bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

A phenomenological description of crystallographic slip and pencil glide in single crystals is outlined, with emphasis on the behavior under prescribed strains Theoretical relations are established between these single-crystal properties and the behavior of quasi-homogeneous, quasi-isotropic polycrystals deforming uniformly on a macroscopic scale, at subdiffusive temperatures Experimental comparisons between single crystals and polycrystals are reviewed, considering flow stress, work hardening, temperature and strain rate effects, and various effects of grain size

The relation between polycrystal deformation and single-crystal deformation

Zur plastischen verformung des vielkristalls

High-entropy alloys (HEAs) are a relatively new class of materials that have gained considerable attention from the metallurgical research community over recent years They are characterised by their unconventional compositions, in that they are not based around a single major component, but rather comprise multiple principal alloying elements Four core effects have been proposed in HEAs: (1) the entropic stabilisation of solid solutions, (2) the severe distortion of their lattices, (3) sluggish diffusion kinetics and (4) that properties are derived from a cocktail effect By assessing these claims on the basis of existing experimental evidence in the literature, as well as classical metallurgical understanding, it is concluded that the significance of these effects may not be as great as initially believed The effect of entropic stabilisation does not appear to be overarching, insufficient evidence exists to establish the strain in the lattices of HEAs, and rapid precipitation observed in some HEAs sug

https://www.tandfonline.com/doi/pdf/10.1080/09506608.2016.1180020

High-entropy alloys: a critical assessment of their founding principles and future prospects

The yield stress of multicomponent nickel solid solution alloys has not been modeled in the past with respect to the effects of composition and temperature. There have been investigations of the effect on the yield stress of solutes in binary systems at a fixed temperature, but the effects on the yield stress of multiple solute elements and temperature changes have not been investigated. In this article, two different forms of the trough model are considered for nickel-base alloys to determine the most applicable model for solid solution strengthening in the system. The yield stresses of three binary nickel-chromium and three ternary nickel alloys were determined at a range of temperatures. The yield stress of the alloys was then modeled using the Feltham equation. The constants determined in fitting the Feltham equation to the experimental data were then applied to other experimental solid solution alloys and also to published information on commercial solid solution nickel alloys. It was found that the yield stress of the nickel solid solution alloys could be modeled successfully using the Feltham equation.

Modeling solid solution strengthening in nickel alloys

For copper single crystals with axis orientations [110], [111], and [100] work-hardening curves and surface phenomena were studied at various temperatures. Analogous to single glide crystals we succeeded in dividing the curves into different stages and correlating them with characteristic slip line patterns. [110] crystals show the same work-hardening behaviour as single glide crystals, contrary to [111] and [100] ones where multiple glide is fundamental. The result of a quantitative analysis of the slip line photographs is that apparently for all orientations the same work-hardening mechanisms are operating. However, in [111] and [100] crystals cross-slip seems to occur in a way different from that in single glide crystals.



Fur zugverformte Kupfereinkristalle mit Achsenorientierungen [110], [111] und [100] wurden bei verschiedenen Temperaturen die Verfestigungskurven aufgenommen und die Oberflachenerscheinungen im Licht- und Elektronenmikroskop studiert. Es gelang, in Analogie zu mittelorientierten Kristallen die Verfestigungskurven in Bereiche einzuteilen und ihnen charakteristische Gleitlinienanordnungen zuzuordnen. [110]-Kristalle zeigen dasselbe Verfestigungsverhalten wie mittelorientierte, im Gegensatz zu [111]-und [100]-Kristallen, bei denen Mehrfachgleitung die entscheideude Rolle spielt. Quantitative Aus-wertungen der Gleitlinienaufnahmen ergeben, das offenbar bei allen Orientierungen dieselben Verfestigungsmechanismen vorliegen. Dagegen scheint sich bei [111]- und [100]-Kristallen die Quergleitung auf andere Weise zu vollziehen als bei mittelorientierten Kristallen.

Work-Hardening and Surface Investigations on Copper Single Crystals Oriented for Multiple Glide†

Es wird eine kinematographische Methode beschrieben, mit deren Hilfe es gelingt, die Abgleitgeschwindigkeit ȧ neutronenbestrahlter Cu-Kristalle zu messen. Da die Abgleitung inhomogen erfolgt, ergeben sich fur ȧ viel grosere Werte als bei homogener Gleitung. Bei einer Bestrahlung von rund 2 × 1018 nvt (> 0,1 MeV) finden wir bei Zimmertemperatur ȧ = 1 bis 4 s−1, verglichen mit ȧ ≈ 5 × 10−4 s−1 bei Annahme homogener Gleitung. Unser Verfahren erlaubt es weiter, den Zusammenhang zwischen Abgleitgeschwindigkeit ȧ und Dehngeschwindigkeit zu untersuchen. Wir finden eine lineare Beziehung zwischen diesen Grosen, keine Proportionalitat, wie bisher stets angenommen wurde. Dieses uberraschende Ergebnis ermoglicht eine zwanglose Beschreibung der in Teil I mitgeteilten Messungen der Geschwindigkeitsabhangigkeit der kritischen Schubspannung. Alle mitgeteilten Messungen sind nur bei Zimmertemperatur durchgefuhrt worden.



A cinematographic method is described for measuring the strain rate ȧ of neutron-irradiated Cu crystals. Since the glide process is inhomogeneous the values for ȧ are much greater than with homogeneous glide. After a radiation dose of about 2 × 1018 nvt (> 0.1 MeV) it is found that at room temperature ȧ = 1 … 4 s−1, to be compared with ȧ ≈ 5 × 10−4 s−1, if homogeneous glide occurs. The present method enables a study to be made of the connection between the strain rate ȧ and the deformation rate of the deforming apparatus. There exists only a linear relationship between ȧ and , not a proportionality, as has been assumed hitherto. With this unexpected result a simple explanation occurs for the measurements of the strain rate dependence of the critical shear stress given in part I. The measurements described have been made only at room temperature.

