An upper-bound evaluation of interactive creep and growth in textured Zircaloy

Journal of Nuclear Materials 90 (1980) 75-88 0 North-Holland PublishingCompany

AN UPPER-BOUND EVALUATION OF INTERACTIVE CREEP AND GROWTH IN TEXTURED ZIRCALOY

B.L ADAMS * and G.S. CLEVINGER Babcock and Wilcox, Lynchburg Research Center, Lynchburg, Virginia24505, USA

and J.P. HIRTH The Ohio State University,Columbus, Ohio 43210, USA Received 1 November 1979

Uniform strain-rate, upper-bounding constitutive modeling techniques are applied to the study of irradiation creep in textured hexagonal metals with a specific application to Zircaloy. A crystallite orientation distribution function is used to describe the distribution of grains in a polycrystal for use in strain and stress averaging. Stress dependent creep compliances are constructed for a system exhibiting slip on a discrete set of crystallographic slip systems. Superposition of creep and stress independent sources of internal strain, such as those produced by the irradiation growth mechanism, is studied. A model for irradiation growth is proposed which accounts for grain shape and size effects as weB as non-uniform dislocation densities arising from variable fabrication history. The internal strains are shown to interact with those dependent on applied stress in a manner which alters the constitutive behavior.

1. Introduction

and growth processes can be modelled, once suitable mechanisms have been assumed, by statistically averaging single crystal behavior over the range of orientations exhibited by the polycrystal. Crystallite behavior must be weighted by the volume fraction of crystallites found in that orientation. The weighting function is known as the “crystallite orientation distribution function” (CODF). Three varieties of statistical averaging are commonly applied in estimating bulk polycrystalline behavior lower-bound, upper-bound, and self-consistent estimates [ 11. In a rough categorization, lower-bound estimates satisfy equilibrium conditions, but do not include considerations of strain compatibility among the crystal&es. Upper-bound estimates preserve strain compatibility, but allow stresses to be discontinuous. Self-consistent estimates preserve both stress equilibrium and strain compatibility. Because

Plastic deformation behavior of hexagonal polycrystals is strongly dependent upon crystallographic texture as well as microstructural variables such as gram size and grain shape anisotropy. Hexagonal materials exposed to in-reactor environments (e.g., Zircaloy fuel cladding) exhibit inelastic behavior even in the absence of externally applied stresses. Such behavior, which is linked to flux activated microstructural strain sources, is termed “irradiation growth”. Both irradiation growth and classical inelastic creep are known to depend upon crystallographic orientation. The effects of crystallographic texture upon creep * Present address: Hanford Engineering Development Laboratories, P.O. Box 1970, Richland, Washington 99352, USA. 15

76

B.L. Adams et al. /Evaluation of creep and growth in Zircnloy

strain ~compatib~ties would not be observed during homogeneous deformation, the upper-bounding schemes intuitively seem closer to the physical situation. They are, however, more difficult to apply in modelling efforts. Self-consistent methods have been applied to power-law creep in isotropic hexagonal polycrystals [2,3], but extension of the method to materials with preferred orientations has not been attempted because of its complexity. Uniform strain rate upper-bound estimates for power-law creep are more readily applied to real textures, and have the advantage of mode~ng ~~~-t~~~n interaction on a first-order level. Creep in Zircaloy has been modelled using a lowerbound approach by Holickjf and Schroeder [4,S]. They assumed that slip occurs only on { lO‘i0) (1120> prismatic slip systems, and employed X-ray polefigure data to obtain constants for phenomenolo~c~ creep equations. Franklin and Franz [6] have extended these lower-bound estimates to include basal and pyramidal slip modes. These methods have been applied by several authors [7,8] in studies of both thermal and irradiation creep behavior. Each of these schemes accounts for the preferred orientation of (0002) basal planes in the specimen. It is assumed, however, that {lOiO} prism planes are isotropically distributed about the c axis. The importance of the true rational crystallographic distribution of prism planes has not been evaluated for lower-bound averaging. Additional stresses arising from strain incompatibility are not treated in these lower-bound models. These stresses are expected to occur as a natural consequence of non-uniform crystallite orientation distributions, and from additional strain sources such as irradiation growth. While the mechanism(s) of irradiation growth in Zircaloy can be quite complex, it is generally understood that strain ~otropies occur in the stress-free condition as a result of the partitioning of radiationinduced vacancies and interstitials to various sinks. Examples of sinks commonly considered are voids, clusters, dislocations, loops, cell boundaries, and free surfaces. Since the orientation of dislocations, loops, and cell boundaries can be a~sotro~ic~y ~stributed, the strains are correspondingly anisotropic. Several authors [9,10] have reviewed the microscopic evidence indicating the relative importance of the various sinks in irradiation growth Further discussion of the various

