NUCLEAR ENGINEERING AND DESIGN 29 (1974) 360-369. © NORTH-HOLLAND PUBLISHING COMPANY

A LARGE DEFORMATION ANALYSIS OF CRYSTALLINE ELASTIC-VISCOPLASTICMATERIALS*

D.R. BHANDARIWestillghouse ElecTric Corporation. Pensacola. Florida. USA

and

J.T.ODEN

Department of Aerospace Engineering al/d Engineeril/g Mechanics, The University of Texas at Austin. Allstin. Texas 787 J 2. USA

Received 10 September 1973

111ethermodynamics of crystalline elastic-viscoplastic materials developed earlier by the authors is extended toaccount for finite strains. TIle present theory utilizes the basic physical concepts derived from the theory of disloca-tions in crystals and the thermodynamics of continua with internal state variables. A crystaUine simple solid is con-sidered to be a homogeneous elastic-plastic continuum containinl\ dislocations which constitute the intrinsic. internalmechanism of plasticity phenomena. The appropriate internal state variables and the basic constitutive relations forthe description of the elastic-viscoplastic behavior are deduced from the consideration of dislocation dynamics. 'Il,etheory proposed in the present paper is based on the assumption that both elastk and inelastic defonnations takeplace at every stage of loading and unloading. '111is assumption has a definite physical basis and is the underlying con-sideration in the lield of dislocation dynamics. Unlike other theories, the present one docs not require the specifica-tion of a yield criterion or the prior determination of whether the material is loading or unloading. It can be shownthat under appropriate assumptions on the constitutive laws the present theory reduces to the case of classical thermo·elasticity or thermoplasticity. TIle general finite-clement formulation of an initial boundary value problem is brieflypresented. A specific example is carried out in detail which involves the solution of equations governing the coupledthermomech:lIlical response of titanium. To demonstrate the effectiveness of the present formulation, a scheme forcalculating the deformations. Ule strcsses and tempcrature distribution is developed and applied to a three-dimensionalstructure subjected to high temperature. surface heat flux. volume heat supply as well as mechanical loading. -nlCtheory and the analysis should be beneficial in the study of the behavior of reactor materials at high temperatures.

l. Introduction

Recent developments in theoretical plasticity havemoved far al1eld from the classical notations of thesubject. Now it is possible to approach plasticity in afairly deductive way so as to simultaneously blend itwith accepted ideas in continuum mechanics. showhow these ideas may be consistent with dislocationtheory, and to extract out of the general theory every-thing essential to classical plasticity. Several versionsof such general continuum theories of plasticity, and

* Invited paper L3/IO· presented at the Second InternationalConference on Structural Mechanics in Reactor Technology.Berlin. Germany. 10-14 September. 1973.

the relation of one theory to another. have been dis-cussed in recent work by Oden and Bhandari [I] andBhandari and Oden [2, 3]. Moreover, the theory hasalready proved to be effective in I1nite-element analysisof certain thermoplastic phenomena (see ref. [4]). Itis our aim in this paper to expand certain technicalaspects of the theory described by the authors so as toaccommodate certain recently reported experimentalresults, to extend the formulation given in ref. [4] tofinite deformations, to develop corresponding finite-element models, and to cite some representative numeri-cal results obtained in specific applications.

The theory described in this investigation utilizesthe basic physical concepts derived from the theory of

D.R. Bhandari. J. T. Oden, Deforma(ion analysis of crysralline materials 361

For fiXed t, it is assumed that eq. (4) is invertible sothat it can be writ len X = X-I(Y)I(=;. TIlen

in which X describes the mapping that carries X in thereference configuration Co onto a place x at time f.

Effectively, X(X. t) is a one-parameter family of map-pings of Co onto current configurations Cr. In additionto eq. (1). we introduce the deformation gradient F ofthe motion X relative to Co, as the second order tensor

where ~ x is the material gradient. and we assume thatdet F> 0 for every X and t. The Green-Saint Venantstrain tensor 'Y is defined by

'Y:: t(FT F -I) (3)

where I is the unit tensor and FT denotes the transposeof F.

