dynamic void collapse in crystals: computational modelling ... · inhibit void nucleation and...

$: Dynamic void collapse in crystals: computational modelling ... · inhibit void nucleation and promote brittle fracture in the experiments of Nemat-Nasser and Chang. An earlier analysis$
PHILOSOPHICAL MAGAZINE A 1998 VOL 78 NO 5 1151plusmn 1174

Dynamic void collapse in crystals computationalmodelling and experiments

By Sia Nemat-NasserT0 moo Okinaka sup2 Vitali Nesterenko andMingqi Liu

Center of Excellence for Advanced MaterialsDepartment of Applied Mechanics and Engineering Sciences

University of California at San Diego La Jolla CA 92093-0416 USA

[Received 15 September 1997 and revised version accepted 17 February 1998]

AbstractLocalization of inelastic macr ow and crack initiation in fcc single crystals are

studied experimentally and by numerical simulations focusing on theanisotropic inelastic response of the crystal and the mechanism of possiblecrack initiation and growth produced upon unloading by the residualinhomogeneous plastic strains Hollow circular cylinders of single-crystalcopper are subjected to externally applied explosive loads which cause thecollapse of the cylinder this procedure is called the thick-walled cylinder(TWC) method Then numerical simulations are performed to understand thedeformation process which leads to localized deformation and tensile crackingwhen partial collapse is followed by unloading Various loads and initialorientations of the lattice are examined in these numerical simulations in orderto study their e ects on the macr ow localization and crack initiation phenomena

sect1 Introduction

The objective of this research is to study the localization of the inelastic macr ow andthe possible subsequent crack propagation in single crystals experimentally and bynumerical simulations focusing on the anisotropic inelastic response and mechan-isms of possible crack initiation and growth produced upon unloading by the resi-dual inhomogeneous plastic strains A detailed study of such inelastic macr owlocalization and crack propagation is important since strain localization in singlecrystals can lead to global instability

Localized deformation and shearbanding of ductile metals have been the subjectof numerous investigations over the past few decades Large plastic deformationsoften lead to failure by strain localization The understanding of the behaviour ofmaterials at large strains and high strain rates is of importance in metal forminghigh-speed machining high-velocity impact crash-worthy design penetrationmechanics explosiveplusmn metal interaction and other similar dynamic phenomena

In a recent paper an experimental technique called the thick-walled cylinder(TWC) methodrsquo has been used by Nesterenko et al (1989 1991) to studystrain localizations at high strain rates The strain rate in the TWC method can bemuch higher than the strain rate which can be achieved by conventional methodssuch as the Hopkinson bar technique Nesterenko and Bondar (1994) observed

0141plusmn 861098 $1200 Ntilde 1998 Taylor amp Francis Ltd

sup2 Also at Department of Civil Engineering Kinki University Japan

shearbanding and cracking under high-strain and high strain-rate conditions invarious polycrystalline materials such as copper and niobium

The dynamic void collapse and void growth in single crystals under uniformfarreg eld stresses were studied analytically by Nemat-Nasser and Hori (1987) showingthat tension cracks can be produced upon unloading in a direction normal to theapplied compression Nemat-Nasser and Chang (1990) followed this analytical workand experimentally illustrated the basic phenomenon using the Hopkinson bar tech-nique They observed that cracks propagate into the recrystallized grains at thenominal strain rate of about 104 s- 1 which corresponds to local strain rates exceed-ing 106 s- 1 near the boundaries of a collapsing void Nemat-Nasser and Changdiscuss the mechanism of the formation of Lomerplusmn Cottrell sessile dislocations duringloading which can then lead to fracturing during unloading Their scanning electronmicroscopy (SEM) seems to support this explanation Cuitino and Ortiz (1996) seekto explore the possibility of vacancy condensation as a void-nucleating mechanism infcc single crystals at large plastic deformations and suggest that high strain ratesinhibit void nucleation and promote brittle fracture in the experiments of Nemat-Nasser and Chang An earlier analysis of Nemat-Nasser and Hori (1987) showeddevelopment of high tensile stresses at the tips of a collapsed void once the appliedcompression is removed

Here we study the localization of inelastic macr ow and the subsequent crack pro-pagation in fcc single crystals under uniform high strain-rate compressive loadingThe TWC method is used in the experiment A hollow cylindrical single-crystalcopper specimen is subjected to high strain-rate uniform compressive loading Theloading condition is carefully chosen to ensure both the full and partial collapse ofthe specimen inducing the macr ow localization and subsequent cracking of the specimenon unloading

