an experimental study of the eect of hardening on plastic...

Journal of the Mechanics and Physics of Solids51 (2003) 1623–1647

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An experimental study of the e(ect of hardeningon plastic deformation at notch tips in metallic

single crystals

W.C. Cronea ;∗, T.W. Shieldb

aDepartment of Engineering Physics, University of Wisconsin, 529 Engineering Research Bldg.,1500 Engineering Drive, Madison, WI 53706, USA

bDepartment of Aerospace Engineering and Mechanics, University of Minnesota, 107 Akerman Hall,110 Union St. SE, Minneapolis, MN 55455, USA

Received 5 June 2002; accepted 29 January 2003

Abstract

Experiments to measure the e(ect of hardening on the plastic deformation 3eld near a notchtip in metallic single crystals were conducted. The specimens were cut from pure Cu and aCuBe alloy (with 1.8–2.0 wt% Be) FCC single crystals. The Cu–2.0wt%Be alloy was selectedbecause its initial hardness and rate of hardening can be modi3ed by heat treatment. The Vick-ers hardness of the specimens ranged from 87 to 353 kg=mm2, while the hardening exponentsranged between 10 and 4.5. The experimental results were compared to analytical and numer-ical solutions from the literature. This comparison shows that the inclusion of elastic regionsin the analytical solutions and anisotropic hardening in the numerical solutions results in betteragreement with the experiments.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: A: Fracture mechanics; Fracture; Single crystal

1. Introduction

The study of the plastic deformation at a crack tip in single-crystal metals is impor-tant to the development of the understanding of polycrystalline material fracture andsingle-crystal failure. Such work extends to the development of models that accuratelypredict micromechanical deformation in multigrained systems and guides the develop-ment of failure analysis techniques for single-crystal engineering components. Previous

∗ Corresponding author. Tel.: +1-608-262-8384; fax: +1-608-263-7451.E-mail address: [email protected] (W.C. Crone).

0022-5096/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0022-5096(03)00021-8

mailto:[email protected]

1624 W.C. Crone, T.W. Shield / J. Mech. Phys. Solids 51 (2003) 1623–1647

work (see below) has focused on the strain 3eld that forms near a stationary crack ina single crystal. The work presented here considers the e(ect of hardening on thesestrain 3elds.

The plastic deformation around a crack tip within a ductile single-crystal materialproduces a plastic 3eld with sectors of deformation and crystallographically dependentradial boundaries. Rice (1987) predicted this type of patchy plastic deformation arounda crack tip in metallic single crystals, and this behavior was con3rmed experimentally(Shield and Kim, 1994; Shield, 1996; Bastawros and Kim, 1998; Crone and Shield,2001) and investigated numerically (Rice et al., 1990; Mohan et al., 1992; Cuitinoand Ortiz, 1996). Rice’s original analytical solution has also been extended to includehardening (Saeedvafa and Rice, 1989) and elastic sectors (Drugan, 2001).

The experimental results have similar features to Rice’s (1987) solution, but thereare signi3cant areas of disagreement. Existing analytical and numerical work providessome insight into the behavior of ductile single crystals in the presence of a crackor notch, but the results available do not completely predict the behavior observed inexperiments. One key area of potential discrepancy is the inFuence of hardening ondeformation 3eld.

The research discussed below involves a detailed experimental exploration of therelationship between hardening and the plastic deformation that develops at a notch.Experimental results for one crystallographic orientation (labelled Orientation II byCrone, 1998) are compared and contrasted to existing analytical and numerical work.

2. Background

This section will outline the continuum theory for single-crystal plasticity. In partic-ular, the yield surface for plane problems based on Schmid’s Law will be reviewed.Several types of hardening models will be presented along with a discussion of themechanisms that lead to hardening of the materials used in the experiments.

In the following, a rectangular cartesian coordinate system centered at the crack tipwill be assumed. As shown in Fig. 1, the x1-axis is in the prospective crack propagationdirection and the x2-axis is normal to the crack plane. Repeated Latin subscripts shouldbe summed over the range of 1–3, while superscripts in parentheses do not follow thissummation convention.

2.1. Plastic deformation in metallic single crystals

In metallic single crystals, slip occurs on close packed planes in close packed direc-tions. For FCC metals such as copper and copper beryllium investigated here, these are{111} planes and 〈110〉 directions. Schmid’s Law relates plastic slip to the resolvedshear stress on these planes in the direction of slip. Slip occurs on a given systemwhen the resolved shear stress on that system reaches a critical level for the material.Multiple slip systems may be at this critical level simultaneously and be undergoingactive plastic slip. The resolved shear stress is given by

�(k) = n(k)i �ijs

(k)j ; (1)

W.C. Crone, T.W. Shield / J. Mech. Phys. Solids 51 (2003) 1623–1647 1625

Fig. 1. The diagram on the left provides an example of slip line traces on a specimen surface for an FCCmaterial with the orientation of (010) crack plane and [J101] crack tip direction. The slip depicted occurs atangles of 55◦, 125◦, and 180◦ from the x1-axis. The slip in the [101] direction at 180◦ from the x1-axisshown as dashed lines corresponds to the line segments BC and FE on the yield surface shown on the right.Adapted from Rice (1987).

where n(k)i are the components of the slip plane normal and s(k)

i are the components ofthe slip direction for the kth slip system, and �ij is the stress state. The slip systems(n(k)i and s(k)

i ) are typically assumed to be independent of loading level except in 3nitestrain (numerical) analyses.

An envelope can be described in stress space that identi3es the stress conditionsunder which yield is reached for a material. For anisotropic materials the yield surfaceor yield locus is dependent on the crystallographic orientation being considered aswell as the active slip systems allowed by the crystallography. For an anisotropicmaterial in plane strain, Rice (1973) showed that the yield surface can be constructedin (�11 − �22)=2 verses �12 stress space. The initial yield locus for the crystallographicorientation of an FCC material (Orientation II with (010) crack plane and [J101] notchtip direction) considered here is shown plotted in this stress space in Fig. 1. Eachline of the yield surface corresponds to slip on one of the plane strain slip systemswhich intersects the plane of the specimen at an angle to the surface as a line. Eachplane strain slip system is comprised of two symmetric slip systems that act togetherto produce planar deformation. These surface slip lines are at angles of 55◦, 125◦, and180◦ from the x1-axis for Orientation II. Another mode of deformation is kink, whichoccurs in the direction of the slip plane normal. For this particular specimen orientationthat would correspond to angles of 145◦, 35◦, and 90◦.