models is not included in this paper. The effect of preferred orientation on stress-free growth has been treated in semi-empirical models which consider the relative numbers of QMO2)basal planes oriented in each of the principal fabrication directions [ 1l-l 51. No attempt has been made to consider the full textural orientation distribution in the materials of interest. Although a limited number of correlations have suggested that creep and growth processes could be separable to first order [ l&16], the possible interactions arising among the various processes, and from grain boundary ~compatib~ties have not been investigated. During the past decade, techniques have been developed for obtaining the full CODF, and using it to statistic~y average behavior in the polyc~st~. The method of Roe [17] and Bunge [18], which utilizes conventional X-ray pole-figure data, has been most widely used. This method, which expresses the CODF in an infinite series of augmented Jacobi polynomials weighted by appropriate coefficients, has been adapted to hexagonal materials by Morris and Heckler [ 191. The accuracy to which the CODF can be determined is limited only by the experimental error in measuring X-ray intensities over the stereographic coordinates. The purpose of this paper is to apply the full characterization of texture to analysis of creep and irradiation growth behavior. Uniform strain-rate upperbounding methods developed by Hutchinson [2,3] are applied to creep behavior using a power-law constitutive equation which embodies slip on prism, basal, and pyramidal systems. Predicted behavior is compared with 673 K thermal data on typical cold-worked, stress-relieved (CW-SR) Zircaloy cladding. The effects of variable fabrication reduction ratio, and cell shape anisotropy on stress-free irradiation growth are modelled for the CW-SRtexture assuming that a net flux of interstitials is annihilated at {lOiO} (1120) prism type edge ~slocatio~, and that a net fhrx of vacancies arrive at cell boundaries. The prism-type dislocations are assumed to be both residual, arising during prior fabrication history, and dyn~ic~y produced at Bardeen-Herring sources operating under the influence of the irradiation flux. The results of the model are compared with data on CW-SRcladding of similar texture, but with varying grain shape anisotropy. Finally, the first-order interaction between creep and growth, which is easily included in the upper-bound estimate, is discussed.

77

B.L. Adams et al. /Evaluation of creep and growth in Zircaloy

to crystallite coordinates. The Greek indices a, 8, r... refer tensor elements to principal fab~cation directions. A Sudan of the coordinate geomet~ used in the analysis is shown in fig. 1. The crystallite coordinate system (0-XYZ) is connected to the fabrication coordinates (o-xvz) by the Euler angIes (13,$, cp).Plane normals, denoted by the reciprocal lattice vector, r, are connected to crystallite and fabrication coordinate axes by sets of sphericalpolar angles (@Ii, $) and (x, V) respectively).

(a)

2. Description of c~s~~~~

+Y

Fig. 1. Shows orientation relationships between the crystallite coordinate system (0-XYZ) and the specimen coordinate system (o-xyz). The reciprocal lattice vector is r. (a) Defines the Euler angles (6, $, #). (b) Defines the sphericat polar angles (C?q,“3 of the reciprocal lattice vector, r, in the crystallite coordinate system IO-XYZ). (c) Defines the sphericalpolar angles (x,q) of the reciprocal lattice vector in specimen coordinates (o-xy.z).

Throughout the paper we make extensive use of the indicial notation for tensors, sties coefficients, and special functions. When tensors are represented, standard matrix/tensor notation is employed. The indices i, j, k . . . * denote tensor elements referred

* For notation, see List of symbols at the end of this paper.

texture

C~t~o~p~c pole-figures are commonly employed in the description of preferred orientation. Each polefigure represents the spatial distribution of a given crystallographic plane in stereographic coordinates. Since the polefigure gives information about the distribution of plane normals, it does not contain any information about the rotation of the crystal&es about these normals; only two angles are specified rather than three which are required for the full description of preferred orienta~on. The CODF, w(B, 9, #), expresses the probab~ty that a crystallite has an orientation 8, 9, c#with respect to the specimen axis. In the method of Roe [ 171 and Bunge [ 181, the CODF can be obtained from a set of conventional pole-figure data. The method generates the CODF from a series of generalized spherical harmonics

w(& tL, #) = 2 t: h W,,, z,,, lzzo m s-2 n z-2 X exp(-inn JI) exp(-in@) ,

(cos 0)

01

where I+,,,, are the appropriate series coefficients and Zf,,r, (cos 6) are the augmented Jacobi poIynomials, The geometry employed in the series formalism is shown in fig. 1. Normalization of the CODF requires that the integral of w(0, Jl, $) over all Euler space be equal to unity. In similar fashion, conventional pole-figure intensity data can be represented as a series of spherical harmonics. The normalized pl~~no~~ orientation distribution for the ith pl~enorm~, qi(x, q), is expressed in

B.L. Adams et al. /Evaluation of creep and growth in Zircaloy

18

the form 4ioG

17) = ,$

m$_l

dm

P;”

(COS Xl w+-~~d

,

(2)

where Qim are the coefficients for the ith plane-normal distribution, and p;” (cos x) exp(-imn) are spherical harmonics. The angles x and q are the sphericalpolar angles of the plane normal as in fig. 1. In terms of the orthogonality properties of spherical harmonics, the series coefficients, QL, can be evaluated, numerically, from the equation

magnitude as the terms associated with such coefficients. Physical and mechanical properties are anisotropic in polycrystals whenever preferred orientations are present. To first order, the bulk property anisotropy is related to the microscopic crystal property (e.g., ~(8, J/, $J))by statistical weighting using the CODF. Hence, the average property, @(0, J/, @)},is 2n we,

44 a

In

1

= J

J

1

0

0

-1

PC~

h

$1. wt

J/, 49

X d(cos 0) dJ/ d$ .

X exp(imn) d(cos x) dn .

