Unorthodox thoughts about deformation, elasticity, and stress
Zeitschrift für Naturforschung 56a, pp.794-808, 2001
Abstract: The nature of elastic deformation is examined in the light of the
potential theory. The concepts and mathematical treatment of elasticity and the choice
of equilibrium conditions are adopted from the mechanics of discrete bodies, e.g.,
celestial mechanics; they are not applicable to a change of 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
classical 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 classical 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.
Linear elasticity and potential theory: a comment on Gurtin (1972)
International Journal of Modern Physics B, 22, pp.5035-5039, 2008
DOI 10.1142/S0217979208049224
Electronic postprint version of published article; © World Scientific Publishing Company
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Abstract: In an exhaustive presentation of the linear theory of elasticity by
Gurtin (1972) the author included a chapter on the relation of the theory of elasticity
to the theory of potentials. Potential theory distinguishes two fundamental physical
categories: divergence-free and divergence-involving problems. From the criteria given
in the source quoted by the author it is evident that elastic deformation of solids falls
into the latter category. It is documented in this short note that the author presented
volume-constant elastic deformation as a divergence-free physical process, systematically
ignoring all the information that was available to him that this is not so.
A point-by-point comparison of Gurtin's text with the source he quotes, and a detailed account of Gurtin's omissions. Best read with both texts on the table.
An approach to deformation theory based on thermodynamic principles
International Journal of Modern Physics B, 22, pp.2617-2673, 2008
DOI 10.1142/S021797920803985X
Electronic postprint version of published article; © World Scientific Publishing Company
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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 at 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 how to apply the new approach to deformation of a discrete sample as a function of loading configuration and sample shape.
The energetics of pure and simple shear for elastic and plastic deformation, as predicted in the paper, can be compared with experimental data by Treloar 1975 (elastic) and Franssen 1993 (plastic).
Vorticity analysis in shear zones: A review of methods and applications. Comment
New script
The 3p-paper is a 'discussion' to a recent review of vorticity analysis methods by Xypolias, J. Structural Geology V.32, December 2010, which is quite exhaustive in the methods, but does not touch continuum mechanics itself. But that's where the gaps are. It is pointed out that bonds are never mentioned in the common theory of elasticity, and that this theory has still all the hallmarks of Newtonian mechanics of discrete bodies in freespace; it should have the structure of thermodynamics instead. The evidence by which vorticity has been assessed, is the fabric dividing line found in porphyroclast systematics in mylonites. What has been found, is the contracting eigendirection predicted by "Approach etc." above.
This paper corresponds strongly with "The kinematics of simple shear" above.
Prediction of the maximum compression direction along the San Andreas
fault and of fabric elements in metamorphic shear zones
New script
Abstract: The maximum compression direction along the San Andreas fault is known to be at 69+/-14° regionally, and at depth in the SAFOD drill hole, inclined against the sense of shear. A
theoretical model predicts a stable direction at 68.4° to the fault. Porphyroclast studies in
mylonites revealed a stable direction which divides sigma-clasts from delta-clasts. The fabric dividers in recent studies are tightly restricted to angles of 66 to 72°, which is indistinguishable from the predicted 68.4° angle. It is suggested that the two phenomena from the brittle and the plastic field are both expressions of the same cause, the contracting eigendirection of the calculated force/displacement field. Elongated porphyroclasts in mylonites accumulate along a direction ca.10° above the bulk foliation plane, inclined against the sense of bulk shear. The theoretical model predicts a stable direction, the extending eigendirection, at 10.7°. The bisector of the two stable directions is at 28.8° to the bulk foliation and inclined in the sense of bulk shear, it should be a maximum shear direction. This direction is observed in S-C mylonites as the direction of C-plane initiation. It has all the kinematic properties predicted by the model. The theoretical model is therefore fully supported by observations from various fields which have been enigmatic so far.