Consider a configuration Cr, to ~ i ~ t. intermedi-ate b~tween Co and C(, and define the place of X attime f by

y = x(X. i). (4)

Formally, y :: x(iC -I(X). i) where k(X) :: X defines theplace of particle X in Ci: thus Ci (or, for that matter,Co and C() need not be a configuration actually occu-pied by Jl during its motion. TIle deformation gradientis then

(7)

(6)

(5)

(2)

(I)

and by chain rule of differentiation,

F:: FF,

x:: x(X(y), i):: x(y· i)

F:: F(X. f) :: ~ xX<X. t).

x = x(X. f),dislocations in crystals and the themlOdynamics ofcontinua with internal state variables. A crystallinesimple solid is considered to be a homogeneous elastic-plastic continuum containing dislocations which con-stitute the intrinsic, internal mechanism of plasticityphenomena. TIle appropriate internal state variablesand the basic constitutive relations for the descriptionof the elastic-viscoplastic behavior are deduced fromconsiderations of dislocation dynamics. The theoryproposed in this paper is based on the assumption thatboth elastic and inelastic deformations take place atevery stage ofloading and unloading. This assumptionhas a definite physical basis and is the underlying con-sideration in the field of dislocation dynamics. Unlikeclassical theories. the present one does not require thespecification of a yield criterion or the prior determina-tion of whether the material is loading or unloading.The approach taken in this paper is quite general andthe techniques described here can be expanded to in-clude virtually any type of constitutive law that fallswithin the framework of our theory. A discussion ofmost of the pertinent ideas, together with completereferences to the related work is given in ref. [3].

Basic thermodynamic preliminaries and the pro-posed constitutive equations for our thermodynamictheory of visco plastic materials are briefly presentedin sections 2-4. In section 5 we review briefly theessential features in which the finite-element modelsof our generalized constitutive laws are formulated toobtain the heat conduction equations and the equa·tions of equilibrium. Finally. in sections 6 and 7, wepresent the application of our theory to solve a repre-sentative example problem in transient, nonlinear.coupled. thermoviscoplasticity of a three-dimensionalbody.

2. ThemlOmechanical preliminaries

where P is given by eq. (5) and F = V yX. Substitutionof eq. (7) into eq. (3) yields

2. I. Kinematics(8)

which can be writ ten in the formMost of the usual kinematical relations are assumed tohold. As is customary in continuum physics, we identifyeach material point X of a continuous body ~ with itsposition vector X in a fixed reference configurationCo at some definite time f:: fo (or T:: f - to = 0). Themotion of the body £j relative to Co is given by therelation

(9)

wherein

2T/=f'TF-l, 2~=FT(Pft-f)F. (iOa,b)

In most crystalline solids, plastic deformation (i.e.yielding in the sense of permanent deformation) is attri-

362 D.R. Bhal/dari, J. J: Odel/. Deformatiol/ al/alysis ofcrysrallil/e materials

2. 2. TIlennodynamic process

which we refer to as the inelastic strain tensor. In eq.(9) the tensor ~ = 'Y- 17shall be referred to as thedifference strain tensor and 'Y'as the total strain tensor.

(15)

(13)

(14)

( 16)

(17)

(I8)

(19)

balance of linear momentum

( ij ) + F - "OXm./.i P m-PUm•

balance of angular momenlum

,jj = ali,

balance of energy. ij . i Jp€ = a 'Yij + q.i + P I,

P<P = oiiiii - psiJ - a*

and

pBS = q!i + ph + 0*,

.p = € - SO.

where P is the mass density and the superimposed dotsindicate time rates. We require also that for each pointX and at every instant of time t. the thermodynamicprocess in ~ satisfies the entropy production inequality(i.e. Clausius-Duhem inequalily) given in the form

It is often convenient to introduce the specific freeenergy .p (Helmholtz free energy per unit mass) by therelation

Then. with the aid of eq. (17). the balance of energy(eq. (15)) can be recast into allernate forms:

(12)