(3)

The series coefficients IV,,, are connected to sets of Qim through the Legendre addition theorem which suggests the solution of a set of simultaneous linear equations

When p(B, $I, $J)is sufficiently continuous and welJ behaved it can be expanded in a series of generalized spherical harmonics with coefficients Plmnexactly like eq. (1). The series coefficients Plmnare evaluated using the expression

Plmn=$jyj 0

Qf,,, = 2r

(5)

de, $,44Zlmn 0

(~0s

f9

-1

X exp(imJI) exp(in$) d(cos 0) d$d$ . X exp(M+) ,

(4)

where Oi and @‘iare the spherical polar angles relating the ith plane-normal to the crystallite coordinate system 0-XYZ (see fig. 1). Using the hexagonal six-fold symmetry of the crystal structure, and the orthotropic (orthogonal biaxial) symmetry of the bulk preferred orientation, one can reduce the number of non-zero IV,,, to even integers of I and m, and integer multiples of 6 in n. In addition it can be shown [ 191 that Wlmn= wMin= W,,,= W,,. The consequences of symmetry reduce considerably the number of sets of coefficients, Qirn, required to calculate the W,,,,. Three sets of planenormal intensity data allow all W,,,up to I< 18 to be computed, as was done in the actual calculations. For 12 Q I < 18, eq. (4) can be solved directly for the coefficients. For I Q 12, eq. (4) becomes redundant in that more relations are given than are needed to solve for the coefficients; in this case a least squares subroutine was used to determine the best values for the WI,,. Coefficients W,,,for I > 20 are not required because typical X-ray data acquisition techniques tend to introduce experimental uncertainties of the same order of

(6)

,Eq (5) can be simplified by applying the orthogonality properties of the augmented Jacobi polynomials to show that the average property is

0

I

I

Co(e, J/, G))= 4n2 C C C ,=c m=-_l n=-l

P/m”

Wlmn

.

(7)

While the plane-normal distributions and the CODF are characteristically continuous, well-behaved functions, some property functions of interest are discontinuous over small increments of Euler space. This necessitates the application of eq. (5) directly. Thus, given sufficient plane-normal intensity data, qi(x, n), the CODF, w(e, $, #), can be constructed in the form of a truncated series of polynomials. Bulk properties can then be obtained by statistically weighting the crystalline property over all Euler space using the CODF. in the present work plane-normal distributions for {0002}, { 1OiO), and { 1Oi 2) planes were determined using a Kula-Lopata [20] specimen sectioned from typical CW-SR Zircaloy fuel cladding. Pole-figures are shown in fig. 2 for the three plane normals used in computing the CODF.

B.L. Adams et al. /Evaluation of creep and growth in Zircaloy

Fig. 2. ShowsX-raypol&figuresfor CW-SR Zircaloy cladding. RD impliesthe rollingdirection, and TD impliesthe transverseor hoop direction in cl@ing fabrication. (a), (b) and (c) show the {0002}basal, (1010) prism, and (1012) pyramidal plane normal distributions respectively.

3.

Irradiation growth model

In this section, a model for irradiation growth (in the stress-free condition) is presented which is

79

mechanistically compatible with current experimental evidence, and which permits investigation of fabrication history (reduction ratio), grain shape anisotropy, and preferred orientation of Zircaloy growth behavior. The following assumptions have been employed. (1) The anisotropy of irradiation growth results from partitioning of irradiation induced interstitials and vacancies to anisotropic distributions of microstructural sinks. More specifically, interstitials are assumed to be annihilated preferentially at edge type { lOiO}Cl 120)prism dislocations, and vacancies are assumed to annihilate preferentially at cell or grain boundaries. (2) Grain shape is assumed to be non-equiaxed in the general case, and describable by an orthorhombic with dimensions a, b, and c. (3) Sink efficiency for vacancies at the cell boundary is assumed to be independent of cell boundary curvature, i.e., the magnitude of the total vacancy sink for a cell is directly proportional to the cell surface area. Mass flow is assumed to be spatially isotropic. (4) Local volume conservation is required. This implies the absence of clusters and voids, and an equilibrium dislocation density for a given cell shape as is shown later. (5) The dislocation densities present in the cells (or grains) are assumed to originate from the final cold reduction process for cold-worked materials, and from the operation of Bardeen-Herring climb sources activated during irradiation under the appropriate material stesses. Dislocation pileups occurring during the operation of such sources are accommodated by equal amounts of shear when steady-state conditions hold. Fig. 3. introduces the spatial arrangement of prism dislocations in crystallite coordinates as utilized in the model. Dislocation densities on the three prism planes, originating from fabrication strains, are calculated from the local shear stresses, 71, TV, and 73 using the criterion pi a $. Shear stresses are estimated from the average Q-ratio (Q = Athickness/ Aehoop ) in the last fabrication pass using the BishopHill upper-bounding,scheme [21] for a rigid/perfectly plastic material. Traditional linear-programming methods developed by Chin and Mammel [22] were employed in this analysis.

B.L. Adams et al. / ~~al~t~~ of creep and growth in Zircaloy strain caused by these two processes, independent of the vacancy contribution, ejTt = &.i

is found to be

,

(8)

where Ar, = a032 +&+:GO*

-P3),

A22=1-All,

Al2

X

Fig. 3. Shows the spatial orientation chosen for the three sets of prism dislocations in the crystallite coordinate system (0-XYZ).

