Problems with Multiple Spatial Scales The Example of Plasticity

2 Problems with Multiple Spatial Scales The Example of Plasticity 

In the previous subsection, we used the example of diffusion to illustrate the proliferation of temporal scales in one of the central problems in the study of materials. The present discussion has a similar aim in that we will briefly review fhe features of plasticity that place modeling demands at many different spatial scales. Though plasticity is also an area of immense importance, the conceptual foundations for its analysis both at the macroscopic level as well as from a reductionist perspective are not nearly as mature as is the study of diffusion. Recall that at the macroscopic scale in the context of diffusion we have the time-honored diffusion equation while at the microseopic scale we have the machinery of transition state theory as the basis of a well-defined scheme for informafion passage. By way of contrast, the macroscopic equations of plasticity are not nearly as robust as the diffusion equafion and there is no clear path for 

As we have emphasized aheady, the study of plasticity is one of the centerpieces of the mechanics of materials. A wide array of technologies depend upon the ability to deform materials into particular desirable shapes, while from a scientific perspective I personally find the subject of great interest because it is an intrinsically dissipative process featuring history dependence and is a strong function of the material s microstructure. In addition, from the effective theory perspective, the study of plasticity is built around the motion and entanglement of dislocations which requires the construction of theories of lines and their interaction thus ushering in a certain nonlocality to the phenomenon right from the outset. 

Like in the case of diffusion, there are many instances in which a separation of scales approach is satisfactory. For example, in the treatment of polycrystal plasticity, the formulation of a constitutive law making no reference to the origins of plasticity in dislocation motions has been very successful. On the other hand, there are particular experimental facts that demand the treatment of several scales simultaneously. As noted above, a simplified view of plasticity 

The case studies considered in the preceding two subsections have attempted to set the stage for the present chapter by concretely arguing for the idea that there are a number of problems in the study of materials that feature several scales (in space or time or both) simultaneously. Though our argument centered on the treatment of diffusion as an example of a proliferation of temporal scales and plasticity as an example of a proliferation of spatial scales, the situation is exacerbated yet further when we consider the action of plasticity induced by mass transport, such as the discussion of creep featured in section 11.3. Our main purpose has been to remind the reader from within the narrowly focused perspective of the study of materials that the proliferation of scales is an everyday challenge. 

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