Then numerical simulations are performed to explore the details of this processA new algorithm to simulate reg nite deformation of fcc single crystals proposed byNemat-Nasser and Okinaka (1996) is implemented into the reg nite-deformation codeDYNA2D and used for the simulation Various loads and initial conreg gurations ofthe lattice are examined in these numerical simulations in order to study their e ectson the macr ow localization and subsequent crack initiation phenomena

sect2 Experimental procedure and results

Externally applied explosives are used to collapse a thick-walled cylinder ofsingle-crystal copper The specimen consists of a hollow cylindrical tube of single-crystal copper encased in a polycrystal copper jacket The single-crystal tube is cutfrom a single-crystal rod using a spark erosion technique Then the sample is loadedby detonating an explosive which surrounds the specimen cell The magnitude of theexplosive is carefully chosen to ensure either full or partial collapse of the hollowcylindrical specimen The size and shape of the hollow cylindrical specimen and theoverall experimental set-up for the TWC method are shown in reg gures 1 and 2respectively The initial conreg guration of the single crystal is measured from theremaining part of the sample using X-ray di raction The axis of the cylinder isfound to be in the [1 3 4] direction

In the reg rst experiment the hollow cylindrical single-crystal specimen is fullycollapsed under conditions which have been discussed by Nesterenko and Bondar(1994) The detonation speed measured in this particular experiment wasD= 4030ms- 1 (explosive ammonite+ sand (10vol) with density 1gcm- 1)

1152 S Nemat-Nasser et al

Dynamic void collapse in crystals 1153

Figure 1 Shape and size of the specimen used in the experiment

Figure 2 Experimental setup

with the diameter of the explosive charge being 60mm The estimated detonationpressure

P =q D2

middot + 1 middot lt 3 (1)

is about 4GPa The initial radial velocity of the inner boundary of the specimen ismeasured by the non-contact electromagnetic method and is about 200ms- 1 lead-ing to a time of collapse of about 8 m s

Optical micrographs of the overall and the central region of the collapsed speci-men are shown in reg gures 3 and 4 respectively Figure 3 shows that the outer


Figure 3 Optical micrograph of the fully collapsed specimen

Figure 4 Optical micrograph of the central part of the fully collapsed specimen The centralcrack is the inner boundary of the collapsed cylinder The lines around the centralcurve indicate macr ow localization zones

boundary of the collapsed specimen is non-circular The lines of the shearbandingrunning from the outer boundary to the collapsed inner boundary are also seen inthis reg gure In reg gure 4 the central curved lines are the inner boundary of the cylinderLines of the macr ow localization are also observed around the central crack in thispicture Since the shape of the outer part of the jacket of the specimen and theloading condition are axially symmetric it can be concluded that the localizationis caused by the anisotropic inelastic response of the single crystal

Next the loading condition is estimated for a partially collapsed specimen and isused to simulate the crack initiation and propagation which occur upon unloadingTo ensure an incomplete void collapse a suitably smaller explosive loading waschosen The outer diameter of the explosive charge was decreased to 55mm result-ing in a smaller detonation speed of about D= 3030ms- 1 According to (1) thisgives a detonation pressure of about 23GPa leading to an incomplete collapse

The overall conreg guration of the deformed specimen is shown in reg gure 5 with thecentral part of the specimen magnireg ed in reg gure 6 for detailed observation Thesepictures are again taken using an optical microscope The outer boundary developsa non-circular shape after unloading The inner boundary which has a circular shapeinitially develops a rectangular shape and cracks are initiated at four corners of therectangle Shearbands develop around the four cracks and also from the outerboundary to the upper and bottom segments of the inner boundary of the cylinderHowever the mechanism of crack initiation and propagation is not revealed by theseobservations Hence numerical simulations are used to examine this process and todevelop a detailed understanding of the phenomenon


Figure 5 Optical micrograph of the partially collapsed specimen

sect3 Numerical simulations

31 Kinematics and constitutive relationsIn this subsection reg rst the fundamentals of the general kinematics on which the

numerical calculation is based are briemacr y reviewed The general kinematics of theelasticplusmn plastic deformation of crystals at reg nite strains is given by Hill (1966) Rice(1970) Hill and Rice (1972) and others Reviews are given by Nemat-Nasser (1983)Asaro (1983) and Havner (1992)

The total deformation gradient F is divided into a non-plastic (elastic plus rigid-body rotation) deformation gradient F and a plastic deformation gradient Fp asfollows

F = FFp (2)

The velocity gradient is dereg ned by

L = CcedilFF- 1 (3a)

where the dot stands for the time derivative Similarly the non-plastic velocitygradient L and the plastic velocity gradient Lp are given by

