1. Microplasticitymitran-lab.amath.unc.edu/courses/MATH768/biblio/...and the basal plane. The minimum stress to cause yield occurs when the tensile axis is 45 to this plane, that is,

1. Microplasticity

1.1 Introduction

This chapter briefly introduces the origins of yield and plastic flow, and in particular,attempts to explain the usual assumptions in simple continuum plasticity of isotropy,incompressibility, and independence of hydrostatic stress. While short, we introducegrains, crystal slip, slip systems, resolved shear stress, and dislocations; the minimumknowledge of microplasticity for users of continuum plasticity.

The origin of plasticity in crystalline materials is crystal slip. Metals are usuallypolycrystalline; that is, made up of many crystals in which atoms are stacked ina regular array. A typical micrograph for a polycrystalline nickel-base superalloy isshown in Fig. 1.1 in which the ‘crystal’ or grain boundaries can be seen. The grain sizeis about 100 µm. The grain boundaries demarcate regions of different crystallographicorientation.

If we represent the crystallographic structure of a tiny region of a single grain byplanes of atoms, as shown in Fig. 1.2(a), we can then visualize plastic deformationtaking place as shown in Fig. 1.2(a) and (b); this is crystallographic slip. Unlikeelastic deformation, involving only the stretching of interatomic bonds, slip requiresthe breaking and re-forming of interatomic bonds and the motion of one plane ofatoms relative to another. After shearing the crystal from configuration 1.2(a) toconfiguration 1.2(b), the structure is unchanged except at the extremities of the crystal.

A number of very important phenomena in macroscopic plasticity become apparentfrom just two Figs 1.1 and 1.2:

(1) plastic slip does not lead to volume change; this is the incompressibility conditionof plasticity;

(2) plastic slip is a shearing process; hydrostatic stress, at the macrolevel, can oftenbe assumed not to influence slip;

(3) in a polycrystal, plastic yielding is often an isotropic process.

As we will see later, the incompressibility condition is very important in macro-scale plasticity and manifests itself at the heart of constitutive equations for plasticity.

Dunne, Fionn, and Nik Petrinic. <i>Introduction to Computational Plasticity</i>, Oxford University Press, 2005. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unc/detail.action?docID=422467.Created from unc on 2019-08-30 11:20:15.

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4 Microplasticity

100 mm

Fig. 1.1 Micrograph of polycrystal nickel-base alloy C263.

τ τ

ττ

(a) (b)

Fig. 1.2 Schematic representation of the crystallographic structure within a single grain undergoingslip.

However, not all plastic deformation processes are incompressible. A porous metal,for example, under compressive load may undergo plastic deformation during whichthe pores reduce in size. Consequently, there is a change of volume and a dependenceon hydrostatic stress. However, the volume change does not originate from the plasticslip process itself, but from the pore closure.

The fact that plastic slip is a shearing process gives more information about thenature of yielding; in principle, it tells us that plastic deformation is independent ofhydrostatic stress (pressure). For non-porous metals, this is one of the cornerstonesof yield criteria. The von Mises criterion, for example, is one in which the initiationof macroscale yield is quite independent of hydrostatic stress. If we take a sample ofthe theoretical material shown schematically in Fig. 1.2(a) and submerge it to an everdeeper depth in an imaginary sea of water, the hydrostatic stress becomes ever largerbut causes no more than the atoms in the theoretical material to come closer together.It will never in itself be able to generate the shearing necessary for crystallographic slip.

Figure 1.1 shows a micrograph of a polycrystal. If we assume that there is no pre-ferred crystallographic orientation, but that the orientation changes randomly from one


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Crystal slip 5

(a) (b)

Fig. 1.3 (a) Photograph of a single zinc crystal and (b) a schematic diagram representing single slipin a single crystal.

grain to the next, and if our sample of material contains a sufficiently large num-ber of grains, we can get a reasonable physical feel that macroscale yielding ofthe material will be isotropic. This is a further cornerstone of the von Mises yieldcriterion.

1.2 Crystal slip

The evidence for crystal slip being the origin of plasticity comes from mechanicaltests carried out on single crystals of metals.

