Stored energy, microstructure, and flow stress of deformed metals

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DURING plastic deformation of metals, a small part of the mechanical energy is stored in the form of dislocations, which may organize in various configurations. The thermomechanical behavior of a metal sample is affected by this stored energy and is therefore an important microstructural parameter. The stored energy can be measured by calorimetry, or it can be estimated through relationships between the stored energy and microstructural parameters or between the stored energy and flow stress.[1,2,3] The relationship between the stored energy (Es) and the deformation microstructure can be expressed by the equation ES  r # E


where  is the total dislocation density, and E is the energy per unit length of dislocation line. Presupposing that the dislocations are arranged in low-energy dislocation boundaries, the energy per unit area of dislocation boundary can be related to the angular misorientation across the boundaries by the Read–Shockley equation. In this work, we use the following formulation of the Read–Shockley equation: g  gm (u/um) [1  ln (u/um)] :u  um g  gm

:u  um

ES  SV #g E (r0)


where E(0) is a contribution from individual dislocations present in the volume between the dislocation boundaries. For medium and high stacking fault energy materials, this contribution is small. In the following, where data for aluminum are presented, E(0) will therefore be neglected (note that we also ignore the relatively small contribution to the stored energy from the original high-angle grain boundaries in the undeformed material). The relationship between the stored energy and the flow stress given in Eq. [1] can be approximated as ES  KrGb2


where G is the shear modulus, b is the Burgers vector, and K is a constant approximately equal to 0.5.[4] Eq. [4], combined with the empirical relationship


s  s0  MaGbr0.5


gives the following relationship between the stored energy and the flow stress:

where m is the energy per unit area of a high-angle boundary,  is the boundary misorientation, and m is the misorientation angle above which the energy per unit area is independent of misorientation angle. The stored energy per unit volume due to a dislocation boundary of misorientation angle is then obtained by mul-

A. GODFREY, W.Q. CAO, and Q. LIU are with the Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China. Contact e-mail: [email protected] N. HANSEN is with the Materials Research Department, Risø National Laboratory, DK-4000 Roskilde, Denmark. Manuscript submitted April 21, 2004. METALLURGICAL AND MATERIALS TRANSACTIONS A

tiplying the energy per unit boundary area with the area per unit volume (SV) for the boundary. The stored energy due to dislocations is therefore given in general by an expression of the form

(s  s0)2  (Ma)2 (G/K)ES



where  is the flow stress, 0 is the friction stress, M is the Taylor factor, and  is a number approximately equal to 0.2 to 0.3.[5] The valid