Following Rice (1987), the yield surface is constructed by identifying the pairs of slipsystems that give plane deformations. These plane strain slip systems have a normalwith components N (k)

� and slip direction with components S(k)� that lie in the x1–x2

plane. It is the directions of S(k) that determine the orientations of the slip line tracesshown in Fig. 1. In addition, the apparent yield stress for the plane strain slip system,�(k)

CR, includes a geometric factor that accounts for the angles the three-dimensional slipsystems make with x1–x2 plane. For these plane strain slip systems the yield condition


reduces to∣∣∣∣−2S(k)1 S(k)

2

(�11 − �22

2

)+(S(k)

1 S(k)1 − S(k)

2 S(k)2

)�12

∣∣∣∣= �(k)CR : (2)

The yield surface is determined by 3nding the minimum values of (�11 − �22)=2 and�12 that satisfy Eq. (2) for some value of k. The details of this calculation for severalorientations are in Crone and Shield (2001).

In the asymptotic perfectly plastic solution for the stress 3eld at a crack in a singlecrystal found by Rice (1987), a patchy stress 3eld with constant stress sectors andcrystallographically dependent radial boundaries was predicted. In order to constructthis solution, Rice found that when the entire 3eld is at yield the stress state withinthe sectors must jump from a corner to a neighboring corner of the yield surface.For instance, the line between corners A and B on the yield surface corresponds toslip along a system which has a trace located at 55◦ from the x1-axis. At a cornerthe stress state may involve slip on both systems of the neighboring Fats, thus in asector at stress state A the slip lines may be at 55◦ and 125◦. In Rice’s theory onlysectors with a 90◦ angular extent may have a stress state that lies on a Fat of the yieldsurface.

For orientation II, Rice (1987) predicts four sectors with boundaries at 55◦, 90◦,125◦. Comparisons have been made with the experimental 3ndings of Shield (1996),Shield and Kim (1994) and Bastawros and Kim (1998). In all of the experimentalwork mentioned, there is good global agreement with this prediction in that there aresectors of deformation that develop with very distinct sector boundaries. However, thereare also signi3cant discrepancies. To extend Rice’s analytical solution, hardening wasincorporated by Saeedvafa and Rice (1989). A comparison of experimental results tothis analytical work and numerical studies that include hardening will be presentedbelow after hardening in general is discussed.

Although Rice’s (1987) work considers the full 3eld to be at yield around a sharpcrack, this assumption is not required. Recent analytical work by Drugan (2001), fol-lows the experimental 3nding of elastic sectors and assumes that only slip (and notkink) is possible. Using the framework developed by Rice (1987) with these modi3edassumptions, Drugan (2001) shows that solutions can be developed that produce thehigh stress triaxiality expected ahead of the crack tip. A comparison of this modi3edtheory to experiments is made elsewhere (Crone and Drugan, 2001).

2.2. Hardening in single crystals

The analytical work discussed above relies on the assumption of perfectly plasticbehavior. For very soft materials and some materials within a limited range of strainor after hardening has saturated, plastic behavior can be idealized as non-hardening.With this type of behavior no increase in stress beyond initial yield is required toinduce further plastic deformation and the current yield surface is 3xed at its initialcon3guration. However, most metallic materials of engineering interest undergo hard-ening after yield and thus theories that include hardening are required to better modelthe observed behavior. With hardening, a larger value of stress is required to produce


Fig. 2. The e(ects of diagonal hardening and isotopic hardening on the yield surface are shown. The initialyield surface is shown in solid lines. The expansion of the yield surface with hardening is shown withdashed lines. In both cases hardening increases the size of the yield surface. In diagonal hardening the shapeof the yield surface is also altered and triple slip arises. Adapted from Cuitino and Ortiz (1996).

additional plastic deformation and the yield surface expands outward from the initialyield surface.

Several theories of latent hardening will be discussed here: isotropic hardening;Taylor power-law hardening; forest hardening; and uncoupled or diagonal power-lawhardening. Examples of these hardening models are provided and related to the problemof deformation about a crack or notch in a single crystal.

A common constitutive form for 3nite plastic deformation assumes a multiplicativedecomposition of the deformation gradient such that

F= FeFp; (3)

where Fe includes only elastic lattice distortions and rotations and Fp is a latticeinvariant plastic deformation which results from dislocation motion.

When hardening occurs in a material, a larger stress is required to further deform thematerial. Hardening therefore changes the size and/or shape of the yield surface. Anisotropic hardening law assumes that the entire yield surface expands in a self-similarway, as shown in Fig. 2. An isotropic power-law hardening model was developed byRice (1971) from the kinematics of dislocation motion. This model relates the plasticdeformation gradient to the shear rate on a slip system through

FpFp−1 =∑k

�(k)S(k) ⊗N(k); (4)

where �(k) is the shear strain rate for the kth slip system. It is generally assumed thatthe shear rate on a particular slip system depends only on the resolved shear stress,�(k), on that system through Eq. (1).


The shear rate is often given the form

�(k) =

�0

[(�(k)=�(k)

CR

)1=M− 1]

if �(k) = �(k)CR

0 otherwise;

(5)

where M is the strain-rate sensitivity exponent and �0 is a reference shear rate. Thehardening relation controlling the evolution of the rate of critical resolved shear stress is

�(k)CR =

∑m

h(km)(�(m))�(m); (6)

where the hardening matrix h(km) is given by

h(km)(�) = h(�)(q+ (1 − q)�(km)): (7)

The hardening modulus, h(�), is a function of the sum of the shear strains over allslip systems. The hardening behavior is characterized by the parameter q which givesisotropic hardening when equal to 1 and has been experimentally found to range be-tween 1 and 1.4 for FCC metals. For analysis, the hardening modulus is often expressedin the form

h(�) =�0

N�0

(��0

)(1−N )=N

; (8)

where �0 and �0 are the reference values and N is the hardening exponent. For thespecial case of Taylor power-law hardening which falls under the category of isotropichardening, q= 1 and

�(k)CR = �0

(�(k)

�0

)1=N

: (9)

Rice et al. (1990) and Cuitino and Ortiz (1996) utilize Taylor power-law hardening intheir numerical models.

