6 - Theoretical Connections between Crystal and Aggregate Behavior

Summary

In this chapter we return to the general theoretical framework of Chapter 3 and extend it to the analysis of characteristics of overall response of macroscopically uniform polycrystalline solids. The objective is the presentation of a rigorous theoretical connection between single-crystal elastoplasticity and macroscopic crystalline aggregate behavior. The development is based upon the original analysis of Hill (1972) and other basic contributions in Hill & Rice (1973), Havner (1974, 1982a, 1986), and Hill (1984, 1985). Central to an understanding of the crystal-to-aggregate transition is the well-known “averaging theorem” introduced by Bishop & Hill (1951a) but only given its final form and initial proof at finite strain in Hill's (1972) seminal work.

Crystalline Aggregate Model: The Averaging Theorem

At the beginning of Chapter 3, the scale of a crystal material point in a continuum model was defined to have linear dimension of order 10⁻³ mm: greater than 10³ lattice spacings but at least an order of magnitude smaller than normal grain sizes in polycrystalline metals. Consider now the choice of physical size of a representative “macroelement” that defines a continuum point at the level of ordinary stress and strain analysis (that is, in structural and mechanical components or materials-forming operations.)

The wall thickness of thin-walled metal tubes used in combined stress tests (say, axial loading and torsion) often is in the range 1−2 mm and 10 to 30 grains (see, for example, Mair & Pugh (1964) or Ronay (1968)).