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Microstructural/Dislocation Mechanics Aspectsof Shear Banding in Polycrystals

R. W. ArmstrongDepartment of Mechanical Engineering

University of MarylandCollege Park, MD 20742

Research Report No. 92-NA-009

March 1992

Carnegie Mellon UniversityCenter for Nonlinear Analysis

Workshop on Shear BandsMarch 23-25, 1992

MICROSTRUCTURAL/DISLOCATION MECHANICS ASPECTSOF SHEAR BANDING IN POLYCRYSTALS

R.W. ArmstrongDepartment of Mechanical Engineering

University of Maryland, College Park, MD 20742

ABSTRACT

At increased tensile strain rates or decreased test temperatures,face-centered-cubic (fee) metals show an increased uniform strainto the maximum load point, in contrast to the decreased uniformstrain behavior exhibited by body-centered-cubic (bec) metals underthe same conditions. The opposite tensile plastic instabilitybehaviors of the two structure-types are explained on a thermalactivation/strain rate analysis (TASRA) basis in terms of strain-dependent dislocation "forest" intersections controlling the feethermal component of stress as compared with a strain-independentintrinsic Peierls stress for individual dislocations beingresponsible for the relatively larger bec thermal component ofstress. Bec metals are differentiated from fee metals, also, byshowing an initial discontinuous yield point behavior that isincreasingly pronounced as the polycrystal grain size is smaller.The yield point behavior is explained by the breakthroughs at grainboundaries of blocked dislocation pile-ups in slip bands. Thebehavior is gauged by the microstructural stress intensity valuedetermined from the dependence of yield stress on the reciprocalsquare root of the average polycrystal grain diameter. Substantialtemperature rises are calculated on a dislocation pile-up avalanchebasis for this instability behavior in metallic, ionic, andenergetic (explosive) materials.

Microstructures and Dislocation Pile-Ups

Figure 1 relates to a number of grain morphology and dislocationmodeling considerations (1-4). RSaumur, in 1722, and Grignon,fifty-three years later in 1775, reported from observing ironcleavage facet reflections on fractured steel specimens thatimproved mechanical properties were obtained with finer grainedmicrostructures (1). The topology of grain structures composing apolycrystalline aggregate remains a subject of interest toresearchers today, for one reason because the dislocation slip anddeformation twinning mechanisms responsible for permanent.deformation and fracturing are tied to the crystal axes within anindividual grain (2) and are limited in movement to the arealextent of planes confined within the grain perimeter (3) . A recentconsideration has involved the role of dislocation pile-ups ineffecting a transition from ductile fracturing to cleavageinitiation at particle clumps in structural steel materials (4).

Discontinuous load drops obtained in titanium (5) and steel (6)materials tested at liquid helium temperature are shown in Figure2. The load drops are attributed to deformation twinning "bursts"that were initiated at micro-slip dislocation pile-ups. Figure 3

gives a sampling of mathematical descriptions of dislocation pile-up behavior, including static and dynamic considerations. Theyield and flow strength/microstructural consequences of the pile-updescription of slip band behavior have been reviewed (7) . Figure4 relates to the predicted dependence of yield and fracturestrength on reciprocal square root of interlamellar spacing forunidirectionally-solidified intermetallic compound materialsdesigned for use in gas turbine engines. The low values offracture strain preceding fracture at ambient temperature areattributed to the relatively large value of the microstructuralstress intensity determined from the slope dependence that isshown. The experimental slope value is near to the computedtheoretical limit for brittle fracturing.

The Thermal Activation/Strain Rate Analysis (TASRA)

Zener and Hollomon (8) described the coupled temperature and strainrate dependence of the flow stresses of various steel materials intheir pioneering investigation of shear banding produced by ablunt-edged punch. Zerilli and Armstrong (9) have given adislocation mechanics description of the thermally-activated flowstress behavior for steel and related body-centered-cubic (bcc)metals as compared with the quite different behavior exhibited byface-centered-cubic (fee) metals such as copper. The TASRAequations, including the strain rate dependent activation areaexpression, are combined with the Hall-Petch stress/grain sizeequation in Figure 5 as applicable to the bcc and fee cases. Thebcc case involves a strain-independent TASRA description of thePeierls friction stress for individual dislocation motion ascompared with strain-dependent dislocation "forest" intersectionscontrolling the thermal friction stress in the fee case.