The single crystal of zinc shown in Fig. 1.3(a) is a few millimetres in width and hasbeen loaded beyond yield in tension. The planes that can be seen are those on whichslip has occurred resulting from many hundreds of dislocations running through thecrystal and emerging at the edge. Each dislocation contributes just one Burger’s vectorof relative displacement, but with many such dislocations, the displacements becomelarge. Figure 1.3(b) shows schematically what is happening in Fig. 1.3(a). The ends ofthe test sample have not been constrained in the lateral directions. It can be seen thatsingle slip in this case leads to the horizontal displacement of one end relative tothe other. Had this test been carried out in a conventional uniaxial testing machine,the lateral motion would have been prevented. In order to retain compatibility, then,with the imposed axial displacement, the slip planes would have to rotate towards theloading direction. The uniaxial loading therefore leads not only to crystallographicslip, but also to rotation of the crystallographic lattice.

1.2.1 Slip systems: slip directions and slip normals

Observations on single crystals show that slip tends to occur preferentially on certaincrystal planes and in certain specific crystal directions. The combination of a slip


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6 Microplasticity

(111)

[11–0]

Fig. 1.4 The particular slip system (111)[11̄0] in an fcc lattice.

(101) (112) (123)

(111–)

Fig. 1.5 Slip systems in bcc materials.

plane and a slip direction is called a slip system. These tend to be the most denselypacked planes and the directions in which the atoms are packed closest together.This is explained in terms of dislocations. In face centred cubic (fcc) materials,the most densely packed planes are the diagonal planes of the unit cell. In fact thecrystal is ‘close-packed’ in these planes; the observed slip systems are shown inFig. 1.4.

The full family of slip systems in such crystals may be written as 〈11̄0〉{111}.There are 12 such systems in an fcc crystal (four planes each with three directions).

In body centre cubic (bcc) crystals, there are several planes that are of similardensity of packing, and hence there are several families of planes on which slip occurs.However, there is no ambiguity about the slip direction, since the atoms are closestalong the [111̄] direction and those equivalent to it. The slip systems observed in bcccrystals are shown in Fig. 1.5.

Thus, there are three families of slip systems operative:

〈111̄〉{101}; 〈111̄〉{112}; 〈111̄〉{123}.Table 1.1 shows the slip systems found in single crystals of some of the important fcc

and bcc pure metals together with the resolved shear stress required to cause slip—thecritical resolved shear stress (CRSS).


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Critical resolved shear stress 7

Table 1.1 Slip systems and CRSS for some pure metals.

Metal Structure Slip systems CRSS (MPa)

Cu fcc 〈11̄0〉{111} 0.64Al fcc 〈11̄0〉{111} 0.40Ni fcc 〈11̄0〉{111} 5.8

α-Fe bcc 〈111̄〉{101}, 〈111̄〉{112}, 〈111̄〉{123} 32Mo bcc 〈111̄〉{101} 50Ta bcc 〈111̄〉{101}, 〈111̄〉{112}, 〈111̄〉{123} 50

Slip direction

Slip plane normal

t

n

s

s

l

�

f

Fig. 1.6 A single crystal containing slip plane with normal n, slip direction s, and loaded indirection t.

1.3 Critical resolved shear stress

Suppose a single crystal in the shape of a rod is tested in tension. The axis of the rodis parallel to unit vector t. The crystal has an active slip plane, normal in directionof unit vector n. It has an active slip direction parallel to unit vector s, as shown inFig. 1.6.

When the applied tensile stress is σ , the shear stress acting on the slip plane and inthe slip direction is τ which may be found as follows: if the cross-sectional area ofthe rod is A, the force in the slip direction is Aσ cos λ and it acts on an area A/cos φ

of the slip plane. Hence the resolved shear stress is

τ = σ cos φ cos λ = σ(t · n)(t · s). (1.1)

Slip will take place on the slip system, that is, the crystal will yield, when τ reaches theCRSS. This is known as Schmid’s law. The data shown in Fig. 1.7 were obtained fromtensile tests on cadmium single crystals which are hexagonal close packed (hcp). Themeasured yield stress depends upon the angle between the tensile loading directionand the basal plane. The minimum stress to cause yield occurs when the tensile axis is45◦ to this plane, that is, when the shear stress is maximized. Schmid’s law providesa good explanation for the observed behaviour.


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8 Microplasticity

7

6

5

4

3

2

1

00 30 60 90

DataSchmid’s law

Angle between tensile axis and basal plane

Yie

ld s

tres

s (M

Pa)

Fig. 1.7 The dependence of yield stress on the angle between the tensile axis and basal plane in an hcpcadmium single crystal.

b

Fig. 1.8 An edge dislocation.

