Dynamic constitutive distortional law of materials—micro mechanism based on dislocation dynamics

CHAPTER SIX
Dynamic constitutive distortional law of materialsdmicro mechanism based on dislocation dynamics The discussion in the Chapter 5 indicate...

Dynamic constitutive distortional law of materialsdmicro mechanism based on dislocation dynamics The discussion in the Chapter 5 indicates that the core problem of what distinguishes the dynamic constitutive distortional law with the quasistatic one is the strain-rate effect. This core problem was discussed in the previous chapter from the view of macroscopic mechanics. Its microscopic physical foundation will be correspondingly discussed from the view of dislocation dynamics in this chapter. The problem of dislocation was raised in 1930s; it is mainly due to that the difference in orders of magnitude exists between the theoretical magnitude of crystal plastic distortion (shear) strength and the actual magnitude. The theoretical magnitude mentioned here is calculated for the perfect crystal as discussed in the Chapter 3, while the actual magnitude is experimentally measured for the actual crystal, which contains various microdefects, e.g., point defects, dislocations, twins, microcracks, phase transformations, and so on. From the microscopic view, the dislocation, i.e., the so-called onedimensional defect of crystals, plays a dominated role for the viscoplastic distortional law, especially its strain-rate sensibility. Consequently, the dislocation dynamics plays a leading role for the dynamic strain-rateedependent constitutive distortional law. Thus in the following, the discussion will be emphatically focused on the dislocation dynamics.

6.1 Theoretical shear strength Recalling the discussions in Sections 3.2 and 3.4 in Chapter 3, if a plastic shear slip occurs along a certain lattice plane in a perfect crystal under the action of shear stress s, the stress s must be large enough to overcome the binding force between the related crystal particles.

Dynamics of Materials ISBN: 978-0-12-817321-3 https://doi.org/10.1016/B978-0-12-817321-3.00006-1

Consider a plastic shear slip between two adjacent rows of crystal atoms under the action of shear stress s, as shown in Fig. 6.1, where a denotes the row spacing and b denotes the atomic spacing. Since a crystal has periodic lattice structure, the distribution of crystal binding energy (potential energy) U and binding force s (¼ vU vX ) must be described by period functions. If the locations A, B, C . in Fig. 6.1 represent balance positions, where the potential energy reaches the minimum (energy barrier valley) and the binding force is zero, then the binding force reaches its maximum when the slip distance equals b/4. Furthermore, when the slip distance equals b/2, the binding force reaches zero again, but the potential energy reaches its maximum (energy barrier peak), which is a substable state. After overcoming this maximum energy barrier, once the plastic slip distance reaches a magnitude of an atomic spacing b from a stable state, the potential energy decreases to the minimum again. Assume that the potential energy can be approximately expressed by a cosine function. U ¼ 2A cos

2pX b

(6.1)

where X denotes the slip displacement and 2A denotes the amplitude, then the corresponding shear stress s can be expressed as: vU 4pA 2pX ¼ sin (6.2a) vX b b When the slip displacement X<< b, Eq. (6.2a) can be simpliﬁed to: s¼

s¼

vU 4pA 2pX ¼ $ vX b b

Figure 6.1 Shear slip in a perfect crystal.

(6.2b)

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Dynamic constitutive distortional law of materials