The fee flow stress equation is shown again in Figure 6 that alsoincludes experimental results obtained on copper over a range oftemperatures, strain rate and grain size by Carreker and Hibbard(10) . The increased temperature dependence of the tensile strengthover that of the yield stress, at all grain sizes, and the strainrate dependence of the flow stress at two grain sizes is inagreement with the combined TASRA and grain size prediction, inthis case including a TASRA contribution to the microstructuralstress intensity. Figure 7 shows counterpart temperature andstrain rate measurements reported by Marchand and Duffy (11) for astructural steel. The roughly parallel shift of the flow stresscurves at different temperatures, for a strain rate of 1000 persecond, and at different strains are in agreement with the bccconstitutive equation. Both equations have been usefully appliedto predicting independent measurements of the full deformationshapes of solid cylinders impacted onto essentially rigid targets.

Plastic Instability for BCC and FCC Metals

The plastic instability property of the bcc and fee equations hasbeen analyzed from the viewpoint of obtaining material constantsfor the strain hardening behavior from reference tests and

comparing the maximum load point strain conditions for the twostructure types. Figure 8 shows the form of the equations fromwhich the effects of temperature and, implicitly, of strain rateproduce opposite influences on the uniform strain to the maximumload point for bcc and fee structures (12) .

Figures 9 and 10 show an example of reference experimental andfitted stress/strain curves for tantalum material, including themaximum load point strain dependence that has been described (13).The constitutive equation constants were employed then to computethe solid cylinder impact results that are shown to be in agreementwith experiment in Figure 11. The opposite dependencies predictedfor bcc tantalum and Armco iron as compared with fee copper areshown in Figure 12. Petch and Armstrong (14) have given atabulation of measurements for various metals and alloys showingagreement with these results. Thus, this average constitutiveequation description of strain rate and temperature influences onthe deformation behavior of bcc and fee metals indicates on auniform strain to the maximum load point basis that the formerstructure type should be relatively shear band prone.

Dislocation Pile-Up Initiation of Shear Banding?

The discontinuous yield point behavior of steel that is associatedwith the propagation of Luders' bands was noted in the pioneeringwork of Zener and Hollomon (8) . Petch (15) has given a recentreview of the behavior that is increasingly pronounced as thepolycrystal grain size is smaller, reflecting the increasingimportance of dislocation pile-ups and the microstructural stressintensity term in the Hall-Petch polycrystal flow stress equation.Armstrong, Coffey and Elban (16) have proposed that veryappreciable heating can occur locally when individual dislocationpile-ups are released from their blocking obstacles, particularlyat stress intensity values near to the theoretical limit forcracking.

The model for this behavior is indicated in Figure 13 along with anequation for the limiting temperature rise that could be expected.In this case, the ratio of the microstructural stress intensity andthe thermal conductivity provides an index of a materialsusceptibility to such heating (17). The results on discontinuousload drops shown in Figure 2 seem in line with the proposedbehavior. New results obtained on shear banding in Ti6A14V alloy,as indicated in Figure 14, show it to be even more sensitive toshear banding than titanium (18). Current research involves theapplication of this proposal to understanding the drop-weightimpact sensitivity of energetic (explosive) crystals where aneffect of crystal size as shown in Figure 15 has been accounted foron a dislocation pile-up avalanche basis (19,20).

References

1. R.W. Armstrong, "Metallurgical Basis for the Strength ofMaterials11, in Centenary of Teaching Metallurgy, (edited by H.B.Bell), University of Strathclyde, Glasgow, 1984, p. Jl; see C.S.Smith, Sources for the History of the Science of Steel: 1532-1786,The Society for the History of Technology and the MIT Press,Cambridge, Massachusetts, 1968, p. 125.

2. R.W. Armstrong, "Strengthening Mechanisms and Brittleness inMetals", Journal of Ocean Engineering, 1, 239, (1968).

3. K. Jagannadham and R.W. Armstrong, "Evidence for DislocationPile-Ups at Grain Boundaries from Slip Band Step HeightObservations", Scripta Metallurgica, 21, 1459, (1987).

4. J.P. Gudas, G.R. Irwin, R.W. Armstrong and X.J. Zhang, "A Modelfor Transition Fracture of Structural Steels from Observations ofIsolated Cleavage Regions", in Defect Assessment in Components -Fundamentals and Applications, (edited by J.G. Blauel and K.-H.Schwalbe), Mechanical Engineering Publications, London, UK, 1991,p. 549.