The forest hardening model has its foundations in dislocation mechanics. Dislocationmotion is impeded by other dislocations or obstacles that intersect the plane on whichthe dislocation is trying to move. The interaction and creation of dislocations have thee(ect of hardening the material. Let n(k) be the density of obstacles to the kth slipsystem, then

n(k) =∑m

a(km)�(m); (10)

where �(m) is the dislocation density in the mth slip system. The interaction matrix a(km)

includes information about the volume of material considered, the stacking fault energy,and a weighting function that takes into account how much each dislocation a(ects aparticular system (Cuitino and Ortiz, 1992). Interaction coeOcients are often classi3edaccording to whether the dislocations belong to the same system, form junctions, orform sessile Lomer–Cottrell locks.


Cuitino and Ortiz (1992) developed an expression for �(k)CR based on the forest hard-

ening model. They also incorporated information about the plastic modulus, Fow stress,and slip strain. Their model has a diagonal hardening matrix, which allows the activesystems to harden faster than other systems, in contrast to an isotropic hardening, whereall systems harden equally. Their hardening rate, which depends on the density of thepoint obstacles, is given by

�(k)CR =

∑l

h(kl)�(l); (11)

where the hardening matrix is

h(kl) =

h(k)

c

(�(k)

CR

�(k)c

)3[

cosh((

�(k)c

�(k)CR

)2)− 1]

if k = l;

0 otherwise

(12)

and

h(k)c =

�(k)c

�(k)c; �(k)

c = ��b√�n(k); �(k)

c =b�(k)

2√n(k)

: (13)

A kinetic equation was determined which relates the dislocation density, �(k), with theshear strain, �(k). When �(k) reaches its saturation value then all slip systems hardenindependently. After saturation, the hardening matrix is

h(kl) ∼

(�(k)

0

)2

2�(k)0 �

(k)CR

if k = l;

0 otherwise:

(14)

Then the hardening relation becomes

�(k)CR ∼ �(k)

0

(�(k)

�(k)0

)1=2

: (15)

Thus in the large strain limit, the material undergoes diagonal, parabolic hardening.In contrast to isotropic hardening where all systems harden equally, a diagonal hard-

ening model, such as the one just mentioned, allows the active systems to harden fasterthan other systems. Cuitino and Ortiz (1996) present evidence against the isotropichardening model by comparisons to uniaxial tension experiments in single crystals.Their research indicates that the metallic crystals they considered undergo diagonaland parabolic hardening. For the situation considered here, the distortions of the yieldsurface due to diagonal hardening may, in extreme cases of hardening, allow the possi-bility of triple slip. Fig. 2 shows how the yield surface could be distorted to allow thispossibility. If triple slip occurs, it may be observed experimentally due to the presenceof three slip line traces at a point on the surface, whereas the initial yield surfacewould only allow single slip (on the Fats) or double slip (on the corners).


2.3. Hardening mechanisms in Cu and CuBe

Two materials are used in the experiments discussed below, copper (Cu) and acopper alloy (Cu–2.0wt%Be). Both of these materials harden during the deformationthrough a process called work hardening. Initially work hardening may provide a rapidrate of hardening, but as saturation occurs the material behaves as if it was a slowerhardening material. Thus, a material may be work hardened by a uniform stress statebefore it is used for notch or crack tip studies to more closely approximate perfectlyplastic behavior (Shield and Kim, 1994). However, no initial work hardening willbe done to the material used in the experiments reported below. Instead precipitationhardening will be used to control the initial hardness level and rate of hardening inthe Cu–2.0wt%Be specimens. The mechanisms of these two hardening processes arediscussed next.

2.3.1. Work hardeningWork hardening of a material is said to occur when additional applied stress above

the yield stress is required to further deform the material. In general, there are threestages to hardening in FCC materials: Stage I has low linear hardening and is indicatedby long 3ne slip lines; Stage II has greater hardening with continued development ofslip lines; and Stage III has a parabolically decreasing hardening rate with coarse slipbands and often cross slip (Honeycombe, 1984). Stage I is associated with easy glidewhere only the most favorably oriented slip systems are active and the hardeningobserved is primarily due to the overlap of dislocation stress 3elds among dislocationsgliding on parallel slip planes and the pile-up of dislocations on particular planes. Inlater stages, other slip systems are activated and interaction between intersecting slipsystems causes greater resistance to dislocation motion.

2.3.2. Precipitation hardeningAlloys may be hardened by allowing the alloying elements to separate into regions

that are rich in one of the constituents. This can be accomplished by holding thetemperature of the material at a point where two (or more) compositions are ther-modynamically preferred over the average composition. This drives di(usion of theconstituents and leads to the formation of small regions rich in one of the constituentscalled precipitates. The precipitates are obstacles to dislocation motion and thus hardenthe material. The amount of hardening obtained is determined by the heat treatment,which controls the size and number of the precipitates formed. A detailed discussionof precipitation in the alloy used in this study is presented next.