L = CcedilFF- 1 (3b)

and

Lp

= CcedilFpFp- 1 (3c)


Figure 6 Optical micrograph of the partially collapsed specimen central part

where the hat is used to denote that the velocity gradient is measured with respect tothe initial conreg guration of the crystal lattice Substitution of (2) (3b) and (3c) into(3a) yields

L = L + FLpF- 1 (3d)

It is assumed in this work that the plastic deformation is solely due to the crystallineslip Hence the plastic velocity gradient L

p is given by the sum of the slips of all slipsystems Since fcc single crystals have four slip planes and three slip directions oneach plane it follows that

Lp

=4

a=1

3

a=1Ccedilg (aa) l (aa)

0 (4)

where Ccedilg (aa) is a slip rate and l (aa)0 is the ath slip direction on the ath slip plane which

is dereg ned by

l(aa)0 = s

(aa)0 Aumln

(a)0 (5)

Here s(aa) and n(a) are the slip direction and the slip plane normal respectively andthe subscript 0 stands for the initial conreg guration of the lattice

Since the elastic strains are generally small it is reasonable to neglect them incomparison with the lattice rotation Hence from (3d) and (4) obtain

L = L +4

a=1

3

a=1Ccedilg (aa) l(aa) (6a)

where

l(aa) = Rl (aa)0 RT (7)

and R is the lattice rotation tensor With small elastic strains the non-plasticvelocity gradient becomes

L = Ccedile + e V - V e + V (8)

where e is the lattice elastic strain measured in the rotated lattice and V is thelattice spin rate dereg ned by

V = CcedilRRT

(9)

The symmetric and antisymmetric parts of (6a) are given by

D = D +4

a=1

3

a=1Ccedilg (aa)p(aa) (6b)

and

W = W +4

a=1

3

a=1Ccedilg (aa)w(aa) (6c)

where D and W are the symmetric and antisymmetric parts of the non-plasticvelocity gradient L respectively Also p(aa) and w(aa) are the symmetric and anti-symmetric parts of the slip system tensor l(aa) respectively

Linear elasticity is used to model the elastic lattice distortion

r = C D (10)


where

r and C are the Jaumann rate of the Cauchy stress and the elasticity tensor inthe rotated lattice respectively

The rate-dependent slip model with the power law is employed to model thecrystalline slip Hence the slip rate of the (aa)th slip system is assumed to begiven by

Ccedilg (aa) = Ccedilg(aa)0 sgn (iquest(aa) )

iquest(aa)

iquest(aa)Y

m

(11)

where iquest(aa) = k r p(aa) l is the resolved shear stress and

sgn (x) = 1- 1

forx sup3 0forx lt 0

(12)

In equation (11) iquest(aa)Y and Ccedilg

(aa)0 are the critical resolved shear stress and the reference

value of the slip rate respectively A linear strain-hardening model is considered Thecritical resolved shear stress iquest

(aa)Y is thus expressed as

iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3

a=1| Ccedilg (aa) |dx (13b)

While thermal softening can be included in the power-law model as in Nemat-Nasser et al (1993) this e ect is neglected in the present case since the macr ow stress ofcopper is relatively small and its heat conductivity is rather high

An e cient algorithm to solve (6b) incrementally for a given constant D has beenproposed by Nemat-Nasser and Okinaka (1996) They exploit the physical fact thatlarge deformations of metals are due to plastic macr ow with only a very small accom-panying elastic contribution when reg ve linearly independent slip systems are activeBased on this an e cient algorithm is proposed by these authors by combining theplastic-predictor elastic-corrector method and a conventional implicit or explicittime-integration scheme In the present work this new algorithm is implementedinto the reg nite-deformation code DYNA2D and is used to simulate the cylinder-collapse experiments numerically

sect4 Numerical simulation of the fully collapsed cylinder

The initial mesh of the hollow cylindrical specimen is shown in reg gure 7 Only thesingle-crystal specimen is simulated The polycrystalline jacket is not included in thisanalysis but its e ect is represented as boundary conditions on the single crystalThe mesh includes 5760 elements with 5904 nodes Although each element has thesame crystal orientation initially it develops its own crystal orientation during thedeformation and hence a very reg ne mesh is required The initial conreg guration of thelattice in this simulation is chosen to correspond to the experiment ie the (1 3 4)plane The x and y axes are arbitrarily chosen to coincide with the [25 3 4] and[0 4 3] directions in terms of the Miller indices In the numerical simulation theinner boundary is dereg ned as a contact surface without friction


