An Approach to Deformation Theory Based on Thermodynamic Principles

Abstract

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.

On the Systematics of Energetic Terms in Continuum Mechanics, and a Note on Gibbs (1877)

“…A proper deformation theory must start with the First Law and the equation of state P V = nRT , and be fully compatible with general thermodynamics. 19 …”

Cauchy's stress theory in a modern light

“…They seem to have sought additional guidance in the common literature, thereby constructing a hybridized approach consisting of incompatible elements. Apparently, they missed the novelty of Helmholtz' concept the core of which is, in fact, the only genuine precursor to this author's work [18].…”

“…The simplest elastic deformation is thus the volume change of an ideal gas due to an isotropic pressure increase. Therefore, standard thermodynamics and the first law must be the starting point for a theory of elasticity applying to denser materials and anisotropic boundary conditions [18]. Without this insight, and without observing it in thought and concept, it is not possible to understand elasticity properly.…”

“…Due to the principle of least work, an elastic deformation will always assume the state of deformation with the highest symmetry properties possible permitted by the boundary conditions, i.e. the spatial gradients are minimized, and forcing the material into a deformation state of lesser symmetry costs extra work [18]. Therefore, the symmetry state of a deformation commonly complies with the properties assigned to Cauchy's tensor which is at least orthogonal.…”

“…The full framework of classical physics became known only after 1850 and the discovery of the first law of thermodynamics. This author has outlined the loopholes before, referring to mid-20th century textbooks [16][17][18][19][20]. It is nonetheless worth following Cauchy's argumentation in his own writings because he is, after all, the ultimate source for the current theory.…”