(1I)x= xx·That is, the tolal deformation X is the composition oftwo deformations X and X, where X is thai part of thedeformalion associated with homogeneous shape dis-tortions rather than homogeneous lattice distortionsand therefore not recoverable according to the hypo-thesis. It is then possible to associate with this part ofthe deformation a strain

buted to a now process of crystalline lall ice defectsnormally described ill terms of develop men t alld propa-gation of dislocations. In such situations il is possibleto interpret the homogeneous deformation X(t) of eq.(I) of the body ~ as consisting of homogeneous latticedistortion and homogeneous shape distortion producedby homogeneous motion of dislocations. The latticedistortion is restorable. and on restoration the latticedistortion disappears completely (except locally at dis-location lines) and the body fA occupies a differentconfiguration Cj. Then in view of eq. (7) it follows that

A thermodynamic process for a body fjJ is describedby nine functions of X and t. whose values have thefollowing interpretations. The functioll x(X. t) whichdescribes the motion of the body..;W, the symmetricsecond Poila-Kirchhoff stress tensor a = a(X. t), thespecific body force F = F(X, t) per unit mass, the speci-fic internal energy € = €(X. t) per unit mass. the heatflux vector q = q(X. t) Oil the boundary 3d1ofdl. theabsolute temperature 0 = O(X, t) > O. the specific heatsupply 11 = I1(X, t) per unit mass and per unit time. thespecific entropy S = SeX. t) per unit mass, and the setof N functions cP> = atil(X. f). i = 1,2, ... , N. called the'internal state variables' which may be scalar. vector ortensor functions of X and time t.

The set of nine functions fx. a, F, €. Q, 8, II. S. a(;)}defined for all X in dI and all t is called a Ihermodyna-mic process in .~, if and only if. it is compatible withthe laws of balance of linear momentum. angular mo-mentum. and energy. Under sufficient smoothnessassumptions, the local forms of these laws are:

where a* is the internal dissipation.In order to define the thermodynamic process in ~

it suffices to specify only seven functions {X. a,.p. Q. S.(}. a(;)}: for the remaining two functions. F and II canuniquely be determined from eqs (13) and (15). Athermodynamic process in .~. compatible with the con-stitutive equations at each point X of f* and all time t.is called an admissible process.

3. Thermoelastic-viscoplastic materials with internalstate variables

The general constitutive relations for an elastic-visco-plastic crystalline simple solid can be represented bythe general forms (see e.g. refs [I J- (5))

.p = <Nt (),K, 17.A (i)). (20)_ ~,,(i») (0- u(c,;. 0, g, 17. A . 21)

S = 9'(t {J, g, 11.A (i»), (22)

D.R. Bhandari. J. T. Odell. Deformatioll allalysis of crysra//ille materials 363

where 3g<l>( . ). 3e<l>(') and 30 <1>( .) denote the partialdifferentiation of <I> with respect to g, ~ and 0, respec-tively.

In view of eq. (18). the internal dissipation inequalityis

and the internal slate variables are assumed to be givenby the functional relationships

il = I (~.0,7], A (i~, (24)

A(i) =a.(i)(~,0.7], A(i», (25)

wherein eqs (20)-(23) g = grad. 8. It should be notedthat in the present formulation one of the independentvariables appearing above is ~ instead of 1- It was. how-ever, pointed out by Green and Naghdi [6. 71 that theuse of the tensor ~ instead of'Y implies no loss ofgenerality in the general theory.

We now require that for every admissible thermo-dymanic process in 94, the response functions appear-ing in Ihe above constitutive equations must be suchthat postulate eq. (16), of positive entropy productionis satisfied at each point X of fM and for all time t. Then,with the aid of eqs (9), (17) and (20). the Clausius-Duhem inequality eq. (16) can be rewritten in the form

[(3<1» .] ( 3<1». 3(1)

tr 0 - P ai ~T - P S + ao 0 - P 3g . g

+ tr [ (0- P ~~ ) ~T ] - tr [p 3~~i)AT(i) ]

I+ 8 q .g ~ O. (26)

We observe that by fixing~, 8, g, 7] and A (i) at time twe also fix il and AU) (as a result of eqs (24) and (25))but ~, iJ and g are left arbitrary. Thus, for the inequalityeq. (26) to hold independent of the signs t iJ and g,their coefficient must vanish. Consequently. we con-clude that the free energy is independent of g and thestress and entropy are derivable from the free energy.Le.

q = f2 (~, 0, g, 7], A (i»,

3g<l>( to. 7], A (i» = 0,

0= P 3t<l>(~, 0, 7], A (i).