The loading condition is dereg ned in terms of the velocity of the nodes on the outerboundary of the single-crystal cylinder Hence nodes on the outer boundary movetoward the centre of the specimen with the prescribed velocity maintaining a circularshape for this boundary The velocity of these nodes is obtained as follows reg rst it isassumed that the hollow cylindrical specimen fully collapses at tc = 85 m s with zerovolumetric strain under plane strain conditions The total travelling distance of theouter boundary is then given by

tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)

The initial velocity is given by

vr (0) = 150ms- 1 (16)

The time-variation of the radial velocity vr (t) is approximated by

vr (t) = A0 exp [- a(t - t0)2] (17)

Then if we require that (14) and (16) are satisreg ed at t0 = 40 m s it follows thata= 0044 and A0 = 0303274 Hence we obtain

vr (t) = 0303274 exp [- 0044(t - t0)2] (18)

This curve is plotted in reg gure 8 In the simulation the points marked in this reg gureare given and they are linearly interpolated to dereg ne the loading curve


Figure 7 Initial mesh used in the numerical simulation

Next the material property used in the simulation is discussed Although thesingle-crystal copper is elastically anisotropic its e ect for large plastic strains androtations is insignireg cant relative to that of the inelastic anisotropy and hence elas-tically isotropic material is assumed The shear modulus and Poissonrsquos ratio aresup1 = 45GPa and t = 033 respectively The rate-sensitivity power in (8) is set atm = 101 The initial value of the critical resolved shear stress is assumed to be025 of the elastic shear modulus so that iquestY0 = 1125MPa For the strain hard-ening the rate of change of the critical resolved shear stress is

Ccediliquest(aa) =4

b =1

3

b=1h(aa)

( b b) Ccedilg ( b b) (19)

Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

a=1Ccedilg (aa) dx (22)


Figure 8 Loading condition used in the simulation

and r = 125 is used This assumes that the latent hardening exceeds self-hardeningby 25 With this strain hardening the critical resolved shear stress levels o at threetimes its initial value after a 200equivalent strain

In the reg rst simulation the cylinder is fully collapsed at 81 m s The overview andthe magnireg ed central part of the deformed mesh are shown in reg gure 9 In reg gure 9 (a)the outer boundary of the collapsed specimen has a circular shape As will be shownfor the partially collapsed case this is due to the plastic macr ow which continues after acomplete collapse The central part of the collapsed mesh in reg gure 9 (b) is shown forcomparison with the results of the experiment reg gure 4 These two reg gures show goodagreement and hence it is concluded that the central curve in the experiment is thecollapsed inner boundary of the cylinder and not cracks originating from the centre


Figure 9 Final deformation state of the fully collapsed specimen at 81 m s (a) overallconreg guration and (b) central part

The lines of the macr ow localization are also observed around the central part of thecollapsed cylinder

In order to study the process of macr ow localization the deformation states at675 m s 725 m s and 775 m s are plotted in reg gures 10(a) plusmn (c) respectively The macr owlocalization initiates at about 674 m s when the initially circular inner boundarydevelops localization at four corners The inner boundary then develops two parallel


Figure 10 Deformation states of the (134) cylinder at (a) 675 m s (b) 725 m s and (c) 775 m s

straight and two semicircular edges It is observed that among the four segmentstwo shrink much faster after initiation of the localization Although the loading onthe outer boundary is uniform the anisotropic inelastic response of the single crystalleads to a non-uniform shrinking speed of the inner boundary and hence localiza-tion at the junction of the resulting segments

The maximum and minimum principal stresses at 81 m s are shown inreg gures 11(a) and 11(b) respectively In these reg gures the unit of stress is MPaand tension is positive It is not possible to obtain the details of the stress distributionfrom the present simulation since various important factors such as thermal soft-ening and frictional contact at the inner boundary are not included in the simula-tion However it is clearly observed that large tensile stresses develop within thelocalized regions These stresses are su cient to cause crack initiation at the innersurface of the collapsed cylinder


Figure 11 Contours of the (a) maximum and (b) minimum principal stresses

Although the inelastic macr ow localization around the inner boundary is well simu-lated the non-circular shape of the outer boundary which is shown in reg gure 3 is notexplored numerically as the uniform loading is continued until the cylinder is fullycollapsed In another simulation the loading on the outer boundary is removed at55 m s rendering the outer boundary traction free Then the cylinder is seen to fullycollapse at 8328 m s The overall conreg guration and central part of the deformed meshare shown in reg gure 12 The inelastic macr ow localization around the central part of thecollapsed cylinder which is shown in reg gure 12(b) is not signireg cantly di erent fromthe last simulation (reg gure 9) while the shape of the outer boundary (reg gure 12(a))shows a signireg cant di erence The experimentally observed non-circular shape of theouter boundary (reg gure 3) is well simulated Here it is noted that the given loadingcondition is uniform on the outer boundary until the loading is removed and hencethis non-circular shape observed here is solely due to the anisotropic inelasticresponse of the single crystal


