Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

Publisher Summary The fundamental assumption of all theories of plasticity—that of time independence of the equations of state—makes simultaneous description of the plastic and rheologic properties of a material impossible. I t is well-known that in many practical problems, the actual behaviour of a material is governed by plastic as well as by rheologic effects. It can even be said that for many important structural materials, rheologic effects are more pronounced after the plastic state has been reached. Every material shows more or less pronounced viscous properties. In some problems, the influence of viscous properties of the material may be negligible, while in others, it may be essential. Both sciences—plasticity and rheology—are concerned with the description of important mechanical properties of structural materials. Each of them has created its own methods of investigation and has developed within the framework of certain assumptions which, unfortunately, cannot always be satisfied in reality. The results of rheology are confined to cases where plastic strain is of no decisive importance.

Fundamental Problems in Viscoplasticity

A model of workhardening is proposed which generalizes the known rules of isotropic and kinematic workhardening by introducing the concept of a ‘field of workhardening moduli.’ This field is defined by a configuration of surfaces of constant workhardening moduli in the stress space. For any loading history the instantaneous configuration can be determined by calculating the translation and expansion or contraction of all surfaces; the material behaviour can thus be determined for complex loading paths, in particular for cyclic loadings. Several examples for a plane stress state are presented.

On the description of anisotropic workhardening

A number of observations are made on the macroscopic behavior of materials subjected to uniaxial random cyclic loadings. These observations are then generalized to construct a model describing the material behavior for complex multiaxial loadings, in particular for cyclic loadings. This generalization introduces the concept of a bounding surface in the stress space which always encloses the loading surface. A parameter defined by the relative position of the loading and the bounding surface, and the plastic work done during the most recent loading, determine the value of the plastic modulus.

A model of nonlinearly hardening materials for complex loading

/pdf/a-modification-of-prager-s-hardening-rule-2di5taue2l.pdf

A Modification of Prager's Hardening Rule

Elastische Systeme konnen im wesentlichen in konservative (und zwar 1. nichtgyroskopische und 2. gyroskopische) sowie nichtkonservative (3. dissipative, 4. zirkulatorische und 5. instationare) Systeme eingeteilt werden. Eine Analyse der funf Kategorien zeigt, dass die bei Stabilitatsuntersuchungen ublichen statischen Verfahren nur in der 1. und in beschranktem Masse in der 3. Kategorie zulassig sind, wahrend die ubrigen Systeme kinetisch behandelt werden mussen.

Linear elastic stability

Wenn ein irreversibler Prozess durch die «Geschwindigkeiten»
 
$$\dot \chi _k $$

 und die «Krafte»X

 i
 (und somit durch die Dissipationsleistung
 
$$X_k \dot \chi _k $$

 ) charakterisiert ist und wenn das phanomenologische Gesetz
 
$$X_i = f_i (\dot \chi _k )$$

 dieses Prozesses linear, das heisst von der Form
 
$$X_i = a_{ik} \dot \chi _k $$

 ist, dann etabliert die Onsagersche Theorie die Symmetriea

 ki
 =a

 ik
 der Matrix (a

 ik
 ). Das phanomenologische Gesetz kann daher in der Form
 
$$X_i = \frac{{\partial \Phi }}{{\partial \dot \chi _k }}$$

 geschrieben werden. Diese Arbeit stellt einen Versuch dar,Onsagers Theorie fur nichtlineare phanomenologische Gesetze zu verallgemeinern. Es stellt sich heraus, dass ein solches Gesetz von der Form
 
$$X_i = \left( {\frac{{\partial D}}{{\partial \dot \chi _k }}\dot \chi _k } \right)^{ - 1} D\frac{{\partial D}}{{\partial \dot \chi _i }}$$

 sein muss und dass sich (2) im speziellen, aber praktisch wichtigen Falle, dass die Vektoren
 
$$\dot \chi _k $$

 undX

 i
 ihre Richtungen gegenseitig unabhangig von den Betragen bestimmen, auf (1) reduziert. Ein wichtiges Anwendungsgebiet ist die nichtlineare Rheologie. Wenn etwa ein ideal-plastischer Korper als Grenzfall eines nichtlinearen Newton-Korpers aufgefasst wird, dann folgt die Konvexitat der Fliessflache und v.Mises' Theorie des plastischen Potentials aus dem rheologischen Gesetz (1).

An attempt to generalize Onsager's principle, and its significance for rheological problems

Um das Verhalten eines Metalls mit dem Spannungs-Dehnungs-Diagramm der Figur 1 unter einem beliebigen raumlichen Spannungszustand zu beschreiben, wird neben der Annahme isotroper Verfestigung, welche aber den Bauschinger-Effekt nicht erklart, die Pragersche Verfestigungsregel [8] verwendent. Es wird hier untersucht, welche Formen diese Regel in den wichtigsten Spannungsraumen von weniger als neun Dimensionen annimmt.

On Prager's hardening rule

The author shows that the so-called standard equation of Chemical Kinetics is compatible with the principle of maximal rate of entropy production, but incompatible with Thermodynamics. The principle, on the other hand, is free of this defect and may be used to establish an improved version of the standard equation.

https://link.springer.com/content/pdf/10.1007/BF00944892.pdf

Chemical reactions and the principle of maximal rate of entropy production

Thin rods or shafts, subjected to thrust and torsion, become unstable under a certain load. Starting from the well knownClebsch-Kirchhoff equations and applyiing three different systems of reference, the author establishes differential equations and boundary conditions for the buckling problem as well as the problem of critical speed. He proves that the purely statical method usually applied is legitimate only in the absence of torsion, the axial moment being non conservative. Treating the simple case of a built-in free rod as a vibration problem, he shows that an axial moment of arbitrary size gives rise to an instability.

Hans Ziegler

Papers

Linear elastic stability

An attempt to generalize Onsager's principle, and its significance for rheological problems

On Prager's hardening rule

Chemical reactions and the principle of maximal rate of entropy production

Stabilitätsprobleme bei geraden Stäben und Wellen