S = -304>Ct 0.7], A (i).

(23)

(27)

(28)

(29)

a* = { t r [( a - P ~: ) ] ilT - t r [ P 3~:~;)A T(i) ] } ~ o.

(30)

and the heat conduction inequality

f2·g~O. (31)

As a consequence of eqs (30) and (31), the generaldissipation inequality takes the form

(32)

The framework of the theory presented thus far aimsat a general description of the fundamental features ofthe elastic-viscoplastic behavior of crystalline simplesolids. Before this mathematical model can be appliedto a given boundary value problem. it is necessary todeduce the explicit forms of the constitutive equations(20)-(25) for a particular material. In what follows.we shall employ a specific model of the theory todeduce the specific forms of the constitutive equationsfor titanium and then present a solution to the initialboundary value problem in transient, nonlinear, coupledthermoelastic-viscoplasticity of a three-dimensionalcrystalline solid.

4. Constitutive equations for titanium

In this section we shall use primarily the results andtheory of Conrad et al. [8-161 to develop the constitu-tive equations which are suilable to characterize theelastic-visco plastic behavior of titanium. In construct-ing these constitutive relations for general deformationprocesses, such as those discussed in section 3. onemain difficulty arises due to the fact that the informa-tion available in the literature on Ihe mechanical pro-perties of tilanium is still incomplete. This lack of in-formation naturally restricts the applications suggestedby the theoretical development. It is. however, suffi-cient in this study to introduce simplified forms of theconstitutive relations obtained by imposing certainheuristic assumptions.

We assume that the material under study is an iso-tropic elastic-viscoplastic crystalline simple solid forwhich the thermoelastic properties arc specified by afree energy function (I> and viscoplastic behavior de-scribed by Conrad's [9) exponential law. We then pos-

364 D.R. Bhandari. J. T. Oden. Deformation analysis of crystalline lIIaterials

(39)

(40)

(42)

_ Ho (T - T*)2).k8 T3

. 2 {H 0 ( T*) 2}TI(s) = ab voexp - kO I - Tt ~

where a is the mobile dislocation density; b is the lengthof Burger's vector; I'n is Debye frequency; H0 is thethermal activation energy: k is Boltzmann's constant;{J is the absolute temperature; and T* is the effectiveshear stress (thermal component of stress) which isgiven by the difference between the applied shear stressT and the internal back stress Tµ (athermal componentof stress, which is proportional to shear modulus µ).TIle so-called 'athermal stress' is of a type known inmany work-hardening theories and is taken to be ofthe form rµ = XµbaJ/2, A being the work-hardeningfactor.

In a recent paper, Kratochvil and Angelis [17] havesuggested that eq. (39) is not satisfactory for smallvalues of T* (for T* = 0, ~(s) = 0) and therefore theyhave employed a slightly modified form of eq. (39) intheir analysis of a simple one-dimensional problem oftorsion of a circular titanium shaft. This form of theequation has been suggested by Alefeld [18] and isgiven by

. _ 2 (Ho[ 0*(O)]2}...§JLTlii - ab voexp - kO I - 03 IS~/21'

where 0*(0) is the thermal stress; o~ is the athermal

In writing eq. (40) Kratochvil and Angelis [17] haveassumed that the plastic flow will occur when T> l'µ,

i.e. the rate of inelaslic strain is controlled by a specialform of the yield function

f= T - AµbaJ/2. (41)

In their approach eq. (40) holds when T > "Aµba1/2, and~(s) = 0 if T ~ XµbaJ/2. However, in our approach it isassumed that the inelastic strain rate is always non-zeroand that both elastic and inelastic deformations takeplace at every stage of loading and unloading. We thenpostulate here that for a general case, the constitutiveequation for the plastic strain-rate is taken to be of theform

(38)

(34)

(35)

(36)

(37)