Figure 12 Final deformation state of 8328 m s (a) overall conreg guration and (b) central part

In order to study the e ect of the initial conreg guration of the lattice on the macr owlocalization two other initial lattice conreg gurations are examined As the reg rst con-reg guration a (0 0 1) cylinder is considered The cross-sectional area is the (0 0 1)plane and the x and y axes coincide with the [1 0 0] and [0 1 0] directions respec-tively Due to the crystal symmetry of fcc single crystals the x y axes and the linesy = 6 x are the axes of symmetry in this cylinder As the second conreg guration a(1 1 1) cylinder is examined The cross-sectional area is the (1 1 1) plane and thex and y axes coincide with the [1 1 0] and [1 1 2] directions respectively The(1 1 1) plane in the fcc single crystal is the plane with the highest symmetry The(1 1 1) cylinder has six axes of symmetry the x y axes and the lines y = 6 3- 12xand y = 6 31 2x In both simulations the mesh size is the same as the one used for the(1 3 4) cylinder and the loading is continued according to the velocity curve shownin reg gure 8 until the cylinder is fully collapsed However only the reg rst quadrant isconsidered because of the symmetry of the deformation


Figure 13 Final deformation state of the (001) cylinder at 815 m s (a) overall conreg gurationand (b) central part

The overall conreg guration and the central part of the deformed meshes at 815 m sare shown in reg gures 13 and 14 for the (0 0 1) and (1 1 1) cylinders respectivelyFigures 13(b) and 14(b) are shown for comparison with reg gure 9 (b) While in thesesimulations fourfold and sixfold symmetries are expected due to inaccuracy in thecalculations of the contact surface this symmetry is not maintained in reg gures 13 and14 Localization of the inelastic macr ow in these simulations is a result of the anisotropicinelastic response of the single crystal Since the anisotropy has a signireg cant depen-dency upon the initial conreg guration of the lattice the di erence in the initial con-reg gurations creates such remarkable di erences in the macr ow localization

In order to study the initiation of the macr ow localization the deformation states at675 m s 725 m s and 775 m s are plotted in each simulation see reg gure 15 for the(0 0 1) cylinder and reg gure 16 for the (1 1 1) cylinder Unlike the (1 3 4) cylinderthe inner boundaries of the (0 0 1) cylinder and the (1 1 1) cylinder develop into anoctahedral and a hexagonal shape respectively These shapes satisfy the required



symmetry associated with the initial lattice orientation since at these stages ofdeformation there are no contact surfaces These shapes are not unique and someother shapes can also satisfy the required symmetry Hence materials other thancopper which have di erent properties may develop di erent shapes upon collapsestarting from a common initial conreg guration

sect5 Numerical simulation of the partially collapsed cylinder

Partially-collapsed cylinders (reg gures 5 and 6) o er unique features whose studyrequires special consideration Crack initiation on unloading in single crystals underuniform compression is one such feature which is examined in this subsection Theinitial mesh for this simulation is shown in reg gure 17 It includes 2800 elements with2940 nodes The initial conreg guration of the lattice coincides with the experimentalsample the cross-sectional area being the (1 3 4) plane Again the x and y axes arearbitrarily chosen so that they coincide with the [25 3 4] and [0 4 3] directionsrespectively Nodes on the outer boundary are moved toward the centre of thespecimen with the given velocity as shown in reg gure 8 until 11 m s The given loading


Figure 15 Deformation state of the (001) cylinder at (a) 675 m s (b) 725 m s and (c) 775 m s




condition is then removed rendering the outer boundary traction-free This leads toan incomplete collapse of the cylinder

The overall conreg guration and the central part of the deformed mesh at 40 m s areshown in reg gure 18 Figure 18(b) shows excellent correlation with the results of theexperiment shown in reg gure 6 It is noteworthy that the similarity between thesimulation and the experiment is observed not only in the macr ow localization aroundthe inner boundary but also in the shape of the outer boundary after unloading Thenon-circular shape of the outer boundary of the unloaded specimen shown inreg gure 5 has been one of our main concerns since it might have been caused bythe variation of the explosive loading although a uniform loading may be expectedHowever comparison with simulation reveals that this geometry is caused by theanisotropic inelastic response of the single crystal