= -42

+ %I&

+ Pd

+ $2

-

p3) ,

otherAjj=O, and B is a factor containing the product of the shear or climb rate, time, and the magnitude of the Burgers vector. The 41 in the above formulation have been normalized such that p1 t p2 t p3 = 1. In macroscopic

Fig. 4 illustrates the contributions to growth strain resulting from climb and glide of prism dislocations formed by interstitial activation of Bardeen-Herring sources, and cell boundary dislocations moving to accommodate the net vacancy fhrx, Bains produced by vacancy a~at~on are normal to cell boundaries, and the strain magnitudes are assumed to be inversely proportional to the semi-major distance, squar@, in each principal direction *. Strains resulting from interstitial annihilation at prism dislocations are positive and perpendicular to the extra half-plane of atoms. With equal rates of dislocation shear and climb, the

coordinates eift is transformed to e:i tion &!j =

using the rela-

BA+ = BT$ T,?‘A, ,

(9)

where cos tj~cos 8 cos Ip sin J, cos f3 cos Q, -sin B cos 4, +cos J, sin @

+sin3/sirld, T=

-cos$costisin#

-sin$cosBsin$

-sin J, cos $J

+cos 9 cos @

cos J, sin

sinGsine

8

(10) From geometric considerations it can be shown * The assumption of isotropic mass flow leads to equivalent vacancy flux densities in all material directions within the cell Thus the ~obab~ty of a vacancy ~~~t~ at an incremental unit of area on the cell boundary will be directly proportional to the solid angle subtended between the vacancy and the area element. The subtended solid angle is directly proportional to the elemental surface area and inversely proportional to the square of the distance separating the boundary element and the vacancy. Strain, as perceived at the cell boundary, is taken to be proportional to the probability of a vacancy arriving there; the strain contribution from a single vacancy will then be inversely proportional to its distance, squared, from the surface element. Calculations involving real cell shapes become highly dependent upon the geometry of the cell. First order calculations for orthorhombic cells lead to strain-rates, in the three principal orthogonal directions, inversely proportional to the square of the cell dimension. These calculations tend to neglect boundary elements at the corner regions of the orthorhombic.

that P2 + P3 = go;2 f 142

f (111

P2 - P3 = 3022.012 7 where (1,2,3)

imply (X, Y, Z). Strains resulting at cell boundaries as a consequence of vacancy annihilation are referred to fabrication coordinates: eri” = B& ,

(12)

where

(13)

B.L. Adams et 111./Evaluation of creep and growth in Zircaloy

81

(C)

Fig. 4. Shows the model for irradiation growth. (a) Depicts the preferential absorption of interstitials (0) at edge type prism dislocations, and the preferred annihilation of vacancies (0) at cell boundaries. Climb strains are accommodated by equal amounts of slip. These strains are referred to the crystalline coordinate system (0-XYZ) as shown in (b). Vacancy annihilation results in strains normal to cell boundaries and referred to fabrication coordinates (o-xyz) as shown in (c).

The rate constant is again B since volume is conserved, and since vacancies and interstitials are created at equal rates. Implicit in eq. (11) is the previous normalization condition, which for zero volume dilatation requires that Pl

+P2

+P3

=

l/a2 + l/b2 + l/c’.

(14)

This last equation compactly expresses the general feature of the present model; steady-state conditions require, for a given cell surface area, a specific total dislocation density under conditions of zero dilatation. As a general trend, a decreasing cell size would require an increasing total dislocation density. This expresses the condition that as cell size decreases, the surface area to volume ratio increases. Combining eqs. (7) and (10) we have total growth strains, E&, of ‘Q = e2; + e&j = B&p

+ l-I,,).

(15)

If cell-to-cell interactions are neglected, a first-order

statistical estimate of bulk anisotropie irradiation growth behavior is given by

0

0

11

(16) Thus, under the assumptions applied in this model, an explicit expression for the total irradiation growth in the specimen, {e&r), is obtained by statistically weighting the sum of climb-plus-slip derived strains, I$-$, and vacancy produced strains, erj. Calculation of the vacancy contribution requires the input of the cell shape parameters, (a, b, c). Evaluation of the climb-plus-slip contribution requires calculation of the relative dislocation densities, pi, on each of the prism slip planes. Effects of anisotropically distributed dislocation densities generated during coldworking operations can be described by the rigid plastic model of Bishop and Hill [21] in terms of

B.L. Adams et al. /Evaluation of creep and growth in Ztrcaloy

82

Fig. 5. Shows dependence on c~rn~pius-slip strain anisotropy on f2brkxi~OXIQ-ratio (Q = A~thi&nes&A&hoop). Dislocation densities were calculated from a Bishop-Hilt analysis employing typical ratio sof critical resolved stresses, rc.

the final-pass fabrication strains. The model was applied to the CW-SR texture shown in fig. 2 using the CODF as in eq. (16). The dependence of e$‘t/eEf on @ratio is shown i.n fig. 5 for a range of 0 < @ < 6 which covers the typical, ratios found in Zircaloy tubing production (6 and z

#

I’ -.I 4 P Ex

2

3

QRAllD

5

--- EQUlAX2D (o=b=e)

-.2-

-.-

z-ELCiWAl2Db=&

-

x.1 ELDNDAlEDk~1/2b=MD)

-.s

1

,)_-__*_---*--

-A

4

/

I*

IIJC)

___&-----

____.-m--4

.~-‘-+--’

-.!I

imply hoop and axial directions respectively). The ratios of critical resolved stress applied in the BishopHill analysis approximate observed ratios for prism slip~p~ami~ twinning [231. Fig. 6 shows the calculated variation of total growth strain ratio (ej/eE) with @ratio for several sets of anisotropic shape factors (a, b, c), The wide variations with shape factors that are predicted result from the vacancy contribution, I$$, in the total radiation growth ratio. The periodic variations caused by variable dislocation density (see fig. 5) are of secondary importance to the grain shape effect. Fig. 7 further illustrates the marked effect of shape a~so~opy in a pl~et~~i~ diagram [ 161 of strain for a wide range of shape factors (a, b, c). Also shown in fig. 7 are measured irradiation growth data for two different CW-SR cladding materials with textures nominally equivalent to that used in the present CODF formulation. The shape factors for these materials (Sl and V2) were taken to be the grain shape factors determined from optical metallography. The measured growth strains are in good agreement with the trends of the calculations.