Copper alloys containing small amounts of beryllium, chromium, zirconium, nickel,silicon, or phosphorus can be heat treated to obtain high hardness and strength(Covington, 1973). A copper–beryllium alloy with 1.8–2.0 wt% Be was chosen forthis research. A combination of solution heat treatment and precipitation heat treatment(aging) is used to obtain the most favorable properties in Cu–2.0wt%Be. A range ofconditions can be employed to achieve the hardness values desired. Prior to aging,solution treatment was performed at 790◦C for 30 min followed by water quench. Thisdissolves alloying elements into the solid solution within the � phase. The result is a


relatively soft condition where the Be is uniformly distributed throughout the Cu ma-trix. Precipitation treatment was performed at 315◦C or 200◦C for 0.5–3:5 h followedby uncontrolled air cooling. During this treatment some of the Be, which is held insolid solution, precipitates in the form of extremely 3ne structures which produce in-creased hardness and tensile strength (Oberg et al., 1985). The precipitation occurs inthe form of 3ne stoichiometric CuBe platelets (� phase) along {100} planes in the Cucrystal matrix called Gunier–Preston zones (Covington, 1973). Hardness in the singlecrystal Cu–2.0wt%Be was increased from approximately 100 kg=mm2 (Vickers) for so-lution treated Cu–2.0wt%Be to a peak value of approximately 350 kg=mm2 after 3 h ofprecipitation treatment. Regulation of the precipitation treatment time allows hardnesslevels between the solution treated level and peak precipitation treatment level to beachieved. Lower precipitation treatment temperatures are used to give 3ner control overthe precipitation process that occurs slower at lower temperatures.

The hardening mechanism of the Gunier–Preston zones was studied by Guy et al.(1948). They proposed that the � matrix has an FCC structure and the equilibrium �phase has a BCC structure. The result is a contraction of the lattice in the directionperpendicular to the planes of beryllium atoms and an expansion within the planes ofthese atoms. They found that the [100] and [011] directions of the � and � phasesrespectively, are nearly parallel. The lattice mismatch between the CuBe (BCC) plateletsand Cu (FCC) matrix leads to accommodation strains that impede dislocation motion.Intermetallic particles known as beryllides, containing Be and Ni, are also formed dur-ing solidi3cation and thermal processing (Murray et al., 1994). The combined e(ectof the precipitates and beryllides is to increase the strength and hardness of the alloy.Characteristically, there is a limit to the increase in the strength and hardness of thematerial that can be achieved through heat treatment and beyond the peak values theydecrease as a result of overaging.

3. Specimen preparation and experimental techniques

For the research conducted, single crystals for Cu and CuBe (with 1.8–2.0 wt%Be) were grown with the Bridgman technique (Crone, 2000). The Cu material wasfrom the same crystal as used by Shield (1996) and Crone and Shield (2001). Thesingle-crystal Cu–2.0wt%Be material was from boule CB10 characterized by Crone(2000) and used by Crone and Shield (2001). Four-point bend notched specimens with(010) crack planes and [J101] notch tip directions were prepared in the same mannerdescribed in Crone and Shield (2001) using X-ray microdi(raction and electricaldischarge machining.

Orientation and machining operations were followed by heat treatment of the speci-men in an argon atmosphere for solution and precipitation treatment. Chemical etchingwith a 50% nitric acid solution was performed to clean the material surface, 3newire EDM cutting was used to create the notch, and mechanical polishing was com-pleted to provide an optically reFective surface. For specimens where moirSe microscopywas to be used, a photolithographic crossed grating of a 5 �m (200 lines/mm) wasapplied.


The four-point-bend specimens used were 6:35 mm×6:35 mm and have a total lengthof 5:1 cm with a central single-crystal section of approximately 2 cm. The 100–200 �mwidth notch is cut to a depth of approximately 2:5 mm. Extensions of polycrystallineCu were attached with epoxy. The copper specimens used in this research have labelsthat are prefaced with ‘C’ and are C4, C8, and C9. These replicate specimens wereloaded to di(erent 3nal displacement levels. The copper–beryllium specimens havelabels that are prefaced with ‘CB’ and are CB1, CB5, CB7, CB8, and CB9. The 3veCu–2.0wt%Be specimens were heat treated using precipitation times ranging from 0to 3:5 h and temperatures of 200◦C and 315◦C to produce a range of hardness levels.All of the specimens used for this study have the same crystallographic orientation;the normal to the notch plane is (010) and the notch tip is along the [J101] direction.

In addition, two other sample types were taken from the Cu and Cu–2.0wt%Beboules in close proximity to each four-point-bend specimen. These were small rect-angular samples used for Vickers hardness testing and small uniaxial tension sampleswith a dumbbell shape. Both were used to evaluate hardness of the Cu–2.0wt%Beafter heat treatment. To provide an accurate measure of the hardness achieved, thesesamples were also heat treated at the same time as the corresponding four-point-bendspecimen.

Mechanical testing and data collection through optical and interferometric meth-ods were conducted on Cu and Cu–2.0wt%Be specimens. An Instron 4502 UniversalTesting Machine was used to apply load to the four-point-bend specimens and tensionsamples. A four-point bending jig was used to apply load to the specimens through6:35 mm hardened steel dowels. The distance between the two sets of dowels was 2.54and 3:81 cm. The tension samples were held by micro-wedge grips. The test controland data collection for load and crosshead displacement were performed with a 486class computer via an IEEE-488 interface board.

In ductile single-crystal copper and copper alloys, persistent strain localization bandsdevelop on the surface of the material with plastic deformation. These bands can beobserved optically and provide a signi3cant amount of information about the activeslip systems within a sector and the sector boundary angles. Optical observation duringtesting was conducted using a macro lens mounted on a CCD camera. Post-test opticalimages were obtained with a Nikon microscope equipped with di(erential interferencecontrast (DIC) objectives. The DIC technique yields an image with false colors relatedto the gradient of the optical path di(erence which shows small di(erences in the tiltof the surface as di(erent colors. Persistent slip bands are readily observable on thesurface of all Cu and some of the Cu–2.0wt%Be specimens tested. In these specimens,the boundaries between sectors can be easily identi3ed.

MoirSe microscopy, an optical interference technique, was used to measure the sur-face components of Almansi strain on the notched bend specimens. A pair of ortho-gonal di(raction gratings on the specimen surface and the moirSe microscope describedin Shield and Kim (1991) and Shield (1996) was used. Slip bands cause problemswhen measuring surface strains using this technique. Thus moirSe measurements wererestricted to the Cu–2.0wt%Be specimens that had been hardened.