Figure 18 Final deformation state of the (134) cylinder at 400 m s (a) overall conreg gurationand (b) central part Loading is removed at 11 m s in this simulation



Figure 20 Expansion of the inner boundary bold and reg ne lines are the inner boundaries at180 m s and 200 m s respectively

In order to study the initiation of the macr ow localization the deformation states at100 m s 125 m s and 150 m s are plotted in reg gure 19 for the (001) cylinder Althoughthe loading is removed at 11 m s the hollow cylindrical specimen keeps shrinkinguntil about 200 m s The macr ow localization is initiated at about 100 m s as shown inreg gure 19(a) Then macr ow localization develops at four corners on the inner boundaryas shown in reg gures 19(b) and 19(c)

At the end of this section crack propagation is discussed The hollow cylinderexperiences a maximum compaction at around 180 m s and then starts to expandThe expansion of the inner boundary is shown in reg gure 20 In this reg gure the thickand thin lines are the inner boundaries at 180 m s and 200 m s respectively Althoughthe total expansion is small when compared with the total deformation it opens thefolds of the inner boundary and hence produces large tensile stresses around thesharp tips of the folded inner boundaries

Next the tensile stresses caused by the expansion are considered by examiningthe maximum principal stresses in elements around the macr ow localization regionsSince the total deformation is almost centrally symmetric only two parts denotedby A and B are considered The positions of the considered parts within the overallmesh are shown in reg gure 21(a) These two parts are magnireg ed and the elementnumbers in each part are shown in reg gures 21(b) and 21(c) respectively


Figure 21 Overall conreg guration of the deformed mesh and elements around the fold


Figure 22 Maximum principal stresses in part A Stresses in the elements around (a) thelarger fold and (b) the smaller fold

The inner boundary in part A has two folds and hence the maximum principalstresses in elements around these two folds are calculated and plotted The maximumprincipal stresses in reg ve elements around the larger fold are shown in reg gure 22(a)and those in three elements around the smaller fold are shown in reg gure 22(b) Inreg gure 22(a) the maximum tension seems to occur in element 714 at 18165 m s and itsmagnitude is 27GPa Such a remarkable tensile stress does not occur in reg gure 22(b)Similarly the maximum principal stresses in part B are plotted in reg gure 23 Themaximum tension of 339GPa occurs in element 22 at 2531 m s Although dynamiccrack propagation under the high-strain high strain-rate conditions has not beenwell studied it may be reasonable to conclude that such large tensile stresses causedby the expansion of the specimen upon unloading initiate and propagate crackswhich are observed in the experiment Similar crack propagations have beenobserved by Nemat-Nasser and Chang (1990) in single-crystal copper where acylindrical hole has been collapsed in uniaxial compression at high strain ratesTensile cracks have been observed to occur normal to the applied compressionpulse These authors conclude that these cracks are developed at the locked disloca-tions upon unloading

sect6 Conclusion

Localization of inelastic macr ow in fcc single crystals is studied experimentally andby numerical simulations The TWC (thick-walled cylinder) method is used in theexperiment Numerical simulations are performed to examine the deformation pro-cess which leads to the observed reg nal conreg guration The new algorithm proposed byNemat-Nasser and Okinaka (1996) is implemented into the reg nite-deformation codeDYNA2D in order to perform these simulations Various initial conreg gurations ofthe lattice are examined in the numerical simulation to study their e ect on the macr owlocalization phenomena


Figure 23 Maximum principal stress in part B

The macr ow localization in fcc single crystals due to their anisotropy inelasticresponse is observed experimentally The process of the localization is studied indetail through numerical simulations and the comparisonof the results with those ofexperiments shows remarkable agreements It is also shown that the initial conreg g-uration of the lattice has a considerable e ect on the macr ow localization phenomena

Crack initiation and propagation in fcc single crystals are observed in experi-ments and successfully simulated by numerical calculations Through the numericalsimulation it is concluded that cracks are produced during the unloading process bythe tensile stresses which are produced by large heterogeneous plastic deformationsthat occur during the loading regime

ACKNOWLEDGEMENTS

This research was supported by the Army Research O ce under contract num-ber ARO DAAL 03-92-G-0108 to the University of California San Diego and theNational Science Foundation under contract number MSS-90-21671 V Nesterenkowas supported by the O ce of Naval Research (Contract N00014-94-1-1040Program O cer Dr J Goldwasser) The authors wish to thank Dr M Bondarand Dr Y Lukyanov for their help with experiments

