an alternate framework to simulations that can adopt experimental parameters and predict numbers that are comparable to experiments. The basic premise of our approach is that describing time evolution of the relevant variables should be adequate to capture most generic features of nano and micro-indentation phenomenon. In the particular case under study, this point of view is based on the following observation. While one knows that dislocations are the basic defects responsible for plastic deformation occurring inside the sample, the load-indentation depth curve does not include any information about the spatial location of dislocation activity inside the sample. In fact, the measured load and displacement are sample averaged response of the dislocation activity in the sample. This suggests that it should be adequate to use sample averaged dislocation densities to obtain load-indentation depth curve. Keeping this in mind, we devise a method for calculating the contribution from plastic deformation arising from dislocation activity in the entire sample. This is done by setting up rate equations for the relevant sample averaged dislocation densities. The ﬁrst problem we consider is the force/load ﬂuctuations in displacement controlled nano-indentation. We devise a novel approach that combines the power of nonlinear dynamics with the evolution equations for the mobile and forest dislocation densities. Since the force serrations result from plastic deformation occurring inside the sample, we devise a method for calculating this contribution by setting-up a system of coupled nonlinear time evolution equations for the mobile and forest dislocation densities. The approach follows closely the steps used in the Anantha krishna (AK) model for the Portevin-Le Chatelier (PLC) eﬀect. The model includes nucleation, multiplication and propagation of dislocation loops in the time evolution equation for the mobile dislocation density. We also include other well known dislocation transformation mechanisms to forest dislocation. Several of these dislocation mechanisms are drawn from the AK model for the PLC eﬀect. To illustrate the ability of the model to predict force ﬂuctuations that match experiments, we use the work of Kiely at that employs a spherical indenter. The ability of the approach is illustrated by adopting experimental parameters such as the indentation rate, the radius the indenter etc. The model predicts all the generic features of nano-indentation such as the Hertzian elastic branch followed by several force drops of decreasing magnitudes, and residual plas¬ticity after unloading. The stress corresponding to the elastic force maximum is close to the yield stress of an ideal solid. The predicted values for all the quantities are close to those reported by experiments. Our model allows us to address the indentation-size eﬀect including the ambiguity in deﬁning the hardness in the force drop dominated regime. At large indentation depths where the load drops disappear, the hardness shows decreasing trend, though marginal. The second problem we consider is the load controlled mode of indentation where sev¬eral displacement jumps of decreasing magnitudes are seen. Even though, the LC mode is routinely used in nano-indentation experiments, there are no models or simulations that predict the generic features of force-displacement curves, in particular, the existence of sev¬eral displacement jumps of decreasing magnitudes. The basic reason for this is the inability of these methods to impose constant load rate during displacement jumps. We then show that an extension of the model for the DC mode predicts all the generic features when the model is appropriately coupled to an equation defining the load rate. Following the model for DC mode, we retain the system of coupled nonlinear time evolution equations for mobile and forest dislocation densities that includes nucleation, multiplication, and propagation threshold mechanisms for mobile dislocations, and other dislocation transformation mechanisms. The commonly used Berkovich indenter is considered. The equations are then coupled to the force rate equation. We demonstrate that the model predicts all the generic features of the LC mode nano-indentation such as the existence of an initial elastic branch followed by several displacement jumps of decreasing magnitudes, and residual plasticity after unloading for a range of model parameter values. In this range, the predicted values of the load, displacement jumps etc., are similar to those found in experiments. Further, optimized set of parameter values can be easily determined that provide a good ﬁt to the load-indentation depth curve of Gouldstone et al for single crystals of Aluminum. The stress corresponding to the maximum force on the Berkovich elastic branch is close to the theoretical yield stress. We also elucidate the ambiguity in deﬁning hardness at nanometer scales where the displacement jumps dominate. The approach also provides insights into several open questions. The third problem we consider is the indentation size eﬀect. The conventional definition of hardness is that it is the ratio of the load to the residual imprint area. The latter is determined by the residual plastic indentation depth through area-depth relation. Yet, the residual plastic indentation depth that is a measure of dislocation mobility, never enters into most hardness models. Rather, the conventional hardness models are based on the Taylor relation for the ﬂow stress that characterizes the resistance to dislocation motion. This is a complimentary property to mobility. Our idea is to provide an alternate way of explaining the indentation size eﬀect by devising a framework that directly calculates the residual plastic indentation depth by integrating the Orowan expression for the plastic strain rate. Following our general approach to plasticity problems, we set-up a system of coupled nonlinear time evolution equations for the mobile, forest (or the SSD) and GND densities. The model includes dislocation multiplication and other well known dislocation transformation mechanisms among the three types of dislocations. The main contributing factor for the evolution of the GND density is determined by the mean strain gradient and the number of sites in the contact area that can activate dislocation loops of a certain size. The equations are then coupled to the load rate equation. The ability of the approach is illustrated by adopting experimental parameters such as the indentation rates, the geometrical quantities defining the Berkovich indenter including the nominal tip radius and other parameters. The hardness is obtained by calculating the residual plastic indentation depth after unloading by integrating the Orowan expression for the plastic strain rate. We demonstrate that the model predicts all features of the indentation size eﬀect, namely, the increase in the hardness with decreasing indentation depth and the linear relation between the square of the hardness and inverse of the indentation depth, for all but 200nm, for a range of parameter values. The model also predicts deviation from the linear relation of H2 as a function of 1/z for smaller depths consistent with experiments. We also show that it is straightforward to obtain optimized parameter values that give a good ﬁt to polycrystalline cold-worked copper and single crystals of silver. Our approach provides an alternate way of understanding the hardness and indentation size eﬀect on the basis of the Orowan equation for plastic ﬂow. This approach must be contrasted with most models of hardness that use the SSD and GND densities as parameters. The thesis is organized as follows. The ﬁrst Chapter is devoted to background material that covers physical aspects of different kinds of plastic deformation relevant for the thesis. These include the conventional yield phenomenon and the intermittent plastic deformation in bulk materials in alloys exhibiting the Portevin-Le Chatelier (PLC) eﬀect. We then provide background material on nano-and micro-indentation, both experimental aspects and the current status of the DC controlled and LC controlled modes of nano-indentation. Results of simulation methods are brieﬂy summarized. The chapter also provides a survey of hardness models and the indentation size eﬀect. A critical survey of experiments on dislocation microsructure that contradict / support certain predictions of the NixGao model. The current status of numerical simulations are also given. The second Chapter is devoted to introducing the basic steps in modeling plastic deformation using nonlinear dynamical approach. In particular, we describe how the time evolution equations are constructed based on known dislocation mechanisms such as nucleation, multiplication formations of junctions etc. We then consider a model for the continuous yield phenomenon that involves only the mobile and forest densities coupled to constant strain rate condition. This problem is considered in some detail to illustrate how the approach can be used for modeling nano-indentation and indentation size eﬀect. The third Chapter deals with a model for displacement controlled nano-indentation. The fourth Chapter is devoted to adopting these equation to the load controlled mode of nano¬indentation. The ﬁfth Chapter is devoted to modeling the indentation size eﬀect based on calculating residual plastic indentation depth after unloading by using the Orowan’s expression for the plastic strain rate. We conclude the thesis with a Summary, Discussion and Conclusions.