Plastic Deformation and Fracture of Materials

1 Introductory Chapter: Microstructure and Mechanical Properties Hael Mughrabi Institut fur Werkstoffwissenschaften, Universitat Erlangen-Niirnberg, Erlangen, Federal Republic of Germany

List 1.1 1.2 1.3 1.4

of Symbols and Abbreviations Introduction Materials and Materials Development The Strength of Materials, Plasticity and Fracture The Stability of Plastic Flow in a Tensile Test, Failure by Necking, Superplasticity 1.5 Deformation Geometry, Crystallography of Slip, Orientation Factors 1.6 Dislocation Concepts of Plasticity and Work Hardening 1.7 Microstructure and Strength: The Major Hardening Mechanisms 1.8 Deformation Mechanisms at Low and High Temperatures, Deformation-Mechanism Maps 1.9 Microstructural Heterogeneities and Instabilities, Localisation of Strain, Damage and Failure 1.10 Microstructure-Based Modelling 1.11 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

2 3 3 4 6 7 7 9 10 12 15 16

2

1 Introductory Chapter: Microstructure and Mechanical Properties

List of Symbols and Abbreviations a a0 b E G ^ K Kc KIc m T Tm

half crack length interatomic distance modulus of Burgers vector Young's modulus shear modulus elastic energy release rate stress-intensity factor critical stress-intensity factor, fracture toughness plane strain fracture toughness strain-rate sensitivity temperature melting temperature

a y y yeff ys s s 0 X Q a ac af cry T T* xG r th Ty <j)

geometrical constant (resolved) shear strain (resolved) shear strain rate specific fracture energy surface energy tensile strain tensile strain rate work hardening rate angle between load axis and slip direction dislocation density tensile stress theoretical cohesive strength fracture stress tensile yield stress (resolved) shear stress effective or thermal component of shear yield stress athermal component of shear yield stress theoretical shear stress, ideal shear strength (resolved) shear yield stress angle between specimen (load) axis and normal to the slip plane

b.c.c. DS f.c.c. ODS PFZ PSB TEM

body-centred cubic directionally solidified face-centred cubic oxide dispersion strengthened precipitate-free zone persistent slip band transmission electron microscopy

1.2 Materials and Materials Development

1.1 Introduction The mechanical behaviour of materials can be described largely in terms of the material properties that govern plastic deformation and fracture. Macroscopically, these properties can be expressed in terms of material parameters, which can usually be measured without detailed knowledge of the microscopic origin of these properties. Microscopically, these properties are related to processes on a microscale ranging from atomic dimensions to typical dimensions of the microstructure of the material, such as the grain size in the case of a polycrystalline material. One important goal of materials scientists is therefore to establish experimentally and theoretically the microscopic mechanisms responsible for a particular material property and to relate the microscopic behaviour to the property that can usually be measured macroscopically. The knowledge and understanding of the relevant material properties is the first step toward improving these properties and/or developing new materials with superior properties. The appropriate consideration of material properties in materials selection during the design of components is one major task of the discipline of materials science and technologyThis volume is devoted to some important aspects of plasticity and fracture. The topics of the individual chapters are related and form a unit, as outlined in the Preface. Together, they represent the effort to establish, based on experiment and theory, the microscopic mechanisms responsible for the mechanical behaviour of different materials and to provide a framework to relate the microscopic behaviour to macroscopic properties that can be measured readily and used by engineers.

In order to prepare the ground for the subsequent chapters which address in detail specific topics in the light of most recent research, it is appropriate to survey first at an elementary level some of the basic concepts of plasticity and fracture. While this chapter addresses mainly crystalline materials, many of the aspects discussed apply, with some modifications, also to glassy solids, a class of materials of increasing importance which is covered in more detail in Chap. 10. In this introductory approach, reference will be made to some of the historical developments that have led to our present understanding. Explicit reference to the vast literature will be confined to a few cases. For further details, the reader is referred to the literature recommended under the heading "General Reading" and to the listed "Proceedings of Conferences".

1.2 Materials and Materials Development A large part of the science of materials stems from early metallurgical developments. Referring to a classical textbook such as CottrelFs Theoretical Structural Metallurgy (1955), one can conclude that most of the concepts that were developed for metals and alloys in that era are still valid today. On the other hand, it is true that these concepts had to be developed further and that they had to be adjusted continually in order to do justice to the growing importance of new and more complex metallic and non-metallic materials. Today's materials can be classified broadly into metallic materials, ceramics, glasses and polymers and include also the growing family of different kinds of composite materials.

1 Introductory Chapter: Microstructure and Mechanical Properties TEMPERATURE CAPABILITY (1000h at 150 MPa)

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OF SUPERALLOYS Engineering ceramics -1500 °C • ODS ALLOYS IN-SITU COMPOSITES^

SINGLE CRYSTAL ALLOYS

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Figure 2-12. Latent hardening ratio, LHR, as a function of primary hardening in single slip, (T — T 0 ), where T 0 is the CRSS of the unstrained crystal and T the final CRSS after the prestrain in the primary system (labelled o, a0 in the figure). Room temperature of 77 K data, Cu and Cu-Al crystals. Lower line: fit to data corresponding to coplanar latent systems. Upper lines: intersecting latent systems. From Basinski and Basinski (1979), compilation from several sources.