The computations reported here were carried out on the CRAY-C90 at the SanDiego Supercomputer Center access to which was made possible by the NationalScience Foundation

ReferencesAsaro R 1983 J appl Mech 50 921Asaro R and Needleman A 1985 Acta Metall 33 923Bondar M and Nesterenko V 1991 J Phys IV (Colloque C3) 1 163Chokshi A and Meyers M 1990 Scripta metall mater 24 605Cuitino A and Ortiz M 1996 Acta mater 44 427Havner K 1992 Finite Plastic Deformation of Crystalline Solids (New York Cambridge)Hill R 1966 J Mech Phys Solids 14 95 1967 J Mech Phys Solids 15 79Hill R and Havner K 1982 J Mech Phys Solids 30 5Hill R and Rice J 1972 J Mech Phys Solids 20 401Meyers M Meyer L Beatty T Andrade U Vecchio K and Chokshi A 1992

Shock-Wave and High-Strain-Rate Phenomena in Metals edited by M MeyersL Murr and K Staudhammer (New York Marcel Dekker) pp 529plusmn 542

Nabarro F 1977 Phil Mag 35 613Nemat-Nasser S 1983 J appl Mech 50 1114Nemat-Nasser S and Chang S-N 1990 Mech Mater 10 1Nemat-Nasser S and Hori M 1987 J appl Phys 62 2746Nemat-Nasser S Li Y-F and Isaacs J 1993 Proceedings of MEETrsquoNrsquo93 the 1st Joint

Meeting of ASCE ASME and SES Experimental Techniques in the Dynamics ofDeformable Solids (VA Charlottesville) p 25

Nemat-Nasser S and Okinaka T 1996 Mech Mater 24 43Nesterenko V and Bondar M 1994 Dymat J 1 245Nesterenko V Bondar M and Ershov J 1993 High-Pressure Science and Technology

(New York AIP Press) p 1173Nesterenko V Lazaridi A and Pershin S 1989 Fiz Goreniya i Vzryva 25 154Rashid M Gray-III G and Nemat-Nasser S 1992 Phil Mag 65 707Rashid M and Nemat-Nasser S 1990 Comput Struct 37 119 1992 Comp Meth appl

Mech Engng 94 201Rice J 1970 J appl Mech 37 728 1975 Constitutive Equations in Plasticity edited by

A Argon (Cambridge MA MIT) pp 23plusmn 79

1174 Dynamic void collapse in crystals

shearbanding and cracking under high-strain and high strain-rate conditions invarious polycrystalline materials such as copper and niobium

The dynamic void collapse and void growth in single crystals under uniformfarreg eld stresses were studied analytically by Nemat-Nasser and Hori (1987) showingthat tension cracks can be produced upon unloading in a direction normal to theapplied compression Nemat-Nasser and Chang (1990) followed this analytical workand experimentally illustrated the basic phenomenon using the Hopkinson bar tech-nique They observed that cracks propagate into the recrystallized grains at thenominal strain rate of about 104 s- 1 which corresponds to local strain rates exceed-ing 106 s- 1 near the boundaries of a collapsing void Nemat-Nasser and Changdiscuss the mechanism of the formation of Lomerplusmn Cottrell sessile dislocations duringloading which can then lead to fracturing during unloading Their scanning electronmicroscopy (SEM) seems to support this explanation Cuitino and Ortiz (1996) seekto explore the possibility of vacancy condensation as a void-nucleating mechanism infcc single crystals at large plastic deformations and suggest that high strain ratesinhibit void nucleation and promote brittle fracture in the experiments of Nemat-Nasser and Chang An earlier analysis of Nemat-Nasser and Hori (1987) showeddevelopment of high tensile stresses at the tips of a collapsed void once the appliedcompression is removed

Here we study the localization of inelastic macr ow and the subsequent crack pro-pagation in fcc single crystals under uniform high strain-rate compressive loadingThe TWC method is used in the experiment A hollow cylindrical single-crystalcopper specimen is subjected to high strain-rate uniform compressive loading Theloading condition is carefully chosen to ensure both the full and partial collapse ofthe specimen inducing the macr ow localization and subsequent cracking of the specimenon unloading

