Victor L. Berdichevsky

Tensors that are invariant with respect to subgroups of three-dimensional group of rotations were constructed first by Lokhin and Sedov ( J Appl Math Mech (PMM), 1963, 27, 597–629). In this paper, we do one more step and derive formulas for the elastic modulus tensors for all symmetry groups from the Lokhin–Sedov results. The tensor basis obtained in this way has an advantage of being built upon the same tensors that are used in construction of non-linear tensor functions. One consequence of the formulas derived is that the number of types of anisotropic elastic behaviors is ten. There is a long-standing controversy concerning this number; many authors claim that the correct number is eight. We discuss this controversy in detail, and add several points to the Fedorov–Khatkevich argument that initiated this controversy. We formulate the tensor relationship which is the underlying cause of the Fedorov–Khatkevich argument. This relationship yields several generalizations (extension to cubic symmetries, the inversed argument). We argue that for crystals, ten types of elastic response cannot be reduced to eight due to built-in intrinsic structure determining the crystal symmetry. For polycrystals with a priori unknown elastic symmetry, the problem of determining the symmetry by elastic moduli is shown to be ill-posed.

Stress and temperature dependence of strain rates in dislocation-mediated plasticity is usually associated with the depinning process. This is an escape-from-the-well process, thus such dependence is of Arrhenius type, Function arises as an asymptotic approximation in the escape-from-the-well problem as or The activation energy is usually accepted to be a linear function of stress sigma. To describe stress-strain curves at high strain rates, Langer, Bouchbinder and Lookman (Acta Mater., 2010, 58, 3718-3732) successfully employed a double-exponential function or with two parameters, and Parameter was associated with the flow stress. The major difference from the usual exponential dependence is the existence of an upper bound for strain rates no matter how high stresses are. For large stresses, the power vanishes and the question arises as to whether the double-exponential stress dependence is consistent with the escape-from-the-well problem. In this paper, we give a positive answer to this question. The key point is that the boundedness of strain rates is caused by the boundedness of dislocation velocities: dislocation velocities cannot exceed the shear wave speed. Accordingly, the velocity-force relation is nonlinear. Incorporation of such nonlinear relation into the escape-from-the-well problem results in a strain rate-stress dependence which has a form of a double-exponential curve.

Energy of a dislocation ensemble depends on dislocation positions, and, since the details of dislocation configuration are not known, should be viewed as a random number. We study numerically statistical characteristics of energy for periodic ensembles of edge dislocations with a large number of dislocations in a periodic cell. We confirm that there is a limit probability density function of energy for randomly placed dislocations as The situation with probability density of energy of equilibrium dislocation states is less clear: probability density keeps evolving as increases. We suggest a plausible hypothesis on the limit behaviour of energy distribution. We also study the probability distribution of internal resistance stresses, as well as the derivatives of energy with respect to dislocation polarisation. Internal resistance stress equilibrates the external shear stress in a loaded dislocation structure. We explain why probability distribution of internal resistance stress for randomly placed dislocations does not have a limit for and confirm this by numerical simulations. Surprisingly, numerical simulations show that such limit exists for equilibrium dislocation structures. A preliminary analysis of geometry of equilibrium dislocation structures is also given.