Cross-slip, twinning and fracture are major deformation modes adopted by loaded materials. It appears sound that these apparently different deformation mechanisms can be analysed on the equal manner!
Cross-slip, twinning, and fracture systems under applied loadings receive the same mathematical theory using continuous distributions of elliptical dislocations in the framework of linear elasticity. Essentially the theory provides a quantity G that is a ratio, defined as the decrease ΔE of the total energy of the system divided by the corresponding change ΔS of the surface of the dislocation distribution, after incremental infinitesimal time dt: G= -ΔE/ΔS. In fracture G is the energy release rate or crack-extension force per unit length of the crack-front. Stationary configurations under which d = 0 are those observed experimentally. is the value of G averaged over all the spatial positions on the defect front. Please refer to the following works for details: Conoidal crack with elliptic bases, within cubic crystals, under arbitrarily applied loadings-I. Dislocation, crack-tip stress, and crack extension force; -II, III, and IV: Application to systems of twinning in copper (II), fracture in CoSi2 (III), and cross-slip in copper (IV). Theory and experiments completely agree.
Cross-slip, twinning and fracture are all deformations occurring in materials under applied loadings. The same math theory is used to derive the solution. The fundamental principle of dynamics and/or static could be used to derive the relevant equations. Hence, each of these mechanisms will have its own final equation. For a fracture of a porous media under high pressure fluid flow, the equation will also be different. The slip follows a line vector when the twin follows an rotation, the combination of slip and twin, the screw deformation will have both linear and angular deformation; the so-called second law of Newton (?) (fundamental principle of dynamic (or static if no movement) could help derive these specific equations; An easier way could be to use the total energy and the generalized entities in the Lagrangian equation. An easy approach is here:
Book Applied Mathematics: Theories, Methods and Practices, From s...
pages 175 or Chap 14 and more.
Hope this helps.
The G value follows the shape of the crack front. This is possible under stationary conditions as demonstrated by Bilby and Eshelby (1968). No acceleration in the equation of motion. We give credit to G, as defined, only in steady states. The G expression we have provided ends the calculations under such conditions! We remember that people as Eshelby first used Lagrange formalism to provide us with the force on an elastic singularity.
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