7. R.W. Armstrong, "The Yield and Flpw Stress Dependence onPolycrystal Grain Size", in Yield, Flow and Fracture ofPolycrystals, (edited by T.N. Baker), Applied Science Publishers,Barking, UK, 1983, p. 1.

8. C. Zener and J.H. Hollomon, "Effect of Strain Rate Upon PlasticFlow of Steel", Journal of Applied Physics, 15, 22, (1944).

9. F.J. Zerilli and R.W. Armstrong, "Dislocation-Mechanics-BasedConstitutive Relations for Material Dynamics Calculations", Journalof Applied Physics, 61, 1816, (1987).

10. R.P. Carreker, Jr., and W.R. Hibbard, Jr., "Tensile Deformationof High-Purity Copper as a Function of Temperature, Strain Rate,and Grain Size", Acta Metallurgica, 1, 654, (1953).

11. A. Marchand and J. Duffy, "An Experimental Study of theFormation of Adiabatic Shear Bands in a Structural Steel", Journalof the Mechanics and Physics of Solids, 36, 251, (1988).

12. F.J. Zerilli and R.W. Armstrong, "Dislocation Mechanics BasedConstitutive Model for Dynamic Straining to Tensile Instability",in Shock Compression of Condensed Matter-1989, (edited by S.C.Schmidt, J.N. Johnson and L.W. Davison), Elsevier SciencePublishers B.V., New York, 1990, p. 357.

13. F.J. Zerilli and R.W. Armstrong, "Description of TantalumDeformation Behavior by Dislocation Mechanics Based ConstitutiveRelations", Journal of Applied Physics, 68, 1580, (1990).

14. N.J. Petch and R.W. Armstrong, "The Tensile Test11, ActaMetallurgica et Materialia, 38, 2695, (1990).

15. N.J. Petch, "Theory of the Yield Point and of Strain-Ageing inSteel", in Advances in Physical Metallurgy, (edited by J.A. Charlesand G.C..Smith), Institute of Metals, London, UK, 1990, p. 11.

16. R.W. Armstrong, C.S. Coffey and W.L. Elban, "Adiabatic Heatingat a Dislocation Pile-up Avalanche", Acta Metallurgica, 30, 2111,(1982) .

17. R.W. Armstrong and W.L. Elban, "Temperature Rise at aDislocation Pile-Up Breakthrough", Materials Science andEngineering, A122, LI, (1989).

18. V. Ramachandran, X.J. Zhang, R.W. Armstrong, W.H. Holt, W.Mock,Jr., W.G. Soper and C.S. Coffey, "Shear Banding in Ti6A14V Alloyvia Reverse-Ballistic Impacts", in Microstructure/PropertyRelationships in Titanium Aluminides and Alloys, (edited by Y.-W.Kim and R.R. Boyer), The Minerals, Metals and Materials Society,Warrendale, Pennsylvania, 1991, p. 647.

19. R.W. Armstrong, C.S. Coffey, V.F. DeVost and W.L. Elban,Crystal Size Dependence for Impact Initiation of RDX Explosive",Journal of Applied Physics, 68, 979, (1990).

20. R.W. Armstrong and W.L. Elban, "Dislocation Roles in EnergeticCrystal Responses", in The Fundamental Physics and Chemistry of theCombustion, Initiation and Detonation of Energetic Materials, LosAlamos National Laboratory, March 3-6, 1992, to appear as aChemical Propulsion Information Agency Publication, 1992.

hexahedral Fe crystals decatetrahedral - hexagon

Grignon (1775)

average grain simple and complex grains

Rhines, Craig and DeHoff

(1974)

Discontinuous Twinning ofTitanium at 4.2 K

Discontinuous Twinning during Essentially ElasticCompression of Steel at 4-2oKf

N. M. MADHAVA AND R. W. ARMSTRONG N . M. MADIIAVA, 1 \ J . W o l l T I I l N a T O N } a n d I t . W . AllMSTUONO

MI - . 1A I I .URCI ICAL T R A N S A C T I O N S V O L U M l i 5, JUNK 1 9 7 4 - 1 5 1 7Reprinted from

PHILOSOPHICAL MAGAZINK, Vol. 25, No. 2, p. 519, February 1972

—1—1—1—1—1—1—1—1

Titanium* Commercial A70 Material

T«4.2°KGrain Dio.« 0.5 mm

Compression

1*12.7 mmd« 4.76 mm

I/ Tension Test —v

/ *sl-3-10" sec./ l«25.4mm

/ dg 2.55 mm _^

1 I

Test —see'

if

1/

y

/

1 1

1

/1

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-

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-

2000 -

1500 -

1000 -

500 -

0 1 2 3

Crossheod Displocement, mmFig. 1—Tension and compression force-displacement curvesfor large grain A-70 titanium material.