Then numerical simulations are performed to explore the details of this processA new algorithm to simulate reg nite deformation of fcc single crystals proposed byNemat-Nasser and Okinaka (1996) is implemented into the reg nite-deformation codeDYNA2D and used for the simulation Various loads and initial conreg gurations ofthe lattice are examined in these numerical simulations in order to study their e ectson the macr ow localization and subsequent crack initiation phenomena

sect2 Experimental procedure and results

Externally applied explosives are used to collapse a thick-walled cylinder ofsingle-crystal copper The specimen consists of a hollow cylindrical tube of single-crystal copper encased in a polycrystal copper jacket The single-crystal tube is cutfrom a single-crystal rod using a spark erosion technique Then the sample is loadedby detonating an explosive which surrounds the specimen cell The magnitude of theexplosive is carefully chosen to ensure either full or partial collapse of the hollowcylindrical specimen The size and shape of the hollow cylindrical specimen and theoverall experimental set-up for the TWC method are shown in reg gures 1 and 2respectively The initial conreg guration of the single crystal is measured from theremaining part of the sample using X-ray di raction The axis of the cylinder isfound to be in the [1 3 4] direction

In the reg rst experiment the hollow cylindrical single-crystal specimen is fullycollapsed under conditions which have been discussed by Nesterenko and Bondar(1994) The detonation speed measured in this particular experiment wasD= 4030ms- 1 (explosive ammonite+ sand (10vol) with density 1gcm- 1)






P =q D2
















F = FFp (2)





and

Lp





L = L + FLpF- 1 (3d)



Lp

=4

a=1

3


0 (4)


is dereg ned by

l(aa)0 = s

(aa)0 Aumln

(a)0 (5)



L = L +4

a=1

3


where

l(aa) = Rl (aa)0 RT (7)




V = CcedilRRT

(9)


D = D +4

a=1

3


and

W = W +4

a=1

3




r = C D (10)


where




iquest(aa)

iquest(aa)Y

m

(11)


sgn (x) = 1- 1


(12)





iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3








tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS
















P =q D2
















F = FFp (2)





and

Lp





L = L + FLpF- 1 (3d)



Lp

=4

a=1

3


0 (4)


is dereg ned by

l(aa)0 = s

(aa)0 Aumln

(a)0 (5)



L = L +4

a=1

3


where

l(aa) = Rl (aa)0 RT (7)




V = CcedilRRT

(9)


D = D +4

a=1

3


and

W = W +4

a=1

3




r = C D (10)


where




iquest(aa)

iquest(aa)Y

m

(11)


sgn (x) = 1- 1


(12)





iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3








tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS





















F = FFp (2)





and

Lp





L = L + FLpF- 1 (3d)



Lp

=4

a=1

3


0 (4)


is dereg ned by

l(aa)0 = s

(aa)0 Aumln

(a)0 (5)



L = L +4

a=1

3


where

l(aa) = Rl (aa)0 RT (7)




V = CcedilRRT

(9)


D = D +4

a=1

3


and

W = W +4

a=1

3




r = C D (10)


where




iquest(aa)

iquest(aa)Y

m

(11)


sgn (x) = 1- 1


(12)





iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3








tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS













L = L + FLpF- 1 (3d)



Lp

=4

a=1

3


0 (4)


is dereg ned by

l(aa)0 = s

(aa)0 Aumln

(a)0 (5)



L = L +4

a=1

3


where

l(aa) = Rl (aa)0 RT (7)




V = CcedilRRT

(9)


D = D +4

a=1

3


and

W = W +4

a=1

3




r = C D (10)


where




iquest(aa)

iquest(aa)Y

m

(11)


sgn (x) = 1- 1


(12)





iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3








tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS












where




iquest(aa)

iquest(aa)Y

m

(11)


sgn (x) = 1- 1


(12)





iquest(aa)Y =

4

b =a

3

b=1h( g )

(aa)( b b) g ( b b) (13a)

where

g =4

a=1

3








tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS













tc

0vr (t) dt = 202acute 10- 3 m (14)

where

vr = dr1

dt (15)


vr (0) = 150ms- 1 (16)


vr (t) = A0 exp [- a(t - t0)2] (17)


vr (t) = 0303274 exp [- 0044(t - t0)2] (18)





Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS













Ccediliquest(aa) =4

b =1

3

b=1h(aa)


Here

h(aa)( b b) = h( g ) (d (aa)

( b b) + r(1 - d (aa)( b b) )) (20)

where

h( g ) =0003125sup1 g pound 032

0003125sup1

10 + 37( g - 032)g gt 032 (21)

g =4

a=1

3

















































sect6 Conclusion






ACKNOWLEDGEMENTS

























































sect6 Conclusion






ACKNOWLEDGEMENTS














ACKNOWLEDGEMENTS












dynamic void collapse in crystals: computational modelling ... · inhibit void nucleation and...

Documents