Since the body in its reference state is assumed tobe unstrained, the linear terms appearing in the freeenergy expansion eq. (33) are neglected. Then, withthe aid of eqs (9), (28) and (29). we obtain stress andentropy:

1 "kl " C 2p<l> = '2 £'1 ~ij~kl - B'1nij - 2To T + va, (33)

where Eijkl and Bij denote the elastic and thermoelasticmoduli; C is the specific heat; To and T denote thereference temperature and the temperature changesrelated by {}= To + T; I' is the energy of dislocation perunit length and a denotes the dislocation density. Thefirst term in eq. (33) describes the elastic energy of thesystem, the second term represents the energy connec-ted with the thermal expansion, the third is a thermalenergy term, and the last term is the energy of all dislo-cations stored in a unit volume. In writing eq. (33), wehave assumed that the dislocation (defect) arrangementsA (i) are characterized by a scalar quantity a, called thedislocation density, which may be considered as anaverage quantity of all defects and is assumed to begiven by the relation

tulate that the free energy function <I> is expressed inthe form

where Kii is the thermal conductivity tensor.The generalized constitulive equation for plastic

strain rate ~ii is developed from the experimental resultsand theory presented by Conrad [9]. It is based on theconcept that a single thermally-activated dislocationmechanism may be rate controlling over a given rangeof temperature and strain-rate and that the plastic shearstrain rate TI(s) is then given by

dj = Eij kt( 'Ykl - TIki) - nii T,

S = BiiC'Yii - Tlij) + C/To.

The internal dissipation 0* is then obtained by usingeq. (30), i.e.

Further, tne heat Ilux is assumed to be given by asimple Fourier law, i.e.

D.R. Bhandari, J. T. Oden, Deformation analysis of crystalline materials 365

5.1. Heat conduction equation

stress component; S1T is the second principal invariantof the deviatoric stress tensor S defmed by the relations

(49)

(54)

eq. (46) can assume the form

f OT[ 0 {B'/(.yij -1]lj) + ;0 T} - ph - a1j1]ij + £lei:] dv

+ f oT,iqi dv - f o Tq ill i dA = 0,V A

where JA dA represents the integration over the surfaceand nl is the unit normal to the surface.

In order to apply the finite element method to eq.(49), we follow a procedure similar to that of Oden etal. [4] and introduce the local approximations for dis-placement and temperature fields in the form

Ui = IjIty{X)ul(. (50)

T= 'Pty{X)'J'Y, (51)

where uf' and TN are the displacement componentsand temperature change at node N of the element, andthe dependence of uf' and TN on t is understood;1jIty{X) and 'Pty{X) are the normalized local interpola-tion functions for displacements and temperaturesrespectively.

The strain-displacement relationship is given by

"roo= l(Uij + u .. + Uk 'Uk .). (52)·II 2. 1.1 ,1.1

Substituting eq. (50) into eq. (52) yields

"rij = A'N1/lf + CNMijuftl;!, (53)

wherein

CNMij = tWM.jIjlM.i' (55)

Finally, introducing eqs (38), (51) and (53) into eq.(49) we arrive at the heat conduction equation for thefinite element.~o:

f ['Pty{To + 'PMyM) ( Bij(At;ijlif + 2CNMipf!uf)

v./ C.;R} 00 M-Ii Tlil + To 'PRr- +'PN,j'PM,i/('IT

- 'PNPh - 'PN,jj1]ij + 'PNvO:] dl) _j'PNqini dA = O.

(56)

(43)

(44)

(45)

(47)

(48)

(46)Ij' . 0- a Tlij + va: = .

S = a - i tr (S)l

and

S1T = t tr (S2),

wherein {3 is a material constant.The constants H 0, T~ or a~, h, and (3 appearing in

the constitutive equations (40)-(45) depend on thetemperature, the grain size, strain rate and the impuritycontents. Some of these dependencies for a: - Ti 'A-70'have been studied by Conrad et al. [9, 14, 15].

In the following sections, we shall make use of theconstitutive equations (35}-(38) together with eqs (42)and (45) in the fmite-element formulation to solve ourboundary value problem.

