The systematics of energetic terms as they are taught in continuum mechanics deviate seriously from the standard doctrine in physics, resulting in a profound misconception. It is demonstrated that the First Law of Thermodynamics has been routinely reinterpreted in a sense that would make it subordinate to Bernoulli's energy conservation law. Proof is given to the effect that the Cauchy stress tensor does not exist. Furthermore, it is shown that the attempt by Gibbs to find a thermodynamic understanding for elastic deformation does not sufficiently account for all the energetic properties of such a process.
The nature of elastic deformation is examined in the light of the potential theory. The concepts and mathematical treatment of elasticity and the choice o f equilibrium conditions are adopted from the mechanics o f discrete bodies, e. g., celestial mechanics; they are not applicable to a change o f state. By nature, elastic deformation is energetically a Poisson problem since the buildup of an elastic potential implies a change of the energetic state in the sense of thermodynamics. In the Euler-Cauchy theory, elasticity is treated as a Laplace problem, implying that no change of state occurs, and there is no clue in the Euler-Cauchy approach that it was ever considered as one. The Euler-Cauchy theory of stress is incompatible with the potential theory and with the nature of the problem; it is therefore wrong. The key point in the understanding of elasticity is the elastic potential.
The Cauchy stress theory has been shown to be profoundly at variance with the principles of the theory of potentials. Thus, a new physical approach to deformation theory is presented, which is based on the balance of externally applied forces and material forces. The equation of state is generalized to apply to solids, and transformed into vector form. By taking the derivatives of an external potential and the material internal energy with respect to the coordinates, two vector fields are defined for the forces exerted by surrounding the system, subject to the boundary conditions, and vice versa, subject to the material properties. These vector fields are then merged into a third one that represents the properties of the loaded state. Through the work function, the force field is then directly transformed into the displacement field. The approach permits fully satisfactory prediction of all geometric and energetic properties of elastic and plastic simple shear. It predicts the existence of a bifurcation at the transition from reversible to irreversible behavior whose properties permit correct prediction of cracks in solids. It also offers a mechanism for the generation of sheath folds in plastic shear zones and for turbulence in viscous flow. Finally, an example is given on how to apply the new approach to deformation of a discrete sample as a function of loading configuration and sample shape.
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