Small-scale plasticity critically needs a new mechanics description

1 Introduction

1 That plastic deformation is a jumpy process is indeed the norm rather than exception for a wide range of materials studied [1–7]. Metallic glasses are well known to deform by discrete shear bands [7], and in crystalline materials, the interactions of groups of dislocations often result in their movements in surges rather than in a continuous fashion, and these could lead to discrete signals such as spin-lattice relaxation detectable in nuclear magnetic resonance measurements [1], lattice waves recordable by an acoustic detector [2–4], or displacement or load excursions recordable by a nanoindenter [5, 6]. Even semicrystalline plastics exhibit discrete nanometric jumps during deformation, and these are likely to arise from the crystalline spherulites in them [8]. In the following, some examples of jumpy deformation in different materials are showcased, and then, suggestions for a replacement mechanics framework are made.

2 Crystalline materials

Nonferrous metals in bulk forms usually do not exhibit discrete yielding, but they were found to do so when in microwhisker forms [9, 10]. The upper yield point was often found to approach the theoretical strength limit and exhibit a strong decreasing dependence on the whisker thickness. These phenomena were attributed to the absence of preexisting dislocations in these small samples, and so at a high enough stress level, dislocations nucleate suddenly and their quick motion leads to the discrete yield. The condition of being initially dislocation-free is also easily achieved in nanometric-sized samples from which any preexisting dislocations can escape easily [11–15]. Another situation in which discrete first yield is often encountered is nanoindentation on well-annealed crystals. As shown by the example in Figure 1, a piece of well-annealed metal (in this case, Ni₃Al) would have an initial dislocation spacing easily reaching 100 μm or beyond, and if this is indented upon by a nanoindenter smaller than 1 μm, then the indenter would often interact with a dislocation-free region of the sample. The sudden generation of dislocations leads to a strain excursion if the experiment is load-controlled or a load drop if displacement-controlled. What is more interesting, however, is that the yield point itself is stochastic – on repeated measurements, the yielding loads from independent experiments scatter, as shown in the example in Figure 1. In addition, as shown in Figure 2, when the load is held within the initial elastic regime, yielding may occur suddenly after some waiting time, and this waiting time is not a constant but may distribute according to some probability density function, as shown in Figure 2. The probabilistic and time-dependent nature of the dislocation generation is thought to be due to the stochasticity of atomic vibrations under the highly stressed elastic state preceding the instability event [16, 17].

Figure 1

Sharp yield point observed during nanoindentation of well-annealed Ni₃Al crystals. Data from ref. [16].

Figure 2

Delayed incipient plasticity observed during nanoindentation of well-annealed Ni₃Al crystals. Data from ref. [16].

After the first yield, the subsequent deformation of small crystals is often found to be jerky, as illustrated by the example in Figure 3. Here, an aluminum micropillar was compressed at a constant load rate followed by unloading and then reloading to a constant stress level in order to study the creep behavior at room temperature. During both the load ramp and the load hold, large strain excursions occurred, as can be seen from the stress-strain plot. These corresponded to discrete slip steps, ranging from a few nanometers to ∼100 nm in size, on the sample’s free surface [18], as can be seen from the micrograph in Figure 3. Such magnitudes of slip steps are usual in bulk-sized specimens, but, as mentioned above, when they occur in micro-sized specimens, the associated strain changes can be as large as a few percent, as shown in Figure 3, and these are not anticipated in any continuum plasticity law. Over the past few years, a large body of literature on such jerky plasticity of small crystals has emerged [6, 11–15, 18–20].

Figure 3

Jumpy deformation in aluminum micropillars during load ramp and creep. Data from ref. [18].

3 Polymers

The jerky mode of deformation is also found in polymers, provided that the crystallinity is high. Figure 4 shows a nanoindentation experiment on polyethylene (PE) with different crystallinity [8]. The displacement of the indenter tip into the sample was monitored during a prolonged load hold, and as shown in the displacement-time plot, the specimen underwent discrete relaxation events on top of the smooth creep deformation. These discrete events, however, do not always happen, and they occur only when the nanoindenter hits specific locations of the sample. In a set of repeated tests in which the indenter hit different locations randomly, the occurrence frequency of these discrete events was found to increase with the crystallinity of the PE but decrease as the test temperature became higher. The positive correlation with crystallinity suggests that the discrete events are due to the crystalline phase of the polymer, and the negative temperature dependence is thought to be due to the enhanced viscous flow of the amorphous phase at higher temperatures, which relieves the stress for the discrete plasticity.

Figure 4

Discrete plastic events in PE during nanoindentation creep tests. Data from ref. [8].

4 Metallic glass

That metallic glass deforms in a jerky manner by the operation of discrete shear zones or bands has become a widely accepted phenomenon, and Figure 5 shows illustrative results from a nanoindentation experiment on a zirconium-based bulk metallic glass [5, 21]. Here, the displacement-load graph is serrated, and after removing the background noise signals that were of high frequencies, the net deformation bursts were found to increase steadily with the depth of indentation. This means a steady-state condition for the deformation bursts, as the indenter used was a pyramidal Berkovich type.

