
The naked continuum mechanicist
Zeitschrift für Naturforschung 56a, pp.794808, 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 EulerCauchy 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.
Cauchy stress in mass distributions
Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) 81, Suppl. 2, pp.S309S310, 2001
Abstract: The thermodynamic definition of pressure P =
¶U/¶V
is one form of the principle that in a given state, the mass in V and
potential are proportional. Subject of this communication is the significance
of this principle for the understanding of Cauchy stress.
Proof showing that the the Cauchy continuity approach as given by Truesdell (1990) is incompatible with the Gauss divergence theorem. The argument is repeated in "Systematics of energetic terms etc." see below.
The postal address in the PDFfile is obsolete.
Linear elasticity and potential theory: a comment on Gurtin (1972)
International Journal of Modern Physics B, 22, pp.50355039, 2008
DOI 10.1142/S0217979208049224
Electronic postprint version of published article; © World Scientific Publishing Company
Publisher's printed version:
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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: divergencefree and divergenceinvolving 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
volumeconstant elastic deformation as a divergencefree physical process, systematically
ignoring all the information that was available to him that this is not so.
A pointbypoint 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.
On the systematics of energetic terms in continuum mechanics,
and a note on Gibbs (1877)
International Journal of Modern Physics B, 22, pp.48634876, 2008
DOI 10.1142/S0217979208049078
Electronic postprint version of published article; © World Scientific Publishing Company
Publisher's printed version:
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Abstract: The systematics of energetic terms as they are taught in continuum
mechanics deviate seriously from standard views 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. 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.
An approach to deformation theory based on thermodynamic principles
International Journal of Modern Physics B, 22, pp.26172673, 2008
DOI 10.1142/S021797920803985X
Electronic postprint version of published article; © World Scientific Publishing Company
Publisher's printed version:
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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).
Cauchy's Stress Theory in a Modern Light
European Journal of Physics, 35, 015010 (15pp), 2014
DOI:10.1088/01430807/35/1/015010
Abstract: The 180 year old stress theory by Cauchy is found to be insufficient to serve as a basis for a modern understanding of material behavior. Six reasons are discussed in detail: (1) Cauchy's theory, following Euler, considers forces interacting with planes. This is in contrast to Newton's mechanics which considers forces interacting with radius vectors. (2) Bonds in solids have never been taken into account. (3) Cauchy's stress theory does not meet the minimum conditions for vector spaces because it does not have a metric. It is not a field theory, and not in Euclidean space. (4) Cauchy's theory contains a hidden boundary condition that makes it less than general. (5) The current theory of stress is found to be at variance with the theory of potentials. (6) The theory is conceptually incompatible with thermodynamics for physical and geometrical reasons.
The kinematics of simple shear
New script
Abstract: The partition of a simple shear into a strain and a rotation is demonstrated to be physically unfounded, it is not helpful to understand flow. Ideal plastic simple shear is predicted to have two nonorthogonal eigendirections, the extending one is close to the bulk shear direction. The predicted eigendirections correlate well with observed fabric elements in rocks. The layering in mylonites is interpreted as a composite fabric element without mechanical significance, it does not mark a shear plane. Flow in mylonites and fluids appears laminar only at a scale larger than the one at which the deformation mechanism works.
Vorticity analysis in shear zones: A review of methods and applications. Comment
New script
The 3ppaper 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.
Strain and displacement: energetic patterns in elastic and plastic deformation
New script
Abstract: The precise details of elastic deformation are more complex than is evident from the common theory of elasticity. Experiments show that elastic simple shear requires more work than pure shear, indicating that strain is not a thermodynamic state function. The natural alternative is displacement. A new approach to elasticity and deformation, a generalized thermodynamic theory for bonded materials, correctly predicts the simple/pure shear energetic difference, and also an elastic dilation under simple shear conditions known as Poynting effect. Experiments in the plastic realm show that simple shear costs substantially less work than pure shear, which is also predicted by the new approach.
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 sigmaclasts from deltaclasts. 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 SC mylonites as the direction of Cplane 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.
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