ORALS
SESSION: MathematicsTuePM2-R8
| 3rd Intl. Symp. on Sustainable Mathematics Applications |
| Tue Nov, 6 2018 / Room: Grego (50/3rd) | |
| Session Chairs: Peter Rowlands; Constantin Udriste; Session Monitor: TBA |
16:45: [MathematicsTuePM211]
Reconsideration of Continual and Statistical Mechanics Based on Scalar Representation of Deformation Valeriy
Ryabov1 ;
1Department of Nuclear Technology, National Research Centre, Obninsk, Russian Federation;
Paper Id: 175
[Abstract] Theory of particle and continuous mechanics is developed, which allows a treatment of deformation in terms of molecular variables "coordinate-momentum-force", instead of the standard treatment in terms of tensor-valued variables "strain-stress". The new concept is based on a representation of the classical mechanics on the surface of Euclidean 6-torus. The six parameters representing the topological dimensions of the torus are responsible for three stretches and three angles related to their orientation in a deformed body. A withdrawal of the strain description from the coordinate system of deformed body to an extended one changes and simplifies essentially all computational basis in particle, continual and statistical mechanics. Instead of stress-strain relation the new constitutive equations contain a dependence of generalized tension forces acting on each atom (or small element of mass) on scalar deformation parameters. In other words, any notions of surface or volume forces could be excluded from statistical and continual mechanics.
The novel concept generates a new type of ensemble with constant tension force NfE. The much simpler principle of virtual work implicit there gives a serious advantage over the widespread isostress ensemble NtE in molecular dynamics (MD) simulations. Besides, the equality of internal tension forces to the boundary forces (traction) enables fully atomistic MD calculations of deformation. In thermodynamic limits, instead of the pressure and volume as state variables, this ensemble employs deformation forces measured in energetic unit and stretch ratios. So the changes might be spread even to formulas in school textbooks.
The governing equations of nonlinear elastostatics for inhomogeneous medium are also formulated. Unlike the standard theory, there is no need for compatibility conditions or the Saint-Venant's principle to justify solutions to boundary value problems in elasticity theory. It suggests a completely different strategy in continual mechanics computations. The conventional algorithms use a finite element analysis for strain-driven constitutive equation. Unlike this, a discretization schema in terms of scalar deformation variables is dealing directly with coordinate dependence of large scale deformation based on fully atomistic foundation. But the most important consequence of derived approach is that the continual mechanics ceases to be a separated and independent branch of theoretical physics, and instead goes over to a part of particle physics. Several key examples illustrate the implementation of the new theory in calculations of statics and dynamics and thermodynamics of deformed solids.
References:
[1] V.A. Ryabov, Mechanics of deformations in terms of scalar variables. Cont. Mech. Thermodyn. Springer, 29, 3, 715 (2017).
[2] V.A. Ryabov, Implementation of isotension ensemble in molecular dynamics. Comp. Meth. Appl. Mech. and Eng., Elsevier, accepted for publication (2018).
