Abstract
Discrete mechanics proposes an alternative formulation of the equations of mechanics where the Navier-Stokes and Navier-Lam\'e equations become approximations of the equation of discrete motion. It unifies the fields of fluid and solid mechanics by extending the fields of application of these equations to all space and time scales. This article presents the essential differences induced by the abandonment of the notion of continuous medium and global frame of reference. The results of the mechanics of continuous medium validated by fluid and solid observations are not questioned. The concept of continuous medium is not invalidated, the discrete formulation proposed simply widens the spectrum of the applications of the classical equations. The discrete equation of motion introduces several important modifications, in particular the fundamental law of the dynamics on an element of volume becomes a law of conservation of the accelerations on an edge. The acceleration considered as an absolute quantity is written as a sum of two components, one soledoidal the other irrotational according to a local orthogonal Helmholtz-Hodge decomposition. The mass is abandoned and replaced by the compression and rotation energies represented by the scalar and vectorial potentials of the acceleration. The equation of motion and all the physical parameters are expressed only with two fundamental units, those of length and time. The essential differences between the two approaches are listed and some of them are discussed in depth. This is particularly the case with the known paradoxes of the Navier-Stokes equation or the importance of inertia for the Navier-Lam\'e equation.
Highlights
The concept of continuous medium developed and used for centuries has led to theoretical predictions which are in agreement with physical observations in a large number of fields of physics, in particular in mechanics [18] and in classical field theory including relativity [19]
The limitations of the classical equations observed for extreme physical phenomena or wider domains of validity are not so much due to the equations themselves as to the notion of continuous medium itself. This concept can be replaced by the notion of discrete medium where the conservation of acceleration on a segment is adopted as a postulate by discrete mechanics
The derivation at a point, the integration and the mathematical analysis must be adapted to the discrete medium, but the differential geometry and the exterior calculus seem to be sufficient to define the framework of discrete mechanics
Summary
The concept of continuous medium developed and used for centuries has led to theoretical predictions which are in agreement with physical observations in a large number of fields of physics, in particular in mechanics [18] and in classical field theory including relativity [19]. Even if the notion of continuous medium has indisputable advantages, the derivation at one point, integration, mathematical analysis, etc., the reduction at one point of the different quantities, variables and physical parameters is not without posing coherence problems solved at the cost of hypotheses and approximations which limit its generalization to certain areas of physics. It is for example the introduction of fictitious forces into the equation of motion to compensate for real forces and translate the mechanical equilibrium. Another drawback of the continuous approach is linked to the transition to discrete equations which requires a spatial discretization step on the basis of numerical methodologies disconnected from physical modeling
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