The strain measure obtained from the analysis of the moirSe fringes is the Almansistrain. Almansi strain, which is a measure of strain in the current con3guration, is


given by

�=12

[I − (FFT)−1]; (16)

where F is the deformation gradient and I is the identity tensor. It is independent of thereference grating orientation and properly invariant to rigid body motions if an initiallyorthogonal specimen grating is used. The moirSe microscope was placed on an elevatorplatform on the Instron testing machine in order to take data in situ without unloadingthe specimen. Loading was halted during testing so that moirSe fringe images could becaptured. Full 3eld strain maps were created from the collected data after testing wascompleted.

4. Results

4.1. Tension tests

The results of Vickers hardness tests on the rectangular samples with the same heattreatment as the four-point-bend specimens are given in Table 1. The pure coppermaterial has a hardness of 87 kg=mm2 on a [010] face, while the solution treated Cu–2.0wt%Be material has a hardness of 134 kg=mm2. The hardest samples prepared withprolonged precipitation treatment had a hardness of 350 kg=mm2.

To further characterize the copper and copper–beryllium used in these experiments,uniaxial tension tests were performed on samples to compare hardening behavior. Al-though di(erences in hardening behavior can be observed for di(erent orientations ofapplied load, the tensile tests provide a relative comparison of the hardening rates thedi(erent materials used in the bend experiments. The tension samples each have atensile axis in a direction 5◦ from [J100] towards [011]. The Cu sample C1S3 and thesolution treated Cu–2.0wt%Be sample CB10S6A were very ductile in their deformationand failure. The slip bands that 3rst appeared on the surface of these samples were 40◦

from the tension direction as shown in Fig. 3. In contrast, Cu–2.0wt%Be samples thatwere precipitation treated (CB10SB, CB10S6B) showed few slip bands and exhibitedbrittle failures. For these Cu–2.0wt%Be samples the fracture surfaces were sharp andoccurred at an angle of 45–55◦ from the tension direction. Of the few slip bands thatdid form in the hardened Cu–2.0wt%Be samples, the predominant ones were also at40◦ from the tension direction.

It is clear from the formation of slip bands on the surface of these samples that therewas one primary slip system active initially (characteristic of Stage I hardening), butadditional slip bands on some samples showed evidence of subsequent activity on thesecond system. For a tensile axis that lies between [100] and [110], slip can occur inone direction on two di(erent planes. The orientation chosen for the tensile specimensfalls into this category, thus more than one slip system may be active. The ratiobetween the slip activity on these two systems cannot be determined from uniaxialmeasurements, and thus it is not possible to calculate parameters for a single-slipsystem. However, the initial strain hardening exponent, N , found from a 3t of thestress–strain curve just after it becomes nonlinear, can still be used to characterize


Table 1Vickers hardness and hardening exponent values for a various heat treatments corresponding to the bendspecimens tested

Heat Hardness Vickers Tension Hardening Bendtreatment sample hardness sample exponent specimen(s)

(kg=mm2) N

As-grown Cu C1S1 87 for [010] C1S3 7.5 C4,C8,C9

Solution-treated CuBe CB10S3B 134 for [010] CB10S6A 10 CB1CB10SC 134 for [101]

Precipitation-treated CuBe CB10S10B 155 for [101] 9:5∗ CB50:5 h at 200◦C

Precipitation-treated CuBe CB10S10A 167 for [101] 9∗ CB71:0 h at 200◦C

Precipitation-treated CB10S11A 184 for [101] 9∗ CB91:5 h at 200◦C

Precipitation-treated CB10S8B 329 for [010] CB10SB 5.5 CB81:0 h at 315◦C CB10SB 353 for [101]

Precipitation-treated CB10S2B 228 for [010] CB10S6B 4.53:5 h at 315◦C CB10SA 338 for [101]

Vickers hardness values are reported for the plane with normal direction indicated in square brackets.Tension samples with a tensile axis of 5◦ from [J100] towards [011] were used to determine values forthe hardening exponent N . All bend specimens have a (010) notch plane and [J101] notch tip direction.Hardening exponents marked with an asterisk were obtained by interpolating from the exponents obtainedfrom experiments on the Vickers hardness.

the material. For isotropic Taylor power-law hardening the strain, �, predicted for ameasured stress value, �, is calculated from the elastic and plastic contributions in theexpression

�=�E

+( �K

)N; (17)

where E is Young’s modulus in the tensile direction, K is a constant, and N is thehardening exponent. An orientation factor can be incorporated into the above expressionthat resolves the stress onto the primary slip plane, but this only alters the value ofK and does not impact the value of N . A typical 3t is shown for C1S3 in Fig. 4and Table 1 gives hardening exponent values for some of the heat treatments used inthe bend specimens. For specimens were good tension data was not obtained, Table1 gives values for N interpolated from the Vickers hardness results. Smaller valuesof N correspond to faster rates of work hardening and these results show that theinitially harder materials work harden faster, presumably due to the increased numberof obstacles to dislocation motion that the precipitates present.


Fig. 3. An optical micrograph is shown for sample C1S3 after tensile testing. The tensile axis is verticaland one sample edge can be seen on the left side. The parallel lines observed are slip lines. Changes incolor/shade indicate small changes in surface tilt. The image has a 3eld of view of 2:2 mm × 1:7 mm.

Fig. 4. Stress/strain curve for the C1S3 since crystal copper tension sample oriented with the with tensileaxis 5◦ from [J100] towards [011]. The dashed line is a power-law 3t with N = 7:5.

4.2. Notched four-point-bend tests

After the material hardening behavior was characterized, the notched four-point-bendspecimens were loaded. During loading optical micrographs were recorded and moirSemeasurements taken. Because of the persistent slip bands in the Cu specimens, opticalmicroscopy can be used to identify the sector boundary angles in these specimens.Additionally, the angles of the persistent slip bands in each sector allow determinationof the active slip systems. These are only active on particular portions of the yieldsurface and thus give information about the stress state. In some cases, the number of


Fig. 5. Optical micrographs are shown for Cu and Cu–2.0wt%Be specimens after deformation. Specimens(a) C4, (b) CB1, (c) CB5, (d) CB7, (e) CB9, and (f) CB8 are shown in order of increasing hardness.Changes in color/shade indicate small changes in surface tilt. The black regions very near the tip are regionswith larger out of plane deformation and thus larger tilt. The lines near the notch tip in the top image areslip lines. The 3eld of view is 2:7 mm × 1:8 mm for all images.

possible stress states for a sector can be reduced to only a few options. For instance,sector 4 (counting sectors counterclockwise from x1 in the +� half plane) displays slipbands at 55◦ and 180◦ from the x1-axis as shown in image (a) of Fig. 5. This reducesthe possible stress states to corners B and E of the yield surface in Fig. 1 becausethese locations are the only points on the yield surface where both of these slip systemscan be active simultaneously. Strain measurements can be used to determine the slipdirection which must have the same sign as the stress, thus further reducing the possiblestress states in each sector. For the case of sector 4 in image (a) of Fig. 5, the moirSeresults below indicate that the stress state is at the corner B.