Fig. 6. Shows ttie total growth stmin ratio dependence on fabrication Q-ratio for several grain shape anisotropies.

Fig. 7. Shows predicted strain-rate behavior for several hypothetical grain shape anisotropies. In each case the Q-ratio was taken as Q = 2. Measured stress-free irradiation growth data for two types of CW-SR cladding [designated as Sl and V2) are shown for comparison. This plane triaxial diagram gives strain ratio information only; ~~~d~ have no meaning.

a3

B.L. Adamset al. f Evaluationof creep andgrowthin Zircaloy

4. Multiaxial creep deformation modelling The anisotropic nature of creep in hexagonal metals is strongly linked to the preferred orientation of crystallites in the bulk polycrystal. Deformation textures arise in these materials during forming operations because of the restricted number of linearly independent slip systems on prism and basal planes. Generally, slip on prism and basal planes is observed to operate at lower shear stresses than those required for slip (or twinning) on the pyramidal planes. Slip (and twinning at lower temperatures) on several pyramidal planes, as well as basal and prism slip, have been observed in the various hexagonal materials [24]. Since the combined activity of both prism and basal slip can supply only four linearly independent strains, whereas the von Mises criterion requires a total of five for an arbitrary grain, pyramidal activity has a controlling effect on general inelastic strain in the crystallite. Indeed, any strain in the c direction in the crystallite must be accommodated by the pyramidal mode, which supplies an additional five linearly independent slip systems. Pyramidal deformation is particularly important in low c/a lattices where prism slip is the primary system but where basal slip is difficult, basal slip not commonly being observed except at higher temperatures e700-800 K m,Zr) and in regions of stress concentration in the crystallite [25, 261. In such materials, pyramidal twinuing occurs at low temperatures and under stress concentrations arising from compatibility constraints [27,28]. Hutchinson has studied creep in polycrystals with isotropic crystal orientation distributions using the uniform strain-rate upper bound for crystallographic slip on prism, basal, and pyramidal planes [3]. This technique involves the assumption that slip on any discrete set of crystallographic planes is proportional to the resolved shear stress on that plane, raised to a power [2]. The stress power-exponent is taken to be identical in each slip mode. A uniform strain-rate perceived at the boundary is applied to each crystallite and the required stress rate is computed from the power-law constitutive relationship. Compatibility is automatically preserved, and discontinuous stresses are allowed in the upper-bound formulation. the power-law model converges to the widely used Bishop-Hill-Taylor bound [2] for rigid/perfectly plastic material as the stress exponent

becomes large. Power-law upper-bound estimates show good agreement with the elaborate self-consistent analysis when the inelastic anisotropy of the crystals is not too large [3], i.e., the reference shear stresses on the participating glide systems do not differ by more than a factor of ten. 4.1. Creep cons& tive behavior of polycrystais The creep behavior of the crystal is described by considering the:‘constituent slip systems and their behavior under applied stress. The unit normal to the kth slip plane is denoted as njk) and the unit Burgers direction as by). The crystal slip tensor for the kth slip system is

/~fi”’= $(bp’+)

+ bfk)njk)) .

(17)

In local crystal coordinates uii is the stress. The resolved shear stress on the system is (k) . r@) = oij/.l,j

(18)

The shear strain rate on the Mh system is denoted as ~(~1,n is the stress exponent, and A as the reference strain-rate. Steady-state creep is assumed to obey a ‘power-law relationship, where r’W = (A/@) 1#d/,-$d 1n - 1 +I .

(19)

Here, rik) is taken to be the reference shear stress on the kth system, and as defined is always a positive quantity. When the resolved shear stress is equal to the reference shear stress (rtk)= ~6~3,the strain rate is equal to the reference strain rate (7’“) = A) inde pendent of the stress exponent, n. The total creep strain rate in the crystal, es, can be expressed as the sum of contributions from each of the active systems such that (20)

where eqs. (18) and (19) give Mij,s = (A/~b”1 IT(~)/T~~) In - ’ @

&) .

Miikr iS the

(21)

tenSOr of creep comphances; further properties of this tensor are described in detail in ref.

PI* For hexagonal crystals attention is confined to basal, prismatic, and pyramidal slip. While several pyramidal systems could be selected, the { 1122)

84

B.L. Adams et al. / Evaluation of creep and growth in Zircaloy

1 an interactive numerical procedure was used in which successive approximations to the actual stress were obtained by means of a form of Newton’s method [2]. If oii + Aoii is the next estimate of the stress, convergence is found to occur when

(22) Since the slip processes result in volume conservative strain rates, Det [Mijkl] is singular. Evaluation of eq. (22) requires the reduction of eq. (21) to an equivalent 5 X5 system by use of the relation ej& = 0, arising from the inherent incompressibility of the material.

a

a2

. al

SLIP SYSTEM

\

PLANE

SLIP DIRECTION

si

Fig. 8. Shows prism, basal, and pyramidal slip systems used in upper-bound creep analysis.

Convergence to a stable stress value results from this application of Newton’s method in a small number of iterations (7 15). it is convenient to use the viscoelastic (n = 1) stress as the initial guess in eq. (22), and then to increment l/n evenly until the stress exponent range of interest has been covered. In the uniform strain rate upper bound the macroscopic strain rate, E&, as measured at the boundary, is imposed independently on each contributing crystallite. Hence e& = Gp) *

(23)

Stresses are discontinuous in the upper-bound approximation, so bulk macroscopic stresses are taken as {Q),

where 2s

+~a~)

=

2s

1

s

s

.[

0

0

-1

w@,$J,

4)

* uap

d(cos

0)

dll

d9.