Figure 5

Jumpy flow in a zirconium-based bulk metallic glass during nanoindentation. Data from ref. [5].

5 Low-density materials

Low-density materials such as those with porous structures deform by discrete unstable events. When metal foams are crushed, these events are shear bands that run at an inclined angle to the compression axis [22], and in the case of honeycomb structures, a cascade collapse of the low-density structure layer by layer usually occurs [23]. As for the nano-regime, Figure 6 shows an example of anodic alumina with a nano-honeycomb structure. Here, both nanoindentation with a pyramidal (Berkovich) tip and crushing of micropillars by a flat-ended punch reveal that the deformation is jerky [24, 25]. In the latter case of uniaxial crushing, collapse in a layer-by-layer fashion can be seen clearly, and the collapse of each layer corresponds to a terrace in the stress-strain curve under a load-controlled mode.

Figure 6

Jerky deformation in anodised alumina with nano-honeycomb structure. Data from refs. [24, 25].

6 A materials mechanics framework for discrete plasticity

The above quick survey illustrates that jerky deformation is very common in small-sized materials of different types. The reason for jerky deformation in different materials has to be different, and the usual “materials science” paradigm is to look at the microstructural or atomic level mechanisms in individual materials to see how these lead to the deformation jumps. Although there is much merit in following such an approach and some progress has already been made for some material systems [26, 27], this is not always useful for engineering practices. Even if the mechanistic processes at a fundamental level are identified and the corresponding governing equations derived, these processes, or the associating material defects, often interact as a huge group and to accurately predict realistic behavior is still a formidable task, even with today’s computational power. Thus, a “materials mechanics” paradigm may be more useful here. As can be apprehended from the concept of yield surfaces in continuum plasticity, the aim in such a paradigm is to make use of results obtained experimentally or otherwise, often from simple load situations, to predict behavior in other load situations that are often more complex. Here, one does not necessarily need to look at the most fundamental mechanisms for the deformation but must know some coarse-grained description of the subject, such as a constitutive law. The results from the simple load situations are often used to identify the material parameters in such a law, and then the law is ready to be used to predict the behavior in other loading situations.

When adopting the above “materials mechanics” approach to discrete plasticity, one has to consider the stochastic nature of the plasticity. As the standard paradigm to deal with stochasticity is to consider an ensemble, the framework is similar to that in Figure 7. Here, a set (i.e., an ensemble) of repeated measurements under identical macroscopic conditions are used to tune an ensemble model so that it can be used to predict the mean-field and scatter behavior in other loading conditions. The key point in this framework is that a single measurement as input or a single predicted behavior as output serves little purpose because of the stochastic nature of the problem.

Figure 7

Materials mechanics framework for discrete plasticity.

An example of how such an ensemble model can be derived may be useful here. Suppose we have an ensemble of stress-strain curves, all obtained under identical macroscopic conditions, such as one of the sets shown in Figure 6. In each of these curves, identify all the discrete events and label them according to their ordinal m on loading, i.e., m=1 for the first event, m=2 for the second, and so on. For a given event ordinal m, the stresses at which this event occurs within the experiments in the ensemble (say, M of these) would scatter, and these yield stresses are then ranked in ascending order, i.e., σ_i,m, i=1, 2,…, M. Then, of this ensemble, the survivability of a sample without exhibiting the m^th event at a certain stress σ=σ_i,m is F_m=1-i/M. The top-right panel in Figure 8 shows an example of plots of F_m vs. σ generated from a set of compression experiments on aluminum micropillars (cf. Figure 3). Here, for each m (known as burst order in Figure 8), F_m drops with σ because the chance of not giving out that particular event should become lower as the applied stress increases. The sizes of the events (i.e., the bursts) were also measured in those experiments, and these are shown in the bottom-right panel of Figure 8. With the F_m(σ) plot determined for each event ordinal, a normalised emission rate of that event is given as , where is the (constant) loading rate used. In this equation, (F_m-F_m-1) is the population of the experiments in the ensemble, which are qualified to give out the m^th event. The model now assumes that is an intrinsic property of the material and applies equally well to other load situations. In a general loading scheme, the emission of the bursts can be predicted from a Monte Carlo scheme [28] such as that shown in Figure 8. Figure 9 shows that the jumps during a constant-load (i.e., creep) segment of the load schedule can be predicted rather satisfactorily by this Monte Carlo scheme, using the F_m(σ) gathered from a preceding load-ramp segment.

Figure 8

A Monte Carlo scheme of implementing the framework in Figure 7 [28].

Figure 9

Predicted discrete plasticity vs. experimental observation in compression of aluminum micropillars. Data from ref. [28].

The above shows just one example of how an ensemble model for discrete plasticity can be developed, and further discussion on this framework is available [29]. Other frameworks are possible, and it will be a worthwhile challenge for the materials mechanics community to come up with satisfactory ensemble models for discrete plasticity, especially for three-dimensional stress states.