Minimal persistent slip banding was observed on the precipitation-treated specimens,thus analysis was conducted primarily from the moirSe results. The moirSe data col-lected from Cu–2.0wt%Be specimens during loading provides full 3eld in-plane strainmeasures for a region of the specimen surface surrounding the notch tip as shown in


Fig. 6. Strain maps for specimen CB10B5 are shown for load of 1700 N. Image (a) shows �11, image (b)shows �12, and image (c) shows �22. The notch is shown coming down from the top.

Fig. 6. Excellent correlation is found between the observed persistent slip band an-gles in Cu specimens and the angle of maximum shear strain calculated from themoirSe data taken on the solution treated Cu–2.0wt%Be specimen. Analysis of thestrain data also indicate that deformation is sometimes occurring on more than one slipplane within a sector although slip bands associated with only one of these planes areobserved.

For quantitative assessment, strain data has been taken from annular regions ofthe full 3eld strain maps. This annulus is chosen following Shield and Kim (1994)


Fig. 7. The strain components are plotted in strain space for specimen CB5 (a) and CB8 (b) at a radius of350 �m averaged radially over 60 �m. The numerical values identi3ed on the plot indicate the angle fromthe x1-axis at which the strain data was measured.

to avoid material close to the notch which is dominated by the notch geometry andregion of the far 3eld where the proximity of the specimen boundaries has an inFu-ence. A radius of 350 �m is used with radial averaging performed over a range of±30 �m. The results of this calculation are plotted in the strain space of �12 versus(�11 − �22)=2 as shown in Fig. 7. From the normality principle of plasticity, the in-cremental plastic strain vector must be normal to the yield surface. If it is assumedthat the strain increment is in the same direction as the total strain, then the normalityprinciple allows determination of the stress state from the measured strains. Addition-ally, elastic sectors can also be identi3ed as regions with small measured strains. Somesectors appear to remain elastic even after large amounts of deformation have occurredelsewhere.

5. Discussion

Prior to delving into more quantitative analysis based on the strain 3eld measure-ments, some basic conclusions about hardening obtained from the slip band observationson pure copper specimens are presented. Although copper is assumed to have fairlylow hardening that compares well with perfectly plastic analytical solutions, clearlysome work hardening does take place in the material during deformation.

One example is the small regions of triple slip that are observed in the Cu specimensafter signi3cant deformation has taken place. Initially sectors display single slip, i.e.persistent slip bands form in a sector along one particular slip direction (even thoughslip on another slip system may also be occurring). After sectors are fully developedtwo sets of slip lines are observed. As shown in Fig. 2, isotropic hardening results in3elds consisting of sectors of single or double slip, whereas diagonal hardening allows3elds that also contain sectors of triple slip. Experimental evidence indicates that tripleslip develops late in the loading and occurs at sector boundaries. Fig. 8 shows details of


Fig. 8. Optical micrographs are shown for specimen C9 after severe deformation. The lines in the sectorsemanating from the notch tip are slip lines. Changes in color/shade indicate small changes in surface tilt.The black regions very near the tip are regions with larger out of plane deformation and thus larger tilt. Thetop image has a 3eld of view of 2:2 mm × 1:7 mm. The bottom image shows a magni3ed region near thesector 1-2 boundary where triple slip is observed.

one region of triple slip at a sector boundary that developed after a signi3cant amountof deformation. Triple slip is indicated in the micrograph by the existence of persistentslip bands along all three crystallographic slip directions shown in Fig. 1. (The 180◦

slip that appears as one of the three active slip systems in this region, comes formboth sectors 1 and 3.)

Because a small amount of sector boundary motion can be anticipated as high loadlevels are approached, it is relevant to question whether or not the three coincidentslip plane traces occurred simultaneously rather than separated in time. The experi-mental data provides evidence for neighboring stress states that either traverse from


B → A → F or jump from B → F in the region highlighted in Fig. 8. Neither ispermitted analytically however. The former because such transitions can only occurat angles associated with S and N that are not coincident with this region and thelatter because these two corners are not adjacent on the yield surface. In the later case,one could propose an elastic jump from B → F, but such a jump is not likely withthe large strains evident in this region. What would permit such behavior is diagonalhardening where corners B and A of the yield surface are moved to a point at whichthey coincide as in Fig. 2. At such a stress state, triple slip would occur.

Cuitino and Ortiz (1996) investigated the dependence of the near tip 3elds computednumerically on the hardening laws employed. Using near tip experimental data as adiscriminating test for validating the choice of constitutive assumptions, they concludethat a forest hardening model is appropriate based on the results of Shield and Kim(1994) and Shield (1996). Their dislocation-based model had also been shown toaccurately predict experimental stress–strain curves for copper in tension (Cuitino andOrtiz, 1992).

The experimental results reported here for copper also support the use of a diagonalhardening model such as forest hardening. The experimentally observed triple slipdiscussed above can be compared to numerical calculations conducted by Cuitino andOrtiz (1996) for orientation II with hardening exponent of N=2. Numerically predictedand experimentally observed triple slip activity are compared in Fig. 9. The number ofactive slip systems is shown at radii of 700 and 350 mm. Although the experiments didnot display as much evidence of triple slip throughout the deformation 3eld, it shouldbe noted that the absence of persistent slip bands does not preclude activity on a slipsystem. It is not possible to determine if triple slip occurred on the Cu–2.0wt%Bespecimens with higher hardening exponents because slip lines are not visible on theharder specimens.