Untersuchungen des Fließbereichs neutronenbestrahlter Kupfereinkristalle. II. Kinematographische Messung der Abgleitgeschwinidigkeit nichtangelassener und angelassener Proben

Der Begriff „aktives Gleitvolumen” bzw. „aktive Kristallange” wird in die Plastizitatslehre eingefuhrt. Er spielt eine wesentliche Rolle, wenn das aktive Gleitvolumen von den auseren Verformungsparametern, Geschwindigkeit und Temperatur, abhangt. Das ist im allgemeinen dann zu erwarten, wenn Gleitvorgange thermisch aktiviert ablaufen. Aus einer energetischen Uberlegung wird eine Naherungsformel fur die Abhangigkeit des aktiven Gleitvolumens von der Abgleitgeschwindigkeit gewonnen. Sie beschreibt die bisher vorliegenden experimentellen Ergebnisse. Folgerungen fur die Analyse thermisch aktivierter Vorgange, die sich aus der Veranderlichkeit des aktiven Gleitvolumens ergeben, werden kurz besprochen.



The concept of an “active slip volume” or an “active crystal length” is introduced into plasticity. It plays an important role if the active slip volume depends on the external parameters of deformation, i.e. strain rate and temperature. In general, this occurs if the slip processes are supported by thermal activation. An energetic consideration yields an approximate formula for the dependence of the active slip volume on the strain rate which is supported by the experimental results available at present. Consequences arising from the variability of the active slip volume concerning the analysis of thermally processes activated are discussed.

Über den Begriff „aktives Gleitvolumen'' und dessen Bedeutung

The occurrence and boundaries of (in-)stabilities along stress–strain curves are investigated for f.c.c. solid solutions of CuMn. There are up to four temperature intervals where the σ(ϵ)-curves over their whole extension are either smooth (stable deformation) or serrated (unstable deformation). In between, three transitional intervals, named α, β, and γ, are constituted. Here two or more regions of stable and unstable deformation alternate along σ(ϵ). In particular, inside interval β a region of stability is embedded between two regions of instability. For two Mn concentrations the variatin of the boundaries with temperature and strain rate is investigated for the intervals α and β. The results allow a comparison with theoretical “scenarios” developed by Kubin and Estrin. Possibilities for an improvement and/or extension of this model are discussed.



Das Auftreten und die Grenzen von Instabilitaten langs Spannungs–Dehnungskurven werden am Beispiel des k.f.z. Mischkristallsystems CuMn untersucht. Es ergeben sich bis zu vier Temperaturintervalle, in denen die σ(ϵ)-Kurven durchgehend entweder glatt (stabile Verformung) oder gezackt (instabile Verformung) sind. Dazwischen stellen wir drei Ubergangsintervalle, α, β und γ, fest, in denen zwei oder mehr Abschnitte stabiler und instabiler Verformung langs σ(ϵ) einander abwechseln. Eine Besonderheit stellt Intervall β dar, wo ein Stabilitatsbereich zwischen zwei Instabilitatsbereichen eingebettet ist. Fur zwei Mn-Konzentrationen wird die Abhangigkeit der Instabilitatsgrenzen der Intervalle α und β von Temperatur und Dehnungsgeschwindigkeit untersucht. Die Ergebnisse lassen sich mit den „Scenarios” der Theorie von Kubin und Estrin vergleichen. Moglichkeiten zur notigen Verbesserung und/oder Erweiterung der bestehenden Vorstellungen werden aufgezeigt.

On Sequences of Alternate Stable and Unstable Regions along Tensile Deformation Curves

Investigations of the easy glide region in single crystals of well-known f.c.c. Cu alloys between 85 and 400°K are reported. From careful measurements of the strain rate dependence of the flow stress and from microscopic observations (static and cinematographic) of the development of the glide bands it is concluded that the “active slip volume” [1] of these crystals depends on strain rate and temperature. A detailed analysis of the results yields values for the number, the activation energy, and the activation volume of the obstacles responsible for the thermally activated process. We suggest that the obstacles are statistically distributed clusters of the atoms alloyed to copper.



Es wird uber Untersuchungen des Fliesbereichs an Einkristallen bekannter k.f.z. Cu-Legierungen zwischen 85 und 400°K berichtet. Aus sorgfaltigen Messungen der Geschwindigkeitsabhangigkeit der Fliesspannung und mikroskopischen Untersuchungen (statisch und kinematographisch) der Entwicklung der Gleitbander schliesen wir, das das „aktive Gleitvolumen” [1] dieser Kristalle von Verformungsgeschwindigkeit und Temperatur abhangt. Die nahere Analyse der Ergebnisse liefert Werte fur die Zahl, die Aktivierungsenergie und das Aktivierungsvolumen der fur den thermisch aktivierten Prozes verantwortlichen Gleithindernisse. Wir vermuten danach, das es sich bei ihnen um statistische Ansammlungen der jeweils dem Kupfer zulegierten Atome handelt.

Ch. Schwink

Papers

Work-Hardening and Surface Investigations on Copper Single Crystals Oriented for Multiple Glide†

Untersuchungen des Fließbereichs neutronenbestrahlter Kupfereinkristalle. II. Kinematographische Messung der Abgleitgeschwinidigkeit nichtangelassener und angelassener Proben

Über den Begriff „aktives Gleitvolumen'' und dessen Bedeutung

On Sequences of Alternate Stable and Unstable Regions along Tensile Deformation Curves

The Active Slip Volume and the Temperature-Dependent Processes in Stage I of Single Crystals of Cu Alloys†