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Continuum mechanics
About: Continuum mechanics is a research topic. Over the lifetime, 5042 publications have been published within this topic receiving 181027 citations.
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TL;DR: In this paper, a 1-d, scalar, time-dependent, Hamilton-Jacobi equation is formulated as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip.
Abstract: Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a travelling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton–Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes–Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.
45 citations
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01 Jan 2001
TL;DR: In this article, the class of thermodynamically compatible systems of balance laws with source terms is considered and every system of this class is hyperbolic and generated by only one thermodynamic potential.
Abstract: The class of thermodynamically compatible systems of balance laws with source terms is considered. Every system of this class is hyperbolic and generated by only one thermodynamic potential. Besides, each equation of such system has a conservative form. For instance, equations of motion of elastic condutors and multiphase media are considered.
44 citations
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TL;DR: This paper extends Hardy's formulas by systematically incorporating both spatial and temporal averaging into the expression of continuum quantities, and the derivation follows the Irving-Kirkwood formalism.
Abstract: In non-equilibrium molecular dynamics simulations, continuum mechanics quantities can be computed from the position and momentum of the particles based on the classical Irving–Kirkwood formalism. For practical purposes, the implementations of Irving–Kirkwood formulas often involve a spatial averaging using a smooth kernel function. The resulting formula for the stress has been known as Hardy stress. Usually results obtained this way still need to be further processed to reduce the fluctuation, e.g., by ensemble or time averaging. In this paper we extend Hardy's formulas by systematically incorporating both spatial and temporal averaging into the expression of continuum quantities. The derivation follows the Irving–Kirkwood formalism, and the average quantities still satisfy conservation laws in continuum mechanics. We will discuss the selection of kernel functions and present several numerical tests.
44 citations
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TL;DR: In this article, a closed form solution for the natural frequencies of a rectangular simply supported nano-plate is obtained by using state-space method in the thickness direction and Fourier series in the in-plane directions.
Abstract: Vibration analysis of a nano-plate, based on three-dimensional theory of elasticity, is studied employing non-local continuum mechanics. By using state-space method in the thickness direction and Fourier series in the in-plane directions, a closed form solution for the natural frequencies of a rectangular simply supported nano-plate is obtained. To verify the accuracy of the present approach, numerical results are compared with the results available in the literature. The effect of the non-local parameter, aspect ratio, thickness-to-length ratio and half-wave numbers in the frequency behavior is examined.
44 citations