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

The universal law of Unified mechanics theory is given by^[1]

Second law

${\displaystyle F(1-\phi )={d(mv) \over dt}}$

where ${\displaystyle m}$ is mass, ${\displaystyle v}$ is velocity, ${\displaystyle \phi }$ is the thermodynamic state index.

Third law

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

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

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

where ${\displaystyle \phi }$ is the thermodynamic index, ${\displaystyle \phi _{cr}}$ is the user defined value of TSI, ${\displaystyle m_{s}}$ the molar mass, ${\displaystyle R}$ is the gas constant.

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

When a material in ground (reference) state, it is assumed to be free of any possible defects. It can be assumed that “damage” in a material is equal to zero. Therefore TSI will be ${\displaystyle \phi =0}$. However ${\displaystyle \phi }$ does not have to be taken as zero initially. In the final stage, material reaches a critical thermodynamic state, such that entropy is maximum. At this stage, entropy production rate will become zero, and ${\displaystyle \phi =\phi _{cr}}$. The function of ${\displaystyle \phi _{cr}}$ is to adequately capture the specimen’s critical states of interest. For example, in the electromigration analysis of microelectronics solder joint, ${\displaystyle \phi _{cr}=0.1}$ is used because 10% change in electrical resistance is considered failure in microelectronics.^[5]

Thermodynamic state index coordinate system

The following example can explain this coordinate system

A 5-year-old boy with terminal illness and a 100-year-old sick person would have different coordinates in the time axis. But they have the same coordinate on TSI axis. Because using Newtonian space-time coordinates without incorporating thermodynamics, the physical state of a system [ a person]  cannot be defined. On the other hand, in the unified mechanics theory their thermodynamic state is represented by the TSI axis in addition to the space-time coordinates. On TSI axis at ${\displaystyle \phi =0.999}$ , 5-year-old boy with terminal illness and a sick 100-year-old man will have the same thermodynamic state index coordinate. Essentially, a person’s age according to calendar does not give any information about that person’s thermodynamic physical state.^[1]

Major difference between Newtonian Classical Mechanics and Unified Mechanics

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.

Applications

Unified mechanics theory based model is a model that uses entropy as a damage metric with a pure physics-based damage evolution function.^[6] It has been extensively studied and experimentally validated in the field of fatigue life prediction in the past 15 years, including the fatigue life under thermomechanical loading,^[3][7][8] electrical-thermal-mechanical loading,^[9][10][11] and fatigue life prediction in particulate composites.^[12][13]

References

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