4000

3500

3000

2500

\ 2000

1500

1000

5 0 0

1010 Steel

2 3 4

Crossheod Dispioccmer.t, mm

A typical force/crosshead displacement curve.

DISLOCATION PILE-UP DESCRIPTIONS OF SHEAR BANDING

I. Static pile-up resultsJ.C.M. Li and Y.T. Chou, "The Role of Dislocations inthe Flow Stress/Grain Size Relationships11,Metallurgical Transactions, 1, 1145 (1970); Wen-Lan Liand J.C.M. Li, "The Effect of Grain Size on FractureToughness", Philosophical Magazine, A59, 1245 (1989);Q. Gao and H.W. Liu, "Characterization of the TipField of a Discrete Dislocation Pile-up for theDevelopment of Physically Based Micromechanics",Metallurgical Transactions, 21A, 2087 (1990).

1.0

0.1 "

0.01

o.ooi -

0.0001

0.00001

SINGLE-ENDED, ESHELBY ET. AL

• SINGLE-ENDED. CHOU & LI

O DOUBLE-ENDED. ARMSTRONG ET. AL.

0.00001 0.0001 0.001 0.01 0.1 10 0 2 0 3 . C M 0-5 0-6 0 7 0 8 0 9 1

Dislocation-frec zone at the crack tip; the effect of lattice friction and barrier distance.The position, Xit of the i-th dislocation in a pileup.

II. Dynamic pile-up resultsA.K. Head, "Dislocation Group Dynamics VI. The Releaseof a Pile-up", Philosophical Magazine , 27, 531 (1973);F.P. Gerstle and G.J. Dvorak, "Dynamic Formation andRelease of a Dislocation Pile-up Against a ViscousObstacle", Philosophical Magazine, 29, 1337 (1974);Hilary Ockendon and J.R. Ockendon, "Dynamic DislocationPile-ups", Philosophical Magazine, 47A, 707 (1983).

i \

The shape of the dislocation distribution as it moves from right to left, at times7 = 0, 0-01, 003, 0-1 and 0-3.

Positions a} of individual dislocations during a constant strain-rate test (< «100 sec4 2 6 )

or

1.00 0.10 0.002 0.001

150 -

100 -

30 70

(211)g,iv8 twin nucleated by (111)y, [ i 0 i ] r slip

Dislocation reaction:

i i ] y , « ^8 - 1/7 (1/2 [1OO]8)

Hall-Petch k€ values for y ory7- Seutectics versus

single phase ordered or disordered H 2 structures

Reference

1

2

3

4

Material

Ni3AI-Ni3Nb(y;- S) eutecticNi{AI}-Ni3Nb

(y- 8) eutectic

Ni3Mn

Ni3Fe

k€,kgf/mm3/2

Ordered

3.9

—

4.7

4.0

Disordered

—

— 2.0

2.8

3.0

References: 1) Thompson, George & Breinan (1973);2) Thompson & Lemkey (1974);3) Johnston, Davies & Stoloff (1965);4) Arko& Liu (1971)

\

bcc case

V a

r\(Z

T « o . B o -fco. case *.

A* = A*{el = db/2

O %

TENSILE DEFORMATION OF HIGH-PURITY COPPER AS A FUNCTIONOF TEMPERATURE, STRAIN RATE, AND GRAIN SIZE

R. P. CARREKER, Jr. and W. R. HIBBARD, Jr.

ACTA MRTALLURG1CA, VOL. 1, NOV. 1953

Tensile strength of copper as a function oftemperature for each of several grain sizes.

tOO 400 tOO tOO 1000 It 00 1400

AN EXPERIMENTAL STUDY OF THE

FORMATION PROCESS OF ADIABATIC

SHEAR BANDS IN A STRUCTURAL STEEL

A. Marchand and J. DuffyDivision of Engineering

Brown UniversityProvidence, RI 02912

Office of Naval ResearchContract No. N00014-85-K-0597

Materials Research LaboratoryGrant DMR-8316893

April, 1987

R.T.