(24) Since the crystal constitutive behavior is a function of r1\, ru, and TV, {uap} is a function of the reference stresses as well as the CODF. Thus, a power-law relationship between the shear strain rates and the applied shear stresses on a set of crystallographically feasible slip planes leads to the formulation of a constitutive equation which links the creep strain rate, E& to the local stress, cij, through the tensor of creep compliances, Mijk,m Application of the upper-bound scheme imposes boundary strain-rate conditions, E&, on each individual crystallite. Local stresses are computed from the constitutive equation. Finally, bulk averages of non-uniform crystallite stresses are obtained by weighting single crystal behavior with the CODF. The uniform strain-rate upper-bound analysis was applied for CW-SR tubing texture over a range of strain-rate ratios commonly observed in multi-axial creep testing of the thin-walled tubing. Stress exponents up to n = 6 were investigated using the powerlaw constitutive relations - eqs. (17)-(24). The strainrate anisotropy of the principal strain-rates was found to be only slightly sensitive to the stress exponent, n, for 1 < n Q 6. In each case, 486 texture orientations were used to evaluate numerically eq. (24). Reference stress values were chosen to emphasize behavior when extremes of relative activity on the three slip systems are imposed.

B.L. Adurnset al. / Evahation of creep and growth in Zircaloy

Figs. 9 and 10 show the thermal results for highly cold-worked CW-SR thin wall tubing (n = 6). In addition, experimental data are superimposed for comparison. These data are the results of numerous creep tests conducted at 673 K under multi~i~ loading imposed by adjusting combinations of uniaxial tension and internal pressurization [29]. Hoop and axial strain rates were monitored using high temperature extensometry (es and e”, respectively). The third principal strain rate in the radial direction, es, was deduced from the incompre~ibi~ty con~tion. A limited amount of data was obtained at temperatures more characteristic of m-reactor environments (615 K); no detectable difference in strain anisotropy was observed. The exper~ental creep data for CW-SR tubing is most closely approximated with a set of reference shear stresses which correspond either to case A (easy slip on basal planes relative to other systems) or case F (easy slip on basal and pyramidal planes). This result is consistent with the expectation that pyramidal slip should be required in any case in order to fulfdl the requirement for five indepenent slip sys-

85

0.75 li 0.50 0.25 0 -0.25 f

-1.21~ -0.1

1 0

, 0.8

1

, I.5

2

I 2.5

[ii{ z

F‘ig. 10. Shows overall stress ratio behavior (averaged over the distribution of cry&Bites for typical CW-SR texture), (~0) / {ur}, as a function of applied creep strain-rate ratio, eg/e& Experimental data for 673 K creep under multiaxial loading conditions are shown for comparison. (- - -) Isotropic texture, (&) experimental creep data at 673 K. In terms of reference critical resokd shear stresses, r, calculated data are for: case D (e), > 10 Turfs = > 10 ?bas = rpyt ; case E (u), >lO 7pris = >lo 7pYr = 7b& case F (A), >lO rbas = >lO rpyr = 7pris; caseG(v), Tpris = rbas = 7pyr.

I.25 I

II

0.75 0.60

rc 0

0.25 0

6

r

-0.25 -0.50 -0.75 -I -1.25 : -0.5

1

I

0.5

I

t

I.5

2

2.5

+_; oi

Fig. 9. Shows overall stress ratio behavior (averaged over the distribution of crystal&es for typical CW-SR texture), Ml1 u, 1 , as a function of applied creep strain-rate ratio, e&F. Experimental data for 673 K creep under multiaxial loading conditions are shown for comparison. (- - -1 Isotropic texture, (*) experimental creep data at 673 K. In terms of reference critical resolved shear stresses r, calculated data are for: case A (e), >lO rbas = rPYr = Ipris; Case B (n), >I0 rpyr = rbas = 7p&; case C (41, >lO rpris = fbas = 7pyr.

terns because it can accommodate strains in all c directions. indeed, previous mode~g approaches [2,3] have also illustrated the importance of pyramidal deformation. The results are also consistent with the results of slip studies, mentioned earlier, which indicate increased basal and pyramidal slip activity at elevated temperatures 125-281 (700-800 K) in the range of that (673 K) of the experimental data in figs. 9 and 10. Prior experimental observations of slip in the zirconium lattice have suggested that prism slip plays a dominant role in deformation up to temperatures 400 K [25-281. App~cation of the power-law upper-bounding scheme to CW-SR tubing at 673 K, however, strongly suggests that prism slip plays a secondary role in the observed deformation. Preferred activity on the basal or basal plus pyramidal slip systems most closely fits the experimental data; preferred prism slip or cornbi~~o~ of prism slip with some basal or pyramidal activity show the greatest deviation from measured creep performance at this temperature.

86

B.L. Adums et al. /Evaluation of creep and growth in Zircaloy

Thus, the model indicates that basal and pyramidal dislocations are dominant in slip processes during creep at 673 K and presumably multiplying during transient creep. The prism dislocations, already present as a consequence of prior cold work during forming, do not have a major role in creep although they are important sinks for interstitials produced in irradiation. The reason for the lack of operation of the prism system is not clear, but it may simply indicate that the system has been hardened by the prior deformation sufficiently that the other systems become preferred during high temperature slip. Alternatively, the temperature dependence of the slip systems may be such that basal slip is favored at elevated temperatures. Further work is needed to resolve this issue.