Aside from the observation of triple slip, there is other evidence that work hardeningoccurs in the Cu specimens. As shown in Fig. 8, slip activity is observed ahead ofthe notch tip. Due to the symmetry of the material and the loading, the value of �12

should be zero on the x1-axis (at � = 0). If this is a constant stress sector, then �12

should also be zero in the entire sector. Thus there should be no persistent slip bandsobserved parallel to the crack in the sector ahead of the crack. Although there is adi(erence in the geometry between a crack and a notch, the observation of persistentslip bands near �=0 is most readily explained by hardening. Saeedvafa and Rice (1989)showed that even a low level of hardening results in a signi3cant value of �12 verynear �= 0.

Optical observations of the Cu–2.0wt%Be specimens show that as the hardness ofthe material increases, the presence of persistent slip bands decreases. The solutiontreated Cu–2.0wt%Be specimen, with hardness nearest to Cu, displays similar slipband features to that observed in Cu, although there are some subtle changes in thelocation of the sector boundaries. Precipitation-treated Cu–2.0wt%Be specimens, havinghigher hardness that the solution-treated specimen, display fewer persistent slip bands asshown in Fig. 5. Clearly, persistent slip band formation is impeded by the precipitationtreatment, whether it is the increased stress levels or the increased rate of hardeningcannot be determined from these results.


Fig. 9. Numerical calculations by Cuitino and Ortiz (1996) of slip activity are compared with using adiagonal hardening model with experimentally observed slip activity in specimen C9 of orientation II. Thevertical axis indicates the number of active slip systems. The graph (a) shows the results of Cuitino andOrtiz (1996) for N=2. Graphs (b) and (c) show experimental results at a radial annulus of 700 and 350 �m,respectively.

In order to compare the features of the notch tip 3elds to the available analyticalsolutions, Table 2 was developed. This table presents both the predicted and measuredstress states in 10◦ intervals around the notch tip. The capital letters at each angularinterval denote locations on the yield surface of Fig. 1. The single letters denote cornersand the pairs of letters indicate stress states that lie on a Fat between two corners on theyield surface. The letters ‘el’ denote regions that are elastic. The experimental resultsshown in Table 2 are derived from a combination of optical observations and analysisof the moirSe strain measurements.


Table 2Progression of the stress state around the yield surface for Orientation II

Start angle 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦ 130◦ 140◦end angle 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦ 130◦ 140◦ 180◦

Analytical

Rice (1987) A A A A A - B B B C C C - D DN = ∞Drugan (2001) A A A A A - B B B B el el - D DN = ∞Saeedvafa and Rice(1989) FA FA FA A A AB B B B BC C C C CD CDN = 5; 8; 20Saeedvafa and Rice(1989) FA FA F FA A AB B B B B BC C C CD CDN = 3

Experimental

Cu (C4, C8, C9) el el el el el A - B B B - el el el DN = 7:5 F A elCuBe (CB5) el el el F F - A A - B B el el el ?N ≈ 9:5CuBe (CB7, CB9) el el el F F FA A A AB B B BC C el ?N ≈ 9CuBe (CB8) DE DE E EF F FA A A AB B B BC C B AN = 5:5

Experimental sector boundary angles for Cu and Cu–2.0wt%Be specimens having the orientation with a(010) notch plane [J101] notch tip direction are compared to the analytical solution from Rice (1987) andSaeedvafa and Rice (1989). The symbol ‘-’ indicates that the boundary between neighboring sectors occursin the middle of the 10◦ segment.

At low hardness levels comparisons to perfectly plastic solutions can be made. Theresults reported here are part of a larger study (Crone et al., 2003) that has motivatedDrugan (2001) to modify Rice’s original formulation to include elastic sectors. Table 2gives both Rice’s (1987) results and Drugan’s (2001) results. An important feature ofDrugan’s results is that the B-type sector persists beyond 90◦ and mates with an elasticsector.

With hardening, the stress 3eld is altered and the perfectly plastic analytical solutionsare no longer a valid comparison. The deformation 3eld that developed at the notch tipin the experimental specimens was dramatically a(ected by changes in initial hardnesslevel. As Table 2 shows, the Cu–2.0wt%Be specimens with higher initial materialhardness displayed much di(erent behavior than the softer Cu and solution treatedCu–2.0wt%Be specimens. The angular extent of plastic deformation increases and theexperimental evidence indicates that the stress state dwells on Fats of the yield surfacein addition to corners of the yield surface.

In the low hardness solution treated Cu–2.0wt%Be specimen, the strain space plotin Fig. 7(a) shows a similar strain increment vector over a large angular extent of


the deformation 3eld. At the lowest hardness levels the strain state, which correlatesto corners of the yield surface, is far from the origin at only a few values of �. Inother words, the strain space plot is irregular with sharp high strain segments focussedin directions where the strain increment vector corresponds to a corner of the yieldsurface. Thus, the stress 3eld is deduced to move from corner to corner of the yieldsurface. Table 2 shows that this behavior is more closely related to perfectly plasticsolutions.

This localization diminished with increasing material hardness, as is shown by com-paring the strain space plots of Fig. 7. The sector boundaries are unidenti3able inthe strain maps of the hardest specimens. Strain space plots become smoother as thematerial hardness increases. In the hardest Cu–2.0wt%Be specimen, the strains be-tween approximately 30◦ and 110◦ are associated with strain increments in directionsspanning over half of the yield surface. The progression around the yield surface iscounterclockwise for increasing � up to 125◦, and includes both corners and Fats ofthe yield surface. In the hardest specimen, however, the progression reverses directionand moves clockwise for larger angles.

Rice’s (1987) solution was also extended to power-law hardening materials bySaeedvafa and Rice (1989) and these results are summarized in Table 2 as well.Their solution incorporated Taylor power-law hardening into a HRR type solution forthe problem of a crack in an anisotropic material. This resulted in a plane straincrack tip stress 3eld displaying non-symmetric spreading of the sector boundaries withincreasing hardening and intermediate sectors corresponding to Fats of the yield sur-face in between sectors corresponding to the corners. Their analysis did not requiresector boundaries to occur in the directions of slip or kink (i.e. S(k) or N(k) direc-tions), which is required of the fully yielded perfectly plastic solution of Rice (1987).As can be seen in Table 2, the stress state progression corresponding to corners andFats of the yield surface agrees generally with the experimental results for hard Cu–2.0wt%Be specimens, however the speci3c correlation at particular angular values is notgood.

