Kinking Nonlinear Elasticity and the Deformation of Magnesium
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INTRODUCTION
AFTER several decades of work on the deformation of single[1–8] and polycrystalline[9–32] Mg at small ( b (Figure 1(a)) and showed that the remote axial stress at the polycrystal level, rc needed to render such a subcritical KB unstable, i.e., to grow spontaneously, is given by[52] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rc 4G2 bcc b s>sc ln ½1 cc w M 2ap2 where sc is the critical shear stress at the single crystal level, G is the shear modulus, b is the Burgers vector,
1742—VOLUME 40A, JULY 2009
where m is the Poisson ratio and D is the distance between dislocations in the IKB. In this work, we assume w = 5b and, therefore, at 0.01 rad or 0.3 deg, cc is quite small. The use of Eq. [2] here implies that the local stress needed to nucleate an IKB is G/170.[34] In our work to date, primarily on the MAX phases, we have shown that 2a is equal to the grain dimension normal to the direction of easy slip.[38,54] Thus, for Mg, similar to the MAX phases, that would be the dimension of the grain along the c-axis. An IKB consists of multiple parallel dislocation loops (Figure 1(a)). As a first approximation, each loop can be assumed to be comprised of two edge and two screw dislocation segments with lengths 2bx and 2by, respectively. The latter are related to the applied stress r and 2a, assuming 2bx
2að1 mÞ r Gcc M
and
2by
2a r Gcc M
½3
The formation of an IKB can be divided into two stages: nucleation and growth.[34] We do not understand the nucleation process. Thus, in our model,[34,38,41,52] only IKB growth from 2bxc and 2byc to 2bx and 2by, respectively, is considered. The dislocation segment lengths, 2bxc and 2bxc, of an IKB nucleus are presumed to pre-exist or are nucleated during prestraining. The values of 2bxc and 2bxc are estimated from Eq. [3], assuming r = rt, where rt is a threshold stress. The latter is, in turn, experimentally obtained (Figure 5(b)). It follows that, for r > rt, the IKB nuclei grow; the IKB-induced axial strain resulting from their growth is assumed to be given by[34] eIKB ¼
DVNk cc 4pð1 mÞNk a3 2 ¼ r r2t ¼ m1 r2 r2t 2 2 k1 3k1 G cc M ½4
where m1 is the coefficient before the term in brackets in the third term. The term Nk is the number of IKBs per unit volume. The product VNk is the volume fraction of the material that is kinked. The factor k1 relates the volumetric strain due to the IKBs to the axial strain along the loading direction. Very recently, Clausen et al.[33] showed that, when oriented Mg samples are uniaxially compressed, the relationship between the volume fraction of tensile twins formed v and the uniaxial engineering strain measured was more or less linear, with a slope of 8. Because that strain associated with a tensile twin is 0.13, it follows that k1 1.04. It is important to note here that, when the stress-strain curves obtained by
METALLURGICAL AND MATERIALS TRANSACTIONS A
Clausen et al. are plotted on the same graph as our results (Figure 4(d)), they are quite comparable. (The stress values of Clausen et al. were divide
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