5. Interactivecreep and growth The simultaneous occurrence of stress dependent (creep) and stress independent (irradiation growth) straining imposes the question of superposition of strain rates. In a previous analysis of creep and growth [ 161, these two processes were found to be. separable. This analysis led to the observation that, to fust-order, in-reactor strain-rate anisotropies are nominally equivalent to thermal anisotropies observed under the same biaxial loading condition. This observation is retained in the present interactive analysis. For irradiation growth, stress effects are assumed to be of second-order in importance in comparison to the primary effect of point defect sink distributions. The tendency of Zircaloy to exhibit irradiation growth strains in the absence of external stresses certainly can be expected to affect the creep constitutive behavior expressed in eq. (20). Under special conditions of imposed stress, favorably oriented crystals should be assisted in their natural creep behavior by complimentary growth tendencies. Under reversed stresses the superposed processes should compensate for one another and reduce creep rates. This effect can be addressed in a natural way in the upper-bound estimate by imposing a strain on each crystal equal to the difference of the total imposed strain, e& and the irradiation growth strain,

eb. Hence, eq. (20) must be modified to E$ - eQ = es = Miikl nil 5

(25)

where u$, the growth affected local stress state, is distinguished from the pure creep consitutive behavior yielding afj. In eq. (25) the local growth strain, efii,is related to the growth strains in fabrication coordinates (expressed in eq. (15)) efa, through

(26) The effects of superposition were modelled using eq. (25) for typical CW-SR 673 K behavior. The relaxed basal reference stress of case A in fig. 9, which best fit pure creep behavior, was used. Growth strains were compared in magnitude to creep magnitudes using their first tensor invariants (i.e., e, + ee t e,). Growth strains were set at onefourth the magnitude of the creep strains. These strains were computed using typical grain shape anisotropies ((at b, c) = (f , 1,4)) and using an isotropic distribution of prism dislocation densities, p r = p2 = p3. The isotropic distribution of dislocations was used because, as shown in fig. 6, the growth strains exhibit small sensitivity to the ratio of dislocation densities on prism planes. Fig. 11 demonstrates the compatibility interaction between growth and creep. For comparison purposes non-interactive behavior, computed by vectorial addition of total principal creep and growth strains, has been included in the graph. -. ----

-

CREEP INTElWllVE CREEP G GROWTH NON-INTERACTIVE CREEP a GROWTH

Fig. 11. Interactive creep-plus-growth constitutive behavior for CW-SR Zircaloy texture. Typical double branch behavior is illustrated for interactive and non-interactive calculations. The magnitude of the creep strains were taken to be four times larger than the growth strains. (+I+) indicates both stress components positive, (-I-) both negative.

B.L. Adams et al. /Evaluation of creep and growth in Zircaloy

Total creep plus growth behavior exhibits a double-branch behavior for either the interactive or non-interactive cases as shown in fig. 11. With both hoop and axial stresses positive the effect over almost all of the range depicted in fig. 11 is to shift the strain ratios to lower absolute magnitudes relative to the pure creep case. This shift is indicated by the arrows in the figure. If both stresses are negative, the strain ratios tend to shift to larger absolute magnitudes. Without the growth bias, the behavior for the t/t and -/- cases is identical as also shown in the figure. The cell-to-cell growth-creep interaction effect, reflected by the difference between the interactive and non-interactive cases in fig. 11, is clearly a function of applied stress ratio as would be anticipated. The interaction effect is appreciable over most of the range of stress ratios investigated. The input parameters chosen for investigating the creep-growth interaction are representative of true m-reactor behavior. Extensive post-irradiation examination data on CW-SR fuel assemblies suggest that, for a nominal compressive load ratio of ue /u, = 2.0, the interactive correction to the non-interactive prediction in fig. 11 is in the correct direction and is roughly the correct magnitude. However, more extensive correlations with a wider range of variables are required to establish the validity of the correlation. Such studies are underway at the present time.

6. Conclusions The CODF provides an important tool for studying anisotropic creep and irradiation growth processes in highly textured hexagonal materials. Once the strain producing mechanisms are identified, statistical procedures employing the CODF give estimates of macroscopic behavior as it relates to the underlying mechanisms and crystallite distribution. Cellto-cell interactions arising from incompatible strain rates can be studied by means of upper-bound estimates in which cell-to-cell strain continuity is preserved. A stress-free irradiation growth model based on preferred orientations of vacancy and interstitial sinks has been developed. The model, which allows preferential absorption of interstitials at prism dislocations in the cell, and preferred annihilation of

81

vacancies at cell boundaries, predicts a strong effect of grain shape on resulting strain anisotropy. For steady-state source operation, the ratio of cell boundary surface area to cell volume was found to dictate overall dislocation densities within the cell. As a general result, finer grain size at a fmed grain shape will increase the stress-free growth rate under steady-state conditions. The non-uniform distribution of dislocation densities arising from prior fabrication history was found to be of secondary importance in affecting the overall strain anisotropy. Comparison of predicted growth strain anisotropies with post-irradiation data shows excellent correlation with measured performance. Creep in Zircaloy was modelled by use of an upperbound estimate, with the assumption of stress powerlaw operation of prism, basal, and pyramidal slip systems. Extremes of operation were modelled by relaxation of the reference stress for slip on one or more of three systems with respect to the others. Strain-rate anisotropies were obtained by statistical averaging using a CODF typical of CW-SR Zircaloy fuel cladding. Thermal data at 673 K for this material indicates that the case where slip on the basal system is relaxed gives the best correlation with the observed strain anisotropy. No significant difference in strainrate anisotropies were noted between 673 and 615 K creep tests. Since prism slip is dominant at lower temperatures, and basal slip is restricted, it is not clear whether this result reflects strong deformation hardening of the prism systems during fabrication of the cladding or that a significant lowering of the reference stress for basal slip has occurred at these test temperatures. Further work is needed to resolve this question. A modification of the upper-bound scheme allows investigation of the cell-to-cell interactions between creep and growth processes. A significant synergism between the two processes was demonstrated by a comparison of the interactive and non-interactive cases. Direct superposition of the two effects is shown to be only approximately correct. The required correction is in reasonable agreement with the observed experimental data; further correlations, however, are required to verify the effect.

