Instytut Podstawowych Problemów Techniki

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The peptide chain is one of the most important biological structures. In this paper it is treated as kinematic chain, the peptide units being regarded as rigid bodies. We begin with a very simple static problem, in which the choice of variables entering the interaction energy follows from a more general theory. Exact solutions are presented for some simple energies and their stability examined. In the case of long chains loaded by a force on its ends, deterministic chaos may occur. A brief account of the continuum theory is given emphasizing its origins in the discrete theory.

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Much research has been done on the dynamics of the peptide chain. The amount of experimental, theoretical and numerical work on structure, stability and folding is staggering. However, there is no widely known analytical model derived from first principles. In this paper we derive equations for the dynamics of a peptide chain in a “quadrilateral chip” model, passing to the continuum from the discrete chain structure and assuming various local interactions. Ours is therefore a kinematic chain [1, 2]appearing in many mechanical structures. However, we include the appropriate constraints peculiar to peptides. Dihedral interactions are included and, when strong, are seen to ensure stability. Thus, they have the same effect as the non-local hydrogen bonds. The celebrated helices are exact particular solutions of our general equations. In order to obtain eigenfrequencies, a simplified model is introduced and peptide oscillations with frequencies of the order of 1013s−1 are obtained, in agreement with observations. Solitons are found in very restricted cases, thus far. External forces, such as those due to the solvent, are included in the model (Section 8).

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A simple dipole chain is the main building block of many physical models in solid state physics. Here, just such a chain of essentially identical dipoles is investigated by considering nearest-neighbor interactions of the individual charges in the dipoles. A three-dimensional model is found. In a restricted, two-dimensional picture, a nonlinear, second-order partial differential equation, more general than sine-Gordon, results. Soliton-kink solutions are found.

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A derivation is given for the force per unit surface of a defect (e.g. a dislocation) in a non-linear elastic medium. The derivation is based on a Lagrangian of the same form as in linear elasticity, introduced in [1]. Principle of energy conservation is deduced on the basis of a particular case of Noether's theorem. Some peculiar properties of the Lagrangian are discussed. In the particular case of a constant Burgers vector the force reduces to the Peach-Koehler force per unit dislocation line, known in the linear theory of elasticity.

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The paper is devoted to a statistical derivation of the equations governing a continuous distribution of dislocations in a linear elastic medium. We begin with a system of infinitesimal Somigliana dislocations moving in an elastic medium in accordance with the laws of dynamics of discrete dislocations. By introducing the classical phase space with its Liouville and transport equations and defining the appropriate expectation values we derive in the usual manner the equations for the density of the “dislocation fluid”, its velocity and the average elastic field. As a result we arrive at a compound continuous medium DR constituting a mixture of a material elastic body and the dislocation fluid. The system of equations constitutes a system of seven quasi-linear partial differential equations which are shown to be hyperbolic under certain definite conditions. Some general features of the system are discussed and a one-dimensional example examined in more detail to demonstrate some properties of the DR medium; thus, shock waves and slip planes are shown to exist. The possibility of constructing in this manner “plastic” or “elastic-plastic” media is briefly considered.

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On the basis of a variational principle we investigate the motion of a concentrated (point) defect with a closed surface (vacancies, interstitial atoms, etc.). We confine ourselves to a quadratic Lagrangian and hence, linear equations of motion. The localization (regularization of the self-part of the Lagrangian) is carried out and leads to explicit differential equations of motion. Three examples are worked out: a free motion of a defect, motion of a defect in the field of a static concentrated force and interaction between the defect and a plane elastic wave.

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The motion of a concentrated (point) defect in an elastic medium is investigated on the basis of a variational principle. The equations of motion and the principles of conservation of energy are derived and examined in some detail. The localization of the Lagrangian makes it possible to regularize its singular part and deduce explicit differential equations of motion. The radiation damping force is introduced by means of the Wheeler-Feynman procedure. In the paper we confine ourselves to the quadratic Lagrangian and hence linear equations of motion.

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