200

300

(2-40)

where the summation is extended to all the dislocation systems of the material. Four different values for the terms of the interaction matrix, aij? are enough to achieve a good description of the f.c.c. behavior. They represent the main interaction types among the 12{lll} f.c.c. slip systems, in order of increasing value: diagonal or self-hardening terms, interaction terms involving no junctions, glissile junctions and sessile junctions. An effect attributable to the SFE of the material is observed on the absolute value of the interaction terms. The strength of attractive junctions increases as SFE decreases, an effect already discussed by Sastry et al. (1974). The shape of the LHRs vs. primary glide flow stress displayed by f.c.c. crystals (i.e., a rather sudden jump followed by a long decreasing period after the maximal value)

400

x [g /mm 2 ]

Figure 2-13. The square root of the ratio of primary to secondary dislocation density in several near [011] Cu crystals as a function of the flow stress of the primary system. Stage I ends approximately for T ~ 100gmm" 2 . From Mecking and Bulian (1976).

44

2 Flow Stress and Work Hardening

appears to be a general feature of slip-controlled plastic behavior. Results are available for b.c.c. and for ionic crystals (Nakada and Keh, 1966, 1969; Franciosi, 1983). For b.c.c. crystals, an analysis of the LHRs in terms of interactions of the mobile and forest dislocations has been performed by Franciosi (1983). Dislocation pair-wise interactions emerge again as the main contribution to the flow stress. As for h.c.p. metals, interaction coefficients for different basal/pyramidal combination situations in Mg and Zn have been calculated in a way akin to that developed by Saada, Schoeck and co-workers and successfully compared with experimental values, as reviewed by Lavrentev (1980). To finish this section, it is important to note that LHRs are very difficult to reconcile with flow stress models relying on a strong contribution from long-range stresses, particularly from long range back stresses associated with coplanar dislocation groups (Seeger's school). As has been repeatedly remarked (Hirsch, 1975; Basinski and Basinski 1979), Stroh (1953) showed that the LHR of a coplanar system on the exclusive basis of the long-range stresses of the primary one would be 0.5 instead of > 1 , as observed. 2.2.6.4 Thermally Activated Glide Through Dislocation Substructures

Section 2.2.2 briefly sketched the glide kinetics and its importance as a tool for analyzing the mechanisms underlying the dislocation glide plasticity was stressed. The temperature and strain rate sensitivity of the flow stress is diagnosed by studying its reversible changes (at a given structure, s) with small changes of temperature and strain rate. The definition of the new flow stress level after the change can be the object of some controversy (see Basinski

and Basinski, 1979) as some degree of back extrapolation to the initial structural state is necessary (the change is only seen after a finite strain, i.e., after some unavoidable structural change). However, since the classic work of Cottrell and Stokes (1955) it is generally accepted that at low stresses - stages I and II - and if a short transient at the beginning of stage I is neglected, the reversible changes, [AT C /AT]^ S or [Azc/Alnf]Ts are proportional to the current flow stress level, TC(S, T,f). This is the so-called "Cottrell-Stokes law". The low-stress deviation from the C-S law is convincingly ascribed to soluble impurities (Basinski and Basinski, 1979) but systematically increasing deviations are observed at large stresses (stage III and beyond) in the form of an upward curvature in the ATC — TC plots ("Haasen plots", Haasen, 1958), Fig. 2-14. The deviation is still more pronounced at very large strains (stage IV), Fig. 2-15 (Korbel et al, 1979; Korbel and Szczerba, 1982; Christodoulou etal., 1982; Alberdi, 1984; Hughes, 1986; Rollet, 1988), the rate sensitivity increasing with strain (stress) and temperature. Solution hardening enhances the rate sensitivity at very low stresses, as referred to above, but decreases it at large strains, when dislocation-dislocation interactions predominate (Kocks, 1979; Mulford, 1979). The latter effect is evident in Fig. 2-15. Two possible explanations were advanced by Cottrell and Stokes for the ATC — TC proportionality in reversible flowstress changes. The first one is strict similitude of the dislocation structure throughout the stress range where the C-S law holds. Only the scale changes, the distribution and proportion of the various kinds of obstacles remains the same. The second one is to assume that the same obstacles are responsible for the long range interactions (quasi-athermal obstacles) and short

2.2 The Flow Stress

45

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range interactions (obstacles which may be overcome by thermal activation). Similitude is assumed by most work hardening models, at least inside a particular deformation stage (Nabarro et al, 1964). It has been given the category of "principle" (Kuhlmann-Wilsdorf, 1968; Hansen and Kuhlmann-Wilsdorf, 1986)

1

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