Eshelbian mechanics of novel materials: Quasicrystals

In this work, the so-called Eshelbian or configurational mechanics of quasicrystals is presented. Quasicrystals are considered as a prototype of novel materials. Material balance laws for quasicrystalline materials with dislocations are derived in the framework of generalized incompatible elasticity theory of quasicrystals. Translations, scaling transformations as well as rotations are examined; the latter presents particular interest due to the quasicrystalline structure. This derivation provides important quantities of the Eshelbian mechanics, as the Eshelby stress tensor, the scaling flux vector, the angular momentum tensor, the configurational forces (Peach–Koehler force, Cherepanov force, inhomogeneity force or Eshelby force), the configurational work, and the configurational vector moments for dislocations in quasicrystals. The corresponding [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocation loops and straight dislocations in quasicrystals are derived and discussed. Moreover, the explicit formulas of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for parallel screw dislocations in one-dimensional hexagonal quasicrystals are obtained. Through this derivation, the physical interpretation of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocations in quasicrystals is revealed and their connection to the Peach–Koehler force, the interaction energy and the rotational vector moment (torque) of dislocations in quasicrystals is established.

Redundancy-Free Integration of Rotational Quaternions in Minimal Form

Redundancy-free computational procedure for solving dynamics of rigid body by using quaternions as the rotational kinematic parameters will be presented in the paper. On the contrary to the standard algorithm that is based on redundant DAE-formulation of rotational dynamics of rigid body that includes algebraic equation of quaternions’ unit-length that has to be solved during marching-in-time, the proposed method will be based on the integration of a local rotational vector in the minimal form at the Lie-algebra level of the SO(3) rotational group during every integration step. After local rotational vector for the current step is determined by using standard (possibly higher-order) integration ODE routine, the rotational integration point is projected to Sp(1) quaternion-group via pertinent exponential map. The result of the procedure is redundancy-free integration algorithm for rigid body rotational motion based on the rotational quaternions that allows for straightforward minimal-form-ODE integration of the rotational dynamics.