S. Finite-element formulation - heat conduction andequilibrium equations

in which tr stands for trace and I a unit tensor.Finally, the rate of change of dislocation density a:

is taken in a simple form

The governing heat conduction equation is derived byintroducing eqs (35}-(37) into eq. (19). We obtain

o {B'/(.yij -1]ij) + ;0 T} - q~i - ph

Multiplying eq. (46) by the temperature variation oTand integrating throughout the volume, we obtain

f c5T[ 8 {Bij(iij -1]ij) + ;0 T}v

- q~i - ph - ,jj1]ij + rxi ] dv = O.

Noting that

f c5Tq~i dv = - f oT,lqi dv + f o Tq ill i dAv V A

366 D.R. Bhandari, J. T. Oden, Deformation analysis of crystalline materials

After some simple algebraic manipulations, eq. (56) canbe rewritten in an alternative form:

where K~ and K5S') are, respectively, the elastic andgeometric stiffness matrices, given by

(68)

(67)

(69)

(65)

K}S'~ = f 2ai/CN!'ij dll,V

(t) (1') (n)

and d~, dJ1., and dJ1., are defined by(I)rkf "k At-rl?dr)\r = {B'/(ANij + 2CN.'l,fijUk )'-PR dr} dll,

V

x {A~ij + 2CNMijUr}] dl'

(a)

+ I 2ai/CNMij duV dv = dJ1.v

Eq. (65) can be rewritten in a compact form:(a) (t) (p) (II)

(Km, + KW) duk = dFt + d~ + dJ1 + d~ (66)

K,(e) - f EijmnAr Ar d .'NJ> - Nij Pmn t"V

(61)

(60)

(59)

(58)

ENM = I Kij'-PN,i<{JM./drl.v

where

DNM = f C'-PN'PMdll.V

f 1 {Eijmn(Apmn d/lr + 2CPRm'P~ du~ - dT/m,.) - Iij dnV

(a)

where Ft is the applied nodal load vector. Then takingthe variation of eq. (63), we obtain the incrementalforms

The approach employed here is essentially that used inthe previous calculations of this type [4]. The high non-linearity is handled by solving the heat conductionequation (57) and the equations of equilibrium (66)iteratively within a small lime increment. As shown inref. [4), a difference operator for a linear variation of

6. Solution procedure

(II)

~ = - I {4Eijmn(At;;ii + Cm'ijut')v

X C!'RmnU~ duf} dv. (71)(n)

TIle contribution of the load vector dFt of eq. (71) tothe overall solution of the problem is negligible (see e.g.ref. 1I9)) and therefore it may be dropped, from thecomputational point of view. It should also be notedthat aij in eq. (68) represents the initial stress or theresidual stress in the structure just prior to a newchange in geometry.

(P)

cJI.t = f {Eijmn(At;;ij + 2CN.\1i/Ur) dT/mn} dt!, (70)v(62)

(63)

(64)f ..a'Y·· f" (a'Y") .~ddl ~ dll + a't d a 'Ie dv = arN'aUN UNv V

Now, introducing eqs (35) and (53) into eq. (64) yields

and

We now write the finite element equilibrium equationsin the form (see e.g. ref. 14]),

f "a'Y" (a)fit ~ dt, = Ft.auNV k

qN = I '-PNqilli dA.A

5. 2. Equilibrium equations

D.R. Bhandari, J. T Oden, Deformation analysis of crystalline materials 367

(72)

temperature within a time increment is combincd withthe iterative solution of equilibrium equation.

The incremental nodal temperature at any timestep kin eq. (57) is solved according to Wilson andNickel (20):

~ (DM•J ENM) -I {ilK)= &+2 GN.(K-I/2)

+ (DNM _ ENM) r.H }f1t 2 (K-i) ,

where M is the small lime increment; (K), (K - 1) and(K - I) represent the value at time step (K). (K - 1)and (K - I), respectively. The reference temperatureand the initial thermal input can easily be incorporatedinto eq. (72), and the temperature change at the endof the first time increment calculated. The results ofthis solution are then used in the assembled incremen-tal equations of equilibrium 10 determine the displace-ments and stresses. The constitutive equation for plasticstrain rate eq. (42) at each time of loading is incorpor-ated into the equilibrium equations (66) and the itera-tive procedures within a time increment are repeateduntil convergence is achieved. With the final values ofthe displacements so determined, it is possible to calcu-late the displacement rates by means of the approxima-tion

(73)

for use ill the heat conduction calculations. Returningto the heat conduction equations for the next timeincrement, the process is repeated, as before. until thedesired length of time has been reached.