As Table 2 indicates, there is a trend in the experimental data with increasing hard-ness; the 3rst appearance of each new stress state shifts to a larger values of � with in-creasing hardness. This shift correlates with the general trend observed in the Saeedvafaand Rice (1989) results, however, it appears that the model developed by Saeedvafaand Rice (1989) is not sensitive enough to the hardening exponent, N . Table 3 showsonly small shift in the angles between di(erent stress states for N between 3 and20, whereas the experimental data in Table 2 displays more signi3cant shifts for Nbetween 5.5 and 9.5. A general observation from the experimental results reported isthat the deformation 3eld observed in very hard materials does not display distinc-tive sector boundaries that are seen in softer materials. This correlates well with theanalytical solutions of Saeedvafa and Rice (1989), which predict a large number ofsectors and transition regions in materials with higher hardness levels. There is alsosome global agreement with the progression around the yield surface in the analyticaland experimental results for the softer (larger N ) specimens; the progression is coun-terclockwise and it includes portions in the lower left quadrant of the yield surface instress space (corners A and B). However, the clockwise progression of the strain space


Table 3Sector boundary angles for the crack tip solution of Saeedvafa and Rice (1989)

N= 3 5 8 20 Yield surface

�1′ 22.220 FA Fat�1′′ 34.015 F corner�1 34.329 33.289 32.858 32.298 FA Fat�2 57.591 56.310 55.609 54.902 A corner�3 60.403 57.620 56.200 54.942 AB Fat�4 98.238 94.891 92.969 91.367 B corner�5 108.964 101.475 96.885 92.367 BC Fat�6 131.385 129.141 127.712 126.221 C corner�7 180.000 180.000 180.000 180.000 CD Fat

This solution, for the orientation with a (010) crack plane and [J101] crack tip direction, includes hardening.The boundary angles are measured in degrees from the x1-axis and the correspondence to the yield surfaceFats and corners are shown. The vertices of the yield surface are given in Fig. 1.

Table 4Comparison of sector boundary angles between experiments, analytical solutions and numerical solutions

Sector boundary Experimental Analytical NumericalC4/C8C9/CB1 Mohan, Ortiz, Shih Cuitino, Ortiz

1–2 50–54◦ 54:7◦ 40◦ 45◦2–3 65–68◦ 90◦ 70◦ 60◦3–4 83–89◦ 125:3◦ 112◦ 100◦4–5 105–110◦ 130◦ 135◦5–6 150◦

Experimental sector boundary angles for C4, C8, C9, and CB1 having a (010) notch plane [J101] notchtip direction are compared to the analytical solution from Rice (1987), the numerical solution of Mohanet al. (1992), and the numerical solution of Cuitino and Ortiz (1996). The origin used for measurementswas 50 �m ahead of notch tip.

plots at larger angles in the experiments is not predicted at the highest hardness levels(smallest N ).

In addition to analytical results, there are numerical results available on the prob-lem under consideration. Rice et al. (1990) present numerical results based on smallstrain theory that agree with Rice’s (1987) perfectly plastic analytical solutions. Thissolution does not agree well with the experiments although the sector boundaries re-ported in Table 4 for the Cu and soft Cu–2.0wt%Be specimens agree better withthe numerical results than with Rice’s (1987) analytical solution. Numerical resultsof Mohan et al. (1992) and Cuitino and Ortiz (1996) include 3nite deformations andagree more closely with the experimental results in the number of sectors observedand the angle of the sector boundaries. When results from the harder materials arecompared to the numerical results of Cuitino and Ortiz’s (1996), good agreement isseen in the results plotted in strain space shown in Fig. 10. The shapes are similar


Fig. 10. The progression around the yield surface in Cuitino and Ortiz’s (1996) numerical model is shownas a dark line overlayed on the experimental results in strain space for specimen CB7. The numericalcalculations used a hardening exponent of N = 2.

and the progression is counterclockwise for both. Although this compares well withthe experimental results for CB7, the observation on the harder specimen CB8 are notcon3rmed.

6. Conclusions

The experiments performed on Cu and a Cu–2.0wt%Be alloy that is hardenable byprecipitation treatment allowed several conclusions to be made about the plastic 3eldthat develops at the tip of a notch in a single crystal. In the material with the lowestrate of hardening (pure Cu and solution treated Cu–2.0wt%Be) the experimental resultsshow that Drugan’s (2001) addition of an elastic sector in the framework of Rice’s(1987) original perfect plasticity asymptotic solution improves the agreement with theexperiments. In addition as the rate of hardening is increased the spreading of the sectorboundaries into transition sectors follows similar trends to that predicted analytically bySaeedvafa and Rice (1989) who added hardening to Rice’s original perfect plasticitysolution. Better agreement might be obtained with the addition of elastic sectors to thehardening solution.

Surface observations of slip bands appear to indicate that triple slip occurs at somelocations in the strain 3eld around the notch. This agrees with the numerical resultsof Cuitino and Ortiz (1996) and validates their conclusion that isotropic hardeningis not a suitable model for these types of materials, indicating that their diagonalhardening model may be more appropriate. In general their numerical results agreedbetter with experiments in the material with a moderate hardening rate. In addition tothe previously noted disagreements with the softest material (Crone and Shield, 2001),


the results from the hardest specimen tested have some features not predicted by theirnumerical solution.

Acknowledgements

The authors acknowledge the support of the National Science Foundation (grantnumber MSS-9257945-2). WCC also gratefully acknowledges the support of a Disser-tation Fellowship from the American Association of University Women and an AmeliaEarhart Fellowship from Zonta International. Portions of this research were conductedin the University of Minnesota Center for Interfacial Engineering, University of Min-nesota Microtechnology Laboratory, and the crystal growth facility of Prof. R.D. James.

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an experimental study of the eect of hardening on plastic...

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