B.L. Adams et al. f Evaluation of creep and growth in Zircaloy

88

List of symbols Tensor indices referred to crystallite coordinates Tensor indices referred to fabrication coordinates Euler angles Crystallite orientation distribution function (CODF) Weighted volume average of ( ) over all crystallite orientations Strain-rate tensors due to interstitial annihilation Strain-rate tensors due to vacancy annihilation Total growth strain-rate tensors Euler coordinate transformations Geometric tensors for vacancy annihilation Geometric tensors for interstitial annihilation Fundamental slip tensor for the kth slip system Creep strain-rate tensors Tensor of creep compliances Crystallite and macroscopic stress tensors Total strain-rate tensors including creep and growth Principal growth strain-rates in radial, hoop, and axial directions Principal creep strain-rates in radial, hoop, and axial directions

Acknowledgements The authors wish to acknowledge helpful discussions and insight provided by Professor J.W. Hutchinson (HarvardUniversity), Dr. W.L Mamrnel(Bell Laboratories) and Dr. J.S. Kallend (Blinois Institute of Technology). Support for this work was provided by the Exploratory Research Division of Babcock and Wilcox.

References [l] A.S. Argon and F.A. McCIintock, Mechanical Behavior of Materials (Addison-Wesley, 1966) p. 359.

[ 21 J.W. Hutchinson, Proc. Roy. Sot. (London) A348 (1976) 101. [3] J.W. Hutchinson, Met. Trans. 8A (1977) 1465. [4] ,J. Schroeder and M.J. Holicky, J. Nucl. Mater. 33 ” (1969) 52. [S] MJ, HoIIcky and J. Schroeder, J. NucL Mater. 44 (1972) 31. [ 61 D.G. Franklin and W.A. Franz, in: Proc. 3rd Intern. Conf. on Zirconium in the Nuclear Industry, Quebec City, 1976, Eds. A.L. Lowe, Jr. and G.W. Parry, ASTM-STP 633, p. 365. [ 71 H. Stehle et aL, in: Proc. 3rd Intern. Conf. on Zirconium in the Nuclear Industry, Quebec City, 1976, Eds. A.L. Lowe, Jr. and G.W. Parry, ASTM-STP 633, p. 486. [8] J.E. Harbottle, Phil. Mag. A38 (1978) 49. [9] G.J.C. Carpenter and D.O. Northwood, J. Nucl. Mater. 56 (1975) 260. [lo] A. Jostsons, P.M. Kelly and R.G. Blake, J. Nucl. Mater. 66 (1977) 236. [ll] C.C. DoBins, J. NucL Mater. 59 (1975) 61. [ 121 W.K. Alexander, V. Fidleris and R.A. Holt, in: Proc. 3rd Intern. Conf. on Zirconium In the Nuclear Industry Quebec City, 1976, Eds. A.L. Lowe, Jr. and G.W. Parry, ASTM-STP 633, p. 344. [ 131 J.E. Harbottle and R.M. Cornell, Nucl. Eng. Design 42 (1977) 423. 1141 R.A. Holt and E.F. Ibrahim, Acta Met., to be published. [ 151 R.A. Holt, J. NucL Mater. 82 (1979) 419. [ 161 G.S. Clevinger, B.L. Adams and K.L. Murty, In: Proc. 4th Intern. Conf. on Zirconium in the Nuclear Industry, Stratford-Upon-Avon, 1978, Ed. T.P. Papazoglou (ASTM). [ 171 R.J. Roe, J. Appl. Phys. 36 (1965) 2024. [18] H.J. Bunge, Z. Metallk. 56 (1965) 872. [ 191 P.R. Morris and A.J. Heckler, Trans. AIME 245 (1969) 1877. [20] S.L. Lopata and E.B. Kula, Trans. AIME 224 (1962) 865. [21] J.F.W. Bishop and R. Hill, Phil. Mag. 42 (1951) 414; 42 (1951) 1298. [ 221 G.Y. Chin and W.L. Mammel, Trans. AIME 345 (1969) 1211. [ 231 G. Dressier, K.H. Matucha and P. Wincierz, Can. Met. Quart. 11 (1972) 177. [24] G.W. Groves and A. Kelly, Phil. Mag. 8 (1963) 877. [ 251 R.E. Reed-Hi& in: Deformation Twinning, Vol. 25, Eds. R.E. Hi& J.P. HIrth and H.C. Rogers (Gordon and Breach, New York, 1964) p. 295. [26] J.L. Martin and R.E. Reed-Hill, Trans. AIME 230 (1964) 780. [27] E. Tenckhoff, Z. Metal&. 63 (1972) 192. [ 281 J.A. Jensen and W.A. Backofen, Can. Met. Quart. 11 (1972) 39. [ 29) K.L. Murty, unpublished data.