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Tham khảo tài liệu 'mechanical behaviour of engineering materials - metals, ceramics, polymers and composites 2010 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 188 6 Mechanical behaviour of metals Fig. 6.22. Straight dislocation line subjected to a shear stress on an area lil2. When the dislocation has moved by I2 the upper half of the crystal has slipped by one Burgers vector b. This requires a work of E Fext b Tli l2b. 6.13 The dislocation has moved by a distance l2. The work needed can also be calculated by E Fdl2 when Fd denotes the force on the dislocation. As both energies are equal the force on the dislocation is Fd Tl1b . 6.14 Here we used the fact that the force is perpendicular on the dislocation line. If the orientation between the stress tensor a the dislocation line li and the Burgers vector b is arbitrary the Peach-Koehler equation Fd a b X li 6.15 holds. Equation 6.15 can be derived in a similar way to equation 6.14 by calculating the energy. If the dislocation line is displaced by 12 the crystal above the covered area has slipped. The normal vector in this area is given by the cross product 11 X l2 li X 12 . The stress in this area is g 11 X 12 11 X 12 resulting in a force of g 11 X 12 . Because the crystal has slipped by a Burgers vector the work is E g 11 X I2 b. Using rules for scalar and vector products this equation can be rewritten due to symmetry of the stress tensor E g b 11 X I2 g b X11 12. This energy equals the force on the dislocation multiplied by 12 E Fd 12 . 6.3 Overcoming obstacles 189 As the energies must agree for arbitrary 12 the Peach-Koehler equation results Fd ơ b X 11 . Peierls force We already saw in section 6.2.3 that atomic bonds have to flip for a dislocation to move. This requires stretching of the bonds and therefore needs energy. The resulting force fixes the dislocation at its momentary position and has to be overcome to move it. Thus if the applied stress is too small no dislocation movement is possible and the crystal cannot deform plastically. Figure 6.13 above illustrates this using the sphere model of atoms. This retaining force is called Peierls force or Peierls-Nabarro
