The effect of boundary spacing on substructure strengthening E. Nes, K. Marthinsen and B. Holmedal The subgrain size and the spacing of high angle boundaries are important parameters used to describe the microstructure in metals deformed to large strains. How the flow stress depends on the boundary spacing is discussed here and it is argued that the best way this is treated is in the work hardening model developed by Nes and co- workers (Progr. Mater. Sci., 1998, 41, 129 – 193; Mater. Sci. Tech., 2001, 17, 376 – 387; Mater. Sci. Eng., 2002, A 322, 176 – 193). The theoretical arguments given are supported by experimental observations. MST/6080 Keywords: Flow stress, Work hardening, Substructure strengthening, Grain boundary strengthening, Modelling The authors are in the Department of Materials Technology, Norwegian University of Science and Technology, N – 7491 Trondheim, Norway (knut.marthinsen@material.ntnu.no). Manuscript received 23 December 2003; accepted 9 June 2004. # 2004 Institute of Materials, Minerals and Mining. Published by Maney on behalf of the Institute. Introduction The problems of work hardening of metals and alloys are many and complex, and how to solve them is still a matter of controversy. A common approach is to divide the problem in two. The first problem is then how to calculate the critical resolved flow stress for a given microstructure resulting from a prior deformation history, and the second problem is how to explain that this deformation history has produced the given microstructure, i.e. the microstructure evolution problem. Different ‘schools’ have approached these problems in different ways. One major difference is the mathematical representation and the experimental measure- ments needed to describe the microstructure. A classical approach relies on a one-parameter microstructural description, i.e. during plastic deformation the flow stress is controlled by the average accumulated dislocation density. The work hardening model developed by Kocks 4 and further refined by Kocks and Mecking, 5 Mecking and Estrin 6 and others relies on such a one-parameter descrip- tion. The way the dislocations are spatially organised into cell or sub-boundary structures is not included explicitly in these treatments. A cell structure as a basis for the modelling of work hardening was first introduced by Kuhlmann-Wilsdorf, 7 and later workers 8 – 13 have com- monly applied a multi-parameter description similar to that pioneered by her. In more recent papers Nes and cow- orkers 1–3 developed a new approach to the problems of work hardening, which relies on a substructure description that is more realistic in terms of what is found in heavily deformed metals. It is well documented that deformation to large strains results in a complex substructure built up of low- and high angle grain boundaries. The objective of this paper is to focus on the first problem identified above, namely how to calculate the flow stress for a given microstructure. A theoretical argumentation is given for how the boundary spacing affects the flow stress. It is explained how Nes and coworkers 1–3 incorporate this into their model and how the theory is supported by a number of experiments. The boundary spacing model During plastic deformation of metals a mobile dislocation will on average travel a slip-distance L~C/dr from the source until becoming stored, where r is the total stored dislocation density and C is a constant of order 100. In other words, when an expanding mobile dislocation loop finally comes to rest (gets stored) it will, on average, have cut through hundreds of subgrains and ten thousand of dislocations within the subgrains. To keep the discussion simple we will ignore long-range elastic interactions between mobile dislocations and the sub-boundaries. It is contro- versial to what extent this may be justified, but nevertheless the discussion here is still valid. The subgrain structure and the expanding loops will interact in three principally different ways. First, an indirect interaction that pertains to the effect of the dislocations stored in the sub-boundaries on the slip length L and through this parameter on the storage rate of disloca- tions. 2,3,14 (In polycrystalline materials the slip length is further restricted by the grain size.) Second, the sub- boundary dislocations provide a direct effect on the flow stress through short-range interactions with the mobile dislocations. Third, we expect a dislocation bow-out contribution to the flow stress. This effect yields the largest boundary-contribution to the flow stress and will be further discussed in the following. An expanding dislocation loop will interact with the subgrain structure as schematically outlined in Fig. 1a and b. When the dislocation segment in subgrains marked A and B migrates as illustrated, a situation as outlined in Fig. 1b will arise. On average such subgrains will in general be misoriented typically by 3–5 degrees. The gliding segments in each of these subgrains will be pinned by the reaction product (debris) left behind in the sub-boundary. A consequence of such pinning reactions is that the boundary spacing provides an Orowan type flow stress contribution given by ^ t~a 2 Gb d : : : : : : : : : : : : : : (1) Here d is the sub-boundary separation and a 2 is a geometric constant of the order of unity (experimentally a 2 #2 is found to be a typical value). Deformation of a polycrystalline metal to large strains results in a complex microstructure built up of low and high angle grain boundaries. A characteristic aspect is a laminated grain boundary structure resulting from the shape change of the original grains and new high angle boundaries created as a result of grain break-up. The schematic drawing in Fig. 1 pertains to a subgrain structure in which the misorientations are represented by a contin- uous crystal lattice over the sub-boundaries. Such a simple picture of a continuous lattice does not apply to high angle boundaries. A bow-out term, however, is relevant also in DOI 10.1179/026708304225022250 Materials Science and Technology November 2004 Vol. 20 1377