Unified mechanics theory

Unified mechanics theory^[1][2][3][4] unifies the laws of newton and the laws of thermodynamics in the ab-initio level.  As the result, the governing equation of any system automatically includes entropy generation [energy loss, dissipation, degradation, damage, etc.].^[1] Unified mechanics theory introduces an additional linearly independent axis into Newtonian space-time coordinate system. The additional axis is called Thermodynamic State Index (TSI) axis. The TSI axis can have values between zero and one. Calculating the TSI axis coordinate of a material point requires deriving the thermodynamic fundamental equation of the material, without use of empirical curve fitting methods, using principals of physics and mathematics. Evolution of the TSI axis follows the Boltzmann entropy formulation of the second law of thermodynamics.

Universal laws[edit]

^[1]

Second law[edit]

${\displaystyle F(1-\phi )={d(mv) \over dt}}$where ${\displaystyle m}$ is mass, ${\displaystyle v}$ is velocity, ${\displaystyle \phi }$ is the thermodynamic state index.

Third law[edit]

${\displaystyle F_{12}={dU_{21} \over du_{21}}={d \over du_{21}}[{1 \over 2}k_{21}[1-\phi ]u_{21}^{2}]}$

where ${\displaystyle F_{12}}$ is the acting force, ${\displaystyle U_{21}}$ is the strain energy of the reacting system, ${\displaystyle u_{21}}$ is the displacement of the reacting system, ${\displaystyle k_{21}}$ is the stiffness of the reacting system.

Thermodynamics state index[edit]

${\displaystyle \phi =[1-e^{({-m_{s}\Delta s \over R})}]}$

where is ${\displaystyle m_{s}}$ the molar mass, ${\displaystyle R}$ is the gas constant.

${\displaystyle \Delta s=\int _{\Delta t}{\dot {s}}dt}$ is the change in entropy

Thermodynamics state index axis

Major difference between Newtonian Classical Mechanics and Unified Mechanics[edit]

1. In Newtonian mechanics there is no axis to define derivative with respect to entropy. As a result all derivatives with respect to entropy are assumed to be zero.
2. In Newtonian mechanics entropy generation must be defined with an empirical function or pseudo-force (like a damping force in structural dynamics). In Unified mechanics entropy generation is included in the universal laws of motion. Hence no empirical curve fitting dissipation/degradation/void/fatigue/failure/damage evolution functions or pseudo-dissipation forces (like damping) are needed. However, the unified mechanics theory does require derivation of the fundamental equation.

References[edit]