In this investigation, a Ihree-dimensiollal linear iso-parametric function for both temperature and dis-placement is used. and the integration is performed byan eight point Gaussian quadrature (see Oden [21)).

7. Results and discussion

Based on the concepts of dislocation dynamics that therate-controlling mechanism during yielding and flow ofQ-titanium at temperatures below 0.4 T.u (where TM isthe melting temperature) is thermally activated, a speci-fic solution to the heat conduction problem has beencarried out. Values for thc mechanical and thermalcharacteristics of commercially pure titanium A-70used in the computations are given in table 1. A three-dimensional titanium solid for which the displacementand thermal boundary conditions are given is shown infig. 1. Here only one-quarter of the symmetrical three-dimensional solid is shown. A temperature change of360°C is applied at the nodes BCDE of the center-leftend element. Effects of elasto-plastic couplings arestudied and the transient temperature distribution inthe X direction with Y = Z = 0 is shown in fig. 2. Thedisplacement W in the Z direction at Y = Z = 100 mm

Table 1.Values of characteristic quantities for Q-titanium A-70.

Characteristic Symbol Value

Young's modulus E 11.5 x 103 kg/mm2

Shear modulus µ 4.3 x) 03 kg/mm2

Density p 4.5 x 10-6 kg/mm3

Poisson's ratio v 0.3Debye frequency vo 1013 sec-t

Specific heat C 0.125 calfC gThermal conductivity " 0.041 cal/cm secoCCoefficient of thermal expansion 0:0 8.4 x 10-6/"CReference temperature To 3000KActivation energy 110 1.25 evAthermal stress 03 = 2T3 147.0 kg/mm2

Boltzmann constant k 1.4 x 10-21 kg/mmoKBurger's vector b 2.95 x 10-8 cmInitial dislocation density 0: 1.5 x 1010 cm2

Disloealion arrangement constant IJ 1011 cm2

Work-hardening factor h 0.97

368 D.R. Bhandari, J. T Oden, Deformation analysis of crystalline materials

TEMPERATURE BOUNDARY AND INITIAL CONDITIONS

DISPLACEMENT BOUNPARY AND INITIAL CONPITION$

It is important to note that the present results arepartly based on the extrapolation of the conclusionsreached by Conrad et aI. The constitutive equations(42)-(45) and the material parameters Ho. o~, Aand(3 used in the computations were deduced from theexperimental results presented by Conrad et aI. In thepresent investigation, the rate of the dislocation energystored in a unit volume has been neglected (the storedenergy is usually at least one order of magnitude lessthan the dissipated energy of the plastic deformation).

The quantity a is identified as the average disloca-tion density as measured by Conrad et al. Kratochviland Angelis [17] in their investigation have pointedout that a better approximation would be to character-ize an arrangement of dislocations by at least two scalar

0.4

200 300 400COORDINATE X(mm)

Fig. 3. Displacement IV in Z direction at Y = 100 mm andZ = 300 mOl.

...... 0.3

! 1 t=0.60hrs.=- t = 0.45 hrs.!;; t = 0.30 hrs.~ 0.2 ~ t = 0.20 hrs.u ~/~t - 0.10 hrs.~p..lI.lH

Q 0.1at Z. 0

at X·O and x·aOOmm

01 Y· 0

~

t1:~I800 mmI"

U(t)·'o'(t)·W(t)·o

'0'(1).0

Vilt)·O

Insulatld an all the surfaces and T (t) • 0 .v.rY\llfltr.txc.pt at th. points 8, C, D, and E which or. k.ptat T(t) • 360·C.

Fig. \. Discretized geometry of a three-dimensional solid withboundary and initial conditions.

for various time levels is shown in fig. 3. In fig. 4, thestress component Oz is plotted for typical elements (0)and (b). As expected the magnitude of Oz for element(0) is greater than (b) at alllevels of !::.T.