1.4 Dislocations

The theoretical shear strength of a crystal, calculated assuming that the shear ishomogeneous (the entire crystal shears simultaneously on one plane), is given by

τth = G

2π.

The expected shear strength is therefore very large; many orders of magnitude greaterthan the CRSS values are shown in Table 1.1. The assumption of homogeneous shear is,of course, wrong and plastic deformation in crystals normally occurs by the movementof the line defects known as dislocations, that are usually present in large numbers.The glide of a dislocation involves only very local rearrangements of atoms close to it,and requires a stress much lower than τth, thereby explaining the low observed valuesof CRSS.

Each dislocation is associated with a unit of slip displacement given by the Burgersvector b. Since the dislocation is a line defect, there are two extreme cases. Figure 1.8shows a schematic representation of an edge dislocation.

In this case the Burgers vector b is perpendicular to the line of the dislocation,and the dislocation corresponds to the edge of a missing half-plane of atoms. Thus b and


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Dislocations 9

b

Fig. 1.9 A screw dislocation.

the dislocation line define a plane: a specific slip plane. If l is a unit vector parallel tothe dislocation line, then

s = b

|b| and n = b × l

|b| . (1.2)

Figure 1.9 shows a schematic representation of a screw dislocation. This is theother extreme case where the Burger’s vector is parallel to the dislocation line. Thedislocation itself corresponds to a line of scissors-like shearing of the crystal. It followsfrom the fact that b is parallel to l, that, although the slip direction s is defined, the slipplane n is not. Hence, a screw dislocation can cause slip on any slip plane containing l.Dislocations are therefore vital in understanding yield and plastic flow.

We have said nothing about the important role of dislocations in strain hardening andsoftening. Nor have we discussed the effect of temperature on diffusivity which influ-ences or controls many plastic deformation mechanisms including thermally activateddislocation climb leading to recovery, vacancy core and boundary diffusion, and grainboundary sliding. All of these subjects can be found in existing materials text books,some of which are listed below. Our aim has been to include sufficient material onmicroplasticity to ensure that the physical bases of at least some of the assumptionsmade in macrolevel continuum plasticity are understood.

There have been many developments over the last 25 years in physically basedmicroplasticity modelling, including the development of time-independent and rate-dependent crystal plasticity. Here, for a given crystallographic lattice (e.g. fcc),using finite element techniques, the resolved shear stresses on all slip systems canbe determined to find the active slip systems. Within either a time-independent orrate-dependent formulation, the slip on each active system can be determined fromwhich the overall total deformation can be found. Such models are being used suc-cessfully for the plastic and creep deformation of single crystal materials which findapplication in aero-engines. The modelling of single crystal components has become


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10 Microplasticity

possible with the development of high-performance computing. More recently, usingfinite element techniques, polycrystal plasticity models have been developed. Here,the understanding of microplasticity discussed above is also employed and again, asin single crystal plasticity, the active slip systems can be identified and the corres-ponding slips determined to give the overall deformation within a given grain. Thisis now done for all the grains in the polycrystal, which all have their own measuredor specified crystallographic orientations, subject to the requirements of equilibriumand compatibility which are imposed within the finite element method. In order to dothis, it is necessary to generate many finite elements within each grain. It is clear thatfor large numbers of grains, such polycrystal plasticity modelling becomes unten-able. The consequence is that while desirable, it is not (for the foreseeable future)going to be feasible to carry out polycrystal plasticity modelling at the engineeringcomponent level. Currently, and for many years to come, the plastic deformation occur-ring in both the processing to produce engineering components, and occurring underin-service conditions at localized regions of a component, will continue to be mod-elled using continuum-level plasticity. This is particularly so in engineering industrywhere pressures of time and cost demand rapid analyses. We will now, therefore, leavemicroplasticity and address, in the remainder of the book, continuum-level plasticity.

Further reading

Dieter, G.E. (1988). Mechanical Metallurgy. McGraw-Hill Book Co., London.Meyers, M.A., Armstrong, R.W., and Kirchner, H.O.K. (1999). Mechanics and

Materials. Fundamentals and Linkages. John Wiley & Sons Inc., New York.


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Documents

1. Microplasticitymitran-lab.amath.unc.edu/courses/MATH768/biblio/...and the basal plane. The minimum stress to cause yield occurs when the tensile axis is 45 to this plane, that is,