HY 100 STEEL

STRAIN RATE3IOOO/S

-800

-600

-400

-200

0000LJQC

00

<LJX00

5 10 15 20 25 30 35 40 45 50 55 60NOMINAL SHEAR STRAIN - PERCENT

LJ

600-

500 j<t

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400-

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HY 100 STEEL

ROOM TEMPERATURE

-

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SHEAR STRAIN

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20 %

.00010.001 0.01 0.1 1 10 100STRAIN RATE - 1/s

1000 10000

1000

800

600CO(X!

400

200

0

I > I i ISTRESS / STRAIN / STRAIN-RATE

- \200Q0

D

O

0.0

4900 s*1

HOGE & MUKHERJEE •HM, LOWER YIELD PTS.MAXIMUM LOAD PTS.EQUATION (1)MAXIMUM LOAD LOCUSico=3OMPaB 0 = 1125MPap0 = -00535 K"

1

Pi = .000327 K"1

K =310MPan =.44

.0001 s"1

0.1 0.2 0.3 0.4

TRUE STRAIN

1000

800

cdP

600COCO

CO

§5400

200

00.0

I ' I ' I 'STRESS / STRAIN / TEMPERATURE

HOGE & MUKHERJEE• HM, LOWER YIELD PTS._O MAXIMUM LOAD PTS.

EQUATION (1)MAXIMUM LOAD LOCUSco=3OMPaB0=1125MPa

o = .00535 K"1

= .000327 K"1

K =310MPa= .44

790 K ~

STRAIN-RATE = .00011 , 1 ,

0.1 0.2 0.3 0.4

TRUE STRAIN

I f

1200o

CL

co 800Q:

co

400a:

LOCUS OF NECKING INSTABILITY

a O Ko 300 K, .001 s"1• 300 K, 1000 s-iA 600 K, .001 s"1A 600 K, 1000 s-lo oo K

OFHC COPPER\TANTALUM

ARMCO IRON

0.2 0.3TRUE STRAIN

Acta mt'tall. Vol. 30. pp. 21 ! J lo 2116. I9K2Printed in Great Britain

ADIABATIC HEATING AT A DISLOCATIONPILE-UP AVALANCHE

R. W. ARMSTRONGS C. S. COFFEY and W. L. ELBANNaval Surface Weapons Cenler. Silver Spring. M D 20910. U.S.A.

I(a) isothermal stress build-up: n-, dislocations

.1 J.

~ ~ *(b) critical stress concentration: n2 T£= TC

.LtLJ

U—*J(c) adiabatic collapse-discontinuous load drop

(d) pressure-time curve for TA, r2, and r3

MODEL OF PILE-UP BREAKTHROUGH

o i a Djwfc*adfcuw\.

X Ow.(a) isothermal stress build-up: rv. dislocations

(b) critical stress concentration: i

-A— lx|rvI3 i \j t<

A____n_x J L A X ^ L-Ux1

(c) adtabatic collapse-discontinuous load drop ~

(d) pressure-time curve for r t, r2, and T3

MODEL OF PILE-UP BREAKTHROUGH

iJL

50 100 150

K 9 thermal .conduc+ivHy

200.

IMPACT SENSITIVITY vs. PARTICLE SIZE FOR RDX

AND OCTANITROBENZIDINE (CL-12)

3 flMfl2 Q135S CKAJ150

10080

60

E 40u6tnc

20

10

5

-

-

nox

-

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I

A - 2

s

i

RDX COMPRESSEDAT 15.000 PSI

I 1 1

o^

1 1 1

10 20 40 60 100 200AVERAGE PARTICLE SIZE, pm

400

A.T. NIELSEN, Working Group Meeting on Sensitivity of Explosives, CETR, New MexicoTech, 1987, pp. 256-276.

Crystal size dependence for impact initiation £ a " £ C ? v . F. DeVostof cyciotrimethyienetrinitramine explosive w. L Eiban

J. Appl. Phys. 68 (3), 1 August 1990 979500

200 -

100 -

" 50 -

20 -

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0.5 1.0 2.0

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CL-12 :