One of the significant results obtained from thesecomputations was that the magnitude of the displace-ments and stresses increase as predicted. This has beenshown in figs 3 and 4 which indicate the affect of in-crease of temperature on the mechanical response oftitanium A-70.

-2.

-6.

Element (s)

0;1 0'.2 0'.3 0.4 0:5 0.6TIME (hrs.)

Fig. 4. Variation at Uz for elements (a) and (b).

-8.

600

t.a 0.60 hrs.t ~ 0.45 hrs.taO.30 hrs.t = 0.20 hrs.t a 0.10 hrs.

Fig. 2. Temperature distribution.

360

U 240o~

D.R. Bhandari, J. T. Odell, Deformation analysis of crystalline materials 369

parameters. The first parameter would represent thedensity of dislocations which are unstable during re-verse loading and the second parameter would describethe density of more complicated structures of disloca-tions which are stable in reverse loading process. Un-fortunately, no such data exists concerning the stabilityof dislocation arrangements at different states ofdeformations.

It is observed that the constitutive equation (42)loses its physical meaning in the regions of high stresses.It is also pointed out by Conrad [9] that at tempera-tures beyond the critical temperatures Tt (i.e. T>OAT",,). eq. (42) cannot be applied in determining theplastic strain rate. In such cases. the plastic strain rateis controlled by other mechanisms which are not in-corporated in eq. (42). Other models for dislocationmotion have been proposed which may be generalizedin a manner similar to eqs (42)-(45). The thermalactivation model is particularly useful because itdescribes the effect of both temperature and stress ondislocation motion in a relatively simple manner.

Acknowledgement

The support of this research by the US Air Force Officeof Scientific Research through Contract F44620-69-C-0124 is gratefully acknowledged.

References

[I] J.T. Oden and D.R. Bhandari, J. Eng. Mech. Div .• ASCE99 (EM I) (1973) 131.

12] D. R. Bhandari and J :r. Oden, Int. J. Non- Linear Mech.(1973) (to appear).

[3] D.R. Bhandari. Ph.D. Dissertation, University of Alabamain Huntsville (1973).

[4J J.T. Oden, D.R. Bhandari. G. Yagawa and T. J. Chung,Nucl. Eng. Des. (to appear).

[51 J. Kratochvil and O.W. Dillon. J. Appl. Phys. 40 (1969)3207.

[6] A.E. Green and PM. Naghdi, Int. J. Eng. Sci. 9 (1971)1219.

[7] A.E. Green and P.~1. Naghdi. Proceedings. InternationalUnion of 'll1eoretical and Applicd Mechanics. Germany(1967).

[8] H. Conrad. Acta Met. 14 (1966) 1631.[9] H. Conrad, Can. J. Rlys. 45 (1967) 58l.

[10] R. Jones and H. Conrad, Trans. AlME 245 (1969) 779.(II) R. Jones and H. Conrad. Scripta Met. 2 (1968) 239.[12J R. Jones and Il. Conrad. Acta Met. 15 (1967) 649.[131 G. Sargent. G. Jones and II. Conrad. Scripta Met. 3

(1969) 4:~ I.(14) H. Conrarl ~nd R. Jones, 11le Science Technology and

Applicarj"rl of Titanium. cd. R. Jaffee and N. Promised,Pergamon Press, Ncw York (1970).

[15] H. Conrad. Prescnted at 2nd General Assembly CentroNacional de Investigaciones Metallurgicas, Madrid, Spain,10-13 June (1969).

[16) R. Orva, G. Stone and H. Conrad, Trans. Amer. Soc.Metals 58 (1966) 171.

[17] J. Kratochvil and R.J. Angelis. J. Appl. Phys. 42 (3)(197!) 109l.

[18] G. Alefeld, Z. Naturforsch 17a (1962) 899.[19] T.J. Chung and G. Yagawa. Proceedings, 13th Midwestern

Mechanics Conference. Pittsburgh, Pa., Aug. (1973).(20] E.L. Wilson and R.E. Nickel. Nucl. Eng. Des. 4 (3) (] 966)

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